the microwave response of ultra thin microcavity arrays

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THE MICROWAVE RESPONSE OF

ULTRA THIN MICROCAVITY ARRAYS

Submitted by

JAMES ROBERT BROWN

To the University of Exeter

as a thesis for the degree of

Doctor of Philosophy

June 2010

This thesis is available for library use on the understanding that it is copyright material

and that no quotation from this thesis may be published without proper

acknowledgement.

I certify that all material in this thesis which is not my own work has been identified and

that no material has previously been submitted and approved for the award of a degree

by this or any other University.

______________________________

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Abstract

The ability to understand and control the propagation of electromagnetic radiation underpins a

vast array of modern technologies, including: communication, navigation and information

technology. Therefore, there has been much work to understand the interaction between

electromagnetic waves and metal surfaces, and in particular to design materials the

characteristics of which can be tailored to produce a desired response to microwave radiation. It

is the objective of this thesis to demonstrate that patterning metal surfaces with sub-wavelength

apertures can afford hitherto unrealised control over the reflection and transmission

characteristics of materials which are an order of magnitude thinner than those employed

historically.

The work presented herein aims to establish ultra thin cavity structures as novel materials for

the selective absorption and transmission of microwave radiation. Experimental and theoretical

approaches are used to elucidate the mechanism that allows such structures to produce highly

efficient absorption via the excitation of standing wave modes in structures that are two orders

of magnitude thinner than the operating wavelength. Also considered is how this same

mechanism mediates transmission of selected frequencies through similarly thin structures.

Later chapters focus on ultra thin cavity structures which, through higher-order rotational

symmetry, exhibit resonant absorption which is almost completely independent of incident and

azimuthal angle and polarisation state. A detailed studied of the absorption bandwidth of these

devices is also presented in the context of fundamental theoretical limitations arising from the

thickness and magnetic permeability of the structure.

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This thesis is dedicated to my wonderful Sharmi: now that it is finished we

get our weekends back!

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It is best to keep an open mind, but not so open that one’s brain falls out.

Richard Dawkins, 2007

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Table of contents

ABSTRACT______________________________________________________________ 2

TABLE OF CONTENTS____________________________________________________5

LIST OF FIGURES AND TABLES___________________________________________9

LIST OF ABBREVIATIONS________________________________________________19

ACKNOWLEDGEMENTS_________________________________________________20

CHAPTER 1:

Introduction ______________________________________________________________23

CHAPTER 2:

The interaction of microwaves with metal surfaces

2.1 Introduction___________________________________________________________25

2.2 The scattering of electromagnetic radiation by matter__________________________25

2.2.1 Radar Cross Section (RCS)____________________________________________26

2.2.2 Electromagnetic scattering regimes _____________________________________30

2.2.2.1 Rayleigh scattering_______________________________________________30

2.2.2.2 Resonant scattering_______________________________________________30

2.2.2.3 Optical scattering________________________________________________31

2.2.3 Scattering from periodically textured surfaces_____________________________33

2.2.3.1 The phenomenon of diffraction _____________________________________33

2.2.3.2 Diffraction gratings_______________________________________________34

2.3 Surface waves_________________________________________________________36

2.3.1 Surface wave excitation ______________________________________________37

2.4 Materials for the absorption of microwave radiation ___________________________40

2.4.1 Underpinning absorption mechanisms ___________________________________40

2.4.2 Conventional absorbing materials_______________________________________44

2.5 Current research in electromagnetic materials ________________________________47

2.6 Important applications for absorbing materials _______________________________49

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CHAPTER 3:

Modelling

3.1 Introduction___________________________________________________________53

3.2 The finite element approach______________________________________________ 53

3.3 An overview of HFSS___________________________________________________ 54

3.3.1 Assembling the structure to be simulated_________________________________ 54

3.3.2 Assigning material properties__________________________________________ 55

3.3.3 Boundary conditions_________________________________________________ 57

3.3.4 Excitations_________________________________________________________60

3.3.5 Meshing___________________________________________________________61

3.3.6 Post-processing _____________________________________________________64

3.4 Modelling approaches used in this thesis ____________________________________65

3.4.1 Mono-grating reflection structures _____________________________________ 65

3.4.2 Mono-grating transmission structures ___________________________________ 68

3.4.3 Bi-grating reflection structures ________________________________________ 70

3.4.4 Tri-grating reflection structures ________________________________________ 72

3.4.5 Broadband structures ________________________________________________ 77

3.4.5.1 Non-parallel slits_________________________________________________77

3.4.5.2 Multi-layer structures _____________________________________________79

3.5 Summary_____________________________________________________________80

CHAPTER 4:

The microwave reflectivity and transmissivity of a low-loss dielectric layer disposed between

two metallic layers perforated periodically by sub-wavelength slits

4.1 Introduction___________________________________________________________81

4.2 Background___________________________________________________________82

4.3 Experimental__________________________________________________________84

4.3.1 Fabrication of samples_______________________________________________ 84

4.3.2 Definition of polarisation state, angles of incidence and azimuth______________ 87

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4.3.3 Measurement of microwave reflectivity and transmissivity___________________87

4.3.3.1 Focused horn____________________________________________________87

4.3.3.2 Long path length azimuthal scan apparatus____________________________ 88

4.4 Results and discussion__________________________________________________ 90

4.4.1 Reflection sample___________________________________________________ 90

4.4.2 Optimisation of resonance depth_______________________________________ 97

4.4.2.1 Optimisation by altering core material properties_______________________101

4.4.2.2 Optimisation by altering core thickness_______________________________103

4.4.2.3 Optimisation by altering slit width__________________________________ 106

4.4.3 Polarisation conversion effects________________________________________ 108

4.4.4 Transmission samples_______________________________________________ 110

4.4.4.1 Aligned slits ___________________________________________________ 110

4.4.4.2 Off-set slits ____________________________________________________114

4.5 Summary____________________________________________________________ 116

CHAPTER 5:

Reduction of azimuthal and incident angle sensitivity and polarisation conversion effects – bi-

gratings

5.1 Introduction__________________________________________________________118

5.2 Experimental_________________________________________________________118

5.3 Results______________________________________________________________119

5.4 Polarisation conversion effects___________________________________________126

5.5 Dispersion___________________________________________________________129

5.6 Conclusions__________________________________________________________130

CHAPTER 6:

Minimisation of azimuthal and incident angle sensitivity and polarisation conversion effects –

tri-gratings

6.1 Introduction__________________________________________________________132

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6.2 Experimental details___________________________________________________133

6.3 Theory______________________________________________________________134

6.4 Results______________________________________________________________136

6.4.1 Tri-grating sample 1________________________________________________ 136

6.4.2 Tri-grating sample 1 – polarisation conversion___________________________ 144

6.4.3 Tri-grating sample 2________________________________________________146

6.4.4 Tri-grating sample 2 – polarisation conversion___________________________ 153

6.5 Summary_____________________________________________________________ 154

CHAPTER 7:

Methods for achieving maximum absorption bandwidth

7.1 Introduction__________________________________________________________156

7.2 Experimental ________________________________________________________ 157

7.3 Theory______________________________________________________________160

7.4 Results______________________________________________________________163

7.4.1 Standard mono-grating______________________________________________163

7.4.2 Structure 1 – multiple discrete repeat periods ____________________________165

7.4.3 Structure 2 – multiple continuous repeat periods__________________________168

7.4.4 Structure 3 - Multi-layering __________________________________________172

7.4.5 Structure 4 - Multiple permittivities ___________________________________175

7.5 Conclusions _________________________________________________________178

CHAPTER 8:

Conclusions

8.1 Summary of thesis_____________________________________________________180

8.2 Ideas for future work___________________________________________________182

8.3 List of publications ____________________________________________________186

REFERENCES___________________________________________________________188

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List of figures and tables

Figure 2.1 RCS for a metallic sphere. The circumference is given in wavelengths and the RCS is

normalised to the actual cross sectional area of the sphere (this figure has been adapted from

Knott (1993))________________________________________________________________32

Figure 2.2 A p-polarised wave incident on a grating structure (a) 3-D projection (b) plan

view_______________________________________________________________________35

Figure 2.3 A p-polarised electromagnetic wave incident on the interface between two media

___________________________________________________________________________37

Figure 2.4 Diagramatical representation of the plasmon dispersion relation_______________39

Figure 2.5 Waves incident on a typical absorbing material____________________________42

Figure 2.6 Plot of the simulated reflectivity of a Dallenbach layer for a p-polarised wave at

different angles of incidence___________________________________________________44

Figure 2.7 A typical Salisbury screen (a) geometry (b) simulated reflectivity for a p-polarised

wave over a range of incident angles_____________________________________________46

Figure 3.1 Plots of typical finite element meshes constructed using HFSS (a) 3-D projection of

the initial mesh for a typical microcavity structure, 1258 tetrahedra (b) cropped 2-D side

elevation of the initial mesh for a typical microcavity structure, 1258 tetrahedra (c) 3-D

projection of the final mesh for typical microcavity structure 43401 tetrahedra (d) cropped 2-D

side elevation of the final mesh for typical microcavity structure 43401 tetrahedra_________62

Figure 3.2 Mono-grating structure as modelled in HFSS (a) Selected Dimensions and materials

(b) Boundary conditions_______________________________________________________66

Figure 3.3 Finite element mesh for the mono-grating reflection structure (a) 3-D projection (b)

Cropped 2-D projection, metal layers marked by black lines___________________________68

Figure 3.4 Diagram of the off-set transmission structure______________________________69

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Figure 3.5 Diagram showing the bi-grating model (a) the model geometry (b) the finite element

mesh_______________________________________________________________________72

Figure 3.6 The tri-grating sample geometries (not to scale) and the co-ordinate system used

(a) 3-D projection of tri-grating 1 (b) 3-D projection of tri-grating 2, tm = 18 μm, tc = 356 μm, ws

= 0.3 mm, 12 gg = 10 mm, is the polar angle, is the azimuthal angle_____________74

Figure 3.7 Forming the tri-grating structures without inputting irrational numbers (a) metal plate

(30 x 30) mm with three slits spaced 10 mm apart (b) second set of slits added and rotated by

60° about the z-axis (c) third set of slits added and rotated by -60° about the z-axis (d)

translation of first set of slits by 5 mm in the z-direction (e) subtraction of all three sets of slits

from the metal layer, two unit cells can be seen_____________________________________75

Figure 3.8 Plots of selected parts of the final finite element mesh for the tri-grating structures

(a) for tri-grating 1 (b) for tri-grating 2____________________________________________77

Figure 3.9 Multiple continuous repeat periods with alternate saw-tooth slits_______________78

Figure 3.10 Multi-layer microcavity structure (a) 3-D projection of multi-layer structure, 2

periods shown (b) end projection of multi-layer_____________________________________79

Figure 4.1 Cross-section through the substrate material The dielectric core is FR4 – a Glass

Reinforced Plastic (GRP) composite material with a permittivity of (4.17 + i0.07) _________85

Figure 4.2 Cross-section through the microcavity samples (a) the reflection sample (b) the first

transmission sample with slits perfectly aligned (c) the second transmission sample with slits

perfectly mis-aligned _________________________________________________________86

Figure 4.3 Definition of polarisation state, incident () and azimuthal () angle (a) TE or s-

polarised – the E-vector is perpendicular to the plane of incidence (b) TM or p-polarised – the

E-vector is contained within the plane incidence ____________________________________87

Figure 4.4 Photograph of the focused horn apparatus The VNA can be seen in the

background, the reference aperture can be seen in the centre. The focal length of the

system is adjusted by a stepper motor attached to the nearest mirror (out of shot to the

left)_________________________________________________________________88

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Figure 4.5 Schematic of the long path length, azimuthal-scan apparatus set up for measurements

of bi-static reflectivity_________________________________________________________89

Figure 4.6 Schematic of the long path length, azimuthal-scan apparatus set up for measurements

of transmission_______________________________________________________________90

Figure 4.7 Experimental Reflected intensities for reflection sample shown as greyscale plots (a)

Rpp data as a function of frequency and azimuthal angle at º16 (b) Rss data as a function of

frequency and azimuthal angle at º16 (c) Rpp data as a function of frequency and azimuthal

angle at º57 , dashed line corresponds to expected position of diffraction edge (d) Rss data

as a function of frequency and azimuthal angle at º57 ____________________________91

Figure 4.8 Line plots of the reflectivity of the reflection sample as measured experimentally at

incident angles of 16 and 4.57 for (a) p-polarisation and 0 (b) s-polarisation

and 90 _______________________________________________________________ 93

Figure 4.9 Reflectivity of the reflection sample as measured experimentally and simulated by

the finite element model: P-polarisation incident at 4.57 and 0 _______________ 94

Figure 4.10 Behaviour of the electric field within the dielectric core of the ultra thin cavities as

simulated by the finite element model (a) Instantaneous magnitude of the electric field at 7.1

GHz, plotted at a phase corresponding to peak field strength, the black line represent the copper

layers, blue corresponds to 0 V/m, red corresponds to 20 v/m, incident wave amplitude 1 V/m

(b) z-component of the electric field along a line through the centre of the core parallel to the x-

axis (c) Instantaneous the electric field vector at 7.1 GHz, plotted at a phase corresponding to

peak field strength, blue corresponds to 0 V/m, red corresponds to 20 v/m, incident wave

amplitude 1 V/m_____________________________________________________________ 95

Figure 4.11 Waves incident on the ultra thin cavity structure__________________________97

Figure 4.12 Reflectivity of the ultra-thin cavity structure as a function of frequency for

different values of imaginary permittivity as simulated using the finite element model (a) for

values of imaginary permittivity (eps’’) between 0.02 and 0.2 (b) for values of imaginary

permittivity (eps’’) between 0.2 and 0.9_________________________________________102

Figure 4.13 Field plots showing the instantaneous magnitude of the electric field for different

values of imaginary permittivity, scale runs from 0 V/m (blue) to 50 V/m (red), incident wave

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amplitude was 1 V/m in all cases (a) imaginary permittivity = 0.02 (b) imaginary permittivity =

0.08 (c)imaginary permittivity = 0.2 (d) imaginary permittivity = 0.4 (e) imaginary permittivity

= 0.9_____________________________________________________________________103

Figure 4.14 Reflectivity versus frequency as predicted by the finite element model for structures

of differing core thickness_____________________________________________________104

Figure 4.15 Field plots showing the instantaneous magnitude of the electric field for different

core thicknesses, scale runs from 0 V/m (blue) to 40 V/m (red) and the incident wave amplitude

was 1V/m in all cases (a) core thickness = 100 μm (b) core thickness = 120 μm (c) core

thickness = 150 μm (d) core thickness = 180 μm (e) core thickness = 250 μm (f) core thickness

= 356 μm __________________________________________________________________106

Figure 4.16 Reflectivity as a function of frequency for the ultra thin cavity arrays with different

slit widths_________________________________________________________________107

Figure 4.17 Experimental polarisation-converted reflected intensities for reflection sample

shown as greyscale plots (a) Rps data as a function of frequency and azimuthal angle at

º16 (b) Rsp data as a function of frequency and azimuthal angle at º16 (c) Rps data as a

function of frequency and azimuthal angle at º57 (d) Rsp data as a function of frequency

and azimuthal angle at º57 ________________________________________________110

Figure 4.18 Transmission as a function of frequency for the aligned slit structure as measured

using the focused horn system and simulated using the finite element model ____________110

Figure 4.19 Plots of the electric field at 7.2 GHz for the aligned transmission structure: blue

corresponds to 0 V/m and red to 20 V/m and the incident wave amplitude was 1 V/m (a) the

instantaneous magnitude of the electric field plotted at a phase corresponding to peak field (b)

the instantaneous electric field vector plotted at a phase corresponding to peak

field______________________________________________________________________111

Figure 4.20 Plots of the instantaneous magnitude of the electric field at different frequencies for

the aligned transmission structure, scale runs from 0 V/m (blue) to 20 V/m (red) and the

incident wave amplitude was 1 V/m in all cases (a) 7.2 GHz (b) 7.4 GHz (c) 7.6

GHz______________________________________________________________________114

Figure 4.21 Transmission as a function of frequency for the off-set slit structure as measured

using the focused horn system and simulated using the finite element model____________114

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Figure 4.22 Plots of the electric field at 13.12 GHz for the off-set transmission structure, scale

runs from 0 V/m (blue) to 20 V/m (red) and the incident wave amplitude was 1 V/m in both

cases (a) the instantaneous magnitude of the electric field plotted at a phase corresponding to

peak field (b) the instantaneous electric field vector plotted at a phase corresponding to peak

field______________________________________________________________________115

Figure 5.1 (a) The mono-grating sample geometry (not to scale) and the co-ordinate system

used: θ is the polar angle, is the azimuthal angle, λg = 10 mm, ws = 0.3 mm (b) 3-D

projection of the bi-grating, λg1 = λg2 (c) Cross-section through the bi-grating structure, tm = 18

μm, tc = 356 μm, ws = 0.3 mm, λg2 = λg1 =10 mm, sample area 500 mm by 500 mm_______119

Figure 5.2 Reciprocal space diagram for the bi-grating______________________________120

Figure 5.3 Bi-grating sample (a) Experimental Rpp data as a function of frequency and

azimuthal angle at = 57° (b) Experimental Rss data as a function of frequency and azimuthal

angle at = 57° (c) Line plot showing comparison of measured data to the predictions of the

numerical model: Rpp = 57°, = 45° (d) Prediction of the electric field vector distribution at

a phase corresponding to peak field strength on the upper surface of the lower metal layer for a

{1, 1} mode at 10.93 GHz: the longest arrows correspond to enhancements of 13 times the

injected field_______________________________________________________________121

Figure 5.4 Bi-grating sample (a) Incident wavevector and electric vectors on the lower surface

of a metal patch, and the resulting charge distribution for: = 0°, p-polarization (b) = 0°, s-

polarization (c) = 45°, p-polarization and (d) = 45°, s-polarization________________123

Figure 5.5 Distribution of the electric field on the upper surface of the lower metal layer plotted

at a phase corresponding to maximum field (a) The (2,0) mode at = 90° and 13.9 GHz (b)

The (0,2) mode at = 0° and 13.9 GHz (c) The degenerate (2,0) and (0,2) modes at = 45°,

at 14.55 GHz (d) The (2,1) mode at = 45°, 16.6 GHz_____________________________125

Figure 5.6 Experimental polarisation-converted reflected intensities for reflection sample





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Figure 5.7 Distribution of the electric field on the upper surface of the lower metal layer plotted

at a phase corresponding to maximum field for the degenerate (2, 0) and (0, 2) modes as

excited by an s-polarised wave = 45°, 4.57 at a frequency of 14.585 GHz________129

Figure 5.8 Dispersion plots determined from the frequency of the modes supported by the bi-

grating sample at = 0° and 45º with (a) p-polarized and (b) s-polarization incident

radiation__________________________________________________________________130

Figure 6.1 The tri-grating sample geometries (not to scale) and the co-ordinate system used:

is the polar angle, is the azimuthal angle, g = 10 mm, ws = 0.3 mm (a) 3-D projection of tri-

grating 1 (b) 3-D projection of tri-grating 2 (c) Cross-section through the tri-grating structure,

one set of slits shown for clarity, tm = 18 μm, tc = 356 μm, ws = 0.3 mm, 12 gg =10 mm,

sample area 500 mm by 500 mm_______________________________________________133

Figure 6.2 Reciprocal space diagrams for the tri-gratings showing: (a) the scattering vectors

and reciprocal lattice points (b) with a series of circles centred on the origin having radii at

which resonant modes are expected_____________________________________________135

Figure 6.3 Tri-grating sample 1: (a) Experimental Rpp data as a function of frequency and

azimuthal angle at 16 ; (b) Experimental Rss data as a function of frequency and azimuthal

angle at 16 ; (c) Experimental Rpp data as a function of frequency and azimuthal angle

at 43 ; (d) Experimental Rss data as a function of frequency and azimuthal angle

at 43 _________________________________________________________________137

Figure 6.4 Tri-grating samples 1 and 2: (a) Line plot showing comparison of measured data to

the predictions of the numerical model for tri-grating 1: Rpp 43 , 30 ; (b) Line plot

showing comparison of measured data to the predictions of the numerical model for tri-grating

2: Rpp 43 , 30 ______________________________________________________139

Figure 6.5 Tri-grating sample 1: predictions of the electric field vector distribution at phases

corresponding to peak field strengths on the upper surface of the lower metal layer for: (a) an

8.35 GHz, p-polarised wave incident at 43 90 ; (b) an 8.35 GHz, s-polarised wave

incident at 43 90 ; (c) a 15 GHz, p-polarised wave incident at 43 90 ; (d)

a 15 GHz, s-polarised wave incident at 43 90 , the longest arrows correspond to

enhancements of 15 times in all cases___________________________________________141

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Figure 6.6 Diagrams showing the incident electric field and resulting charge distribution for a

15 GHz s-polarised wave incident at (e) 90 ; (f) 60 ______________________143



17.3 GHz, p-polarised wave incident at 43 , 90 ; (b) a 17.8 GHz, p-polarised wave

incident at 43 , 90 , the longest arrows correspond to enhancements of 15 times in

both cases_________________________________________________________________144

Figure 6.8 Experimental polarisation-converted reflected intensities for tri-grating sample 1





Figure 6.9 Tri-grating samples 1 and 2: (a) Experimental Rpp data as a function of frequency

and azimuthal angle at 16 for tri-grating sample 2; (b) Experimental Rss data as a function

of frequency and azimuthal angle at 16 for tri-grating sample 2; (c) Experimental Rpp data

as a function of frequency and azimuthal angle at 43 for tri-grating sample 2; (d)

Experimental Rss data as a function of frequency and azimuthal angle at 43 for tri-grating

sample 2 (e) Experimental Rpp data as a function of frequency and azimuthal angle

at 43 for tri-grating sample 1; (f) Experimental Rss data as a function of frequency and

azimuthal angle at 43 for tri-grating sample 1________________________________147



8.1 GHz, p-polarised wave incident at 43 , 60 ; (b) a 8.1 GHz, s-polarised wave

incident at 43 , 60 ; (c) a 13.8 GHz, p-polarised wave incident at 43 , 60 ;

(d) a 13.8 GHz, s-polarised wave incident at 43 , 60 , the longest arrows correspond

to enhancements of 15 times in all cases_________________________________________151

Figure 6.11 Tri-grating sample 2: (a) prediction of the electric field vector distribution at a

phase corresponding to peak field strength on the upper surface of the lower metal layer for: a

16.4 GHz, p-polarised wave incident at 43 , 60 ; (b) diagram showing the incident

electric field and resulting charge distribution for a p-polarised wave incident at 90 ; (c)

diagram showing the incident electric field and resulting charge distribution for a s-polarised

wave incident at 90 ____________________________________________________152

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Figure 6.12 Tri-grating sample 2: (a) prediction of the electric field vector distribution at a

phase corresponding to peak field strength on the upper surface of the lower metal layer for: a

17.1 GHz, s-polarised wave incident at 43 , 60 ; (b) prediction of the electric field

vector distribution at a phase corresponding to peak field strength on the upper surface of the

lower metal layer for: a 18.3 GHz, s-polarised wave incident at 43 , 60 _______153

Figure 6.13 Experimental polarisation-converted reflected intensities for tri-grating sample 2





Figure 7.1 The microcavity structure geometries (not to scale) and the co-ordinate system used:

θ is the incident angle, is the azimuthal angle, ws is the slit width, g is the repeat period of

the structure (a) 3-D projection of a standard mono-grating structure in which all slits run

parallel (b) Cross-section through the standard mono-grating structure (c) 3-D projection of

Structure 1, multiple discrete repeat periods (d) Plan view projection of Structure 2, multiple

continuous repeat periods with alternate saw-tooth slits (e) 3-D projection of Structure 3, multi-

layer structure, 2 periods shown (f) end projection of Structure 3, multi-layer structure, 2

periods shown (g) 3-D projection of Structure 4, multiple refractive

indices___________________________________________________________________159

Figure 7.2 Response of an example Salisbury screen absorber as predicted using the finite

element model (a) reflectivity in decibels versus frequency (b) reflectivity in decibels versus

wavelength________________________________________________________________160

Figure 7.3 Theoretical and experimental data for standard mono-grating (a) Reflectivity in

decibels versus wavelength as predicted by the finite element model and measured

experimentally (b) Reflectivity in decibels versus wavelength as predicted by the finite element

model for mono-grating structures of differing core thickness (c) Percent of narrowband

bandwidth limit versus core thickness for the a series of mono-grating

structures_________________________________________________________________164

Figure 7.4 Multiple discrete period structures (a) Reflectivity in decibels versus wavelength for

structure with dielectric core thickness of 190 μm (b) Reflectivity in decibels versus

wavelength as predicted by the finite element model for multiple discrete period structures of

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differing core thickness (c) Percent of narrowband bandwidth limit versus core thickness for

the a series of multiple discrete period structures (d) Cross-section of modified multiple

discrete repeat period structure, (e) Reflectivity in decibels versus wavelength as predicted by

the finite element model for multiple discrete period structures with different values of t2

_________________________________________________________________________167

Figure 7.5 Multiple continuous period structures (a) Reflectivity in decibels versus wavelength

as predicted by the finite element model and measured experimentally (b) Reflectivity in

decibels versus wavelength as predicted by the finite element model for multiple continuous

period structures of differing core thickness (c) Percent of narrowband bandwidth limit versus

core thickness for the a series of multiple continuous period structures (d) Plot of the

instantaneous electric field vector on the upper surface of the lower metal layer at a wavelength

of 47 mm and a phase corresponding to peak field, the longest arrows correspond to 30 V/m

(an enhancement of 30 times the incident field), dashed lines added to indicate position of slits

(e) Plot of the magnitude of the instantaneous electric field on the upper surface of the lower

metal layer at a wavelength of 40 mm and a phase corresponding to peak field, dark blue areas

correspond to 0 V/m, green areas to 20 V/m (f) Plot of the magnitude of the instantaneous

electric field on the upper surface of the lower metal layer at a wavelength of 36 mm and a

phase corresponding to peak field, dark blue areas correspond to 0 V/m, green areas to 20 V/m

and red areas to 30 V/m______________________________________________________171

Figure 7.6 Multi-layer structures (a) Reflectivity in decibels versus wavelength for structure

with dielectric core thicknesses t1 = 0.13 mm, t2 = 0.12 mm, t3 = 0.1 mm, t4 = 0.075 mm, (b)

magnitude of the electric field at a wavelength of 20.2 mm and at a phase corresponding to

peak field for the N = 1 mode, scale runs from 0 V/m to 90 V/m (c) magnitude of the electric

field at a wavelength of 20.2 mm and at a phase corresponding to peak field for the N = 3

mode, scale runs from 0 V/m to 15 V/m (d) magnitude of the electric field at a wavelength of

20.9 mm and at a phase corresponding to peak field for the N = 1 mode, scale runs from 0 V/m

to 50 V/m (e) magnitude of the electric field at a wavelength of 20.2 mm and at a phase

corresponding to peak field for the N = 3 mode, scale runs from 0 V/m to 20 V/m________173

Figure 7.7 Multiple-permittivity structure (a) Reflectivity in decibels versus wavelength for

structures with a range of dielectric core thicknesses (b) Percent of narrowband bandwidth

limit versus core thickness for the series of multiple-permittivity structures (d) Reflectivity in

decibels versus wavelength as predicted by the finite element model for multiple-permittivity

structures with different values of loss tangent in the cavity with εr = 3.5 _______________177

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Figure 8.1 Hybrid transmission structures (a) array of slits in the upper metal layer, single slit

in the lower metal layer (b) rotation of slits in lower metal layer relative to those in the upper

metal layer, layers shown separately (c) progressive reduction in slit number to concentrate

field_____________________________________________________________________183

Figure 8.2 Pseudo-fractal multi-layer absorbing structure___________________________184

Figure 8.3 Absorbing structures in which each cavity contains dielectric media of different

refractive index____________________________________________________________185

Table 7.1 Resonant wavelengths in millimetres for Structure 4 – Multi-layer structure as

predicted using (7.4) and observed using HFSS___________________________________174

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List of abbreviations

E-Vector – electric field vector

FEA – Finite Element Analysis

GRP – Glass Reinforced Plastic

HFSS – High Frequency Structure Simulator (software)

MathCAD – Mathematical Computer Aided Design (software)

PCB – Printed Circuit Board

Q-Factor – Quality Factor

RCS – Radar Cross Section

RF – Radio Frequency

Rpp – reflection coefficient when both receiver and transmitter are p-polarised

Rps – reflection coefficient when transmitter is p-polarised and receiver is s-polarised

Rsp – reflection coefficient when transmitter is s-polarised and receiver is p-polarised

Rss – reflection coefficient when both receiver and transmitter are s-polarised

TE – Transverse Electric polarisation (s-polarised)

TM – Transverse Magnetic polarisation (p-polarised)

VNA – Vector Network Analyser

m - microns

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Acknowledgements

The successful (if protracted) completion of this thesis owes much to many people other than

myself. That I even contemplated undertaking an MPhil which slowly morphed into a PhD can

be credited to (or should that be blamed on?) Professor Chris Lawrence. Chris is one of the most

encouraging and self-less individuals I have ever been fortunate enough to meet, and marries a

tireless work ethic to his wonderfully inquisitive approach to science, resulting in a breadth of

knowledge that spans fields as diverse bio-inspiration and radio frequency tagging. His support

and enthusiasm is contagious and I would not be here writing this had I not experienced it first

hand during our time together at QinetiQ.

