Band diagrams of layered plasmonic metamaterialsMohammed H. Al Shakhs, Peter Ott, and Kenneth J. Chau Citation: Journal of Applied Physics 116, 173101 (2014); doi: 10.1063/1.4900532 View online: http://dx.doi.org/10.1063/1.4900532 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/17?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rapid and highly sensitive detection using Fano resonances in ultrathin plasmonic nanogratings Appl. Phys. Lett. 105, 161106 (2014); 10.1063/1.4899132 Experimental evidence of exciton-plasmon coupling in densely packed dye doped core-shell nanoparticlesobtained via microfluidic technique J. Appl. Phys. 116, 104303 (2014); 10.1063/1.4895061 Triple plasmon resonance of bimetal nanoshell Phys. Plasmas 21, 072102 (2014); 10.1063/1.4886151 Effect of ultrasonication on properties of sequential layer deposited nanocrystalline silver thin films AIP Conf. Proc. 1447, 695 (2012); 10.1063/1.4710193 Multiple enhanced transmission bands through compound periodic array of rectangular holes J. Appl. Phys. 106, 093108 (2009); 10.1063/1.3254248

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Band diagrams of layered plasmonic metamaterials

Mohammed H. Al Shakhs,1 Peter Ott,2 and Kenneth J. Chau1,a)

1School of Engineering, The University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada2Heilbronn University, Heilbronn, Germany

(Received 30 September 2014; accepted 15 October 2014; published online 3 November 2014)

We introduce a method to map the band diagrams, or equipotential contours (EPCs), of any layered

plasmonic metamaterial using a general expression for the Poynting vector in a lossy layered

medium of finite extent under plane-wave illumination. Unlike conventional methods to get band

diagrams by solving the Helmholtz equation using the Floquet-Bloch theorem (an approach

restricted to infinite, periodic, lossless media), our method adopts a bottom-up philosophy based on

spatial-frequency decomposition of the electric and magnetic fields (an approach applicable to

finite, lossy media). Equipotential contours are used to visualize phase and group velocities in a

wide range of layered plasmonic systems, including the basic building block of a thin metallic layer

and more complex