My thanks must also go to those at QinetiQ who agreed to fund my studies and also pitched-in

with useful suggestions throughout my time there, not the least of which is my long-time office

mate Dr. Pete Hobson. Pete has a unique and very entertaining perspective on life which

becomes magnified when he has consumed even the minutest quantities of alcohol, as many of

us at QinetiQ had the joy of witnessing! Rumours abound that he wrote his PhD thesis by

driving a radio-controlled tank up and down his keyboard; it is that sort of innovation coupled

with that ever-so-slightly messy desk of his that assure me he is a professor in waiting. My

thanks also go to the likes of Dr. Benny Hallam who, during his cameo at QinetiQ imparted

much wisdom on those who met him including me.

Being a part-time student based a long way from the university has the potential to leave one

feeling isolated from the rest of the group, particularly when visits to the university were as

infrequent as mine! However, when I walked back into the department after my usual six-month

absence there was a core of people who not only remembered who I was but welcomed me back

as if one of their own. That always gave me a tremendous feeling and I much appreciate the

friendship that I was shown by several people. At the top of that list is Dr. Matt Lockyear who

as well as providing me with numerous funny moments during my visits, was also incredibly

patient and supportive and always stopped what he was doing in order to help me. I

21

experienced similar altruism from the like of Dr. James Suckling and Dr. Rob Kelly who never

failed to assist me whenever I had forgotten how to set-up the kit correctly, again!

Officially Professor Roy Sambles was my supervisor, and I must thank Roy for being a highly

enthusiastic supervisor: his level of knowledge is quite remarkable and he applies it with energy

and passion. I must also mention the invaluable contribution of Dr. Alastair Hibbins who was

my unofficial oracle throughout my visits to Exeter. Alastair was always something of a role

model to me both in terms of his scientific knowledge and ability to articulate it, and more

recently in terms of hairstyles – as Pete Hobson likes to remind me! Anyway, my thanks to him

for providing knowledge and guidance that proved both accurate and useful.

I would like to formally acknowledge that in addition to the measurements I myself performed,

I was occasionally assisted in data collection by C A M Butler as a consequence of our having

been colleagues at QinetiQ. Specifically she helped me collect data on the reflectivity at 43°

incidence of the bi-grating and tri-grating structures of Chapters 5 and 6 respectively.

Furthermore, the data appearing in the first paper “Squeezing millimetre waves into microns,”

was taken by Dr. Alastair Hibbins after the equivalent data I took was inadvertently deleted

from a shared computer. The data included in Chapter 4 are from subsequent measurements

which I performed myself. The analysis of the form of the resonant modes supported by both

the reflection and transmission structures of Chapter 4 is taken from the first paper as written by

Dr. Alastair Hibbins and this has been acknowledged within the text of that chapter. Later work

in Chapter 4 concerning the effect of changes to the geometry of the reflection structures and

the mechanism by which their response is optimised is entirely my own.

The majority of the work in Chapter 5 appeared in the second paper "Angle-independent

microwave absorption by ultrathin microcavity arrays" and was undertaken and written by

myself with my co-authors acting as internal reviewers and providing essential insights and

guidance. Similarly the work in Chapter 6 and Chapter 7 is entirely my own but was

22

extensively reviewed by Professor J. Roy Sambles, Dr. Alastair Hibbins and Professor Chris

Lawrence, all of whom made useful suggestions which I have incorporated.

23

Chapter 1

Introduction

The work presented in this thesis pertains to the control of electromagnetic radiation and in

particular to the control of microwaves. The myriad applications of microwave radiation range

from detection systems such as radar, both commercial and military, to wireless networks for

communication and asset tracking systems such as RFID (Radio Frequency IDentification). The

primary objective within all these fields is the control of microwave radiation and its

interactions both intentional and unintentional, with matter, and in particular with metal

surfaces. The rapid expansion of microwave-frequency wireless technology has spawned a huge

increase in research in the field with the goal of even finer control of the radio frequency

environment in ever-thinner, lower-cost materials. The goal of this thesis is the realisation of

ultra-thin materials for the control of microwave radiation across the entire wireless application

space.

Chapter 2 presents a theoretical background to the scattering of electromagnetic radiation by

matter, and is intended to establish a context for the detailed discussion of ultra thin absorbers to

follow. The characteristics required for a material to be absorbing are discussed, and the

conventional approaches to designing such materials are covered. This is followed by a review

of current research in the area of absorbing materials. Finally the typical applications for

absorbing materials are presented.

Chapter 3 focuses on the use of finite element modelling as a tool for the design of ultra thin

materials for controlling microwave radiation and specifically on Ansys’s HFSS software. The

basis of the finite element method is described and the manner in which HFSS applies the finite

element approach to simulate electromagnetic problems is detailed. The specific modelling

tactics employed to simulate the behaviour of each variant of the ultra thin microcavities is also

covered in detail.

24

Chapter 4 constitutes the first chapter which is dedicated to exploring in detail the mechanism

which underpins the selective absorption and transmission of microwaves by ultra thin cavity

arrays. A study of the behaviour of these structures as a function of frequency, polarisation state

and azimuthal and incident angles is presented, and the finite element model is used to

investigate the form of the resonant modes excited. Further work considers the effect of

changing both material properties and the physical geometry of the cavities, and leads to the

development of a strategy for optimising the resonance depth as well as a more detailed

understanding of the mode of operation.

In Chapter 5 the possibility of reducing the incident angle and polarisation sensitivity of the

ultra thin cavities is explored through the use of ―bi-gratings;‖ structures which feature two

orthogonal sets of sub-wavelength apertures. It is found that these structures support a higher

number of resonant modes than the equivalent mono-gratings of Chapter 4 and that several of

these modes exhibit incident and azimuthal angle invariance as well as polarisation

independence. The finite element model is used to explore the character of the modes they

support and hence predict their resonant frequencies accurately. In Chapter 6, the concept of

higher-order rotational symmetry is extended to include two hexagonally symmetric ―tri-

grating,‖ structures each of which features three sets of sub-wavelength apertures. These

structures support an even higher number of modes than the bi-gratings in addition to affording

incident and azimuthal angle invariance.

Chapter 7 considers the absorption bandwidth of the ultra thin cavities and presents four

strategies for maximising this bandwidth by exciting multiple resonant mode series through a

multiplicity of cavity lengths and refractive indices. It is found that absorption bandwidth can

be increased significantly but that ultimately the bandwidth-to-thickness ratio is limited by

fundamental limitations imposed by structure thickness and magnetic permeability.

The work presented herein is summarised in Chapter 8 which also includes several ideas for

extending this work through future studies on hybrid structures.

25

Chapter 2

The interaction of microwaves with metal surfaces

2.1 Introduction

An understanding of the interaction of microwaves with metal surfaces is integral to a vast array

of modern technologies which are becoming ever more ubiquitous, including Wi-Fi and cellular

phones, to name but two examples. Unsurprisingly therefore, research into microwaves and

microwave materials constitutes a huge and growing field of interest and covers many different

areas, from high impedance ground planes which improve the performance of cellular phone

handsets (Broas et al (2001)), microstrip antennas (Qian (1998)) and new ultra-small antenna

configurations for radio frequency tagging (Brown et al (2008)). Materials which allow the

passage of microwave radiation to be manipulated are hence of great use in an environment

where radio frequency contamination is an ever-increasing problem.

This chapter aims to provide a context for the following discussion of ultra thin absorbing

materials. The theoretical background to the scattering of electromagnetic radiation by matter is

presented and the characteristics required for a material to be absorbing are discussed. The

conventional approaches to designing such materials are covered including their relative merits

and drawbacks. This is followed by a review of current research in the area of absorbing

materials. Finally the typical applications for absorbing materials are presented.

2.2 The scattering of electromagnetic radiation by matter

Some of the following has been adapted from Knott (1993) and Raether (1988).

According to Knott (1993), scattering can be defined as the dispersal of electromagnetic

radiation by matter. It is due to the interaction of the fields that constitute the radiation (in

particular the electric field) with the electrons of the material being illuminated. The properties

of the material, the frequency of the radiation and the shape of the object being illuminated

combine to determine the form of the scattered field.

26

At radio frequencies, metals behave as near-perfect conductors: the ―nearly-free,‖ electrons

vibrate in sympathy with the incident electric field to produce a scattered field of the same

frequency and amplitude as the incident field - the metal is a ―perfect reflector.‖ This assumes

that there is no dissipation of energy by the metal, which is a valid assumption at microwave

frequencies in most cases.

Non-conducting materials by contrast do not contain free-electrons and hence are generally not

perfect reflectors at radio frequencies. However, certain materials exhibit natural resonances in

their material properties (permittivity and permeability) at particular frequencies and can

therefore behave as highly efficient reflectors despite being non-metallic. Overall however,

metal surfaces and objects generally have the greatest capacity for creating large scattered field

amplitudes.

2.2.1 Radar Cross Section (RCS)

When considering the scattering of radiation from a single object, it is useful to define an

effective area or cross-section based on the scattering efficiency of the object. In a monostatic

radar system, the transmitting and receiving antennas are co-located by definition. If the

transmitting antenna emits a total power Pt, the resulting power density, St, is inversely

proportional to the distance from the antenna, R:

24 R

PS t

t

(2.1)

Some proportion of the power from the transmitting antenna is then intercepted by the object in

question, located at a distance R from the radar system. This intercepted proportion can be

found from the product of the incident power density, St and the effective capture area of the

object, Ar. Some fraction of this intercepted power is then converted to heat and the remainder is

re-radiated. If all the intercepted power is converted to heat and none is re-radiated then the

amplitude of the field scattered from the object must be zero and its scattering cross section, σ,

would be zero even though its effective capture area or cross-section, Ar, is > 0. If however, the

object was a near-perfect reflector (e.g. a metal) then there would be very little dissipation,

27

almost all of the intercepted power would be re-radiated and the average scattering cross

section, σ, would be the same as the effective capture area. Note the difference between these

two cross-sections: the former considers all of the power extracted from the incident wave by

the object, the latter considers only that which is scattered back towards the radar system.

This introduces the idea of an effective area or scattering cross-section that an object may

posses. However, it is not simply one number. Any object other than a sphere will tend to

scatter more power in one direction and less in another: the scattering cross-section is therefore

dependant on the orientation of the object relative to the incident wave. The dependence of the

scattering cross-section on orientation, i.e. the distribution of the re-radiated power, is

dependant on the shape and material properties of the object: it may re-radiate most power back

towards the receiving radar antenna in which case the signal received by the radar system will

be relatively large. Alternatively, it may re-radiate most power in other directions such that the

signal received by (mono-static) radar system is relatively small. In fact one key strategy in

RCS reduction is shaping: deliberately designing an object such that it scatters very little power

in the retro-direction (back towards the radar system) but instead scatters it into other ―Non-

threat‖ directions. This is only useful however, if the radar system’s transmitting and receiving

antennas are co-located: for bi-static radar systems this is not the case. For maximum reduction

of bi-static RCS, the object must absorb and dissipate (convert to heat) as much of the incident

radio wave energy as possible.

Returning to the power density as a function of distance from the transmitted (equation (2.1)), it

is possible to create a definition for the scattering cross-section of an object in general. The total

power intercepted by an object, PI , is the product of this power density with the effective cross-

sectional area of the object, Ar.

rII ASP (2.2)

28

Note that this effective area, Ar, is not the same as the object’s actual physical area, it is the area

the object appears to have by virtue of the total power it is able to extract from a wave of given

power density, although in many cases the effective area is greater for larger objects.

Consider now only that fraction of the intercepted power which is re-radiated by the object in

the direction of the radar system: this excludes any power dissipated as heat or scattered in non-

threat directions, Pr. This is found from the product of the incident power density SI and the

scattering cross-section, σ:

Ir SP (2.3)

This scatter power results in a scattered power density received back at the radar system:

22 44 R

S

R

PS Ir

r

(2.4)

Consider that the power density at the object, SI, is a result of power initially radiated by the

radar system Pt, therefore the power density resulting from scatter by the object is Sc:

22 44 RR

PS t

c

(2.5)

Multiplying this by the effective area of the receiving antenna, At, gives the total power

received, Pc:

22 44 RR

APP tt

c

(2.6)

The scattering cross-section, σ, is the Radar Cross Section or RCS. It is the effective size the

object appears to have by virtue of its ability to scatter radiation back to the radar receiver. A

29

large, thick, flat metallic sheet normal to the direction of the incident radiation would have a

large RCS since the metal is a near-perfect reflector. However, if this metal sheet were coated

with a near-perfect absorber it would scatter very little radiation and would have a low RCS

despite having the same physical area.

Consider again the scattered power, Pr, which results in a power density at a distance R, Sr as

given by (2.4).

Hence the RCS, σ, can be found from:

I

r

S

SR24 (2.7)

It would appear that σ therefore depends on the distance R, but in fact Sr =Pr/4πR2 hence:

I

r

S

P (2.8)

Therefore the RCS can be considered as the ratio of the scattered power to the incident power

density: the larger the RCS the more power the object scatters for a given incident power

density. This is also consistent with the IEEE definition of RCS, which states that RCS is: 4π

times the ratio of the power per unit solid angle scattered in a specified direction to the power

per unit area in a plane wave incident on the scattered from a specified direction (Knott et al

(1993)):

2

2

24lim

incident

scattered

r E

ER

(2.9)

The power per unit solid angle multiplied by 4π steradians gives the total power, the ratio of this

total power to the power per unit area, or power density, of an incident plane wave is the RCS –

as was shown above. Note that on the basis of the Poynting vector (Grant et al (1995)) power

densities can be replaced with the squares of the respective electric field amplitudes.

30

2.2.2 Electromagnetic scattering regimes

The manner in which electromagnetic radiation is scattered from an object is dependant on the

size of the object relative to the wavelength () of the radiation and is therefore classified into

three regimes:

1) Rayleigh scattering: wavelength much greater than object size

2) Resonant scattering: wavelength of the order of object size

3) Optical scattering: wavelength much smaller than object size

2.2.2.1 Rayleigh scattering

In cases where the wavelength is much greater than the object size the phase of the incident

wave does not vary significantly over the extent of the object and the problem reduces to one of

electrostatics. The whole object contributes to the scattering process making the overall shape

far more important than the detailed geometry. The most important feature of scattering in the

Rayleigh regime is that the RCS is proportional to the frequency raised to the fourth power: it

increases very quickly with frequency, see Figure 2.1 which has been adapted from Knott

(1993).

2.2.2.2 Resonant scattering

Whilst there does not exist a strict definition, resonant scattering is generally taken to occur for

objects that are between 1 and 10 in size. Resonant scattering can be sub-divided into two

scattering mechanisms: optical mechanisms and surface wave mechanisms. In this context,

optical mechanisms refers to specular re-radiation.

In this regime it is possible for the energy from incident waves to become bound to the surface

of metallic objects in the form of surface waves (in the field of optics such surface waves are

referred to as surface plasmon polaritons or SPPs). Surface waves can be classified into the

following types:

31

Travelling waves – these propagate along a metal surface or more formally along the

boundary between two materials that have permittivities of opposite sign, until they encounter a

discontinuity whereupon they can be reflected. The subsequent re-radiation of these surface

waves can increase the specular and non-specular RCS of the object.

Edge travelling waves – these propagate along edges or ridges such as the leading edge of

aerofoils

Creeping waves – these are surface travelling waves that are launched into shadow regions –

regions that are not directly illuminated by the incident radio wave. Creeping waves can

propagate around curved surfaces in the process of which they progressively re-radiate and

hence contribute to non-specular RCS.

The RCS due to surface waves is proportional to the square of the wavelength (Knott (1993))

hence the significance of surface wave scattering is much less at higher frequencies. For this

reason surface wave scattering is not significant in the optical regime although the phenomenon

still occurs. It should be noted that surface wave scattering is independent of the size of the

body – it depends only on wavelength.

2.2.2.3 Optical scattering

In this regime the wavelength is much smaller than the object size and details of an object’s

geometry become important. The optical scattering regime can be sub-divided into four

mechanisms:

Specular scattering – in this case the angle of reflection is simply equal to the incidence

End-region scattering – non-specular sidelobes result from scattering by the end regions of

objects

32

Diffraction – this is scattering due to end regions but is in the specular direction and is

typically caused by leading and trailing edges, tips or tightly curved surfaces

Multi-bounce rays – rays that are scattered by one surface can be subsequently scattered by

another surface and be directed back in the direction of the source thereby increasing the mono-

static RCS.

The variation in RCS of a metallic sphere as it increases in size is shown in Figure 2.1 (adapted

from Knott (1993)). The RCS has been normalised to the cross-sectional area of the sphere (the

area of a circle of radius equal to that of the sphere) and the circumference of the sphere has

been normalised to the wavelength. Hence as one moves along the x-axis to the right the

wavelength is decreasing and/or the sphere circumference is increasing.

Figure 2.1 RCS for a metallic sphere

The circumference is given in wavelengths and the RCS is normalised to the

actual cross sectional area of the sphere (this figure has been adapted from

Knott (1993))

The three scattering regimes have been marked in Figure 2.1. It can be seen that the degree of

scattering is small in the low frequency (long wavelength) limit and increases steadily

throughout the Rayleigh regime. When the sphere circumference reaches a wavelength the RCS

levels off and remains at this mean level. Interference between re-radiated creeping waves and

33

waves reflected off the front face gives rises to the interference fringes seen throughout the

resonance region. These fringes become less significant in the optics region where the

wavelength is only a small fraction of the sphere circumference.

2.2.3 Scattering from periodically textured surfaces

As was shown in the previous section, the RCS of a sphere (or an object in general) starts to

become significant when its circumference (or in the case of a more general object its

characteristic dimension) is equal to the wavelength of the incident radiation. Any textured

surface, which has a characteristic dimension that is approximately equal to the wavelength of

the incident radiation, will scatter that radiation significantly.

Surfaces that have random texture will tend to produce random or diffuse scattering: the

incident radiation is dispersed over a range of angles with no particular angle preferred.

Surfaces that have periodic texture (for example a series of grooves of equal width, equally

spaced) scatter radiation into a series of discrete directions. Such surfaces constitute diffraction

gratings and by concentrating the power they re-radiate into beams or orders, they have the

potential to increase the non-specular RCS significantly.

2.2.3.1 The phenomenon of diffraction

According to Huygen's principle, propagating waves, electromagnetic or otherwise can be

visualised as consisting of an infinite number of point sources distributed along the wavefront

(surface of constant phase). Each source emits what are known as secondary wavelets, such that

the wavefront at some later time is the envelope of these wavelets (Hecht (1998)). This concept

is rather oversimplified: if correct, one would observe both forward and backward propagating

waves. The concept was later modified by Fresnel who applied the idea and mathematics of

interference to obviate the need for a backwards propagating wave, and later still by Kirchoff

who demonstrated that it was a direct consequence of the differential wave equation. Thus the

modified Huygens-Fresnel principle, which considers both amplitude and phase, can be applied

to understand the phenomenon of diffraction on a qualitative basis (Hecht (1998)).

34

If a wave encounters an obstacle, either opaque or transparent, which retards the phase or serves

to reduce the amplitude of some segments of the wavefront more than others, then the secondary

wavelets will no longer all be in phase in the forward direction. There will exist other directions

in which the wavelets will interfere in-phase to produce beams or orders and still other

directions in which the wavelets will be completely out of phase and in which there will exist

nulls.

Reflection of a wave from a textured surface has the effect of retarding the phase of some

portions of the wavefront relative to others and hence creates the phase conditions whereupon

constructive and destructive interference can occur. In cases where the wavelength is much

larger than the characteristic dimension of the texture (the Rayleigh regime), the phase shift

created between different segments of the wavefront is only a small fraction of the wavelength

and hence the interference is almost totally constructive.

2.2.3.2 Diffraction gratings

Consider the grating structure shown in Figure 2.2: the surface is textured periodically by slits a

distance g apart, a p-polarised wave (that is one in which the electric field is contained within

the plane of incidence and the magnetic field is transverse to the plane of incidence) is shown,

incident on the grating at an azimuthal angle and incident angle , any diffracted wave occurs

at an angle to the surface normal.

35

(a) (b)

Figure 2.2 A p-polarised wave incident on a grating structure

(a) 3-D projection (b) plan view

In considering the interaction of the incident wave with the grating and the possible creation of

diffracted orders, momentum must be conserved. The momentum, p can be found from the

product of the wavevector 0k with Planck’s constant divided by 2π, :

0kp (2.5)

It transpires that is common to all terms and therefore it cancels, the problem then becomes

one of matching wavevectors.

Begin by considering the momentum parallel to the plane of incidence in the plane of the

grating (the xy plane). The sum of the momentum from the incident wave plus that due to

scattering from the grating, which is equal to an integer multiple, N of the grating vector kg,

must equal to that of the diffracted order, which occurs at an angle to the surface normal and

at an angle to the plane of incidence, hence:

36

cossincossin 00 kNkk g (2.6)

Now consider the in-plane x- and y-components:

gx kkk cossin0 (2.7)

sinsin0kky (2.8)

The total in-plane momentum available, ksum is therefore:

222

yxsum kkk (2.9)

20

2

0

2sinsincossin kNkkk gsum (2.10)

22

00

222sincossin2 kkNkkNk ggsum (2.11)

Real, propagating diffracted orders only occur for 90 , therefore set 90 and

consequently 1sin , and substitute in (2.11):

22

00

222

0 sincossin2 kkNkkNk gg (2.12)

0cossin2sin122

0

22

0 gg kNNkkk (2.13)

This quadratic in k0, (2.13), can be solved to yield the limit frequency at which diffracted orders

will occur for any incident and azimuthal angle and any grating period.

2.3 Surface waves

As stated earlier a surface wave is the microwave analogue of a surface plasmon polariton or

SPP: a localised surface charge density oscillation that can propagate along the boundary

between the metal and the dielectric medium from which the wave was incident (typically air).

37

2.3.1 Surface wave excitation

Consider a p-polarised electromagnetic wave incident on the boundary between two media of

permittivity 1 and 2 as shown in Figure 2.3

Figure 2.3 A p-polarised electromagnetic wave incident on the interface

between two media

If the two media possess permittivities of opposite sign, then any component of electric field

normal to the interface will change direction as it crosses the interface. This results in the

formation of a sheet of charge at the interface. Note that this can only occur for p-polarised

incident radiation, as with s-polarised radiation the electric field has no component normal to

the interface. The SPP is essentially a longitudinal oscillation of this trapped electric charge.

Interestingly, it is also possible to excite magnetic surface plasmons on the interface between

two media which have magnetic permeabilities of opposite sign, see Sarychev et al (2006) and

references therein.

The component of the incident wavevector that is parallel to the metal surface is:

sin0kkx (2.14)

Raether (1998) demonstrates that by solving Maxwell’s equations for the electromagnetic wave

at the interface, by applying the boundary condition that the tangential components of the

electric field E and the electric displacement D are continuous across an interface (recalling that

38

the field inside a perfect conductor is everywhere zero) the wavevector of the SPP is given by

(2.15) which is also referred to as the dispersion relation:

2

1

21

21

ckSPP (2.15)

In order excite an SPP, the momentum given by the product of (2.15) with must be supplied.

The form of dispersion relation can be plotted if the frequency dependence of the permittivities

1 and 2 is known. The dispersion of most dielectrics is negligible hence 1 can be considered

to be frequency independent. However, the permittivity of the metal 2 , undergoes huge

changes with frequency, changes which can be described using the Drude free electron model

(Ashcroft (1976)). For a typical metal at 10 GHz the real part of the complex permittivity

around (-104) whilst the imaginary part is around (10

6), by contrast at optical frequencies the

values of the real and imaginary components are typically (-10) and (0.1) respectively. Using

models such as the Drude model allows the dispersion relationship to be plotted.

The dispersion relation (2.15) for the SPP has been plotted in Figure 2.4: the curve describes the

relationship between the wavevector of the surface plasmon and its frequency. Also plotted is

the frequency versus in-plane wavevector for the incident wave – the ―Light line;‖ the constant

of proportionality is the speed of light hence the line is straight with a gradient c.

39

Figure 2.4 Diagramatical representation of the plasmon dispersion relation

In the high frequency limit the frequency of the SPP tends towards the plasmon frequency: at

this frequency the real part of 2 and the real part of 1 are equal in magnitude and opposite in

sign, therefore kSPP is purely imaginary. Close to the plasmon frequency the group and phase

velocities of the mode tend to zero, the SPP is tightly bound to the surface and is described as

being ―plasmon like,‖ its behaviour being predominantly that of a longitudinal oscillation of the

sheet of charge trapped at the interface. By contrast, in the low frequency limit the SPP

frequency tends towards that of the incident wave. This latter limit arises since the behaviour of

the metal very closely resembles that of a perfect conductor, resulting in the dispersion relation

of (2.15) reducing to 1 ckx which is identical to that of the incident radiation, in this

limit the SPP is described as ―photon like,‖ as the SPP is only very loosely bound to the surface.

However, at all frequencies the plasmon momentum is slightly greater than the incident wave

momentum hence some extra momentum is needed in addition to that supplied by the incident

wave, even if the wave is at grazing incidence. This momentum deficit must be satisfied for the

SPP to be excited. A grating which supplies a momentum equal to the product of and the

grating vector kg and can satisfy the deficit:

40

g

gk

2 (2.16)

At microwave frequencies the momentum deficit is very small, at optical frequencies it is much

larger, hence surface plasmons are more easily excited at microwave frequencies than at optical

frequencies. In fact an entire grating is not required: a single ridge, crack or other discontinuity

is all that is required to excite SPPs.

In the microwave regime the conductivity of metals is so high that SPPs can continue to

propagate for distances of hundreds of metres with very little attenuation. This is in stark

contrast to the optical regime where, due the metal having much lower ac conductivity, typical

propagation distances are of the order of microns. On striking a discontinuity such as a gap in a

metal surface, an SPP may be re-radiated and thus contribute to an object’s RCS. Furthermore,

this characteristic permits the design of surfaces which can excite surface plasmons and then re-

radiate them in a tailored manner, for example see the work of Lockyear et al (2004).

2.4 Materials for the absorption of microwave radiation

2.4.1 Underpinning absorption mechanisms

Regardless of the specifics of their design, all absorbing materials rely on the fact that certain

substances absorb energy from electromagnetic waves propagating through them. Materials

which absorb have two components to their refractive index: a real part and an imaginary part,

and it is the imaginary part which accounts for the absorption. The real part is related to the

storage of energy rather than its dissipation as heat, for example the storage of energy in a

capacitor is proportional to the real part of the permittivity of the dielectric between the plates.

The refractive index is related simply to a product the permittivity ( ) and the permeability

( ), and absorption can result from electric loss mechanisms or magnetic loss mechanisms or

both. The loss due to electric effects is analogous to Joule heating of resistance in a circuit,

whereas magnetic losses are generally attributed to the rotation of domains. In either case, the

41

net result is the conversion of energy to heat, this is what is meant by the term loss, or more

formally non-radiative loss, in this context. The degree of absorption exhibited by a particular

structure depends on its configuration in addition to the inherent electromagnetic properties of

its constituent material, although for conversion to heat to occur the refractive index must have

some imaginary component.

It is convenient to deal in terms of the relative permittivity and permeability which are both

expressed as complex numbers as described above:

''' rrr i (2.18)

''' rrr i (2.19)

Note that the permittivity and permeability are relative to those of free space, having values of

8.854 x 10-12

F/m (to 4.s.f.) and 4 x 10-7

H/m respectively.

By considering Maxwell’s equations and applying the requisite boundary conditions and

trigonometric identities it can be shown (Grant et al (1995)) that the refractive index n is given

by:

rrn (2.20)

Furthermore the intrinsic impedance Z, of material, is given by:

Z (2.21)

Thus demonstrating that the refractive index has both electric and magnetic components.

In many conventional absorbing materials, a dielectric layer is used over a metal backing.

Consider a wave incident on the boundary between the outer surface of the dielectric and free-

space or air. When the incident wave impinges on the outer surface of the dielectric some

proportion of it is coupled into the dielectric and the remainder is reflected: the proportion that

is reflected determined by the mis-match between the intrinsic impedance of free-space and the

42

input impedance of the material: the greater the difference in impedance the larger the

reflection.

Figure 2.5 Waves incident on a typical absorbing material

The wave coupled into and propagating through the dielectric eventually reaches the metal

backing whereupon it is wholly reflected (assuming a perfect metal) and propagates back

towards the dielectric-air interface. In accordance with reciprocity the same proportion of the

wave energy originally reflected at the outer surface of the dielectric is reflected internally at the

dielectric-air boundary, the remainder couples back into the surrounding air. The wave coupled

back into air then interferes with that initially reflected at the air-dielectric interface.

The internal reflection process within the dielectric is subsequently repeated and the waves

propagating back and forth interfere. If the dielectric is a quarter-wavelength thick, then the

waves reflected from the metal surface will interfere constructively with those internally

reflected at the dielectric-air interface. This situation arises since the metal plate constitutes a

perfect electric reflector and has a reflection coefficient of -1 – the wave suffers a phase change

of radians on reflection and the waves incident on and reflected from the metal will therefore

interfere constructively. Furthermore, since the permittivity of the dielectric is greater than that

of the incident medium (air) the dielectric-air boundary has a positive reflection coefficient: the

waves do not suffer any phase change on reflection from it hence interference between waves

incident on the dielectric-air boundary and those internally reflected is again constructive.

43

Constructive interference results in a standing wave inside the dielectric, the amplitude of which

increases progressively over time as more energy from the incident wave becomes trapped

within the dielectric. If the dielectric is lossy i.e. if the imaginary component of the permittivity

and/or permeability is non-zero, some proportion of the energy of the standing wave is

converted to heat. It can be shown (Grant et al (1995)) that the loss of energy per unit volume

occurs at a rate, DP , given by (2.22):

*.*.Re21 HEikJEPD (2.22)

Where:

E is the electric field strength in V/m

H* is the complex conjugate of the magnetic field strength in A/m

J * is the complex conjugate of the current density in A/m2

From the definition of intrinsic impedance, it is apparent that the value of (2.22) is proportional

to the square of the standing wave amplitude, hence as the wave amplitude increases so does the

rate of loss. Eventually a point is reached where the rate of loss is equal to that at which energy

from the incident wave is coupling into the dielectric – the standing wave amplitude reaches a

steady state.

The amplitude of the wave re-emitted from the dielectric back into air depends on the

impedance mis-match at the dielectric-air interface and also on the standing wave amplitude:

enhancement of the standing wave amplitude results in the amplitude of the re-emitted wave

also increasing. This in turn affects the resulting interference between the re-emitted wave and

the wave initially reflected at the air-dielectric interface. If the amplitudes of these two waves

are equal then they may interfere destructively, hence the net radiation from the material or

rather its reflectivity will be zero. This is the mechanism underpinning the operation of all

resonant absorbers.

44

If the material properties of the dielectric are such that the amplitude of the initially reflected

wave is too high, or the steady state amplitude of the standing wave is too low, then although

there will be destructive interference the net reflectivity will not be zero. Conversely, if the

amplitude of the re-emitted wave is higher than that of the initially reflected wave then the net

reflectivity will not be zero. For the reflectivity to be zero then radiative loss (re-emitted wave

amplitude) must equal the non-radiative loss (the absorption by the dielectric).

2.4.2 Conventional absorbing materials

Examples of resonant absorbing materials that can be described as above include Salisbury

screens (Salisbury (1952)) and Dallenbach layers (Ruck (1970)). The latter structure is as

described above and shown in Figure 2.5. Shown in Figure 2.6 is a plot of the reflectivity of a

hypothetical Dallenbach layer for a p-polarised wave incident over a range of angles, as

simulated using the finite element model to be detailed in the next chapter. In this case the

dielectric material was non-magnetic 01 ir , with permittivity 4.04 ir and 3.75 mm

thick.

Figure 2.6 Plot of the simulated reflectivity of a Dallenbach layer for a p-

polarised wave at different angles of incidence

For normal incidence the resonance is just above 10 GHz a consequence of the quarter-

wavelength condition detailed above. Minimum reflectivity is –22 dB indicating a close match

45

between the radiative and non-radiative losses. As the incident angle is increased the resonance

becomes shallower and shifts to higher frequencies. Note that the minimum thickness of a

Dallenbach is a quarter-wavelength at the frequency of interest allowing for the refractive index

of the core. This restriction can be of significant inconvenience and is not encountered with the

ultra thin absorbers which are the focus of this thesis.

The Salisbury screen consists of a very thin lossy layer spaced a quarter-wavelength from a

metal backing typically by a low loss material such as air or foam. The conventional approach

to designing a Salisbury screen is to consider the structure as a quarter-wave section of

transmission line and then adjust the surface impedance of the thin lossy layer until the input

impedance of the structure matches that of free-space. However, since this structure constitutes

a resonant absorber it can also be thought about in the terms described above. A diagram of a

typical Salisbury screen Shown in Figure 2.7, along with the corresponding reflectivity spectra

as simulated using the finite element model.

Again the resonance occurs close to 10 GHz for normal incidence due to the spacing of the

lossy layer above the ground plane being a quarter-wave wavelength. It is clear that the

performance of the Salisbury screen is highly sensitive to incident angle with a frequency shift

of 4 GHz occurring between normal incidence and 45° incidence. Note that the measured

physical thickness of the Salisbury screen is double that of the Dallenbach layer due to the

former using air as a dielectric.

46

(a)

(b)

Figure 2.7 A typical Salisbury screen

(a) geometry (b) simulated reflectivity for a p-polarised wave over a range of

incident angles

In order to increase the absorption bandwidth of a Salisbury screen, a multiplicity of thin lossy

layers can be used instead of just one: such a structure is referred to as a Jaumann absorber

(Conolly (1977)). Each lossy layer is typically spaced a quarter-wavelength from the adjacent

lossy layers thus increasing the overall thickness of the structure by a factor of n if there are n

layers. As will be demonstrated later in this thesis, increasing absorption bandwidth beyond a

47

certain point cannot be achieved without proportional increases in overall thickness, this is true

for all types of absorber, irrespective of their geometry.

2.5 Current research in electromagnetic materials

Research concerning the interaction of electromagnetic radiation and patterned metal surfaces

has produced a series of extraordinary and unexpected breakthroughs in recent years. The

pioneering work of Ebbesen et al (1998) has revealed that two-dimensional arrays of cylindrical

holes in a thin metal film can support the transmission of more radiation that is directly incident

on the apertures themselves. This phenomenon can be explained in terms of the excitation of

surface plasmon polaritons on the metal surface, the frequency of which is determined by the

hole spacing in accordance with momentum conservation as detailed above.

Others, such as Suckling et al (2007) have corroborated these findings and have extended the

work to consider hexagonally symmetric hole arrays to create azimuthally-independent surface-

plasmon mediated transmissive structures. Yang (2008) demonstrated that the SPP dispersion

curve can be tailored by altering the geometry of the hole arrays. Specifically, Yang (2008)

demonstrates that the size of the holes in a transmissive structure can be modified in order to

alter the plasma frequency and asserts that rectangular holes result in more tightly bound

plasmons than otherwise equivalent arrays of circular holes. Others including Pendry (2004)

have termed these tightly bound surface waves as ―Spoof‖ surface plasmons since their

dispersion can be manipulated through changes to structure rather than to constituent material

properties.

This type of structure has evolved into much thinner forms via the work of Sievenpiper (1999)

who analysed a ―mushroom‖ structure: an array of hexagonal metal patches connected to a

ground plane using vias. Using a circuit model, Sievenpiper demonstrates that this structure

supports propagating TM surface modes below its resonance frequency and TE modes above its

resonant frequency. Lockyear (2009) exploited prism coupling to experimentally determine the

48

dispersion of a Sievenpiper mushroom structure that was only 0.8 mm thick despite operating at

frequencies around 20 GHz or 15 mm free-space wavelength.

The ability to tailor the response of this class of metamaterial has been demonstrated

numerically by Navarro-Cia et al (2009). Navarro-cia presents numerical results for split-ring

resonator structures and demonstrates their capacity to excite broadband surface plasmons, and

hence overcome many of the problems associated with the preceding structures. Navarro-cia

also demonstrates that these split ring structures can be applied to create a planar waveguide for

the guiding of spoof plasmons.

The challenge of creating absorbing materials which are much thinner and exhibit greater

stability under azimuthal and incident angle variations than those used conventionally is the

primary focus of this thesis. However others have also pursued similar goals including Lockyear

(2003) who realised the use of a dual-period bi-grating structure for the excitation of surface

plasmon modes which are highly independent of incident and azimuthal angle. Although these

structures are much thicker than the absorbing materials considered herein, they still afford

significant reductions in reflectivity by virtue of dielectric losses in the material used to fill the

grating grooves. The same author has also recently published work (2009) pertaining to an

absorbing structure formed from an array of discs, which operate via a similar principle to the

structures considered in this thesis. This structure combines the advantages of being very thin

(total thickness approximately 1% of the operating wavelength) and producing absorption which

is largely independent of incident and azimuthal angles and polarisation state.

Others including Zhang (2003), Sievenpiper ((1999) and (2003)) and Gao (2005) considered

metal-dielectric-metal laminates into the upper metal layer of which are cut two mutually

orthogonal sets of sub-wavelength slits, and thus dividing the upper metal layer into an array of

square patches. These bi-grating structures are capable of guiding electromagnetic waves and

therefore have myriad applications in communications and electronics. Zhang et al (2003)

analyzed such a square patch array as a perfect magnetic conductor (see Sievenpiper (1999))

49

and using a transmission line model applied the structure to suppress side lobes from an antenna

array. Zhang also presents normal incidence reflectivity data for this structure but does not

present a study of its angle or polarisation dependence.

Work by Sievenpiper et al (2003) involved the introduction of varactor diodes to a square patch

array similar to that used by Zhang (2003) but with vias between the centre of the patches and

the lower metal plane. The diodes were connected between adjacent patches and the application

of a bias voltage across them alters the phase of the reflected waves thus creating an electrically

tuned device for use in beam-steering that can also be used as an absorber (Gao (2005)).

Other approaches include those of Tennant and Chambers, (2004) who created a hybrid

Salisbury screen type absorber in which, in place of the conventional resistive layer, was used a

frequency selective surface (FSS). The FSS consisted of small dipole-like elements at the centre

of which was connected a PIN diode. Varying the diode bias current permits resistive tuning of

the structure and hence allows the structure’s reflectivity to be varied. In this case the overall

structure thickness was less than 5 mm despite being formed using a spacer with permittivity

very close to that of air. Significant reductions in reflectivity were measured between 8 GHz and

14 GHz, therefore, whilst being substantially thicker than the structures considered in this

thesis, this hybrid absorber is thinner than a conventional Salisbury screen.

2.6 Important applications for absorbing materials

Historically, the primary application for absorbing materials has been in reducing the RCS of

military vehicles. Whilst the military demand remains, the advent of mobile communications

and wireless networking has resulted in a huge growth in the number of radio frequency

transmissions across many frequency bands. This has in turn led to a requirement to ―clean-up‖

the radio frequency environment and to be able to screen-out specific signals and frequencies.

Materials which can selectively absorb and transmit radio frequency radiation, whilst being low

cost and low profile are therefore increasingly in demand in commercial applications.

50

One example of a commercial application for absorbing materials is RFID (Radio Frequency

IDentification). RFID is a wireless system used in the tracking of assets, inventory management

and access control. A series of tags attached to the assets in question can be read automatically

by readers which transmit and/or receive encoded radio frequency signals from the tags. The

tags consist of an antenna and an integrated circuit (IC) which contains a unique identifier or

Electronic Product Code (EPC). Modulation of the IC’s input impedance encodes its unique

EPC onto the signal sent back to the reader. Such systems are the natural successor to barcodes

and offer greater speed, range and accuracy in addition to removing the line-of-sight

requirement inherent to optical systems.

The adoption of RFID as a tool for improving the efficiency of numerous business processes is

increasing rapidly, with sectors including IT (e.g. the Sarbanes-Oxley Act (2002)), retail (RFID

Journal (2003)) and Oil and Gas producing the greatest demand. However, the bands used by

most longer-range RFID systems such as passive UHF (Ultra High Frequency) are sub 1 GHz

and hence correspond to large wavelengths which make the use of conventional absorbers

impractical. Furthermore, RFID systems typically operate over small bandwidths e.g. the EU

UHF RFID band is only 2 MHz wide, spanning 865.6 MHz to 867.6 MHz

(http://www.epcglobalinc.org/), which, as shall be demonstrated later in this thesis, makes ultra

thin absorbers a logical choice for signal control since the required operating bandwidth is

small.

There are several situations where better control of the radio frequency signal can dramatically

improve the performance of RFID systems. For example, large depots or warehouses feature

multiple points of exit and entry through which tagged goods pass, are scanned and their

passage recorded on a database for tracking purposes. However, there is the potential for loss of

traceability if a reader located at one exit point reads a tag passing through another adjacent

entry point. Control of the radio frequency environment by the introduction of compact

absorbing materials greatly reduces the probability of these sorts of errors commonly referred to

as ―Bleed-over.‖ In other situations, reflections off walls and ceilings can result in tags some

51

distance from the exit point being read spuriously – the database will then record them as

having passed out of the warehouse when in fact they have not. Again the application of low-

profile absorbing material provides a cost efficient, unobtrusive solution.

Another important application for absorbing materials is the field of renewable energy

generation. The widespread deployment of wind turbines has hitherto been impeded by the large

RCS such structures present to both civilian and military radar systems (Tennant and Chamber

(2006)). Furthermore the ability to distinguish between moving aircraft and static objects by

using Doppler filtering is negated since the tip of the turbines can easily achieve speeds of 100

miles per hour. This has precluded the development of many otherwise preferred potential wind

farm sites due to their proximity to airports. The application of absorbing material can

significantly reduce the RCS of wind turbines and affords much greater freedom in the location

of wind farms.

The explosive growth of wireless networks and mobile telecommunications over the past

decade, has created a desire to control the radio frequency environment within buildings. For

example, a study by Houliston et al (2009) revealed that RFID systems have the potential to

cause significant interference with drug infusion equipment in hospital environments. Another

study by Butterworth et al (1997) considered the propagation of radio frequency signals within a

building in relation to the use of a mobile communication system. In both cases there is the

potential to apply thin materials which selectively absorb and/or transmit to either screen out

unwanted signals or to reduce problems arising from reflections and interference. Selective

transmission and absorption of particular frequencies can be applied to prevent the use of

cellular phones whilst permitting the use emergency TETRA band radio. A similar requirement

exists in the creation of ―Quiet zones‖ in public places such as on trains, where the ability to

continue to use TETRA band frequencies is paramount. Thin absorbing and transmitting

materials such as those detailed in this thesis are of particular value in these situations.

Conventional frequency selective screens (Munk (2000)) could also be applied in this instance

52

but might not afford the same degree of frequency selectivity, without requiring multiple layers

and hence increased thickness.

53

Chapter 3

Modelling

3.1 Introduction

The theoretical results presented in this thesis have been produced using Ansys’s High

Frequency Structure Simulator (HFSS) software. HFSS is a finite element code that can be

applied to simulate the electromagnetic behaviour of various systems from antennas, to radar

absorbers, frequency selective screens and even complex PCB geometries. In this chapter the

basis of the finite element method is described and the manner in which HFSS is used to

simulate the behaviour of the microcavity arrays is presented. Some of the work in this chapter

has been adapted from Huebner (2001) and Zienkiewicz et al (2005), the HFSS user manually

has been used extensively to provide information specific to HFSS.

3.2 The finite element approach

According to Huebner (2001), Finite element analysis is a numerical method of finding

approximate solutions to partial differential equations, such as Maxwell’s equations. The

method consists of dividing the problem geometry into a series of small elements which

combine together to form a mesh representing the entire structure.

Finite element analysis can be applied across a diverse range of problems in physics and

engineering including: fluid dynamics, mechanical vibrations and stress, thermodynamics and

electromagnetism. In each of these cases the governing equations and boundary conditions are

generally well-known but very often the geometries under consideration are arbitrary or feature

irregularities, thus precluding analytical solutions. Furthermore, when considering the behaviour

of a field quantity within a continuous body, there are an infinite number of unknowns since the

field quantity can possess any number of values at each generic point within the body (Huebner

(2001)).

54

These problems can be transformed into ones of finite unknowns by dividing the geometry into

a series of elements within each of which the unknown field quantity is expressed as an

approximating function, the form of which is assumed. The approximating functions are defined

in terms of the values of the field quantities at specific points or nodes. Nodes lie either on the

boundaries between adjacent elements or within an element. Hence the nodal field values and

the approximation functions define the behaviour of the field quantity within an element. In

finite element analysis the nodal values are the unknowns and once found the approximation

functions then describe the behaviour of the field quantity throughout the entire geometry, this

is the essence of the finite element method (Zienkiewicz et al (2005)).

Within HFSS the mesh elements take the form of tetrahedra, and the field quantities of interest

are the electric and magnetic fields. The accuracy of the HFSS solution depends on the size of

the elements with smaller elements producing more accurate answers. However, the smaller the

average element size the greater the number of elements that is needed to represent the entire

structure and hence the larger the computational resource needed to solve the problem. HFSS

deals with the compromise between problem size and accuracy by using an iterative method in

which the element size is progressively reduced and hence problem size and accuracy

progressively increased. The solution resulting from each successive iteration is compared to

that from the previous iteration. Once the difference in successive solutions is sufficiently small

one can be confident that any further refinements to the mesh are superfluous and the solution is

said to have converged. Typically, a convergence criterion equivalent to a change in the solution

of less than 1% is applied.

3.3 An overview of HFSS

3.3.1 Assembling the structure to be simulated

Within HFSS there exists a CAD program which allows the user to assemble the structure to be

simulated by using a number of commands for the construction, to-scale, of one-, two- and three

dimensional objects. For regular forms such as cuboids and cylinders there exist integral

drawing commands where the user simply specifies the location of one corner of a cuboid for

55

example, then enters its x-, y- and z-dimensions. For more complex and/or irregular shapes, the

user must execute a series of drawing commands in a certain sequence in order to form the

desired geometry. Such commands might include rotating objects, mirroring an object about a

plane, splitting objects or creating a surface from the intersection of two object faces. Such a

surface can then be rotated about an axis or swept along a vector in order to create a three-

dimensional object. There exist myriad combinations of these commands such that almost any

desired structure can be assembled. The series of drawing commands that have been executed

by the user are stored within HFSS but can be modified retrospectively as required, for example

one may wish to change the radius of a sphere or the location of a plane.

3.3.2 Assigning material properties

When initially created, each three-dimensional object is automatically assigned to have the

electromagnetic properties of vacuum by default. Two-dimensional objects are defined in terms

of a boundary condition rather than a material property as will be described later. The material

properties of a three-dimensional object can be assigned in terms of a number of quantities

including:

o Relative permittivity – the real component, 'r and dielectric loss tangent ''' rr

where ''r is the imaginary component, are specified

o Relative permeability – the real component, 'r and magnetic loss tangent ''' rr

where ''r is the imaginary component, are specified

o Bulk conductivity in Siemens/metre

The user may themselves enter the values of the requisite quantities or they may choose from an

integral database which lists typical values for well-known materials. Furthermore, it is possible

to represent anisotropic materials by entering values corresponding to those in the three

Cartesian axial directions. The frequency dependency of materials can also be considered by

56

either entering the known quantity values at specific frequencies or by using integral fitting

models including: linear, Debye and Djordjevic-Sarkar (Hall et al (2009)).

Within HFSS, metal objects that have high but finite conductivity are represented by a surface

approximation: the field solution is generated only at the surface of the metal not within its

volume. This finite conductivity boundary approach reflects the fact that metals are imperfect

conductors – they are to some extent, lossy. At the surface of these imperfect conductors the

following boundary condition applies:

tan

tan

H

EZ s (3.1)

Where:

Etan is the component of the electric field tangential to the boundary or surface

Htan is the component of the magnetic field tangential to the boundary or surface

ZS is the surface impedance

In cases where the thickness of the metal is much larger than a skin depth, the surface

impedance can be found from equation (3.2):

2

1

21

jjZs (3.2)

Where:

is the angular frequency of the excitation,

is the conductivity of the metal,

is the permeability of the metal

And:

2

(3.3)

This approach is only valid if the metal’s thickness is significantly greater than the skin depth at

the solution frequency. If this condition is not met then it is necessary to apply what is termed a

layered impedance boundary. A layered impedance boundary represents the thin metal layer as a

57

two-dimensional surface which has a complex impedance resulting from the intrinsic properties

of the material in question (typically copper for the models used in this thesis) and its thickness.

The user is required to select the surface in question, its constituent materials an also enter a

value for surface roughness, h: the root-mean-square deviation of the surface of the metal from a

plane.

The greater the surface roughness the lower the conductivity. HFSS uses the following formula

to modify the conductivity to allow for roughness:

2

w

cK

(3.4)

Where:

c is the modified conductivity

And:

6.1

2exp1

hKw

(3.5)

And all other terms have the meanings given above.

Using the user-defined thickness, conductivity and roughness, HFSS is then able to calculate the

complex surface impedance using analytical formulae which are not detailed in the user manual.

3.3.3 Boundary conditions

The behaviour of the electromagnetic fields at the surfaces of the user-defined geometry and at

the interfaces between objects are specified by one of the following boundary conditions:

o Layered impedance

o Perfect E

o Perfect H

o Radiation

o PML

58

o Finite conductivity

o Symmetry

o Master and Slave

o Lumped RLC

Layered impedance boundaries have been detailed above. A perfect E boundary represents a

surface which is perfectly conducting: the component of electric field tangential to the surface is

zero hence the electric field is normal to the surface. A perfect H boundary is one across which

the tangential component of H is continuous. For surfaces internal to the model this represents a

boundary that is ―natural,‖ and the field propagates through with no distortion. For external

model surfaces this boundary condition simulates a perfect magnetic conductor: one on which

the tangential component of the H-field is zero.

In situations where the geometry being simulated radiates or re-radiates, HFSS applies a

radiation boundary condition to truncate what would otherwise be an unbounded problem. This

boundary condition is designed to remove the radiated energy from the model but without

causing any unphysical reflection that would distort the solution. There are two types of

radiation boundary condition that HFSS applies: absorbing boundary conditions (ABCs) and

perfectly matched layers (PMLs).

ABCs can only absorb waves which are incident normal or near-normal to the boundary surface:

whilst there is no strict definition, in this case near-normal would generally mean angles up to

approximately 20°. Furthermore ABCs must be spaced at least a quarter-wavelength away from

radiating structures. These restrictions arise because the ABC is an approximate boundary

condition the accuracy of which decreases rapidly as the distance between it an any radiating

structure decreases to less than a quarter wavelength.

PMLs are three-dimensional structures which consist of artificial materials, the properties of

which are automatically tailored by HFSS such that waves incident upon them are wholly

59

coupled into the PML with minimal reflections. The permittivities and permeabilities of the

PML are complex and anisotropic in order to provide material which is absorbing and which has

an intrinsic impedance matched to that of free-space i.e. the relative permittivity and

permeability are equal, hence the term perfectly matched layer. In contrast to an ABC, a PML

can absorb waves incident over a wide range of angles and can therefore be applied to problems

in which waves are incident on a surface at angles significantly off-normal. The disadvantage of

PMLs is that they require greater computational resource than do ABCs.

Finite conductivity boundaries are automatically applied to the faces of three-dimensional

objects which are assigned the properties of a metal but can also be applied to two-dimensional

surfaces. A description of the finite conductivity boundary condition has been given above.

A symmetry boundary can be applied to exploit a plane of electromagnetic symmetry which a

problem geometry possesses. Using a symmetry boundary condition allows only half or, if two

symmetry boundaries are applied a quarter, of the problem geometry to be modelled and the

solution can be mirrored about the plane or planes to produce the solution for the entire object.

This approach has the advantage of significantly reducing the required computational resource

without compromising the accuracy of the solution.

Periodic boundary conditions, also known as master and slave pairs, can be applied to periodic

structures in order to represent an infinite array of unit cells whilst only drawing and solving the

field equations for one unit cell. These boundary conditions force the electric field on the slave

boundary to be identical to that on the master boundary to within a phase difference. This type

of boundary condition has been used extensively throughout this thesis and is detailed with

reference to microcavity arrays in the next section.

A lumped RLC boundary can be applied to designate a surface as having a certain resistance

and/or inductance and/or capacitance. This type of boundary is often applied to model problem

60

geometries which include circuit components e.g. RFID tags. This boundary condition has not

been used within models presented this thesis.

3.3.4 Excitations

Within HFSS there exist several methods for exciting i.e. injecting electromagnetic energy into,

the structure. These include:

o Wave ports

o Lumped ports

o Incident waves

o Voltage sources

o Current sources

o Magnetic bias

Wave ports are two-dimensional objects applied to an external face of a problem geometry that

represent a cross-section through a waveguide. The problem geometry is therefore excited by

the modes naturally supported by such a waveguide. Wave ports can only be applied to external

faces in contrast to lumped ports which can be applied to faces internal to the modelled

geometry. Lumped ports also feature a reference impedance that is defined by the user. Note

that neither wave nor lumped ports have been used in models presented in this thesis.

Incident wave excitations have been used exclusively throughout this thesis and consist of

sinusoidal radiation incident upon the problem geometry. Within HFSS it is possible to define

the form of the incident wave as being plane or cylindrical, and to define the wave as being

evanescent if desired. An evanescent wave is a plane wave but one whose amplitude decays

with distance in the propagation direction.

Throughout the models presented in this thesis, plane incident wave excitations have been used

to excite the microcavity array structures. The angles of incidence and azimuth can all be

tailored as required by entering the appropriate values for the propagation vector in spherical

61

polar co-ordinates. The polarisation state can be varied from TE to TM or the electric field

vector can be orientated at any angle between these two extremes.

3.3.5 Meshing

After the model geometry has been input, material properties specified, boundaries and

excitations applied, the solution process begins with the generation of what is termed the initial

mesh. This coarse mesh of tetrahedral elements then undergoes optional ―lambda refinement,‖

whereby the maximum tetrahedral size is reduced to a user-defined fraction of a wavelength,

typically 0.33. This ensures a basic level of resolution taking into account the materials used

within the model. HFSS then uses this mesh to create an initial field solution at a frequency

specified by the user. Typically, an initial mesh contains between 500 and 2000 elements.

The objective of the mesh refinement process is to create an array of elements each of which is

small enough to represent the field accurately but not so large that the overall problem size

becomes prohibitive. The required tetrahedral size is determined locally by selectively refining

the mesh in accordance with the field gradient: in areas of high field gradient the tetrahedral size

must be smaller in order to accurately represent the field. The generation of the initial mesh

allows these high error density areas to be identified and they become the initial focus of the

mesh refinement.

The refinement of the mesh is monitored by comparing the solution it yields to that yielded by

the previous mesh iteration. As the tetrahedral size decreases the level of detail increases,

eventually a point is reached where the mesh affords sufficient detail and further decreases in

tetrahedral size serve only to increase the computational resource required. At this point the

solution is said to have converged and the mesh refinement process stops. Typically, the

criterion for the refinement process to be stopped is a change in the field solution between

successive iterations of less than 1%. For the microcavity array structures considered in this

thesis, convergence typically occurs once the mesh contains 30, 000 to 40, 000 elements. Such a

model might take

62

The figure below contains images of typical initial and final meshes. In this case the structure

being simulated is a mono-grating type microcavity array with the slit located at the centre of

the unit cell. The metal layers are shown by the thick black lines added to the 2-D elevations;

Figures 3.1(b) and (d). The decrease in the size and increase in the number of tetrahedra is

readily visible.

(a) (b)

(c) (d)

Figure 3.1 Plots of typical finite element meshes constructed using HFSS

(a) 3-D projection of the initial mesh for a typical microcavity structure, 1258

tetrahedra (b) cropped 2-D side elevation of the initial mesh for a typical

microcavity structure, 1258 tetrahedra (c) 3-D projection of the final mesh for

typical microcavity structure 43401 tetrahedra (d) cropped 2-D side elevation

of the final mesh for typical microcavity structure 43401 tetrahedra

63

Once solution convergence has been realised a frequency sweep can begin. Rather than create a

new mesh HFSS uses the last mesh created during the refinement process during the frequency

sweep. There is an assumption that this mesh is sufficient to produce accurate solutions across a

range of frequencies. However, this assumption is validated by the results in later sections.

Furthermore, by meshing or re-meshing at the resonant frequency of the structure the highest

field gradients will already have been encountered and accommodated by the time other

frequencies, including higher frequencies, are simulated. The total run time for a model

including meshing, convergence and a frequency sweep is typically between four and eight

hours depending on the number of points required to accurately describe the resonant modes

present.

The above process can be set-up to run automatically and the user will often not need any

knowledge of the detailed form of the mesh. However, in certain circumstances it is

advantageous to instruct HFSS to create a finer mesh in specific areas of interest. For example

when considering the detailed behaviour of the field in the vicinity of a sub-wavelength

aperture, a very fine mesh might be required such that the field plots have sufficient detail.

The process of manually creating regions of finer mesh is known as seeding. Seeding can be

done one object at a time by specifying the maximum length of the tetrahedra within the object.

However, in some cases an even finer level of control over the mesh is required and the use of

what are known as virtual objects is required. A virtual object is a small object drawn entirely

inside a larger object and has identical material properties to the larger object. The virtual object

is located where the greater level of detail is required and it alone is seeded as required. This

leaves the majority of the volume of the larger object with a coarser mesh and thus uses less

computational resource than seeding the whole object would require.

64

3.3.6 Post-processing

The field solution produced by HFSS can be used to determine many of the electromagnetic

characteristics of the problem geometry. For the microcavity array structures considered in this

thesis, the focus is on determining the reflectivity and transmissivity, however other

characteristics such as the near field and far field radiation pattern can also be studied. In order

to understand the interaction of microwaves with the microcavity arrays, the electric and

magnetic fields of the resonant modes supported can be plotted and studied in three dimensions

and can be animated with respect to phase, frequency and position.

The HFSS fields formulation divides the problem into three sources: incident fields, total fields

and scattered fields. The incident field is that arising from the incident wave as defined by the

user. The total field is that which exists by virtue of the incident field and its interaction with the

structure in question: it is the total field that the HFSS field solution determines. The scattered

field is determined from the difference between the incident and total fields. To illustrate this

approach, consider the following example.

Consider a model of a metal plate, infinite in extent within the xy plane and that is several tens

of skin depths in thickness in the z-direction. In a laboratory measurement, a wave incident on

the surface of this plate in the z-direction would be near-perfectly reflected; there would be no

transmission of the wave through the plate. Within HFSS the incident field is simply the wave

incident on the plate, but the manner in which HFSS is formulated places this exciting field

everywhere within the model. Therefore, even though there can be no transmission of the field

through the plate, the incident field does exist in both half spaces – both above and below the

plate.

The total field is the actual field solution that would exist above and below the plate in a

laboratory measurement. So, below the plate the total field is zero and above the plate the total

field is the superposition of the incident wave and that reflected from the metal plate. The

scattered field is determined by subtracting the total field from the incident field, allowing for

65

both amplitude and phase. In the upper half-space this corresponds to the wave reflected from

the metal plate – but only the reflected component. Intuitively one might expect the scattered

field in the lower half-space to be zero since no energy can propagate through the plate.

However, the scattered field amplitude in the lower half-space is high-valued, but the wave is in

anti-phase with the incident field in this region. This results in destructive interference and the

net amplitude (i.e. total field) in the lower half-space is zero – as expected.

The above description can be applied to structures which do transmit radiation. In such cases the

total fields can be used to find the fractional transmission coefficient whilst scattered fields must

be selected to find the fractional reflection coefficient. In both cases the time-averaged Poynting

vector must be integrated over a plane which cuts through the model. The ratio of this

calculation for scattered fields to that for incident fields yields the reflection coefficient when

performed over a plane located in the upper half-space. Similarly, total fields yields the

transmission coefficient when the total fields calculation is performed over a plane in the lower

half-space.

3.4 Modelling approaches used in this thesis

3.4.1 Mono-grating reflection structures

Chapter 4 details the angle- and polarisation-dependant reflectivity of microcavity array

structures in which only one of the two metal layers is perforated by sub-wavelength slits. The

total thickness of these structures is less than 1% of their operating wavelength and the width of

the slits is of the same order of magnitude. These structures support resonant modes which

exhibit field strengths around 100 times greater than those of the incident wave. These factors

combine to produce very high field gradients and consequently a dense finite element mesh.

The metal and dielectric layers were formed from cuboids drawn using the HFSS primitive

object commands. The metal layers were designated as copper with a conductivity of 5.8. 107

S/m as obtained from the HFSS materials database. The permittivity of the FR4 core was set to

(4.17 +i0.07) in accordance with analysis of free-space analysis of a plain FR4 sheet using the

focused horn apparatus as detailed in Chapter 4.

66

One period of the structure in the y-direction (a length of 10 mm) was drawn using the HFSS

user interface with an infinite array of these unit cells being modelled by application of periodic

boundary conditions. The x-dimension was set to 4 mm and the application of a second set of

periodic boundary conditions resulted in an infinite array in both x- and y-directions being

modelled. The sub-wavelength slit was located at the centre of the unit cell between y = 4.85

mm and 5.15 mm, thus it has a width of 0.3 mm. In later models the effect of altering the slit

width and the dielectric core thickness were investigated. These dimensions are easily

manipulated in the model using the same commands as were used to draw the model described

above. However, every time a dimension of material property is changed, the solution must be

generated again from the beginning.

(a) (b)

Figure 3.2 Mono-grating structure as modelled in HFSS

(a) Selected Dimensions and materials (b) Boundary conditions

Above the microcavity structure a box designated as vacuum was added. This box was 40 mm

high – approximately one wavelength at the fundamental resonant frequency. Whilst this is

67

larger in the z-dimension than is generally regarded to be adequate, experience has

demonstrated that a larger vacuum box produces more accurate values when integrating the

Poynting vector over the plane located as shown: if the plane was closer to the surface of the

microcavity array, then the integration may include reactive field components rather than just

those of the propagating reflected wave.

To the top of the vacuum box was added the PML, the thickness of which was set to 7.5 mm.

The top face of the PML was not explicitly assigned any boundary condition, therefore HFSS

assigns it a perfect E condition by default. Again 7.5 mm is thicker than the minimum value the

HFSS PML macro recommends, but experience demonstrates that a greater thickness produces

more accurate solutions, particularly at higher incident angles. This is due to the anisotropy of

the PML: the imaginary permittivity and permeability are much greater in the z-direction than

the x- and y-directions, hence the wave amplitude decreases much more rapidly in the z-

direction than the other two directions. However, as the incident angle is increased from 0° the

specular reflected wave propagates through the PML at angle to the z-direction, hence the rate

of attenuation with distance in the z-direction is diminished. A thicker PML is therefore

required in order to ensure that total attenuation is sufficient to eliminate any reflections.

Models were run at incident angles of 16° and 57.4° to allow comparison with experimental

data. Models were meshed at the resonant frequency of the fundamental model: 7.1 GHz in this

case. The initial mesh had approximately 1, 300 tetrahedral elements and after 20 iterations this

had increased to approximately 32, 000, by which point the variation in the solution between

iterations was approximately 0.6%. The final mesh has been plotted in the figure below. It can

be seen that the typical element size is much less in the core and in particular around the slit

than it is in the vacuum box above the microcavity structure. This demonstrates the selective

nature of the mesh refinement process as described above.

68

(a) (b)

Figure 3.3 Finite element mesh for the mono-grating reflection structure

(a) 3-D projection (b) Cropped 2-D projection, metal layers marked by black

lines

In later versions of HFSS, including V.11 as was used to produce the mesh diagram above, the

need to perform the Poynting vector integration manually was obviated by the addition of an

optional "Reference for FSS" command to the PML set-up. When selected by the user, this

command automates the calculation of the reflection coefficient by using the boundary between

the vacuum box and the PML as the integration surface.

3.4.2 Mono-grating transmission structures

Chapter 4 also details the microwave reflectivity and transmissivity of two microcavity array

structures in which both of the metal layers are perforated by sub-wavelength slits of equal

spacing. In the first structure the slits in the upper layer are aligned with those in the lower

layer, whereas in the second structure the two sets of slits are off-set by half the slit spacing.

The slit spacing was set to 10 mm, the slit width to 0.3 mm and the core thickness to 0.356 mm.

The structure was located at the centre of a vacuum box which had a total height of 80 mm, and

thus divided the vacuum box into upper and lower half-spaces for the consideration of reflected

and transmitted fields respectively.

69

In all cases an incident wave excitation was used and the radiation was incident normally upon

the structure. Two pairs of periodic boundaries were applied as per the reflection models such

that an infinite array of microcavites was modelled. That the radiation was incident normally

meant that a two-dimensional absorbing boundary condition could be used instead of a PML.

This boundary condition was applied to both the x-y faces of the vacuum box and the reference

for FSS option was selected thus allowing the reflection and transmission coefficients to be

calculated automatically.

Figure 3.4 Diagram of the off-set transmission structure

The aligned structure was meshed at its fundamental frequency of 7.2 GHz, the initial mesh

contained approximately 900 tetrahedra increasing to approximately 30, 000 tetrahedra after 20

iterative passes. After the 20th pass the difference between successive solutions had decreased to

0.3%. The off-set structure has a fundamental resonant frequency of 13 GHz and was therefore

meshed at this frequency. The initial mesh for this structure contained approximately 1, 500

tetrahedra - more than the aligned structure due to the higher meshing frequency and the lambda

refinement criterion. After 18 passes the mesh had grown to approximately 38, 000 tetrahedra

and the difference between successive solutions was less than 1%.

70

3.4.3 Bi-grating reflection structures

In Chapter 5 two sets of mutually orthogonal sub-wavelength slits are cut into the upper metal

layer thus dividing it into an array of square patches. This bi-grating structure therefore

possesses 90° rotational symmetry and the focus of Chapter 5 is exploring the incident and

azimuthal angle dependence of the modes this structure supports. Extensive modelling of this

structure was undertaken in order to understand the character of these modes.

In all cases the microcavity structure lies parallel to the x-y plane and the inter-slit spacing is 10

mm for both sets of slits, the slits themselves are 0.3 mm wide. The dielectric core material is

0.356 mm thick FR4 with complex permittivity (4.17 + i0.07) and the metal layers are 0.018

mm thick copper with conductivity 5.8.107 S/m. The microcavity structure resides at the bottom

of a 40 mm high vacuum box, to the top of which is added a 7.5 mm thick PML. The structures

were excited with an incident wave over a variety of incident and azimuthal angles, hence the

use of a PML boundary condition.

The model was constructed such that the slit was to the periphery of the unit cell and the square

metal patch in the centre, this is in contrast to the mono-grating structure which has the slit

located at the centre of the unit cell. However, the application of the periodic boundary

conditions merely requires that the material properties of the object which has an interface with

the master boundary are the same as those which has an interface with the slave boundary,

hence there is a choice in where to locate the boundaries.

Several models were solved over a range of incident angles, including 57.4°. Initial results for

this structure were poor with the reflection coefficient values exceeding unity at frequencies

below 6 GHz. These non-physical results were eliminated by seeding the mesh. Three separate

mesh seeding criteria were applied: to the vacuum box a target maximum tetrahedral length of 1

mm was specified within the bounds of the total number of tetrahedra within the vacuum box

not exceeding 1, 000. Within the dielectric core the target maximum tetrahedral length was set

to 0.5 mm, again with the restriction that the total number of tetrahedra did not exceed 1, 000.

71

Finally the surface of intersection between the vacuum box and the PML was set a target

maximum element length of 1 mm but with no restriction on the number of triangles rather than

tetrahedra as this is a surface rather than a volume. For this model the initial mesh contained

approximately 32, 000 - a consequence of the strict mesh seeding and the addition of the second

set of slits. The field gradient is highest within the slits hence the mesh density is also high

therein. The bi-grating model has approximately 20 mm of combined slit length - five times that

of the mono-grating model, hence more tetrahedra are required. After five passes the number of

tetrahedra had reached approximately 55, 000 and the difference between successive solutions

was less than 0.1%.

Shown in Figure 3.5 below, is the model geometry, Figure 3.5(a) and the finite element mesh on

the vacuum box only, Figure 3.5(b). The mesh exhibits a high degree of regularity and the

results of seeding the intersection of the vacuum box and PML are also apparent: there is a very

fine mesh in this area. This fine mesh improves the accuracy of the field solution and therefore

of the reflection coefficient values as this surface is used to perform the automated Poynting

vector integration.

72

(a)

(b)

Figure 3.5 Diagram showing the bi-grating model

(a) the model geometry (b) the finite element mesh

3.4.4 Tri-grating reflection structures

Chapter 6 considers the microwave reflectivity of structures in which there are three sets of sub-

wavelength slits in the upper metal layer, each orientated at 60° to the other two. These

73

structures therefore exhibit 60° rotational symmetry and support a greater number of modes than

either the mono-grating or bi-grating structures.

Two hexagonally symmetric configurations were considered: on the first sample all three sets of

slits intersect at common points creating an array of equilateral triangles of side equal to 32l

where l is the repeat period of each set of slits measured parallel to the grating vector, see Figure

3.6(a). In the second structure each set of grooves is off-set by half its repeat period relative to

the other two sets hence all points of intersection feature slits from only two of the three sets

and a pattern of hexagons interspersed with equilateral triangles both of side 3l is formed, see

Figure 3.6(b).

Inputting these geometries into HFSS is somewhat more complex than either the mono-grating

or bi-grating structures which could be formed from simple primitive drawing commands since

they feature only parallel or perpendicular sides. The 60° rotational symmetry of the tri-grating

structures can be dealt with by using spherical rather than Cartesian co-ordinates or equivalently

by using the rotate command rather than translate command within HFSS. Using Cartesian co-

ordinates introduces a degree of inaccuracy since the projection of the unit cell side onto the

axes involves taking sine of the 60° angle which is an irrational number. This irrational number

must be rounded to a finite number of decimal places before it can be entered into the model but

the accumulation of these input rounding errors can result in a meshing error and the model may

fail to solve.

74

Figure 3.6 The tri-grating sample geometries (not to scale) and the co-ordinate

system used (a) 3-D projection of tri-grating 1 (b) 3-D projection of tri-grating

2, tm = 18 μm, tc = 356 μm, ws = 0.3 mm, 12 gg = 10 mm, is the polar

angle, is the azimuthal angle

Figure 3.7 shows the various stages in the construction of the upper metal layer of the tri-

gratings. Start with a copper layer 0.018 mm thick, set the x- and y-dimensions to 30 mm

arbitrarily. Then create an array of slits 0.3 mm wide, 0.018 mm deep and spaced by 10 mm -

Figure 3.7(a). Create a copy of the slits then rotate these by 60° about the z-axis using the rotate

command - Figure 3.7(b). Repeat this process but rotate by -60° - Figure 3.7(c). Now all three

sets of slits have been created, at the correct spacing and orientation and no irrational numbers

have been input.

75

Figure 3.7 Forming the tri-grating structures without inputting irrational numbers

(a) metal plate (30 x 30) mm with three slits spaced 10 mm apart (b) second set of

slits added and rotated by 60° about the z-axis (c) third set of slits added and

rotated by -60° about the z-axis (d) translation of first set of slits by 5 mm in the z-

direction (e) subtraction of all three sets of slits from the metal layer, two unit

cells can be seen

Subtracting the slits from the copper will form the upper metal layer of tri-grating sample 1 as

shown in Figure 3.7(c). Before performing the subtraction, translate the original set of slits by 5

76

mm in the x-direction, Figure 3.7(d), a subsequent subtraction process yields the upper metal

layer of tri-grating 2, Figure 3.7(e). Note that because the translation was entirely parallel to one

of the Cartesian axes no irrational numbers were input. A similar albeit simpler process can be

followed to form the lower metal layer, dielectric core and vacuum box. The excess cells can be

removed by the use of further rotate, translate, and split about axis or subtract commands, and

will yield accurate results without the need for rounded irrational numbers to be input, provided

that all translations take place parallel to the x- or y-axis and parallel to one of the k-vectors of

the structure.

In a similar manner to the preceding mono-grating and bi-grating structures, periodic boundary

conditions were applied and a 7.5 mm thick PML was added to top of the 40 mm high vacuum

box. The established permittivity and conductivity values for FR4 and copper respectively were

applied. As with the bi-grating, it was found that seeding the mesh helped to improve the

accuracy of the reflectivity values. The same seeding criteria were applied to both tri-gratings as

were applied to the bi-gratings, typically starting meshes contained approximately 40, 000

tetrahedra and converged after 5 - 6 passes whereupon the mesh had grown to around 70, 000

tetrahedra. Plots of selected sections of the final meshes are shown in Figure 3.8: it is clear that

seeding has resulted in a highly dense mesh in both cores and on the integration surfaces.

77

(a) (b)

Figure 3.8 Plots of selected parts of the final finite element mesh for the tri-

grating structures (a) for tri-grating 1 (b) for tri-grating 2

3.4.5 Broadband structures

Chapter 7 details the response of several microcavity array configurations designed to afford

increased absorption bandwidth over the structures hitherto investigated. Four different

strategies were pursued in an attempt to maximise absorption bandwidth whilst minimising the

thickness of a the mono-grating structure. The first strategy involved alternating between two

inter-slit distances such that the modes excited in adjacent cavities would correspond to

different frequencies - two fundamental resonances would result. Another similar strategy with

the same goal was to alternate between two different values of core material permittivity. Both

these strategies used modelling approaches that have been covered in detail in earlier sections

hence no repeat of that treatment is given here.

3.4.5.1 Non-parallel slits

Of the two remaining strategies, the first introduces the concept of non-parallel slits with the

intention being to excite a continuous albeit finite range of resonant frequencies. Every other slit

has a saw-toothed shape, the period of the saw-tooth being 40 mm and its amplitude being 2

mm. Thus the distance between adjacent slits varies from 8 mm to 12 mm and back to 8 mm

78

every 40 mm along the y-axis, see Figure 3.9. Successive saw-tooth slits are half a period out of

phase with each other, thus making the overall period of the structure 40 mm in the x-direction.

Figure 3.9 Multiple continuous repeat periods with alternate saw-tooth slits

The saw-toothed slits were formed in the upper metal layer by using the polyline command in

HFSS, whereby the user inputs the locations of the corners or vertices of a two-dimensional

object. This two-dimensional object is then swept along an axis to form a three-dimensional

structure. In this case the polyline was swept along a vector 0.018 mm along, parallel to the z-

axis (out of the page) to form an object which was then subtracted from the upper metal layer,

thus leaving a saw-toothed aperture. The application of two pairs of periodic boundaries then

results in an infinite array of the unit cell being modelled. This takes account of the periodicity

in both the x- and y-directions. All other aspects of this design were formed by use of the

primitive drawing commands for forming sheets and cuboids. Material properties were as per

the standard mono-grating: dielectric core material was FR4 with relative permittivity (4.17 +

i0.07), all metal layers were copper with conductivity 5.8.107 S/m.

No mesh seeding was undertaken for the models of this structure as good convergence and

reliable, accurate results were obtained without such intervention. Since the unit cell for this

structure measures (40 x 40) mm rather than (4 x 10) mm as for the standard mono-grating, a

79

much larger initial mesh was required: typically 4, 000 tetrahedra were required initially,

increasing to approximately 48, 000 after 18 passes, whereupon the solution converged.

3.4.5.2 Multi-layer structures

The third method for achieving additional absorption bandwidth consisted of stacking a series of

mono-grating transmission structures, each of identical period but differing core material

properties, one upon the other. Three transmission structures were backed by one reflection

structure thus forming a four-layer, stacked microcavity absorber, see Figure 3.10.

Figure 3.10 Multi-layer microcavity structure

(a) 3-D projection of multi-layer structure, 2 periods shown (b) end projection

of multi-layer

The multi-layer structure consists of 4 layers each with a period of 10 mm and slit width of 0.3

mm. The four layers are stacked one upon the other such that their slits are aligned: the top three

layers have slits in both copper layers to allow penetration of the incident wave to the lower

layers. Starting with the lowermost layer and working upwards, the dielectric materials used

were: alumina (Al2O3) FR4, polyester (PET) and air. The complex permittivities of PET and

alumina are εr = (3.2 + 0.0096i) and εr = (9.4 + 0.0564i) respectively and were obtained from

the materials database within HFSS. All metal layers were assigned the conductivity value of

copper.

The period of the structure in the y-direction is 10 mm, the structure is not periodic in the x-

dimension, therefore the x-dimension of the unit was set to 4 mm. Two pairs of periodic

boundaries were applied to represent an infinite array. The structure was excited with a

80

normally incident wave hence an absorbing boundary condition rather than a PML was used.

Whilst no mesh seeding was found to be necessary, the inclusion of four layers resulted in a

starting mesh of approximately 7, 000 tetrahedra. Furthermore the model was slow to converge

and required 18 passes and approximately 102, 000 tetrahedra for the variation between

successive solutions to fall to 1%.

3.5 Summary

This chapter has described the basis of the finite element method and the manner in which

HFSS can be applied to simulate the behaviour of the microcavity arrays. The method of

inputting the geometries of the various microcavity structures has been detailed and the types of

boundary which can be applied have been described and explained. The manner in which HFSS

deals with dielectric materials is covered as is the use of a surface impedance approximation in

simulating the behaviour of metals in a computationally efficient manner. The ability to tailor

the finite element mesh to suit a particular problem geometry has also been presented with

reference to the microcavity bi-grating and tri-grating structures.

81

Chapter 4

The microwave reflectivity and transmissivity of a low-loss dielectric

layer disposed between two metallic layers perforated periodically by

sub-wavelength slits

4.1 Introduction

Conventional materials for the attenuation of centimetric electromagnetic radiation have

historically been limited to a minimum thickness of a quarter-wavelength, with examples of

such materials including Salisbury screens (Salisbury (1952)) and Dallenbach layers (Knott

(1993)), this limitation can present significant practical problems in terms of thickness and

weight penalties incurred. In this chapter ultra-thin cavities are studied for use as low-profile

resonant absorbers: two metal layers are spaced by a dielectric whose thickness is grossly sub-

wavelength, the uppermost of which is perforated periodically by a single set of continuous sub-

wavelength slits. Hibbins et al (2006) have applied this type of geometry to produce absorption

in the optical regime and have shown that the structure supports a series of resonant modes

which can be likened to those supported by a Fabry-Perot etalon (Takakura (2001)). This

structure can readily be scaled to produce absorption at radio frequencies. Furthermore, by

perforating both metal layers, it is possible to selectively transmit radiation through the

structure, the wavelength of which is more than an order of magnitude greater than the structure

thickness.

This first experimental chapter shall focus on the reflectivity and transmissivity of the ultra-thin

cavities as a function of the incident (or polar) angle, and the azimuthal angle of the structure

with respect to the incident wave. The specular reflectivity of and polarisation conversation

produced by the structures is studied experimentally. The nature of the resonant modes

supported by the structures is investigated by using a finite element model. The model is also

used to determine how the absorption depth can be maximised by altering structure geometry

and material properties. The transmission of radiation through structures in which both metal

layers are perforated is also studied and explained by reference to the finite element model. A

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significant proportion of the work contained within this chapter was published by Hibbins et al

(2004) and reference is hereby directed to that publication.

4.2 Background

That metal layers perforated with sub-wavelength apertures can transmit more radiation than is

incident upon the apertures themselves, has been reported by many including Ebbesen et al

(1998) and Suckling (2007). Both multiple diffraction and the excitation of surface plasmon

polaritons have been demonstrated as being important in mediating this effect (see Hibbins

(2004)). According to classical waveguide theory (Collin (1991)) a slit in a metal surface can

support the propagation of TEM modes that do not exhibit a minimum cut-off frequency, in

contrast to a hole which can support either TE or TM modes only above a certain frequency.

Two closely-spaced metal layers, the uppermost perforated periodically with sub-wavelength

slits of spacing L, constitutes an array of open-ended cavities. These cavities will support a

series of pseudo Fabry-Perot resonances, the approximate frequencies, fN of which can be

obtained from:

nL

NcfN

2 (4.1)

Here, N is the order of the Fabry-Perot mode, c is the speed of light in vacuo and n is the

refractive index of the dielectric material used to separate the two metal layers. This resonance

condition (equation (4.1)) arises from constructive interference between waves within the cavity

being successively reflected from the open ends of the cavity, due to the impedance mis-match

present at these interfaces. This constructive interference results in resonant enhancement of the

electromagnetic fields and the steady-state amplitude of the resonant mode can be more than an

order of magnitude greater than the injected field.

Resonant enhancement plays a key role in the high transmission efficiency of the structures in

which both metal layers are perforated. The sub-wavelength dimensions of the slits means that

83

they intercept only a small proportion of the radio wave energy that is incident on the structure

as a whole. This small proportion of the total incident energy is coupled through the slits in the

upper metal layer into the resonant mode within the cavity. The impedance mis-match at the

slits in the lower metal layer results in most of the energy associated with the mode being

internally reflected back along the cavity and only a small proportion coupling out through the

slit into the surrounding medium (typically air). Hence the initial proportion of energy incident

on the upper slits which is transmitted through the structure and out of the slits in the lower

metal layer is low.

However, the resonant enhancement that occurs within the cavity results in the amplitude of the

mode incident on the slits in the lower metal layer being much larger than that of the incident

wave. High transmission efficiencies can therefore result from a small proportion of a highly-

enhanced field being coupled through the slits in the lower metal layer.

Equation (4.1) predicts the resonant frequencies of the modes supported by the microcavities at

normal incidence and with the electric field of the incident wave perpendicular to the slits, and

an azimuthal angle 0 . However, rotation of the sample away from 0 and or irradiation

at any angle other than normal incidence results in equation (4.1) becoming increasingly

inaccurate. In such cases the resonant frequencies can be predicted by considering the

momentum of the incident photons and resolution of this incident momentum into the x- and y-

directions.

Assume infinitesimal slit width and set λy = 2λg for the fundamental standing wave resonance. It

can then be shown using the equation:

222

0

2

yx kkkn , (4.2)

Where n is the refractive index of the incident medium (n = 1 for vacuum), k0 is the wavevector

of the free-space wave, kx is the wavevector of the wave in the core in the x-direction, and ky is

the wavevector of the wave in the core in y-direction, that:

84

2222

0 sinsin12 gn , (4.3)

Where λ0 is the free-space wavelength.

One important distinction between the responses of the mono-grating to p-polarized and s-

polarization radiation concerns the frequency-stability of the fundamental mode with changing

polar angle. At an azimuthal angle of 0 the frequency given by equation (4.3), is

independent of the polar angle, : hence, the resonance is invariant. At this angle, coupling to

the mode from p-polarized radiation is maximized. There can, however, be no coupling from s-

polarized radiation. By comparison, the s-polarized mode is optimally coupled at an azimuthal

angle of 90 at which angle the frequency given by equation (4.3), is strongly dependent on

the polar angle. The primary consequence of this is a difference in the frequency dependence of

the p- and s-polarized absorption bands at their respective optimally-coupled azimuthal angles.

Any periodic surface will support diffracted orders, the frequency of which can also be

determined by considering the momentum of the incident photons and that supplied by the

grating. The superposition of the incident momentum with that supplied by the grating results

in a quadratic equation in k0:

0cossin2sin12

022

0 gg kkkk . (4.4)

The solution of equation (4.4) yields the limit frequency at which diffraction will occur.

4.3 Experimental

4.3.1 Fabrication of samples

The experimental samples were fabricated from double-sided PCB: a thin, low-loss, dielectric

layer sandwiched between two copper layers. The dimensions and properties of this substrate

material are detailed in Figure 4.1. These dimensions were obtained from the datasheet of the

blank FR4 material and verified to within ±10 μm using a micrometer. It was later found that

these dimensions produced a good fit between modelled and experimental data, although it

85

should be emphasised that within narrow limits (approximately 1% – 2%) there are a number of

combinations of the principal dimensions and material properties which when input to the HFSS

model that produce a good match to the theoretical data. This may be due in part to the

uncertainties in the HFSS solution which are typically quoted as ±2% (Ansys).

Figure 4.1 Cross-section through the substrate material. The dielectric core is

FR4 – a Glass Reinforced Plastic (GRP) composite material with a permittivity

of (4.17 + i0.07)

Parallel slits 0.3 mm wide spaced every 10 mm were formed in the copper layers by using a

chemical etchant to selectively remove the copper. This approach was used to fabricate one

sample which had slits in only the uppermost copper layer, and hence did not transmit any

radiation but served only as an absorber, and two samples which had slits in both copper layers.

These latter samples are capable of transmitting and absorbing radiation. In the first of these two

transmission samples, the slits in the upper and lower copper layers are aligned with each other,

the second transmission sample the slits are off-set by 5 mm – half the repeat period of the

structure. These two configurations represent the two extreme cases: perfect alignment and

perfect misalignment, see Figure 4.2.

With reference to Figure 4.2, ws is the slit width and is equal to 0.3 mm in all cases, λg is the

repeat period of the structure and is everywhere equal to 10 mm. The thickness of the metal

layers, tm, is 18 μm, and the thickness of the core tc is fixed at 356 μm.

86

(a)

(b)

(c)

Figure 4.2 Cross-section through the microcavity samples (a) the reflection

sample (b) the first transmission sample with slits perfectly aligned (c) the

second transmission sample with slits perfectly mis-aligned

87

4.3.2 Definition of polarisation state, angles of incidence and azimuth

Figure 4.3 defines the terms incident angle, azimuthal angle and polarisation state.

(a) (b)

Figure 4.3 Definition of polarisation state, incident () and azimuthal () angle

(a) TE or s-polarised – the E-vector is perpendicular to the plane of incidence

(b) TM or p-polarised – the E-vector is contained within the plane incidence

4.3.3 Measurement of microwave reflectivity and transmissivity

Two measurement systems were used during the course of this investigation:

• Focused Horn

• Long path length azimuthal scan apparatus

The focused horn system is located at QinetiQ Farnborough; the long path length azimuthal

scan apparatus is located in the School of Physics at the University of Exeter.

4.3.3.1 Focused horn

The focused horn apparatus consists of two vertical, parabolic mirrors to focus the radiation

from horizontally mounted, conical, horn antennas on to a planar sample. The sample is

mounted vertically on a reference aperture located between the mirrors. This system has the

advantage of simultaneous measurement of transmission and reflection but only at normal

incidence. It is also possible to rotate the sample mounting allowing characterisation at any

azimuthal orientation. This apparatus operates from 5.38 GHz to 18 GHz; RF energy is supplied

to the horns via co-axial lines from an Anritsu 37397C VNA. A photograph of the focused horn

apparatus is shown in Figure 4.4

88

Figure 4.4 Photograph of the focused horn apparatus The VNA can be seen in

the background, the reference aperture can be seen in the centre. The focal

length of the system is adjusted by a stepper motor attached to the nearest

mirror (out of shot to the left)

It is possible to calculate the complex permittivity and permeability of a dielectric sample using

the reflection and transmission data from the focused horn. This is routinely done at QinetiQ

Farnborough using the theory due to Nicholson and Ross (1970) and Weir (1974) written in the

form of a MathCAD program. This technique was applied to calculate the permittivity and

permeability of the FR4 core material of the VTRAM. These data are prerequisites for

modelling the VTRAM using the finite element model.

4.3.3.2 Long path length azimuthal scan apparatus

Bi-static reflectivity measurements were carried out using the long path length apparatus at the

University of Exeter. An HP8757D scalar network analyser is connected to Narda microwave

horns via co-axial cables. Spherical mirrors focus the microwave radiation onto the sample and

ensure that the degree of wavefront curvature is minimal. The sample is mounted on a non-

conducting pedestal. This pedestal is rotated in azimuth in 0.5-degree steps by a stepper motor

controlled via a PC. This set-up therefore has the advantage of fine control of azimuthal

89

orientation and semi-automatic data collection. The polarisation state of the horns can be

adjusted independently. This allows the sample to be characterised for TE or TM polarisation

and also allows any polarisation conversion effects to be quantified. The height of the mirrors

and their distance from the pedestal can be altered to allow characterisation at different incident

angles. A diagram detailing the set-up of the long path length system for reflection

measurements is shown in Figure 4.5: this set-up is similar to that of the Czerny-Turner

spectrometer (James et al (1969)).

Figure 4.5 Schematic of the long path length, azimuthal-scan apparatus set up

for measurements of bi-static reflectivity

The long path length kit can also be modified to measure transmission, see Figure 4.6.

90

Figure 4.6 Schematic of the long path length, azimuthal-scan apparatus set up

for measurements of transmission

4.4 Results and discussion

4.4.1 Reflection sample

The greyscale plots of Figs. 4.7(a) and 4.7(b) show the respective specular Rpp and Rss reflected

intensities from the reflection sample of Figure 4.2(a) with 16 . The subscripts refer to the

incident and detected polarizations, respectively. The data are plotted as a function of frequency

and azimuthal angle, the magnitude of the reflected intensity is shown by the shade of grey,

with black corresponding to a reflected intensity of 0 equivalent to 100% absorption (neglecting

any polarisation conversion effects which are fully quantified later).

91

Figure 4.7 Experimental Reflected intensities for reflection sample shown as

greyscale plots (a) Rpp data as a function of frequency and azimuthal angle at

º16 (b) Rss data as a function of frequency and azimuthal angle at º16

(c) Rpp data as a function of frequency and azimuthal angle at º57 , dashed

line corresponds to expected position of diffraction edge (d) Rss data as a

function of frequency and azimuthal angle at º57 .

The dark bands at 7 GHz and 14.6 GHz in Figure 4.7(a) indicate strong absorption. Using (4.1),

with (εr = 4.17 +0.07i) for the relative permittivity of the dielectric core one can identify the

mode at ≈ 7 GHz as the fundamental resonance. Similarly, the mode at ≈ 14.6 GHz is the

second order mode. The fundamental resonance is strongest at = 0° where the component of

the incident electric field vector perpendicular to the slits is greatest. It becomes progressively

shallower as the grating is rotated; with no resonance occurring at = 90° since there is no

component of the electric field across the slit. Conversely, the fundamental absorption band in

Figure 4.7(b), for s-polarized radiation shows greatest absorption at = 90°and no absorption

at 0. Again this is due to the electric field lying parallel to the slits for 0 with s-

polarization. Figures 4.7(c) and 4.7(d) are Rpp and Rss respectively for the mono-grating at =

92

57°. Now the bands demonstrate a much higher degree of curvature in order to satisfy

conservation of momentum (equation (4.3)). With reference to Fig. 4(c), the faint, highly

curved band centred on = 0° is due to conical diffraction. The solution of equation (4.4) has

been plotted as a dashed line on Figure 4.7(c) and closely follows the diffraction feature as

expected.

As noted above, at an azimuthal angle of 0 the frequency given by equation. (4.3), is

independent of the polar angle: hence, the resonance is invariant. At this angle, coupling to the

mode from p-polarized radiation is maximized. There can, however, be no coupling from s-

polarized radiation. By comparison, the s-polarized mode is optimally coupled at an azimuthal

angle of 90 at which angle the frequency given by equation (4.3), is strongly dependent on the

polar angle. The primary consequence of this is a difference in the frequency dependence of the

p- and s-polarized absorption bands at their respective optimally-coupled azimuthal angles and

this is indeed evident from the greyscale plots of Figure 4.7 and the line plots of Figure 4.8.

The finite element model was used to simulate the reflectivity of the reflection sample between

5 GHz and 18 GHz, at an incident angle of 4.57 and an azimuthal angle of 0 , for p-

polarised radiation. A comparison between this simulated reflectivity and that measured at the

same angles is shown in Figure 4.9. There is very good agreement between the measured and

modelled data with the fundamental and second order modes appearing as resonances at 7.1

GHz and 13.9 GHz respectively. The diffraction edge is also visible at 16.5 GHz although it is

more pronounced with the experimental data than the theory. This may be a finite size effect: if

the diffracted beam misses the mirrors and therefore the receiving horn antenna in the

experimental set-up then it makes no contribution to the overall received signal. In the model

however, the received signal is calculated by integrating the time-averaged Poynting vector over

the appropriate plane, hence even though the diffracted beam is oblique to the integration plane

there is still some contribution to the integral thus increasing the overall received signal.

93

(a)

(b)

Figure 4.8 Line plots of the reflectivity of the reflection sample as measured

experimentally at incident angles of 16 and 4.57 for (a) p-

polarisation and 0 (b) s-polarisation and 90

94

Figure 4.9 Reflectivity of the reflection sample as measured experimentally

and simulated by the finite element model: P-polarisation incident at 4.57

and 0

The finite element model can also be used to investigate the nature of the resonant modes. In

Figure 4.10(a) the instantaneous magnitude of the electric field within the dielectric core has

been plotted at a phase corresponding to peak field strength and a frequency of 7.1 GHz – that

of the fundamental mode. In this case the slit is shown at the centre of one period or unit cell of

the structure with half a cavity to each side of the slit. The colour represents the strength of the

field with red equating to a field strength of 20 V/m (20 times the incident field) and blue to 0

V/m. The field decreases from a maximum of over 20 V/m in the vicinity of the slit, to zero

mid-way between slits or in this case at the edges of the unit cell: it is apparent that between

adjacent slits, the mode completes half a cycle.

Also evident from Figure 4.10(a), is a region of zero field strength on the inner surface of the

lower metal layer immediately below the slit. The origin of this zero field region and the

detailed behaviour of the electric field within the slit can be elucidated by using the finite

element to plot the value of the z-component of the field along a line through the centre of the

95

FR4 core parallel to the y-axis see Figure 4.10(b) – and to plot the electric field vector within

the dielectric core in the vicinity of the slit – see Figure 4.10(c).

(a)

(b)

(c)

Figure 4.10 The electric field within the dielectric core of the microcavities as

simulated by HFSS: scale runs from 0 V/m (blue), to 20 V/m (red) incident

wave amplitude 1 V/m (a) Instantaneous magnitude of the electric field at 7.1

GHz, at a phase corresponding to peak field strength, the black lines represent

the copper layers (b) z-component of the electric field along a line through the

centre of the core parallel to the x-axis (c) Instantaneous the electric field

vector at 7.1 GHz, at a phase corresponding to peak field strength

96

As described by Hibbins et al (2004) the incident wave creates regions of enhanced charge

density on either side of the slit, a consequence of which is that the mode within the core must

complete another half-cycle within the width of the slit. This phase reversal within the slit

results in the field in the centre of the slit being parallel to the plane of the sample or in this case

parallel to the x-axis – see Figure 4.10(c). As the tangential component of the electric field

across an interface must be continuous, and the field inside a perfect conductor (metals

approximate perfect conductors at microwave frequencies) is zero, a region of zero field

strength is exhibited on the lower metal layer below the centre of the slit.

In this structure the slit width ws = 0.3 mm, hence the mode has suffered ―phase compression;‖

(Hibbins et al (2004)) being forced to undergo a π-radians phase change in only 0.3 mm,

whereas in the space between the slits the same phase change occurred over 9.7 mm, this is

demonstrated graphically in Figure 4.10(b). Hence within one complete period of the structure

the mode has undergone one complete cycle of 2π radians: π radians between adjacent slits as

would be expected, and an additional π radians within the slit. The period of the mode and that

of the structure are equal for the fundamental, this structure would therefore be expected to

support any modes for which N in equation 4.1 is odd but not those for which N is even.

The second-order mode for s-polarized radiation, shown as the dark bands between 14 GHz and

17 GHz in Figs. 4.7(b) and 4.7(d), cannot be coupled to at = 90° (unlike the fundamental

mode). Again this can be understood by considering the behaviour of the standing wave fields

within the core. The second order mode undergoes a phase change of 2π-radians between the

slits: when added to the enforced π-radians phase change across the slit, this results in a total

phase change of 3π-radians within one period of the structure. Therefore, at any given slit the

field resulting from the wave coupled in through that slit will be π-radians out of phase with the

field coupled in through the adjacent slits. This results in destructive interference and prevents

propagation of the mode. To permit propagation of the second order mode some additional

phase off-set is needed. At normal incidence the fields are in-phase at all slits regardless of the

orientation of the electric vector, and hence the second order mode can never be excited. Off-

97

normal there is a phase difference between slits for p-polarized radiation at all azimuths but for

s-polarization there is no phase difference between slits at = 90°, regardless of the polar

angle, and hence no mode is excited at these orientations. Furthermore, there exists an optimum

incident angle which results in the second order mode having just the right amplitude to balance

the radiative and dissipative losses, and hence maximise the depth of the second-order

resonance.

4.4.2 Optimisation of resonance depth

Consider a wave incident on the surface of the ultra thin cavity structure. A proportion of the

wave incident is reflected in the specular direction from the top metal surface and the remainder

is coupled through the slits into the core. One might expect that the proportion of the incident

energy that is initially coupled through the slits into the core is very small since the slits

themselves constitute a very small proportion of the period. The wave coupled into the core

travels along it until it reaches the next slit whereupon the impedance mismatch at this interface

results it being internally reflected back along the cavity. Interference between waves

successively reflected from the ends of the cavity will result in resonant enhancement when the

inter-slit distance is an integer number of half-wavelengths.

Figure 4.11 Waves incident on the ultra thin cavity structure

98

If say, 3% of the power incident on the cavity structure is coupled through the slits and into the

core then by reciprocity c.3% of the power that got into the core is coupled back out of the slits

or equivalently the total power re-emitted from the core each time would be 3% of 3% or 0.09%

of the total incident power. If this re-emitted wave were in anti-phase with the wave reflected

from the top surface then there would be a degree of destructive interference. However, as the

amplitudes of the specular reflection and the re-emitted wave are 97% and 0.09% respectively,

the net amplitude would be high.

Returning to the waves propagating inside the dielectric core, every cycle c.3% of the power in

the core is re-emitted back out of the slits (0.09% of the total incident energy). Every cycle also

sees another 3% of the incident power coupling into the core. For wavelengths in the core that

are exactly twice the inter-slit distance then modes running back and forth through the cavity

will set-up standing waves - this is the Fabry-Perot etalon resonance condition. At other

wavelengths the waves running back and forth do not interfere constructively and the standing

waves are not set-up.

Regardless of the proportion of the incident energy which initially couples through the slits into

the core, the amplitude of the mode in the core increases progressively over time. The standing

waves are ―fed‖ every cycle by the power coupled through the slits and into the core hence the

amplitude of the waves in the core progressively increases. The greater amplitude of the waves

in the core results in the amplitude of the waves re-emitted through the slits also being greater.

There comes a point after several cycles where the amplitude of waves in the core has been

sufficiently enhanced that the re-emitted wave has an amplitude that is of the same order of

magnitude as the specular reflected wave: in which case there is the potential for totally

destructive interference and hence the overall reflectivity of the material to be very low. When

the radiative loss from the system (the power coupled back out through the slit) exactly matches

the dissipative loss from the system (conversion of energy from the propagating mode to heat

via Joule heating of the dielectric core and metal layers) the net reflectivity of the material will

be zero.

99

Extending this argument still further it can be seen that as the amplitude of the waves in the core

continues to grow the amplitude of the re-emitted waves will eventually become greater than

that of the waves reflected at the top surface (this is known as ―Over-coupling‖). In this case the

two waves cannot interfere in a manner which is completely destructive. The net reflectivity of

the material then starts to rise again.

Consider for a moment that the mechanism for resonant field enhancement and hence

reflectivity reduction as described above, requires a series of cycles in order for the requisite

amplitude to build up. This then raises the question of whether the microcavity arrays and

resonant absorbers in general be circumvented by employing very short pulse times? Another

approach for mitigating absorption and trying to maximise the probability of detection, would

be to use a radar system with a very wide bandwidth – wider than that of the absorber – so that

at some frequencies outside the band of absorption the material’s reflectivity would be high.

These two approaches are in fact equivalent since applying Fourier analysis it can be seen that a

short time pulse is equivalent to a wideband frequency pulse.

The amplitude of the waves in the core cannot continue to rise indefinitely. As the amplitude

increases so does the power loss density PD, (power absorbed per unit volume by the core):

every time the wave propagates back and forth between the slits some power is absorbed by the

dielectric core and by the walls of the cavity. The amount of power absorbed increases as the

square of the amplitude of the wave increases (see (4.5) and (4.6)) and eventually a point is

reached where the power supplied to the wave every cycle (by the proportion of the incident

wave coupling through the slits) is equal to the power absorbed by the wave in the core

propagating between the slits – the net power change is zero and the system has reached a

steady state.

100

With reference to the theoretical model (Ansoft), the power loss density, PD in W/m3 is given by

(4.5):

*.*.Re21 HEikJEPD (4.5)

Where:

E is the electric field strength in V/m

H* is the complex conjugate of the magnetic field strength in A/m

J * is the complex conjugate of the current density in A/m2

Furthermore:

H

EZ (4.6)

Where:

Z is the intrinsic impedance of the medium through which the wave is propagating

E is the electric field in V/m

H is the magnetic field in A/m

From (4.5) and (4.6) it can be seen that the power loss density PD, is proportional to the square

of the electric field.

If, when the system has reached a steady state the amplitude of the re-emitted wave is equal to

that of the specular reflected wave then perfect cancellation occurs and reflection coefficient =

0. In most cases however there is a mismatch but this mismatch can be reduced or removed by

optimising the geometry.

Consider: if in a steady state the amplitude of the wave re-emitted from the slits is less than that

of the specular reflected wave, then to maximise the resonance depth the amplitude of the re-

emitted wave needs increasing. This can be done by increasing the steady state amplitude of the

mode in the core, which in turn is accomplished by reducing the amount of absorption by the

core and metal layers. It might also be expected that increasing the width of the slits to let more

proportionately power into and out of the cavity in (and also reduce the amplitude of the

101

specular reflected wave) would increase the amplitude of the resonant mode and permit

optimisation of resonance depth, but as shall be demonstrated later, this is not the case.

4.4.2.1 Optimisation by altering core material properties

The core thickness was maintained at 356 μm and the imaginary component of the permittivity

of the core was varied from 0.02 to 0.9. The reflectivity of structure as simulated using the finite

element model is shown in Figure 4.12. Minimum reflectivity is obtained for an imaginary

permittivity of 0.2: based on the preceding it argument it can be inferred that imaginary

permittivity values less than 0.2 will result in reduced non-radiative loss, a higher steady state

amplitude of the mode in the core and hence an over-coupled state – see Figure 4.12(a). For

values of imaginary permittivity greater than 0.2 the non-radiative loss is too high, the steady

state amplitude is reduced and again the depth of the resonance is reduced – see Figure 4.12(b).

(a)

102

(b)

Figure 4.12 Reflectivity of the ultra-thin cavity structure as a function of

frequency for different values of imaginary permittivity as simulated using the

finite element model (a) for values of imaginary permittivity (eps’’) between

0.02 and 0.2 (b) for values of imaginary permittivity (eps’’) between 0.2 and

0.9

The above description of change in the non-radiative loss with imaginary permittivity can be

verified by using the finite element model. Shown in Figure 4.13 are a series of field plots

showing the steady state amplitude of the resonant mode in the vicinity of the slit for different

values of imaginary permittivity. In all cases the magnitude of the electric field is shown at the

resonant frequency for the structure in question (typically 6.95 GHz), at a phase corresponding

to peak field, and using a colour scale where blue corresponds to 0 V/m and red to 50 V/m, the

incident wave amplitude was 1 V/m in all cases. The steady state amplitude decreases as the

imaginary component of permittivity increases as anticipated.

103

Figure 4.13 Field plots showing the instantaneous magnitude of the electric

field for different values of imaginary permittivity, scale runs from 0 V/m

(blue) to 50 V/m (red), incident wave amplitude was 1 V/m in all cases (a)

imaginary permittivity = 0.02 (b) imaginary permittivity = 0.08 (c)imaginary

permittivity = 0.2 (d) imaginary permittivity = 0.4 (e) imaginary permittivity =

0.9

4.4.2.2 Optimisation by altering core thickness

The finite element model was used to simulate the reflectivity of the structure for core

thicknesses between 356 μm and 100 μm, core permittivity was maintained at 4.17 + 0.07i,

results are shown in Figure 4.14.

104

Figure 4.14 Reflectivity versus frequency as predicted by the finite element

model for structures of differing core thickness

As tc is decreased from 356 μm, the resonance becomes deeper and shifts to higher frequencies

and maximum absorption depth is achieved with a core 180 μm thick. In accordance with the

explanation given above, this structure affords the closest match between the non-radiative loss,

arising from dissipation by both the metal layers and the dielectric core, and the radiative loss,

arising from re-radiation by the cavity, and hence absorption depth is a maximum.

By analogy with changing the imaginary permittivity of the core, it would appear that as core

thickness is decreased below 180 μm, the non-radiative loss increases, becoming greater than

radiative loss and resulting in a net increase in reflectivity. For core thicknesses greater than 180

μm, the non-radiative loss becomes too small to perfectly cancel the radiative loss and again the

net reflectivity begins to rise. This description can be verified by using the finite element model

Shown in Figure 4.15 are a series of field plots showing the electric field of the resonant mode

in the vicinity of the slit for structures with core thicknesses between 100 μm and 356 μm. In

all cases the magnitude of the electric field is shown at the resonant frequency for the structure

in question, at a phase corresponding to peak field, and using a colour scale where blue

105

corresponds to 0 V/m and red to 40 V/m, again the incident wave amplitude was 1 V/m in all

cases.

The steady state amplitude increases as the core thickness decreases, indicating an increase in

the value of (4.5), the loss density in W/m3. However, the total non-radiative loss in Watts is

calculated by the integral of (4.5) over the volume of the structure, a volume which decreases in

direct proportion to core thickness. Hence as the core thickness decreases there are two

competing factors in determining the change in the non-radiative loss: the increasing amplitude

and the decreasing absorbing volume. However, using (4.6) it is apparent that the loss density is

proportional to the square of the electric field, therefore the value of (4.5) is increasing more

quickly than the volume is decreasing and the net effect of reducing core thickness is to increase

the non-radiative loss.

106

Figure 4.15 Field plots showing the instantaneous magnitude of the electric

field for different core thicknesses, scale runs from 0 V/m (blue) to 40 V/m

(red) and the incident wave amplitude was 1V/m in all cases (a) core thickness

= 100 μm (b) core thickness = 120 μm (c) core thickness = 150 μm (d) core

thickness = 180 μm (e) core thickness = 250 μm (f) core thickness = 356 μm

4.4.2.3 Optimisation by altering slit width

The finite element model was used to explore the effect of changing slit width from 0.025 mm

to 5 mm; results are shown in Figure 4.16. Core permittivity and thickness were fixed at 4.17

+0.07i and 356 μm respectively. The length of the metal patch between slits was kept constant

at a value of 9.7 mm.

107

Figure 4.16 Reflectivity as a function of frequency for the ultra thin cavity

arrays with different slit widths

For slits widths between 0.025 mm and 0.35 mm there is no apparent change in resonance

depth. As slit width increases from 0.35 mm to 5 mm the resonance does begin to deepen but

the rate of change is very small. As slit width is initially increased from 0.025 mm there is a

rapid shift up in frequency, the rate of change in resonant frequency decreases however, and

above 1 mm the frequency barely changes at all.

That there is little change in resonance depth indicates there is no change in the steady state

amplitude of the mode and that width of the slit is not a key factor in determining the proportion

of power from the incident wave which couples into the core. It is an oversimplification to

suggest that only the radiation incident directly on the slit itself couples through it into the core.

As the work of many including Ebbessen (1998) and Suckling (2007) has shown, the slit is

transmitting more power than is directly incident upon it. This can be attributed to the excitation

of surface currents on the metal patches and consequent channelling of power through the slit.

The decrease in resonant frequency with slit width is analogous to work by Suckling et al

(2004) who showed that the transmission frequency of a series of half-wavelength cavities

decreases rapidly as cavity thickness is reduced below a certain value. This shift is a

108

consequence of coupling between surface currents on the metal patches on opposite slides of the

slits. The strength of coupling decreases with increasing slit width and becomes almost

negligible for widths greater than 1 mm; hence there is no further frequency shift for widths

greater than this.

As the slit width is increased the proportion of the dielectric core which is beneath the metal

patches decreases. If the cavities are beginning to act as independent resonators, then any

dissipation will be confined to the region within the cavities – under the metal patches. As the

slit width is increased the proportion of the volume which effectively absorbs is hence reduced

which can affect the structure’s overall reflectivity. Furthermore, the specular reflection will

consist of two components: one from the top surface of the metal patches and one from the

lower metal plane. The latter component will be come more significant as the slit width

increases. The increase in the latter component can alter the relative phases of the reflected and

re-emitted signals which therefore alters the net reflectivity of the structure.

4.4.3 Polarisation conversion effects

It is only the component of the incident electric field that is perpendicular to the slits which can

couple through them and excite the resonant mode in the core. By reciprocity any re-radiation

from the cavity, howsoever excited, will have its electric field aligned perpendicular to the slits.

For p-polarisation maximum coupling occurs for 0 and decreases as cos . If the structure

is irradiated with p-polarisation for 0° < < 90° then coupling to the mode will be sub-

optimum but the mode will still be excited. Furthermore, for this azimuthal range, any re-

radiation by the cavity will have components both parallel and perpendicular to the plane of

incidence: the re-radiated signal contains a mixture of both p- and s-polarized components.

Therefore, whilst the minima observed in the reflectivity spectrum of the reflection sample

typically result from Joule heating of the metal and dielectric layers of the sample, it is also

possible for polarisation conversion by the sample to produce significant reductions in

reflectivity (see: Bryan-Brown (1990) Hallam (2004)).

109

In their standard configuration, a polarisation-converted signal is undetectable by the linearly

polarised antennas of both the focused horn and long path length azimuthal scan systems.

However, additional measurements were performed using the latter, in which the transmitting

and receiving antennas were cross-polarized, hence any polarisation conversion effects could be

detected and quantified. Two configurations were investigated: that in which the transmitting

antenna was p-polarised and the receiver s-polarised so as to measure the p- to s- conversion

coefficient Rps, and similarly the opposite configuration so as to measure Rsp. Results are

presented in form of greyscale contour plots in Figure 4.17 for 16 and 4.57 .

For the lower incident angle, polarization conversion is occurring but only for the more

strongly-coupled fundamental. Peak conversion efficiency is around 50% and occurs at

45 for both Rps and Rsp. For Rps, as increases from 0° towards 90°, the component of the

re-radiated field which is converted is increasing, but this is off-set by a reduction in coupling to

the mode from the incident field, hence conversion peaks at 45° and then decreases again. For

Rsp coupling to the mode increases upon rotation from 0° to 90° azimuth, but the component of

the re-radiated field that is converted decreases so as to off-set this, and again the peak

conversion occurs for 45°.

110

Figure 4.17 Experimental polarisation-converted reflected intensities for

reflection sample shown as greyscale plots (a) Rps data as a function of

frequency and azimuthal angle at º16 (b) Rsp data as a function of

frequency and azimuthal angle at º16 (c) Rps data as a function of

frequency and azimuthal angle at º57 (d) Rsp data as a function of

frequency and azimuthal angle at º57

At the higher incident angle both modes exhibit polarisation conversion: for the fundamental

mode peak conversion efficiency is around 30% whilst for the second order mode it does not

exceed 20%. At the higher angle, the polarisation conversion features appear asymmetric with

peak conversion occurring at less than 45° azimuth for the fundamental and above 45° azimuth

for the second order mode. This may be a consequence of anisotropy in the FR4 core material.

4.4.4 Transmission samples

4.4.4.1 Aligned slits

The transmission through the aligned transmission structure of Figure 4.2(b) was studied by

using the focused horn system. The fractional transmission or transmissivity as a function of

111

frequency is shown in Figure 4.18 for both the focused horn measurement and the

corresponding finite element simulation.

Figure 4.18 Transmission as a function of frequency for the aligned slit

structure as measured using the focused horn system and simulated using the

finite element model

Both experiment and theory demonstrate that the structure transmits approximately one-third of

the radiation incident upon it at its resonant frequency of 7.2 GHz. Using the finite element

model, the instantaneous electric field has been plotted in order to determine the nature of the

resonant mode, see Figure 4.19: in both plots the colour indicates the strength of the field with

blue corresponding to 0 V/m and red to 20 V/m, the incident field amplitude was 1 V/m.

The structure supports an N = 1 resonant mode at 7.2 GHz which undergoes a phase change of π

radians between slits and another π radians within the slit, as would be expected based on the

behaviour of the reflection sample. Note that as documented by Hibbins et al (2004), the field in

the upper slit is π radians out of phase with that at the lower slit and that the net field midway

between the slits at the mid-height of the dielectric core is zero. By further analogy with the

reflection sample, at normal incidence the structure will only support modes for which N in

(4.1) is odd since the sum of the phase change between slits (Nπ) and that within the slit (π)

112

must be an integer multiple of 2π. However, when irradiated off-normal, the change in phase of

the incident field along the surface permits coupling to modes for which N is even also.

(a)

(b)

Figure 4.19 Plots of the electric field at 7.2 GHz for the aligned transmission

structure: blue corresponds to 0 V/m and red to 20 V/m and the incident wave

amplitude was 1 V/m (a) the instantaneous magnitude of the electric field

plotted at a phase corresponding to peak field (b) the instantaneous electric

field vector plotted at a phase corresponding to peak field

At frequencies below 7.2 GHz, transmission efficiency decreases to less than 10% but remains

around this value rather than falling to zero. Hibbins et al (2004) asserts that this behaviour can

be likened to that of a low pass filter such as a frequency selective screen (Munk (2000)). In a

basic low pass filter (Bowick (1982)), a capacitor is connected parallel with the load and forms

a frequency-dependant voltage divider. The inverse dependence of impedance with frequency of

a capacitor, results in the voltage across the load remaining high in the low-frequency limit,

hence transmission remains elevated.

113

At 7.6 GHz there is a pronounced minimum in the transmission efficiency. This results from the

aforementioned region of zero field that occurs in the slits at the mid-height of the core, moving

towards the lower aperture at frequencies just above the resonant frequency of 7.2 GHz - an

observation first made by Hibbins et al (2004). When the zero field region is located within the

aperture as occurs at 7.6 GHz, the transmission efficiency is zero. The progressive movement of

the zero field region from the mid-height of the core to the lower aperture has been plotted using

the finite element model – see Figure 4.20: in all cases the fields have been plotted at a phase

corresponding to peak field and using a scale where blue corresponds to 0 V/m and red to 20

V/m, the incident field amplitude was 1 V/m.

(a)

(b)

114

(c)

Figure 4.20 Plots of the instantaneous magnitude of the electric field at

different frequencies for the aligned transmission structure, scale runs from 0

V/m (blue) to 20 V/m (red) and the incident wave amplitude was 1 V/m in all

cases (a) 7.2 GHz (b) 7.4 GHz (c) 7.6 GHz

4.4.4.2 Off-set slits

The second transmission structure, shown in Figure 4.2(c), in which the slits in the upper metal

layer are off-set by 5 mm from those in the lower metal layer, was measured using the focused

horn system and simulated using the finite element model: results are shown in the figure

below.

Figure 4.21 Transmission as a function of frequency for the off-set slit

structure as measured using the focused horn system and simulated using the

finite element model

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In contrast to the first transmission structure, the lowest frequency resonance occurs at 13.2

GHz. This is in fact the N = 2 mode – see Figure 4.22. The introduction of the off-set means that

each period of the structure now contains two slits within each of which a π radians phase

change occurs. Hence within one period of the structure (10 mm) the total phase change is 2π

radians from the slits plus Nπ radians between slits (in the same layer) and this sum must be an

even integer multiple of π radians, hence the modes for which N is even-numbered can be

excited.

(a)

(b)

Figure 4.22 Plots of the electric field at 13.12 GHz for the off-set transmission

structure, scale runs from 0 V/m (blue) to 20 V/m (red) and the incident wave

amplitude was 1 V/m in both cases (a) the instantaneous magnitude of the

electric field plotted at a phase corresponding to peak field (b) the

instantaneous electric field vector plotted at a phase corresponding to peak

field

According to Hibbins et al (2004), for resonant transmission to occur there must exist regions of

high field in both sets of slits: those in the upper metal layer and those in the lower metal layer.

For modes in which N is odd this condition cannot be satisfied: for an N = 1 mode, if the electric

field was high the vicinity of the upper slit then it would be low in the vicinity of the lower slit

116

since the slits are separated by a quarter of the wavelength of this mode, rather than by half a

wavelength. Similarly for N = 3 mode the distance between the slits is three-quarters of a

wavelength, and so on for all odd-numbered modes.

One might expect this structure to support modes for which the electric field at the upper slit

was high and the magnetic field at the lower slit was also high. However, the lower slit prevents

the flow of surface currents and hence precludes the formation of the requisite region of high

magnetic field.

4.5 Summary

In this chapter it has been demonstrated that ultra thin microcavity arrays can be used to produce

highly efficient absorption and transmission of microwave radiation. The cavity arrays have

been shown to operate at wavelengths more than 100 times their thickness via the excitation of

standing wave modes which can be tailored to produce the required absorption or transmission

characteristics. Furthermore, it has been demonstrated using theoretical modelling that that the

standing wave modes exhibit the remarkable effect of phase compression within the slits: the

wave undergoes a phase change of π radians in a distance which is many times smaller than half

a wavelength.

The incident and azimuthal angle dependence of the modes of has been studied experimentally

and explained with reference to the nature of the modes as determined using the theoretical

model, and by momentum conservation of the incident photons. The mechanism responsible for

the determining the depth of the resonance has been described and verified by using the

theoretical model to vary factors including: core imaginary permittivity, core thickness and slit

width. The polarisation conversion produced by the ultra thin cavity structures has been studied

experimentally as a function of both azimuthal and incident angles and has been shown to be

highly sensitive to azimuthal angle in particular.

117

The transmission spectra of two structures which have both their metal layers perforated by slits

have also been presented and their detailed behaviour explained by reference to the theoretical

model.

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Chapter 5

Reduction of azimuthal and incident angle sensitivity and polarisation

conversion effects – bi-gratings

5.1 Introduction

The work presented in the previous chapter demonstrates that a one-dimensional array of

slits formed in the upper metal layer of the microcavity structure results in resonant

absorption which is strongly dependant on incident and azimuthal angles and on the

polarisation state of the incident wave. This chapter considers the microwave response of an

ultra-thin microcavity array which has two orthogonal sets of slits formed in the upper

metal layer but is otherwise identical in geometry to the one-dimensional structure studied

in Chapter 4. This "bi-grating" maintains the advantage of being an order of magnitude

thinner than conventional absorbing materials, but exhibits strong absorption bands that are

almost completely independent of the angle of incidence and sample orientation. The

resonant modes produced by the structure are identified and their variation in response with

azimuthal (rotation) and polar incident angles is investigated. Experimental data are

compared to the predictions of HFSS thus allowing the nature of the resonant modes to be

examined.

5.2 Experimental

The experimental co-ordinate system and sample geometries are shown in Figure 5.1. The

samples are formed from 356 μm thick FR4 and bounded top and bottom by 18 μm thick copper

layers. The period of each sample is 10 mm, each slit having a width of 0.3 mm, being formed

using standard print and etch techniques as were used to produce the mono-grating structure of

the previous chapter (Eurotech). The specular reflectivity of the bi-grating was measured using

the long path length azimuth scan apparatus as detailed in Chapter 4. Measurements were taken

at polar angles ( ) between 16 and 73 and the sample rotated between 0 and 90 azimuth

( ), with the full azimuthal behaviour being obtained from the rotational symmetry of the

sample.

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Figure 5.1 (a) The mono-grating sample geometry (not to scale) and the co-

ordinate system used: θ is the polar angle, is the azimuthal angle, λg = 10

mm, ws = 0.3 mm (b) 3-D projection of the bi-grating, λg1 = λg2 (c) Cross-

section through the bi-grating structure, tm = 18 μm, tc = 356 μm, ws = 0.3 mm,

λg2 = λg1 =10 mm, sample area 500 mm by 500 mm

5.3 Results

In order to fully understand the modes supported by the bi-grating sample, it is useful to

represent its periodicity on a reciprocal space diagram. Two arrays of slits of identical spacing,

etched orthogonally to one another, yields a two-dimensional array of lattice points (Figure 5.2).

The lattice points can be grouped into sets, for example the set {1, 1} includes the (1, 1), (-1, 1),

(1, -1) and, (-1, -1) individual points, note the different parentheses used to represent a set of

points versus an individual point. In addition to the {0, 1} set of lattice points associated with

each mono-grating, a set of {1, 1} points also exist. This makes it possible to couple to a set of

modes that are inaccessible for the mono-grating.

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Figure 5.2 Reciprocal space diagram for the bi-grating

A greyscale plot showing Rpp at = 57° for the bi-grating is presented in Figure 5.4(a). There

are four dark bands indicating absorption at ≈ 7 GHz, ≈ 11 GHz, ≈ 14.5 GHz, and ≈ 17 GHz. A

cross-section through the greyscale plot for a fixed azimuthal angle of 45 is shown as a line

plot in Figure 5.3(c). Also shown is the reflectivity as predicted by HFSS for the same incident

angles. Excellent agreement between the measured data and the HFSS prediction is obtained

using the previously determined values of permittivity for FR4 (εr = 4.17 + 0.07i) with a period

of 10 mm, and slit width of 0.3 mm.

121

Figure 5.3 Bi-grating sample (a) Experimental Rpp data as a function of

frequency and azimuthal angle at = 57° (b) Experimental Rss data as a

function of frequency and azimuthal angle at = 57° (c) Line plot showing

comparison of measured data to the predictions of the numerical model: Rpp

= 57°, = 45° (d) Prediction of the electric field vector distribution at a phase

corresponding to peak field strength on the upper surface of the lower metal

layer for a {1, 1} mode at 10.93 GHz: the longest arrows correspond to

enhancements of 13 times the injected field

Returning to Figure 5.3(a), the addition of the second set of slits enables coupling to the

fundamental {1, 0} set of modes at 7 GHz at all azimuthal angles as there is always a

component of the electric field perpendicular to at least one set of slits. At = 90° the (1, 0)

mode is coupled to, whereas at º0 the coupling is to the (0, 1) mode, for 0 < < 90 both

modes are excited. Strikingly, unlike in the mono-grating case, there is no significant curvature

of the band. This is due to the interaction of these two modes.

122

As was demonstrated for the mono-grating of the previous chapter, the sample supports a series

of TEM waveguide modes (Grant et al (1995)) within the dielectric layer. These modes

resonate in a region beneath the metallic regions of the illuminated surface in a similar manner

to the Fabry-Perot-like resonances of a metallic slit (Takakura (2001)). Consider rotation of the

sample away from = 0° and resolution of the incident momentum into the x- and y-directions.

Assume infinitesimal slit width and set λy = 2λg for the fundamental standing wave resonance

(Hibbins et al (2004)):

222

0

2

yx kkkn , (5.1)

Where: n is the refractive index of the incident medium (n = 1 for vacuum), k0 is the wavevector

of the free-space wave, kx is the wavevector of the wave in the core in the x-direction, and ky is

the wavevector of the wave in the core in y-direction.

The mode observed at ≈ 11 GHz in Figure 5.3(a) is due to (1, 1) scattering and has no

equivalent on the mono-grating sample. Considering (5.1), kx and ky must be equal, giving:

22

2 xo knk , (5.2)

and hence the frequency of the {1, 1} modes should be 2 times greater than that of the {1, 0}

modes. Inspecting Figure 5.3(a) and taking the ratio of the resonant frequencies returns a value

of ≈ 1.5. The total momentum available to this mode does not change with azimuthal angle and

the mode is flat-banded. Also note that the {1, 1} modes cannot be coupled to for = 0°, 90°.

This characteristic can be explained by using HFSS to examine the mode’s field distribution.

Figure 5.3(d) is a plot of the electric field vector at a phase corresponding to maximum field on

the upper surface of the lower metal layer for a {1, 1} mode: the strongest electric field and

therefore the greatest charge densities occur at the corners, with diagonally opposite corners

having charge accumulations of the same sign. In order to couple to a {1, 1} mode this charge

distribution must be created by the electric field of the incident wave. For a p-polarized wave

123

incident at = 0°, this charge distribution cannot be set-up at any polar angle: the incident

electric field is parallel to the y-axis and results in the accumulation of a net positive charge

along one side of the patch and net negative charge on the opposite side, see Figure 5.4(a).

Figure 5.4 Bi-grating sample (a) Incident wavevector and electric vectors on

the lower surface of a metal patch, and the resulting charge distribution for:

= 0°, p-polarization (b) = 0°, s-polarization (c) = 45°, p-polarization and

(d) = 45°, s-polarization

An s-polarized wave incident at the same azimuth ( = 0°) creates a net positive charge on one

corner and a negative on the adjacent corner – the left side of the patch in Figure 5.4(b) - but on

the right-hand side the electric field direction and therefore the charge distribution is reversed

due to phase delay across the patch if the wave is incident off-normal. This reversal creates the

requisite charge distribution and permits coupling.

A p-polarized wave incident off-normal and at = 45°, Figure 5.4(c), has components of its

electric field (shown by the dashed lines) parallel to all sides of the patch. Again, the phase

124

reversal across the patch drives charges of like sign to diagonally opposite corners, making

coupling to the {1, 1} mode possible. As for = 0°, the requirement for a phase change across

the patch prevents coupling at normal incidence. On first inspection, an off-normal, s-polarized

wave at = 45° has components of electric field parallel to the edges of the patch and should

therefore couple to the {1, 1} mode. However, the field direction shown in Figure 5.4(d) drives

charges of opposite rather than like sign towards diagonally opposite corners and hence the

requisite charge distribution is not created.

The second order (2, 0) mode which is visible between 14 GHz and 15 GHz in Figure 5.3(a)

shows significant curvature due to the change in momentum available to the mode as the grating

is rotated, as is the case for the mono-grating: see (5.2). However, on rotation of the bi-grating

away from = 90°, coupling to the (0, 2) mode associated with the second, orthogonal set of

slits becomes possible. At = 45° the two modes become degenerate and the two curved bands

intersect, upon reaching = 90° the mode is wholly the (0, 2) mode.

Plots of the electric field vector on the lower metal layer for the {2, 0} modes are shown in

Figures 5.5(a) - (c). At 0 azimuth the (2, 0) mode [Figure. 5.5(a)] has electric field antinodes at

the edges of the patch parallel to the x-axis and also across the centre of the patch. At = 90°

the (0, 2) mode has anti-nodes along the patch edges parallel to the y-axis and in the centre of

the patch, [Figure 5.5(b)]. The degeneracy of the (2, 0) and (0, 2) modes at = 45° means

that the field pattern at this azimuth is simply the superposition of the fields from the (2, 0) and

(0, 2) modes [Figure 5.5(c)]. The curved bands between 16.5 GHz and 17.5 GHz are also absent

from the mono-grating being due to the excitation of {2, 1} modes; a plot of the electric field

vector for a {2, 1} mode is shown in Figure 5.5(d).

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Figure 5.5 Distribution of the electric field on the upper surface of the lower

metal layer plotted at a phase corresponding to maximum field (a) The (2,0)

mode at = 90° and 13.9 GHz (b) The (0,2) mode at = 0° and 13.9 GHz (c)

The degenerate (2,0) and (0,2) modes at = 45°, at 14.55 GHz (d) The (2,1)

mode at = 45°, 16.6 GHz

A greyscale plot of Rss at a polar angle of 57 is shown in Figure 5.3(b). As for p-polarization,

the fundamental is non-dispersive and the {1, 1} modes appear at ≈ 11 GHz. In this case the

mode cannot be coupled to at 45 azimuth, as explained above. The second order mode appears

between 14 GHz and 15 GHz and behaves in a very similar manner to the second order of the

mono-grating, the main difference being the degeneracy of the (2, 0) and (0, 2) modes at 45

azimuth. Note that the {2,0} modes are not excited for = 90° since for this polarisation state

there is no difference in the phase of the wave at adjacent slits as detailed in Chapter 4. The {2,

1} modes are visible between 16.5 GHz and 17.5 GHz.

126

With reference to Figure 5.3(a), the faint, highly curved band centred on = 0° is due to

conical diffraction. As shown in Chapter 2, the superposition of the incident momentum with

that supplied by the grating results in a quadratic equation in k0:

0cossin2sin12

022

0 gg kkkk (5.3)

The solution of (5.3) yields the limit frequency at which diffraction will occur and this solution

has been plotted as a dashed line in Figure 5.3(a).

5.4 Polarisation conversion effects

It was shown in Chapter 4 that the mono-grating structure exhibits conversion of radiation from

one polarisation state to the other and that this effect served to reduce the actual absorption

produced by the structure to much less than it appeared to be from an inspection of either the Rpp

or Rss greyscales. Hence, it is only by inspecting both the reflectivity greyscales and the

polarisation conversion greyscales that the true absorption efficiency of the structure can be

ascertained.

As was undertaken for the mono-grating, the transmitting and receiving antennas of the long

path length azimuthal scan apparatus were set to be cross-polarised and both possible

configurations were investigated: that in which the transmitting antenna was p-polarised and the

receiver s-polarised so as to measure the p- to s- conversion coefficient Rps, and similarly the

opposite configuration so as to measure Rsp. Results are presented in form of greyscale contour

plots in Figure 5.6 for 16 and 4.57 .

127

Figure 5.6 Experimental polarisation-converted reflected intensities for

reflection sample shown as greyscale plots (a) Rps data as a function of




frequency and azimuthal angle at º57 .

It can be seen from Figure 5.6 that the bi-grating does produce polarisation conversion from p-

to s- and vice-versa. However the levels of conversion at the fundamental frequency are on the

very limit of detection: conversion is barely visible for º16 as shown in Figures 5.6(a) and

(b) where the scale runs from 0 (black) to 0.01 (white), and for º57 (Figure 5.6(c) and (d))

no conversion is apparent at the fundamental frequency, although here the graph has been re-

scaled to better show the stronger features and consequently white corresponds to 0.1. In all

cases the degree of polarisation conversion is much less than that exhibited by the mono-grating

which ranged from 0.3 to 0.5 at the fundamental frequency.

There is some indication of polarisation conversion for the {1, 1} modes at approximately 11

GHz but only for the higher incident angle: poor signal-to-noise ratio precludes detection for the

128

lower angle. The observed conversion peaks at approximately 0.03 for azimuthal angles of =

22.5° and = 67.5° and falls to zero at = 45°. This behaviour can be explained with reference

to the Rpp and Rss greyscales of Figure 5.3(a) and (b) respectively. In order for conversion to

occur at any given azimuthal angle it must be possible for both p-polarised and s-polarised

waves to excite the {1, 1} modes at this angle. For = 0° the {1, 1} modes can be excited with

a p-polarised wave but cannot be excited with an s-polarised wave. Hence any {1,1} mode

excited with a p-polarised wave cannot couple out to an s-polarised wave in accordance with

reciprocity and therefore the conversion at this azimuthal angle is zero. The same argument can

be applied to explain the other nulls in the polarisation conversion at = 45° and = 90°.

The highest levels of polarisation conversion are exhibited by the higher order {2, 0} modes:

peak conversion efficiencies reach 0.1 with the shape of the features reflecting the curvature of

the {2, 0} modes as shown in Figure 5.3(a) and (b). Again there are nulls in the converted signal

at = 0° and = 90° which are a direct result of the fact that s-polarisation cannot excite {2,

0} modes at these azimuths. However, there is a sharp null in the converted signal for = 45°.

This is of course expected from mirror symmetry arguments. However, it may otherwise be

thought unexpected since the both p- and s-polarisation can couple to {2, 0} modes for this

azimuth. However, an inspection of the electric field of the {2, 0} modes as excited via an s-

polarised wave, reveals that they are rather different in character to that of the {2, 0} modes

excited by p-polarised waves. The latter field plot is shown in Figure 5.5(c) for = 45° and

appears to be simply the superposition of the degenerate (2, 0) and (0, 2) modes as would be

expected. However, since the {2, 0} modes cannot be excited for 90 with s-polarisation,

it is rather more difficult to anticipate the form of the field at = 45° as excited by this

polarisation. Figure 5.7 reveals that the form of the {2, 0} modes at = 45° as excited by an s-

polarised wave is the inverse of that excited by a p-polarised wave at the same azimuth: the field

in the centre and at the corners of the patch is zero and the anti-nodes occur along the edge of

the patch mid-way between the corners. The inverse nature of the two field distribution patterns

129

thus explains why there is no conversion at this azimuth: the two field patterns cannot be

supported simultaneously.

Figure 5.7 Distribution of the electric field on the upper surface of the lower

metal layer plotted at a phase corresponding to maximum field for the

degenerate (2, 0) and (0, 2) modes as excited by an s-polarised wave = 45°,

4.57 at a frequency of 14.585 GHz

5.5 Dispersion

To conclude this chapter, the resonant frequency of each of the modes as a function of the

incident angle is tracked in order to give their dispersion curves. This has been done for

azimuthal angles of 0 and 45 for both p-polarized and s-polarized radiation, the dispersion

curves are shown in Figures 5.8(a) and (b) respectively. It can be seen that each mode is

relatively flat-banded. However, the resonant frequency of the {1, 0} mode increases very

gradually with increasing angle. By contrast, the frequency of the {1, 1} modes decreases

slightly. This occurs for both polarization states at 0 and 45 azimuth. The {2, 0} modes do not

vary in frequency at 45 azimuth for either polarization but do decrease in frequency with

increasing polar angle at 0 azimuth for p-polarization.

130

(a)

(b)

Figure 5.8 Dispersion plots determined from the frequency of the modes

supported by the bi-grating sample at = 0° and 45º with (a) p-polarized and

(b) s-polarization incident radiation

5.6 Conclusions

Experimental measurements have demonstrated that the mono-grating ultra-thin microcavity

array structure studied in Chapter 4, which selectively absorbed one polarization of incident

131

radiation, is readily improved by patterning in two dimensions to form a bi-grating which

strongly absorbs any polarization. Furthermore, the degree of polarisation conversion exhibited

by the bi-grating is more than an order of magnitude lower than that exhibited by the mono-

grating. In addition, some of the resonant modes supported by the bi-grating exhibit a high

degree of azimuthal and polar-angle independent electromagnetic responses thus enabling the

absorption of both TE and TM polarized radiation over a wide range of angles. This

characteristic can clearly be exploited to create lightweight, thin, low-cost absorbers which are

independent of the polarisation state and orientation of the incident radiation.

It has also been demonstrated that the behaviour of these ultra-thin absorbing structures can be

fully predicted, even at off-normal incidence, using finite element modelling. This approach has

been used to examine the electromagnetic character of the modes and has revealed that in

contrast to the previously studied structure, the two-dimensional array supports coupled modes

having both x- and y-components. The two-dimensional array also supports higher-order modes

that cannot be excited by the one-dimensional structure which results in a series of discrete

absorption bands.

132

Chapter 6

Minimisation of azimuthal and incident angle sensitivity and

polarisation conversion effects – tri-gratings

6.1 Introduction

The previous chapter demonstrated that by introducing a second set of slits and thus creating a

90° rotationally symmetric bi-grating, the sensitivity of the response of the micro-cavity

structure to polarization and incident and azimuthal angles could be reduced. This chapter

extends the concept of higher order rotational symmetry to consider microcavity arrays which

feature three sets of slits each orientated at 60° to the other two sets, and thus possess hexagonal

symmetry. This higher order of rotational symmetry improves the azimuthal independence of

the resonant modes and also provides more reciprocal lattice vectors of the same magnitude than

either the mono-grating or bi-grating structures hence allowing access to a greater number of

modes.

Hexagonal symmetry has been exploited by Lockyear et al (2005) to reduce the azimuthal

dependence of surface plasmon resonances in dual pitch metal gratings and by Suckling et al

(2007) to create sub-wavelength hole arrays that support azimuthally-independent, surface

plasmon-mediated transmission. Sievenpiper (1999) has used hexagonally symmetric arrays of

metal patches to create high impedance ground planes that suppress the propagation of surface

currents, whilst Broas (2001, 2005) has applied such materials to create low profile antennas for

cellular phone handsets and novel phased array antennas. Chandran et al (2003, 2004) have used

fractal configurations of triangular metal patches to create efficient absorbers of both TE

(Transverse Electric) and TM (Transverse Magnetic) radiation at microwave frequencies.

Two hexagonal microcavity structures are studied in this chapter and are shown to exhibit

highly efficient absorption at microwave frequencies. The azimuthal and incident angle

independence of their response is demonstrated experimentally and the results are compared to

the predictions of an HFSS model. The model is then used to explore the nature of the resonant

modes excited.

133

6.2 Experimental details

The experimental co-ordinate system and sample geometries are shown in Figure 6.1. The

samples are formed from 356 μm thick FR4 PCB (printed circuit board) made from laminated

glass cloth infused with resin and bounded top and bottom by 18 μm thick copper layers. The

repeat period of each set of slits, (the perpendicular distance between them) on both samples is

10 mm, each slit having a width of 0.3 mm, being formed using standard print and etch

techniques (Eurotech).

(a) (b)

(c)

Figure 6.1 The tri-grating sample geometries (not to scale) and the co-ordinate

system used: is the polar angle, is the azimuthal angle, g = 10 mm, ws =

0.3 mm (a) 3-D projection of tri-grating 1 (b) 3-D projection of tri-grating 2 (c)

Cross-section through the tri-grating structure, one set of slits shown for

clarity, tm = 18 μm, tc = 356 μm, ws = 0.3 mm, 12 gg =10 mm, sample area

500 mm by 500 mm

The specular reflectivity of each sample is measured using the long path length azimuthal scan

apparatus as detailed in Chapter 4. Measurements are taken at polar angles of 16 and 43

134

and the sample rotated between 0 and 60 azimuth , with the full azimuthal behaviour being

obtained from the rotational symmetry of the sample.

On the first sample all three sets of slits intersect at common points creating an array of

equilateral triangles of side equal to 32l where l is the repeat period of each set of slits

measured parallel to the grating vector, see Figure 6.1(a). In the second structure each set of

grooves is off-set by half its repeat period relative to the other two sets hence all points of

intersection feature slits from only two of the three sets and a pattern of hexagons interspersed

with equilateral triangles both of side 3l is formed – see Figure 6.1(b). These two structures

represent the two possible hexagonally symmetric configurations. Comparison of their relative

responses should elucidate whether it is the repeat period of the slits and their orientation, or the

shape of the metal patches that determines the behaviour.

6.3 Theory

As demonstrated for both the mono-grating structure in Chapter 4 and bi-grating structure in

Chapter 5, the tri-gratings support TEM (Transverse Electric Magnetic) waveguide modes

within the dielectric layer. These modes resonate in a region beneath the metallic regions of the

illuminated surface in a similar manner to the Fabry-Perot-like resonances of a metallic slit

(Takaura (2001)). In the preceding chapters it was shown that the mono-grating and bi-grating

structures support a series of modes the frequencies of which could be accurately predicted by

considering the conservation of momentum of the incident photons.

Both tri-gratings feature three arrays of slits of identical spacing, etched at 60° to one another,

which yields a two-dimensional array of lattice points in reciprocal space which is hexagonally

symmetric (see Figure 6.2). The directions of the k-vectors are shown by the grey arrows

centred on the origin of Figure 6.2(a). Each lattice point on the reciprocal space diagrams gives

a diffractive centre for the light cone.

135

Figure 6.2 Reciprocal space diagrams for the tri-gratings showing: (a) the

scattering vectors and reciprocal lattice points (b) with a series of circles

centred on the origin having radii at which resonant modes are expected

The lattice points can be grouped according to the distance they are located from the origin. In

Figure 6.2(b), a series of concentric circles have been added to the reciprocal space diagram, the

radii of which correspond to the distances from the origin to the groups of points. It can be seen

that the (1,0) (1,1) (0,1) (-1,0) (-1,-1) and (0,-1) points are all at a reciprocal lattice distance kg

from the origin in reciprocal space. The modes supported by the tri-gratings, are expected to

occur in reciprocal space on radiation cones centred on reciprocal lattice points shown in Figure

6.2(b) with groups occurring at distinct frequencies.

Strictly speaking, these reciprocal space diagrams are oversimplified. The Fourier transform of a

pure sine wave will yield a line of spots in reciprocal space, but a cross-section through even the

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mono-grating of Chapter 4 is not sinusoidal but rather is a square-wave. To re-produce the

profile of the mono-grating would require a Fourier series with numerous higher order terms in

addition to the base period sine wave, hence the true reciprocal space diagram is more complex

than that shown in Figure 6.2.

This structure factor allows the square-wave type profile of the mono-grating to be reproduced.

However, because the slits of tri-grating 2 are off-set relative to those of tri-grating 1, sections

through the two gratings do not produce the same profile, so in the strictest sense the two

gratings have differing reciprocal space diagrams and hence one might not expect them to

support identical sets of modes.

6.4 Results

6.4.1 Tri-grating sample 1

The greyscale plots of Figure 6.3 show the specular (a) Rpp and (b) Rss reflected intensities from

the tri-grating 1 sample, where the subscripts refer to the incident and detected polarisations,

respectively. The data are plotted as a function of frequency and azimuthal angle with 16 .

Figure 6.3 (c) and (d) again show Rpp and Rss respectively for tri-grating sample 1 but

with 43 .

137

Figure 6.3 Tri-grating sample 1: (a) Experimental Rpp data as a function of

frequency and azimuthal angle at 16 ; (b) Experimental Rss data as a

function of frequency and azimuthal angle at 16 ; (c) Experimental Rpp

data as a function of frequency and azimuthal angle at 43 ; (d)

Experimental Rss data as a function of frequency and azimuthal angle

at 43

The dark bands at 8.35 GHz, 15 GHz, 17.3 GHz and 17.8 GHz in Figure 6.3(a) and (c) indicate

strong absorption. The mode at 8.35 GHz is the fundamental mode: that at 15 GHz is the first

harmonic. The ratio of these two frequencies is 1.80 (to 2 d.p.) which is close to 3 . The

fundamental and the first harmonic exhibit azimuthal independence, whilst the higher order

modes at 17.3 GHz and 17.8 GHz do exhibit some variation in coupling strength and also a

degree of curvature. As the grating is rotated the component of in-plane momentum in the

direction of propagation of the mode changes, hence the frequency shifts to satisfy momentum

conservation. 17.3 GHz is close to double the fundamental frequency, however the mode at 17.8

GHz is 2.13 times the fundamental frequency (to 2 d.p.) which is significantly less than 7 . It is

also a very strong mode. Note there are no exact simple numerical relationships between the

138

frequencies of these modes as the triangular microcavities have rather complex electromagnetic

mode patterns by comparison with those for square microcavities. The presence of these modes

lower in frequency than is expected based on the reciprocal space diagrams of Figure 6.2

demonstrates that these diagrams are oversimplified as discussed above.

The data in Figure 6.3(b) and (d) show that for Rss the 15 GHz mode is not excited at 16

whilst the other three modes are still present, whereas for 43 a mode does appear at

approximately 15.5 GHz but its strength exhibits a high degree of azimuthal dependence being

optimally coupled at 30° and not being excited at 0° and 60°. This character can be explained by

using HFSS to examine the mode’s field distribution, as we see later.

A cross-section through the greyscale plot showing Rpp at 43 and 30 for tri-grating 1

is shown in Figure 6.4 (a). The dark bands from the greyscale plots appear as resonances on the

cross-section. The resonances at 8.35 GHz, 15 GHz and 17.8 GHz are strongly coupled and

result in minimum reflected intensities of less than 0.2. The mode at 17.3 GHz is less strongly

coupled: exhibiting a minimum reflectivity of c. 0.45, it appears to overlap the 17.8 GHz mode.

These two modes may in fact be a splitting of the single mode expected to occur at twice the

fundamental frequency. Note that the predictions of resonant frequencies based on momentum

matching have assumed infinitesimal slit width, an assumption which is slightly violated with

increasing incident angle as the phase difference across the slit increases as the sine of the

incident angle.

139

(a)

(b)

Figure 6.4 Tri-grating samples 1 and 2: (a) Line plot showing comparison of

measured data to the predictions of the numerical model for tri-grating 1: Rpp

43 , 30 ; (b) Line plot showing comparison of measured data to the

predictions of the numerical model for tri-grating 2: Rpp 43 , 30

Also shown is the reflectivity as predicted by HFSS for the same incident angles. The finite

element prediction was obtained using the previously determined values of permittivity for FR4

ir 07.017.4 with a period of 10 mm and slit width of 0.3 mm and is an excellent match

for the fundamental resonance: both the depth and frequency of the resonance closely match the

measured data. For the higher order modes, the model continues to accurately predict the

resonant depth, but there is a slight disagreement in frequency with the model underestimating

the resonant frequencies by approximately 0.05 GHz to 0.1 GHz.

140

Note also from both parts of Figure 6.4 that the model predicts a reflectivity greater than unity

at low frequencies: typically below 7 GHz and significantly higher than that measured between

9 GHz and 12 GHz. The former feature is obviously non-physical and indicates an error in the

model. Several modifications to the mesh were made but although the magnitude of the error

was reduced it could not be eliminated. This error is likely to be a consequence of the non-

orthogonal faces of the unit cell walls combined with the lower frequency excitation resulting in

a longer wavelength and thus reducing the relative spacing between the periodic boundaries.

The mismatch between theory and experiment between 9 GHz and 12 GHz is more likely a

consequence of normalisation errors during the measurement than a problem with the model.

HFSS has been used to plot the electric field vector on the upper surface of the lower metal

layer of tri-grating 1 at the resonant frequencies of: 8.35 and 15 GHz for 43 and 90

in Figure 6.5 (note that due to the hexagonal symmetry of the structure, 90 is equivalent

to 30 , 30 etc).

In Figure 6.5(a) the incident wave has a frequency of 8.35 GHz and is p-polarised having an

electric field component parallel to the y-axis: this results in a resonant mode with a field pattern

that is similar to that of the half-wavelength modes supported by the bi-grating and with regions

of strongest field along the edges of the patches which are parallel to the y-axis. An 8.35 GHz,

s-polarised incident wave also results in a half-wave resonance, with a field distribution again

determined by the direction of the electric field, which is parallel to the x-axis, see Figure

6.5(b). For both the p-polarised and s-polarised incident waves, coupling to these half-wave

modes is due to Bragg scattering associated with the reciprocal lattice points: (1, 0), (1, 1), (0,

1), (-1, 0), (-1, -1), (0, -1) which lie at kg from the origin.

Figures 6.5(c) and (d) show the electric field vector at 15 GHz for waves p-polarised and s-

polarised respectively. In both cases resonant modes are excited in which the anti-nodes appear

at and mid-way between, the corners of the triangular patches: however, in neither case are the

141

strongest fields confined to the sides of the patches that are parallel to the y-axis. In contrast to

the mode at 8.35 GHz, the fields appear to be approximately equal in strength along all edges,

indicating that rather than a regular one-wavelength mode, this is a coupled mode resulting from

scattering associated with the reciprocal lattice points: (2, 1), (1, 2), (-2, 1), (-2, -1), (-1, -2), (1, -

1) which lie at 3gk from the origin. These modes are the direct analogue of the {1, 1} coupled

modes supported by the bi-grating. (Note that the parentheses {} denote a set of modes whereas

the regular parentheses () denote an individual mode).

(a) (b)

(c) (d)

Figure 6.5 Tri-grating sample 1: predictions of the electric field vector

distribution at phases corresponding to peak field strengths on the upper

surface of the lower metal layer for: (a) an 8.35 GHz, p-polarised wave

incident at 43 90 ; (b) an 8.35 GHz, s-polarised wave incident at

43 90 ; (c) a 15 GHz, p-polarised wave incident at

43 90 ; (d) a 15 GHz, s-polarised wave incident at

43 90 , the longest arrows correspond to enhancements of 15 times

in all cases

142

For s-polarisation, the antinodes at the vertices of one triangle are half a cycle out of phase with

those at the vertices of the other triangle: the charge accumulations are of opposite sign. To

couple to the 15 GHz mode, the electric field of the incident wave must create this charge

distribution: see Figure 6.6(a). However, for s-polarisation, at 60,0,60 etc the electric

field lies parallel to one of the sets of slits. In Figure 6.6 (b) 60 for example, it is parallel

to the slit between the two triangles shown. The electric field of the incident s-polarised wave is

in-phase everywhere along this slit, therefore the charge accumulations at adjacent vertices of

the two triangles will be the same. Upon rotation in azimuth through another 60 the electric

field will be in-phase everywhere along the next set of slits and owing to the 60 rotational

symmetry of the structure this same behaviour will occur every 60 , hence this mode cannot be

excited at 60,0,60 etc as shown by the greyscale plot of Figure 6.3(d). The formation of

this charge distribution requires a difference in the phase of the incident electric field across the

unit cell. For small incident angles there is only a very small phase difference leading to weak

coupling, hence the mode is visible at 43 incidence but is not visible at 16 incidence.

143

(a)

(b)

Figure 6.6 Diagrams showing the incident electric field and resulting charge

distribution for a 15 GHz s-polarised wave incident at (a) 90 ; (b)

60

The higher order resonant modes between 17 and 18 GHz are associated with the (2, 0), (2, 2),

(0, 2), (-2, 0), (-2, -2), (0, -2) reciprocal lattice points which lie at 2kg from the origin. The

electric field vector has been plotted on the upper surface of the lower metal layer at the

resonant frequencies of: 17.3 and 17.8 GHz for a p-polarised wave incident at 43 and

60 in Figures 6.7(a) and (b) respectively. It is apparent that the field distribution of the

two modes is very similar which again indicates that this is in fact a splitting of one mode.

144

(a)

(b)



surface of the lower metal layer for: (a) an 17.3 GHz, p-polarised wave

incident at 43 , 90 ; (b) a 17.8 GHz, p-polarised wave incident

at 43 , 90 , the longest arrows correspond to enhancements of 15

times in both cases

6.4.2 Tri-grating sample 1 – polarisation conversion

Shown in Figure 6.8 are greyscale contour plots containing polarisation conversion intensity

data for tri-grating sample 1.

145

(a) (b)

(c) (d)

Figure 6.8 Experimental polarisation-converted reflected intensities for tri-

grating sample 1 shown as greyscale plots (a) Rps data as a function of





At the lower incident angle there is no discernable conversion occurring for either configuration.

Noise is apparent particularly between 10 GHz and 12 GHz and appears as a grey speckled

pattern. All plots feature a white line at 10 GHz, this is simply due to a discontinuity in the data:

two data sets were taken using two sets of antennas, one from 5 GHz to 9.98 GHz, the other

from 10 GHz to 20 GHz, hence there is a small gap between them. All plots also exhibit another

narrowband feature at 19.2 GHz. This is due to a resonance inherent to the system set-up: the

horn antennas used for 10 GHz – 20 GHz are technically only valid for use at frequencies up to

18 GHz: at 19.2 GHz the antennas support a higher order mode which appears as a resonance,

146

the magnitude of which is reduced by normalisation but becomes more apparent here since the

scale has been so greatly expanded.

The higher incident angle does exhibit very low levels of conversion – no more than 2%. This

conversion is produced by the higher order (2, 0), (2, 2), (0, 2), (-2, 0), (-2, -2) and (0, -2)

modes. The conversion appears as two distinct features, one centred on 13° the other on 42°.

With reference to Figure 6.1(a), at 0° azimuth the incident electric field is perpendicular to one

set of slits (that parallel to the y-axis) and at 30° to the other two sets. Upon rotation to 15° the

electric field vector is then at 45° to one set of slits. By analogy to the mono- and bi-gratings,

when the vector is at 45° to the slit this constitutes the best compromise between coupling to

and from the mode to and from p- and s-polarisation, and hence conversion is a maximum. Due

to the hexagonal symmetry of the tri-grating the electric field vector is at 45° to one set of slits

every 30° of rotation, hence the polarisation conversion intensity is periodic with respect to

azimuth, with the period being 30°. That the features occur 2° lower in azimuth than expected is

due to slight errors in the experimental set-up: the sample was not quite aligned with one sets of

slits parallel to the y-axis when the rotation began.

6.4.3 Tri-grating sample 2

The greyscale plots of Figure 6.9 show the specular (a) Rpp and (b) Rss reflected intensities from

the tri-grating 2 sample. The data are plotted as a function of frequency and azimuthal angle

with 16 . Figure 6.9(c) and (d) again show Rpp and Rss respectively for tri-grating sample 2

but with 43 . Shown in Figure 6.9(e) and (f) are the Rpp and Rss data respectively for tri-

grating sample 1 with 43 - these have been shown again here to facilitate a comparison of

the responses of the two samples.

147

(a) (b)

(c) (d)

(e) (f)

Figure 6.9 Tri-grating samples 1 and 2: (a) Experimental Rpp data as a function

of frequency and azimuthal angle at 16 for tri-grating sample 2; (b)

Experimental Rss data as a function of frequency and azimuthal angle at

16 for tri-grating sample 2; (c) Experimental Rpp data as a function of

frequency and azimuthal angle at 43 for tri-grating sample 2; (d)

Experimental Rss data as a function of frequency and azimuthal angle at

43 for tri-grating sample 2 (e) Experimental Rpp data as a function of

frequency and azimuthal angle at 43 for tri-grating sample 1; (f)

Experimental Rss data as a function of frequency and azimuthal angle

at 43 for tri-grating sample 1

148

The dark bands at 8.1 GHz, 13.8 GHz, 16.4 GHz and 18.3 GHz in Figure 6.9(a) and (c) indicate

strong absorption. The fundamental 8.1 GHz mode exhibits a high degree of azimuthal

invariance, whereas the first harmonic (the mode at 13.8 GHz) shifts to slightly higher

frequencies upon rotation towards 30 azimuth. As the grating is rotated the component of in-

plane momentum in the direction of propagation of the mode changes, hence the frequency

shifts to satisfy momentum conservation. The 16.4 GHz mode also exhibits some azimuthal

variation, shifting towards lower frequencies on rotation towards 30 azimuth. At 43 the

highest frequency mode appears to split with two azimuthally dependant modes appearing

between 17.5 GHz and 18.7 GHz.

Both the measured data and the HFSS model demonstrate that the fundamental mode for tri-

grating 2 occurs slightly lower in frequency than that of tri-grating 1 despite both samples

having identical repeat periods and material properties: see Figures 6.3, 6.4 and 6.9. This

indicates that in addition to the repeat period of the slits, the off-set between the sets of slits and

therefore the size and shape of the patches, also affect the resonant frequencies, as was

highlighted previously during the discussion of the grating profile and its Fourier components.

As expected on the basis of momentum conservation, the higher order modes occur at

frequencies which approximate simple multiples of the fundamental: the ratio of 13.8 GHz to

8.1 GHz is 1.70, which is very close to 3 ; the ratio of 16.4 GHz to 8.1 GHz is 2.02, which is

close to 2. However, the next mode is expected to occur at 7 times the fundamental frequency,

approximately 21.4 GHz, whereas modes are observed at 17.5 GHz and 18.7 GHz.

The data in Figure 6.9(b) and (d) show that for Rss the 16.4 GHz mode is not excited for

either 16 or 43 , this can again be explained by considering the field distribution of the

mode. The other three modes are still present and occur at: 8.1 GHz, 13.8 GHz and 18.3 GHz as

for p-polarisation. The fundamental demonstrates azimuthal invariance, whereas the first

harmonic mode exhibits behaviour similar to that for p-polarisation, except that it shifts down in

149

frequency rather than up upon rotation towards 30 azimuth: this effect is very subtle but can be

discerned by viewing the plots from the side at a grazing angle. For this mode, rotation towards

30 azimuth increases the component of in-plane momentum in the direction of the mode;

hence there is a shift down in frequency to off-set this increase. Another mode appears at 17.2

GHz and is highly sensitive to azimuth, being optimally coupled at 0 and 60 azimuth, and

not being excited at 30 azimuth. The 18.3 GHz mode remains largely stable in frequency with

only a minor shift down upon rotation to 30 azimuth.

Comparing the greyscale plot of Figure 6.9(c) to that of Figure 6.9(e) and similarly part (d) to

part (f), reveals the striking differences in the structure of the higher frequency modes. The

greyscales plots for tri-grating sample 1 exhibit a more complex, more intricate structure than

those of tri-grating sample 2. This demonstrates very clearly the effect of the extra Fourier

components.

A cross-section through the greyscale plot showing Rpp at 43 and 30 for tri-grating 2 is

shown in Figure 6.4 (b). The dark bands from the greyscale plots appear as resonances in this

cross-section. All four resonances are strongly coupled and result in minimum reflectivities of

less than 0.2. Also shown is the reflectivity as predicted by HFSS for the same incident angles.

The model predicts the frequency and depth of the fundamental resonance very accurately: for

the higher order modes, the model continues to accurately predict the resonant depth, but there

is a slight disagreement in frequency with the model underestimating the resonant frequencies

by approximately 0.05 GHz to 0.2 GHz.

HFSS can be used to examine the field distribution of the modes shown in Figures 6.4 and 6.9:

it has been used to plot the electric field vector on the upper surface of the lower metal layer of

tri-grating 2 at the resonant frequencies of 8.1 and 13.8 GHz for 43 and 60 in Figure

6.10. In Figure 6.10(a) the incident wave has a frequency of 8.1 GHz and is p-polarised: the

region beneath the central hexagon supports a half-wave resonant mode. 8.1 GHz s-polarised

radiation incident at the same angle also excites a half-wave mode beneath the hexagonal patch,

150

although the mode appears transverse to the plane of incidence due to the polarisation of the

incident wave, see Figure 6.10(b). For both the p-polarised and s-polarised incident waves,

coupling to these half-wave modes is due to Bragg scattering associated with the reciprocal

lattice points: (1, 0), (1, 1), (0, 1), (-1, 0), (-1, -1), (0, -1) which lie at kg from the origin.

Figures 6.10(c) and (d) show the electric field vector at 13.8 GHz for waves p-polarised and s-

polarised respectively: both polarisations excite modes in which there are field anti-nodes on

opposing sides of the central hexagon which are in-phase with each other. Half-wavelength

modes would exhibit antinodes that were half a cycle out of phase with each other, and

wavelength modes would require an additional anti-node in the centre of the hexagon.

Furthermore, in accordance with momentum conservation, wavelength modes would be

expected to occur at twice the fundamental frequency. Therefore, the field plots in Figures 6.10

(c) and (d) do not show degenerate one-wavelength modes but rather coupled modes that result

from scattering associated with any of the following reciprocal lattice points: (2, 1), (1, 2), (-2,

1), (-2, -1), (-1, -2), (1, -1) which lie at 3gk from the origin, as was shown to be the case for

tri-grating sample 1.

151

(a) (b)

(c) (d)



surface of the lower metal layer for: (a) an 8.1 GHz, p-polarised wave incident

at 43 , 60 ; (b) a 8.1 GHz, s-polarised wave incident at

43 , 60 ; (c) a 13.8 GHz, p-polarised wave incident at 43 ,

60 ; (d) a 13.8 GHz, s-polarised wave incident at 43 , 60 , the

longest arrows correspond to enhancements of 15 times in all cases

Shown in Figure 6.11 (a) is a plot of the electric field vector for a 16.4 GHz p-polarised wave

incident at 43 , 60 . The fields are again confined to the region beneath the hexagon

and exhibit a one-wavelength resonance: the anti-node at the centre of the hexagon is half a

cycle out of phase with that which appears around the perimeter. That there appears to be an

anti-node around the entire perimeter of the hexagon, indicates that degenerate modes are

152

excited at these incident angles due to simultaneous scattering associated with two or more the

lattice points located at 2kg from the origin.

(a)

(b) (c)

Figure 6.11 Tri-grating sample 2: (a) prediction of the electric field vector

distribution at a phase corresponding to peak field strength on the upper

surface of the lower metal layer for: a 16.4 GHz, p-polarised wave incident at

43 , 60 ; (b) diagram showing the incident electric field and

resulting charge distribution for a p-polarised wave incident at 90 ; (c)

diagram showing the incident electric field and resulting charge distribution for

a s-polarised wave incident at 90

The excitation of this mode for p-polarisation is possible as the phase change across the patches

causes a reversal of the direction of the electric field and the resulting components of the electric

field drive charges of like sign towards the centre of the hexagon; see Figure 6.11(b). However,

it is not possible to create this charge distribution with s-polarisation at any azimuth because the

components of the electric field do not drive like charges towards the centre Figure 6.11(c).

153

Plots of the electric field vector for modes excited by an s-polarised wave incident at

43 , 60 for frequencies of 17.1 GHz and 18.3 GHz are shown in Figures 6.12(a) and

(b) respectively, these modes are associated with the (2, 0), (2, 2), (0, 2), (-2, 0), (-2, -2), (0, -2)

reciprocal lattice points which lie at 2kg from the origin. It is apparent that the field distribution

of the two modes is similar, however that for 18.3 GHz, Figure 6.12(b), appears to show a

mode confined to region beneath the triangular patch with very little excitation of any mode

under the much large hexagonal patch. The field distribution appears similar to that of the

fundamental modes of tri-grating sample 1 which featured only triangular patches. This may

therefore be the fundamental mode for the small triangle.

(a) (b)

Figure 6.12 Tri-grating sample 2: (a) prediction of the electric field vector

distribution at a phase corresponding to peak field strength on the upper

surface of the lower metal layer for: a 17.1 GHz, s-polarised wave incident at

43 , 60 ; (b) prediction of the electric field vector distribution at a

phase corresponding to peak field strength on the upper surface of the lower

metal layer for: a 18.3 GHz, s-polarised wave incident at 43 , 60

6.4.4 Tri-grating sample 2 – polarisation conversion

Shown in Figure 6.13 are greyscale contour plots containing polarisation conversion intensity

data for tri-grating sample 2.

As with tri-grating sample 1, the lower incident angle does not produce any measurable

polarisation conversion. However, the higher incident angle does exhibit more significant levels

of conversion: up to 8% for both configurations. In this case it is the: (2, 1), (1, 2), (-2, 1), (-2, -

1), (-1, -2), (1, -1) modes which support conversion. Again the polarisation-converted intensity

is periodic with respect to azimuth due to the hexagonal symmetry of the sample.

154

(a) (b)

(c) (d)

Figure 6.13 Experimental polarisation-converted reflected intensities for tri-

grating sample 2 shown as greyscale plots (a) Rps data as a function of





6.5 Summary

It has been demonstrated by both theory and experimental measurement, that microcavity arrays

with hexagonal symmetry can be tailored to produce efficient microwave absorption that is

independent of azimuthal and incident angle. The structures studied herein, support a series of

resonant modes for both p-polarised and s-polarised incident radiation, the frequencies of which

are determined predominantly by the repeat period of the slits, and can be predicted by

considering the conservation of momentum of the incident photons. However, to fully represent

155

the detailed shape of the grating profile requires that higher order Fourier components be

considered. These components are responsible for the otherwise unexpected differences

between the structure of higher order modes for the two tri—grating samples.

These microcavity absorbing structures could be applied to reduce the level of backscattered

radiation in environments where the direction and polarisation of incident radiation is varying or

unpredictable. Examples of such environments include: airports where reflections from

buildings can cause significant interference problems and the interior of buildings where

screening from electromagnetic radiation is required. Furthermore, the thin, flexible and

lightweight nature of the material makes it ideal for EMC (Electromagnetic Compatibility)

applications where space is critical such as on or within the housing of sensitive instruments.

156

Chapter 7

Methods for achieving maximum absorption bandwidth

7.1 Introduction

The preceding experimental chapters have demonstrated that ultra-thin microcavity arrays offer

a novel approach for the control and in particular the absorption of microwave radiation.

However, the results hitherto presented feature narrow resonances which only absorb efficiently

over a very limited range of frequencies. This chapter constitutes efforts to increase the

absorption bandwidth of the microcavity array structures. To this end, the absorption bandwidth

of a series of optimised microcavity array absorbers is studied and compared to theoretical

expectations based on cavity thickness and magnetic permeability. It is found that even basic

microcavity array structures, when optimised, afford absorption bandwidths that exceed 90% of

the theoretical limit. By introducing a multiplicity of cavity configurations including multiple-

period, multiple-permittivity and multiple-layer designs, total absorption bandwidth is increased

slightly and greater control over the frequency distribution of the absorption is obtained.

One of the greatest challenges in the design of absorbing materials, is the bandwidth-to-

thickness ratio. The work of Brewitt-Taylor (1999, 2007) and that of Rozanov (2000) has

demonstrated that there is a fundamental limit on the maximum absorption bandwidth that can

be achieved from a material, and that this limit depends on the thickness of the material and its

magnetic permeability. The primary consequence of this is that a given material can only exhibit

efficient absorption over a finite range of frequencies and that outside of this range the material

will not absorb efficiently.

The purpose of this chapter is to compare the total bandwidth product of the microcavity arrays

to the theoretical limits based on thickness and permeability, and to present four strategies for

maximising the total absorption bandwidth. These four strategies are embodied in the

microcavity array structures presented herein, and can be summarised as:

Multiple discrete repeat periods

Multiple continuous repeat periods

157

Multi-layering

Multiple permittivities

7.2 Experimental

The experimental co-ordinate system and structure geometries are shown in Figure 7.1. The

structures are formed from various thicknesses of non-magnetic (i.e. μr = 1) dielectric materials

including: FR4 PCB (printed circuit board) made from laminated glass cloth infused with resin,

polyester (PET) sheet, alumina (Al2O3) and air, bounded top and bottom by 18 μm thick copper

layers. The dimensions of the structures are as annotated in Figure 7.1. In all cases the slits have

a width of 0.3 mm, being formed using standard print and etch techniques (Eurotech); all

structures measure 450 mm by 450 mm. Their reflectivity at normal incidence was measured

over 5.4 GHz to 18 GHz (55.6 mm to 16.7 mm wavelength) using the focused horn apparatus

as detailed in Chapter 4. In all cases the electric field vector of the incident radiation was

orientated perpendicular to the slits.

Structure 1 is a combination of two repeat periods (inter-slit distances): L1 = 10 mm and L2 = 9

mm giving a total repeat period of 19 mm. All slits run parallel to the y-axis and perpendicular

to the electric field of the incident wave: see Figure 7.1(c). In Structure 2, every other slit has a

saw-toothed shape, the period of the saw-tooth being 40 mm and its amplitude being 2 mm.

Thus the distance between adjacent slits varies from 8 mm to 12 mm and back to 8 mm every 40

mm along the y-axis, see Figure 7.1(d). Successive saw-tooth slits are half a period out of phase

with each other, thus making the overall period of the structure 40 mm in the x-direction. In

Structures 1 and 2, the dielectric core material is FR4 with complex permittivity εr =4.17 + 0.07i

as detailed in Chapter 4.

158

(a) (b)

(c) (d)

(e) (f)

159

(g)

Figure 7.1 The microcavity structure geometries (not to scale) and the co-

ordinate system used: θ is the incident angle, is the azimuthal angle, ws is

the slit width, g is the repeat period of the structure (a) 3-D projection of a

standard mono-grating structure in which all slits run parallel (b) Cross-section

through the standard mono-grating structure (c) 3-D projection of Structure 1,

multiple discrete repeat periods (d) Plan view projection of Structure 2,

multiple continuous repeat periods with alternate saw-tooth slits (e) 3-D

projection of Structure 3, multi-layer structure, 2 periods shown (f) end

projection of Structure 3, multi-layer structure, 2 periods shown (g) 3-D

projection of Structure 4, multiple refractive indices

Structure 3 consists of 4 layers each with a period of 10 mm and slit width of 0.3 mm, but with

differing material properties and thicknesses. The four layers are stacked one upon the other

such that their slits are aligned: the top three layers have slits in both copper layers to allow

penetration of the incident wave to the lower layers, see Figure 7.1(e) and (f). Due to the

complexity of this structure, the high probability of inter-layer misalignment, and consequent

distortion of results, fabrication of a physical sample was not undertaken. Instead HFSS was

used to simulate its performance. The complex permittivities of PET and alumina are εr = (3.2 +

0.0096i) and εr = (9.4 + 0.0564i) respectively and were obtained from the database within

HFSS.

Structure 4 has a fixed period of 20 mm, each period containing two slits spaced apart by 10

mm and both having widths of 0.3 mm, the properties of the core material alternate between

adjacent cells. In the first cell the core material has a complex permittivity of εr1 =3.5 + i0.07, in

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the second unit cell the complex permittivity is εr2 = 4.17 + i0.07, see Figure 7.1(g). Again due

to the complexity of fabrication HFSS was used to simulate the performance of this structure.

7.3 Theory

Work by Bode (1945) and later work by Fano (1950) demonstrated that there is a fundamental

limit to the bandwidth over which a source impedance can be matched to a load impedance

within an electrical circuit. Both Brewitt-Taylor (1999, 2007) and Rozanov (2000) have applied

this work to the field of absorbing materials and demonstrated that there is a limit to the

maximum absorption bandwidth which is determined by the material’s thickness and

permeability.

Rozanov (2000) formulates the bandwidth-to-thickness relationship in terms of wavelength,

whereas Brewitt-Taylor’s (1997) formula is written in terms of frequency. Both formulations

use reflectivity in decibels and describe a limit to the area beneath the curve on a plot of

reflectivity in decibels versus wavelength or frequency respectively, see Figure 7.2.

Figure 7.2 Response of an example Salisbury screen absorber as predicted

using the finite element model (a) reflectivity in decibels versus frequency (b)

reflectivity in decibels versus wavelength

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For a Salisbury screen (Knott et al (1993) and Salisbury (1952)) in which all the constituent

materials have unity permeability, the result due to Brewitt-Taylor (1999) is:

0 00 10ln

3201

hdffR

fdB (7.1)

Where:

RdB is the reflectivity in decibels

f0 is the centre frequency of the absorption band

λ0 is the centre wavelength of the absorption band

h is the thickness of the spacer within the Salisbury screen

Note that this formula applies to structures illuminated by plane waves at normal incidence and

considers only the lowest-frequency absorption band. This latter restriction is a consequence of

performing the integral over frequency, in which regime results an infinite series of absorption

bands and hence an infinite integral, see Figure 7.2. In a more recent publication Brewitt-Taylor

(2008) extends this concept to determine the bandwidth limits on magnetic conductor surfaces

at oblique incidence.

Rozanov (2000) presents the following formula for broadband absorbers:

iiiidB hhdR

17210ln

40 2

0

(7.2)

Where:

μi is the relative magnetic permeability of the ith

layer at zero frequency

hi is the thickness of the ith layer

and all other terms have the same meaning as in (7.1).

In the case of narrowband absorbers, Rozanov (2000) presents an alternative formula:

iiiidB hhdR 13910ln

320

0

(7.3)

162

In this case, a narrowband absorber is one in which 1.00 , where is the operating

bandwidth. It can be shown that the narrowband limit (7.3) is equivalent to (7.1) and as such

applies only to the lowest-frequency absorption band. Taking the ratio of (7.3) to (7.2) returns a

value of 0.81 (to 2 d.p.) indicating that the lowest frequency absorption band accounts for no

more than 81% of the total bandwidth. Typical values for 0 for the microcavity absorbers,

are < 0.05 (see preceding chapters) which classifies them as narrowband, therefore their total

absorption bandwidth has been compared to the narrowband limit (7.3). Note that the

formulations (7.1) - (7.3) all exclude the thickness of the ground plane or lowermost metal

layer.

The microcavity array structures support a series of standing-wave modes similar to the Fabry-

Perot resonances of a metallic slit (Takakura (2001)), the frequencies of which, fN, are given by

(7.4):

nL

Ncf N

2 (7.4)

Where:

fN is the frequency of the Nth order Fabry-Perot mode

c is the speed of light in vacuum

n is the refractive index of the dielectric material within the cavity

L is the length of the cavity

N is a positive integer

As was demonstrated in Chapter 4, the incident angle θ and the azimuthal angle , influence

which modes the mono-grating microcavity structure will support, for example: at normal

incidence the structure will support only those modes where N is odd, whereas off-normal the

grating can also support the modes where N is even. Additional modes will increase the total

absorption bandwidth, however, this is not useful when considering the limits (7.2) and (7.3)

since this applies only to structures illuminated at normal incidence. Note also that upon rotation

163

away from º0 and θ = 0°, the resonant frequencies given by (7.4) become increasingly

inaccurate. In such cases the resonant frequencies can be found by considering the momentum

of the incident photons, details can be found in Chapter 5.

From (7.4) it can be seen that the two factors which determine the frequencies of the series of

modes are the slit-spacing L and the refractive index n. By introducing a multiplicity of slit

spacings, and/or refractive indices, multiple series can be excited with the potential of

increasing the overall absorption bandwidth.

7.4 Results

The total absorption bandwidth for each structure studied, was calculated by dividing the area

beneath the 0 dB line into a series of trapezia and finding their individual areas, the sum of these

giving the total area under the curve to a good approximation. The total bandwidth product of

each structure is compared both to the theoretical limit (7.3) and to that afforded by the

standard, single layer mono-grating with an FR4 core, period, λg, of 10 mm and slit width ws of

0.3 mm, see Figure 7.1(a).

7.4.1 Standard mono-grating

The reflectivity of the standard mono-grating in decibels versus wavelength both as measured

experimentally and predicted by the finite element model, is shown in Figure 7.3(a) for

radiation incident normally and with its electric field vector perpendicular to the slits. There is

excellent agreement between the two datasets both exhibiting a deep resonance at 43.7 mm (6.9

GHz). From (7.4) it is apparent that this resonance is the mode for N = 1: the N = 2 mode is not

excited at normal incidence (see Chapter 4) and the N = 3 mode lies just above 20 GHz (15 mm

wavelength) and is therefore inaccessible using this measurement set-up. The total bandwidth

products for the experimental and theoretical datasets are 23.4 dBmm and 24.4 dBmm

respectively, equating to 45% and 47% of the narrowband limit (7.3).

164

(a)

(b)

(c)

Figure 7.3 Theoretical and experimental data for standard mono-grating (a)

Reflectivity in decibels versus wavelength as predicted by the finite element

model and measured experimentally (b) Reflectivity in decibels versus

wavelength as predicted by the finite element model for mono-grating

structures of differing core thickness (c) Percent of narrowband bandwidth

limit versus core thickness for the a series of mono-grating structures

165

The experimental microcavity structure was fabricated from an available thickness of FR4: tc =

356 μm. This does not necessarily constitute an optimum structure or therefore the maximum

possible absorption bandwidth. By altering the FR4 thickness using HFSS the depth of the

resonance can be maximised and the largest absorption bandwidth obtained.

Shown in Figure 7.3(b) and (c) are the results of varying the FR4 thickness, tc, in the HFSS

model: as tc is decreased from 356 μm, the resonance becomes deeper and shifts to shorter

wavelengths and maximum absorption depth is achieved with a core 180 μm thick. This

structure affords the closest match between the non-radiative loss, arising from dissipation by

both the metal layers and the dielectric core, and the radiative loss, arising from re-radiation by

the cavity, and hence absorption depth is a maximum. This structure also exhibits the greatest

percentage absorption bandwidth: 25.6 dBmm equivalent to 92.9% of the narrowband limit

(7.3), see Figure 7.3(c).

As core thickness is decreased to less than 180 μm, resonance depth begins to decrease as does

the total absorption bandwidth. However, as the core thickness decreases so does the theoretical

bandwidth limit (7.3) hence the rate of percentage bandwidth decrease is relatively slow. Above

180 μm, the resonance becomes shallower, but this is off-set by an increase in the width of the

resonance. Overall, the total bandwidth product remains at approximately 25 dBmm for

thickness up to 356 μm, however since the value of (7.3) is increasing with core thickness the

percentage bandwidth decreases rapidly.

7.4.2 Structure 1 – multiple discrete repeat periods

The overall repeat period of this structure is 19 mm, comprising a 10 mm inter-slit distance

alternated with a 9 mm inter-slit distance. All slits are parallel to each other and are 0.3 mm

wide, the dielectric core material is FR4 with permittivity 4.17 + i0.07. The first structure

investigated had a core thickness 190 μm, see Figure 7.1(c). The reflectivity of this structure as

a function of wavelength is shown in Figure 7.4(a): this structure supports resonances at 41 mm

166

and at 37 mm wavelength. From (7.4) is it clear that these correspond to the fundamental, N = 1

modes for the 10 mm and 9 mm sections respectively.

(a)

(b)

(c)

167

(d)

(e)

Figure 7.4 Multiple discrete period structures (a) Reflectivity in decibels versus

wavelength for structure with dielectric core thickness of 190 μm (b)

Reflectivity in decibels versus wavelength as predicted by the finite element

model for multiple discrete period structures of differing core thickness (c)

Percent of narrowband bandwidth limit versus core thickness for the a series of

multiple discrete period structures (d) Cross-section of modified multiple

discrete repeat period structure, (e) Reflectivity in decibels versus wavelength as

predicted by the finite element model for multiple discrete period structures with

different values of t2

As with the standard mono-grating, the dielectric core thickness was varied in order to derive

the maximum absorption depth and maximum percentage bandwidth of (7.3). It can be seen

from Figure 7.4(b) that a core thickness of around 250 μm maximises the depth of the longer-

wavelength resonance. Upon close inspection, the depth of the shorter-wavelength resonance

changes in the same manner with greatest depth being reached at a core thickness of 250 μm,

but the magnitude of the change on a decibel scale is much smaller for this resonance.

168

A core thickness of 230 μm - 250 μm also produces optimum absorption bandwidth: this

structure achieves 96% of the narrowband limit (7.3) and therefore does constitute a small

improvement over the standard mono-grating. Note that the combined area of both resonances

has been summed in order to produce the 96% figure. The variation in absorption bandwidth

with core thickness is summarised in Figure 7.4(c): there is a gradual increase in percentage

bandwidth (7.3) up to 250 μm after which begins a much faster decline. For thicknesses greater

than 250 μm, the resonances are becoming shallower but the bandwidth limit (7.3) is

simultaneously increasing, hence the rate of decrease in the percentage bandwidth is large.

If the depth of both resonances could be simultaneously optimised, then overall absorption

bandwidth might be increased. To this end the thickness of the dielectric core within the 9 mm

long cavity, t2 in Figure 7.4(d), was progressively decreased from 240 μm to 180 μm, whilst the

thickness of the dielectric core within the 10 mm long cavity was fixed at 240 μm, the results of

this modification are shown in Figure 7.4(e). The depth of the shorter-wavelength resonance has

been increased beyond values seen with the previous structure, Figure 7.4(b), although it still

has not exceeded -10 dB. However, the longer-wavelength resonance has been perturbed and is

sensitive to any change in t2. The total absorption bandwidth has actually decreased. This is

because there has been a reduction in total absorbing volume: the fields are excluded from all

but top few microns of the metal and hence by decreasing, t2 the fraction of the total structure

volume available to absorb is reduced. Note also that this design creates an ambiguity as to what

value of h should be used in (7.3), making a determination of the percentage bandwidth very

difficult.

7.4.3 Structure 2 – multiple continuous repeat periods

In Structure 2, successive slits alternate between being straight and having saw-toothed shape.

The period of the saw-tooth slits is 40 mm and their amplitude is 2 mm. In this structure the

inter-slit distance varies from 8 mm to 12 mm and back to 8 mm every 40 mm along the y-axis,

refer to Figure 7.1(c). In Structure 2 the dielectric core material is FR4 with complex

permittivity εr =4.17 + 0.07i, the core thickness of the measured structure was 356 μm.

169

The experimentally measured reflectivity of this structure versus wavelength is compared to that

predicted by HFSS in Figure 7.5(a). There is good agreement between the two datasets: both

predict a deep resonant absorption at 47 mm and a number of shallower resonances at shorter

wavelengths. The HFSS model predicts that these shallower features occur at slightly shorter

wavelengths than those observed experimentally. This may be due to imperfections in the saw-

toothed slits of the experimental structure; around the vertex of the saw-tooth the accuracy

achievable with the etching process may be less than elsewhere on the structure.

Although by contrast to basic mono-grating, there are no simple numerical relationships for

predicting the resonant wavelengths of structures in which the slits are not parallel, on the basis

of (7.4) one might that the response of this structure would be a broad and possibly shallow

resonance lying between 6.1 and 9.2 GHz. However, instead a series of narrow resonances

result. This is further evidence that the simple approximation of (7.4) is indeed too simple.

Nevertheless, the nature of the resonant modes can be explored by using HFSS.

From Figure 7.5(d) it is apparent that the mode at 47 mm is a half-wave mode, the strength of

which varies in the x-direction: strongest at points where the inter-slit distance is 12 mm and

weakest at points where the inter-slit distance is 8 mm. Using a value for L of 12 mm in (7.4)

returns a resonant wavelength of 49 mm which is close to the 47 mm resonance observed. The

shorter wavelength modes have more complex field patterns which cannot readily be likened to

regular modes, nor do their resonant wavelengths correspond to any apparent period or inter-slit

distance of the structure – see Figure 7.5 (e) and (f). These complex field patterns are

reminiscent of modes supported by hexagonally symmetric microcavity structures studied in

Chapter 6 which also featured metal patches the sides of which were not everywhere

perpendicular.

170

(a)

(b)

(c)

171

(d)

(e) (f)

Figure 7.5 Multiple continuous period structures (a) Reflectivity in decibels

versus wavelength as predicted by the finite element model and measured

experimentally (b) Reflectivity in decibels versus wavelength as predicted by

the finite element model for multiple continuous period structures of differing

core thickness (c) Percent of narrowband bandwidth limit versus core thickness

for the a series of multiple continuous period structures (d) Plot of the

instantaneous electric field vector on the upper surface of the lower metal layer

at a wavelength of 47 mm and a phase corresponding to peak field, the longest

arrows correspond to 30 V/m (an enhancement of 30 times the incident field),

dashed lines added to indicate position of slits (e) Plot of the magnitude of the

instantaneous electric field on the upper surface of the lower metal layer at a

wavelength of 40 mm and a phase corresponding to peak field, dark blue areas

correspond to 0 V/m, green areas to 20 V/m (f) Plot of the magnitude of the

instantaneous electric field on the upper surface of the lower metal layer at a

wavelength of 36 mm and a phase corresponding to peak field, dark blue areas

correspond to 0 V/m, green areas to 20 V/m and red areas to 30 V/m

172

The dielectric core thickness was progressively decreased from 356 μm to 180 μm. As per

Structure 1, the optimum absorption depth and bandwidth occurs for a core thickness of 230 μm

to 250 μm: see Figures 7.5(b) and (c). For this core thickness, the structure achieves over 96%

of the narrowband limit (7.3): this level of performance is also very similar to that of Structure

1. Despite the additional modes being excited, the total absorption bandwidth still does not

exceed the limit set by (7.3).

7.4.4 Structure 3 - Multi-layering

HFSS was used to study the behaviour of the multi-layer structure shown in Figure 7.1(e) and

(f) with: t1 = 0.13 mm, t2 = 0.12 mm, t3 = 0.1 mm, t4 = 0.075 mm, the copper layers each had a

thickness of 0.018 mm giving a total thickness excluding the lowermost copper layer, of 0.497

mm. This structure’s reflectivity in decibels versus wavelength is shown in Figure 7.6(a). Over

the 15 mm to 70 mm (20 GHz to 4.29 GHz) range the structure supports five resonances, all of

which exhibit minimum reflectivities of less than -10 dB which corresponds to more than 90%

of the incident radiation being absorbed.

173

(a)

(b) (c)

(d) (e)

Figure 7.6 Multi-layer structures (a) Reflectivity in decibels versus wavelength

for structure with dielectric core thicknesses t1 = 0.13 mm, t2 = 0.12 mm, t3 =

0.1 mm, t4 = 0.075 mm, (b) magnitude of the electric field at a wavelength of

20.2 mm and at a phase corresponding to peak field for the N = 1 mode, scale

runs from 0 V/m to 90 V/m (c) magnitude of the electric field at a wavelength

of 20.2 mm and at a phase corresponding to peak field for the N = 3 mode,

scale runs from 0 V/m to 15 V/m (d) magnitude of the electric field at a

wavelength of 20.9 mm and at a phase corresponding to peak field for the N =

1 mode, scale runs from 0 V/m to 50 V/m (e) magnitude of the electric field at

a wavelength of 20.9 mm and at a phase corresponding to peak field for the N

= 3 mode, scale runs from 0 V/m to 20 V/m

By using (7.4), the approximate theoretical resonant frequencies can be found for different

values of N, these are presented in Table 7.1. The correlation between the resonant frequencies

predicted by HFSS and those obtained from (7.4) is excellent and demonstrates that each layer is

174

behaving as a largely independent resonator. Equation (7.4) also allows each of the resonances

in Figure 7.6(a) to be identified: the three longest-wavelength resonances are due to N = 1

modes in the alumina, FR4, and PET layers, the shortest-wavelength resonance, occurring at

20.2 mm is the N = 1 mode for the air layer. The mode at 20.9 mm is in fact the N = 3 mode in

the alumina layer which partially overlaps with the N = 1 air mode.

Alumina FR4 PET Air

N Eqn. (7.4) HFSS Eqn. (7.4) HFSS Eqn. (7.4) HFSS Eqn. (7.4) HFSS

1 61.3 62.3 40.8 41.6 35.8 35.8 20 20.2

2

3 20.4 20.9

Table 7.1 Resonant wavelengths in millimetres for Structure 4 – Multi-layer

structure as predicted using (7.4) and observed using HFSS

The identification of each mode using (7.4), can be corroborated by using HFSS to plot the

fields within the core: this is particularly useful for the closely-spaced N = 1 air and N = 3

alumina modes. Since these modes partially overlap, field plots at either wavelength (20.2 mm

or 20.9 mm) will show a resonance in both layers. However, the field strength exhibited by any

one mode will be a maximum at its centre resonant wavelength. Therefore, by plotting the

electric field and adjusting the phase to determine maximum field strength, it is possible to

determine which resonance on Figure 7.1(a) corresponds to which mode.

The instantaneous magnitude of the electric field within the dielectric layers of Structure 3 is

plotted in Figure 7.6(b) - (e). In Figure 7.6(b) and (c), both plotted at 20.2 mm wavelength, the

N = 3 and N = 1 modes are visible in the alumina and air layers respectively. The phase selected

in Figure 7.6(b) is that at which the N = 1 mode has its maximum field strength: approximately

90 V/m. In Figure 7.6(c) the phase selected is that at which the N = 3 mode has its maximum

strength: approximately 15 V/m.

175

By plotting the fields at 20.9 mm wavelength and adjusting phase, it can be seen that the peak

field of the N = 1 mode is approximately 60 V/m – much weaker than it was at 20.2 mm, see

Figure 7.6(d). Furthermore the peak field strength for the N = 3 mode is 20 V/m - higher than

was the case for 20.2 mm wavelength. This confirms that the 20.9 mm resonance in Figure

7.6(a) is due predominantly to the N = 3 mode in the alumina layer, whilst that at 20.2 mm is

predominantly due the N = 1 mode in the air layer.

The total absorption bandwidth afforded by this structure over the 15 mm to 70 mm wavelength

range is 65.4 dBmm which according to (7.3) is 95% of the theoretical limit, and is more than

double that of the optimised standard mono-grating. However, the reflectivity profile includes

the N = 3 resonance for alumina, the bandwidth of which should be excluded from the

summation for valid comparison to the standard mono-grating and to the narrowband limit (7.3)

which includes only the lowest order resonances. However, the partial overlap of the N = 1 air

and N = 3 alumina modes precludes this additional bandwidth being excluded from the

summation, and creates a small degree of ambiguity in the result that the structure achieves 95%

of the limit (7.3).

7.4.5 Structure 4 - Multiple permittivities

Structure 4 has a fixed period of 20 mm, each period containing two slits spaced apart by 10

mm and both having widths of 0.3 mm, the properties of the core material alternate between

adjacent cells. In the first cell the core material has a complex permittivity of εr1 =3.5 + i0.07 in

the second unit cell the complex permittivity is εr2 = 4.17 + i0.07, see Figure 7.1(g). The former

material is ficticous but might be achieved in practice by using a polymer, the latter material is

FR4 as per the previous chapters.

176

A series of structures spanning a range of core thicknesses was investigated. The reflectivity of

these structures as a function of wavelength is shown in Figure 7.7(a). Each structure supports

two resonances: one at approximately 41.5 mm and the other at approximately 38 mm

wavelength. From (7.4) it is clear that these correspond to the fundamental, N = 1 modes for the

cavities with permittivities εr1 =4.17 + i0.07 and εr1 =3.5 + i0.07 respectively.

(a)

(b)

177

(c)

Figure 7.7 Multiple-permittivity structure (a) Reflectivity in decibels versus

wavelength for structures with a range of dielectric core thicknesses (b)

Percent of narrowband bandwidth limit versus core thickness for the series of

multiple-permittivity structures (d) Reflectivity in decibels versus wavelength

as predicted by the finite element model for multiple-permittivity structures

with different values of loss tangent in the cavity with εr = 3.5

A core thickness of around 240 μm provides both greatest absorption depth and the also greatest

percentage of the theoretical limit (7.3) – see Figure 7.7(b). This structure achieves over 97% of

the narrowband limit (7.3) and therefore does constitute a small improvement over the standard

mono-grating but broadly equivalent performance to that of the other single-layer structures -

Structures 1 and 2.

Whilst a core thickness of 240 μm appears to provide optimum absorption bandwidth per unit

thickness, the depth of the shorter-wavelength resonance is only -8 dB, corresponding to 84% of

the incident radiation being absorbed. By contrast, the cavity in which εr = 4.17 absorbs over

99.9 % (-30 dB) of the incident radiation. Furthermore, the depth of the shorter-wavelength

resonance remained largely unaffected by alterations to the overall core thickness. This may be

due to the limited range of core thicknesses explored.

If the depth of both resonances could be simultaneously optimised then overall absorption

bandwidth might be increased. To this end the imaginary component of the core material in the

cavity with real permittivity equal to 3.5, was progressively decreased from 0.07 to 0.046, see

178

Figure 7.7(c). The depth of the shorter-wavelength resonance has increased from -8 dB to -12

dB whilst that of the longer-wavelength resonance has also increase from -40 dB to -50 dB.

However, the total absorption bandwidth has not changed significantly, remaining between 96%

and 98% of the limit (7.3).

By analogy with a series resonant circuit (Grant et al (1995) and Bowick (1982)), decreasing the

loss tangent within the cavity in which εr = 3.5, is the equivalent of decreasing the overall circuit

resistance. This decreased resistance results in an increase in the quality factor of the cavity: the

depth of the cavity may be increased but this is off-set by minor reductions to the width of the

resonance, rendering the overall bandwidth product largely unchanged.

7.5 Conclusions

This chapter has demonstrated that the standard mono-grating structure, when optimised by

altering core thickness, can achieve absorption bandwidths equivalent to approximately 93% of

the theoretical limit for narrowband absorbers. A series of optimised hybrid designs afford

greater absolute absorption bandwidth and allow the reflectivity profile to be tailored. However,

the hybrid designs are only a marginal improvement over the standard mono-grating in terms of

their bandwidth-to-thickness ratio.

Each of the structures investigated has its own advantages: the multilayer design, whilst thicker

than the others, is still grossly sub-wavelength and consequently of much lower profile than

conventional radar absorbers. It also offers the greatest opportunity to tailor the reflectivity

profile and ensure that absorption is produced at the desired frequency. Unlike some of the other

structures, its period is no greater than that of the standard mono-grating structure, meaning that

diffraction will be less of a problem. However, the overall bandwidth-to-thickness ratio for the

multi-layer design is likely to be slightly inhibited by the inclusion of numerous metal layers,

each of which contribute to the thickness of the structure but only absorb energy within a small

fraction of their thickness due to skin effects.

179

Consider that the skin depth of the copper at 10 GHz is approximately 0.6 μm (Grant et al

(1995)), therefore only the outer 0.6 μm of the copper will absorb energy and contribute to the

overall absorption bandwidth: the volume of metal more than 0.6 μm from the surface of the

metal will not contribute to absorption but will contribute to the overall thickness of the

structure and is hence superfluous. In Structure 3 the copper layers are each 18 μm thick hence,

allowing for absorption in the outer 0.6 μm of the copper means that almost 90% of the copper

volume is not absorbing any significant amount of energy.

Both the multiple permittivity (Structure 4) and multiple discrete period (Structure 1) structures

offer two resonances from a single-layer. However, both structures have a period significantly

longer than the standard mono-grating and will begin to diffract at longer wavelengths than

would otherwise be the case. The multiple-period structure is significantly easier to fabricate

than the multiple-permittivity structure, but the latter might prove easier to optimise. Both

structures have just one metal layer that contributes to overall thickness for bandwidth limit

calculations: hence they are able to achieve a higher percentage of the theoretical limit than the

multi-layer structure.

The multiple continuous period structure (Structure 2) affords several closely-spaced absorption

bands and affords the same bandwidth per unit thickness Structures 1 and 4. However, as a

consequence of the non-parallel slits, the frequencies at which the modes are excited are not

readily predictable which is a significant obstacle to tailoring the reflectivity profile.

In summary, this chapter constitutes efforts to increase the absorption bandwidth of the

microcavity array structures. The results demonstrate that the existing bandwidth-to-thickness

theories cannot be bypassed and that when attempts are made to increase the bandwidth, the

response of the structure changes in such as way as to the limit set by the bandwidth theories

and that these limits cannot be exceeded or violated in any way. The primary advantage of the

structures studied herein is that they permit the bandwidth to be re-distributed as required rather

than increasing the total bandwidth product.

180

Chapter 8

Conclusions

8.1 Summary of thesis

The work presented in this thesis has demonstrated that contrary to conventional thinking, ultra-

thin structures can exhibit highly efficient absorption of microwaves despite being more than

two orders of magnitude thinner than the operating wavelength. This creates the possibility of

designing low profile, lightweight and even flexible materials and claddings which can help

ensure a much higher degree of control over radio frequency signals across a vast range of

applications. Furthermore, frequency-selective transmission can also be mediated by variants of

the absorber structure thus affording precise control over the radio frequency environment.

The work detailed in Chapter 4 reveals that the ultra-thin cavity structures support a series of

pseudo-Fabry-Perot resonant modes, the frequencies of which are determined by the inter-slit

distance and the refractive index of the dielectric material filling the core. By considering the

momentum of the incident photons, the resonant frequencies of these modes under azimuthal

and polar incident angle variation has been accurately predicted. Further experimental work has

demonstrated that these mono-grating structures are highly efficient at converting one

polarisation state into the other with up to 50% of the incident field being re-emitted in the

opposing polarisation state.

Finite element modelling has elucidated the mechanism which underpins the frequency-

selective absorption, and has revealed that the resonance depth can be optimised by adjusting

the steady state amplitude of the resonant mode; this in turn is altered by changes to the

geometry and material properties of the structure. The modelling work also allows the form of

the resonant modes excited to be determined. This leads to the startling and counterintuitive

discovery that the modes undergo ―phrase compression,‖ being forced to undergo a phase

change of half a wavelength in a distance several orders of magnitude smaller than the

wavelength.

181

Further work considered the use of the ultra thin cavities as frequency-selective filters of

microwave radiation. Again it was found that the nature of the mode and in particular the phase

compression within the sub-wavelength apertures is key to determining the resonant frequencies

of the structures.

Chapter 5 focused on two-dimensional ―bi-grating,‖ structures in which two orthogonal sets of

sub-wavelength apertures have been formed in the upper conducting layer. These structures

support more modes than their mono-grating equivalents due to the greater number of scattering

vectors that are accessible. They also exhibit a remarkably high level of azimuth and incident

angle independence with the fundamental appearing completely invariant for both polarisation

states. Furthermore, the bi-grating structures exhibit very little polarisation conversion, in

contrast to the mono-grating equivalent. Again the resonant frequencies of the modes are

accurately predicted by considering the momentum of the incident photons. The finite element

model is used to explore the character of the modes and reveals that both degenerate and

coupled modes are supported, the former resulting from simultaneous scattering from

orthogonal lattice vectors and the latter from lattice vectors indigenous to two-dimensional

gratings.

Chapter 6 details the investigation of two hexagonally symmetric ―tri-grating,‖ structures each

of which features three sets of sub-wavelength apertures each rotated by 60° relative to the other

two. On the first sample all three sets of slits intersect at common points creating an array of

equilateral triangles. In the second structure each set of grooves is off-set by half its repeat

period relative to the other two sets, hence all points of intersection feature slits from only two

of the three sets and a pattern of hexagons interspersed with equilateral triangles is formed.

Comparison of their relative responses demonstrates that it is the repeat period of the slits and

their orientation, rather than the shape of the metal patches that determines the behaviour.

Furthermore, both structures exhibit reflectivity spectra which are incident and azimuthal angle

independent for any polarisation state and contain more resonances than the bi-grating

equivalent.

182

Chapter 7 considered the absorption bandwidth of the ultra thin cavities and presented four

strategies for maximising this bandwidth by exciting multiple resonant mode series through a

multiplicity of cavity lengths and refractive indices. The specific embodiments included a

structure which alternated between two inter-slit distances and hence supported two

―fundamental,‖ resonances. Similarly a structure in which the permittivity of the dielectric core

alternated between adjacent unit cells supported two fundamental modes. Other approaches

included non-parallel slits and multi-layers structures. All these approaches proved successful in

increasing the total absorption bandwidth versus that of the simple mono-grating structure.

However, in all cases the total bandwidth was fundamentally limited by the thickness of the

structures and their magnetic permeability as expected on the basis of established theory.

8.2 Ideas for future work

The work embodied in this thesis has considered the performance of ultra thin cavity structures

as devices for the control and manipulation of microwave radiation. In addition to having many

applications, the range of possible geometries and configurations of these cavities is myriad and

hence constitutes a massive body of potential future work, the scope of which is limited only by

imagination. Presented here is a small sample of the possible interpretations of and extensions to

the work presented herein.

One could also envisage further single layer transmission structures in which the array of slits in

the upper metal layers was accompanied by a solitary slit in the lower layer. The number of slits

in the lower layer could be progressively increased and transmission efficiency as a function of

the number of slits investigated. In fact, basic tests on hand-made samples of this type were

conducted using the focused horn apparatus. The results indicated an increase in transmission

efficiency with number of slits. However, due to the hand-made nature of the samples these

results were not included: further testing with more precise sample formed by etching would

provide more reliable data.

183

(a)

(b)

(c)

Figure 8.1 Hybrid transmission structures

(a) array of slits in the upper metal layer, single slit in the lower metal layer (b)

rotation of slits in lower metal layer relative to those in the upper metal layer,

layers shown separately (c) progressive reduction in slit number to concentrate

field

One might also pursue the effect of rotating the slits in the lower metal layer relative to those in

the upper metal layer hence introducing a degree of chirality and raising the possibility of

polarisation conversion via transmission. This approach creates a periodic structure of longer

period – the period is dictated by the angle of rotation. to create a working device may require

adiabatic rotation: only small twist angles. It might also be possible to create extremely high

field concentrations by combining the multi-layer and single slit ideas. Consider an aligned

transmission sample as per those in Chapter 4, wherein the number of slits could be

184

progressively reduced layer after layer. This might have the effect of channelling the energy

collected by many slits towards a single exit slit.

Multi-layer absorbing structures were shown in Chapter 8, to exhibit increased absorption

bandwidth. It might be feasible to create a pseudo-fractal structure similar to the multi-layer

geometry but in which the period changes by a factor between layers – see Figure 8.2. However,

this may only work if the radiation can progress through the upper layer to the lower layer,

which in turn may require the periods of the lower cavities to be multiples of the upper cavities.

This could combine in one device the ability to manipulate radiation from disparate regions of

the electromagnetic spectrum such as the microwave region investigated by this work and the

optical region investigated by other including Hibbins et al (2006).

Figure 8.2 Pseudo-fractal multi-layer absorbing structure

In addition to altering the physical geometry of the structures, the material properties of both the

conducting and dielectric layers could be considered. Introducing magnetic material to the core

would serve to increase absorption bandwidth in accordance with established theory, Brewitt-

Taylor (1999) and Rozanov (2000), whilst the use of dielectrics which are active rather than

passive would create devices whose bandwidth is not limited by the aforementioned theoretical

results. The addition of liquid crystal material to the core might permit dynamic control over the

microwave reflectivity and transmissivity of the structures if a method could be found for

applying a voltage across the core and hence polarising the material. One might also investigate

the effect of adding two media of differing refractive index to same cavity – see Figure 8.3.

185

Figure 8.3 Absorbing structures in which each cavity contains dielectric media

of different refractive index

A further novel embodiment of the microcavity structures might arise from the use of very thin

metal layers. Materials such as indium tin oxide (ITO) are already in widespread use in

commercial applications such as thermally efficient glazing. If variants of materials such as ITO

could be engineered to have sufficiently high conductivity, then optically transparent micro

cavity array absorbers become possible, widening the application space for the technology even

further.

Chapters 5 and 6 were devoted to the behaviour of microcavity structures with higher order

rotational symmetry and it was demonstrated that these devices afford absorption which is

independent of incident and azimuthal angle and polarisation state. These studies could be

expanded to consider other polygonal shapes into which the upper metal layer could be divided.

Rectangles rather than squares would introduce geometric anisotropy and might be applied to

create a form of orientation sensor. Patterning the upper metal layer in a manner similar to that

of Penrose tiles (Penrose (1979)) is of particular fascination. Penrose tiling patterns have the

extraordinary property of rotational symmetry but without translational symmetry, hence the

response of a microcavity absorber patterned in this manner is difficult to predict.

Several structures for maximising absorption bandwidth were presented in Chapter 7. Each of

these warrants further investigation, in particular the idea of non-parallel slits and hence a

continuum of inter-slit distances. Curved slits or other forms of corrugation besides the saw-

tooth designs considered herein might be of interest. Structures in which the slit width varies

along its length might also provide a means of increasing bandwidth since slit width can affect

186

resonant frequency as shown in Chapter 4. As outlined above, changing the core material

properties could create structures which are not bound by the same bandwidth-to-thickness ratio

as those studied in Chapter 8 and could exhibit hitherto unachieved bandwidths.

One very important conclusion that can be drawn from the work presented herein, is that

viewing the microcavity structures as pseudo Fabry-Perot resonators is too simple a description.

This description is useful only for an estimate of the resonant frequency of the simple mono-

grating structure at normal incidence. Changes to incident angle, polarisation state and

azimuthal angle all serve to alter the resonant frequency for the reasons detailed in Chapter 4.

Furthermore, geometric factors such as slit width, core thickness and the shape of the polygons

into which the upper metal layer is divided, all influence the frequencies of the resonant modes

supported. A further piece of work could be undertaken to establish a firmer theoretical basis for

the dependence of resonant frequencies on these factors.

8.3 List of publications

Squeezing millimetre waves into microns

Alastair P. Hibbins, J.R. Sambles, C.R. Lawrence and J.R. Brown

Physical Review Letters, 92, 143904 (2004)

Angle independent microwave absorption by ultra thin microcavity arrays

J.R. Brown, Matthew J. Lockyear, Alastair P. Hibbins, C.R. Lawrence and J.R. Sambles

Journal of Applied Physics 104, 043105 (2008)

Angle independent resonant absorption of microwave radiation by hexagonally symmetric ultra

thin microcavity arrays

J.R. Brown, Matthew J. Lockyear, Alastair P. Hibbins, C.R. Lawrence and J.R. Sambles

Submitted for publication to J. Phys. D. Applied physics

187

Ultra thin microcavity arrays for optimum absorption bandwidth

J.R. Brown, Alastair P. Hibbins, C.R. Lawrence and J.R. Sambles

Awaiting clearance for publication

188

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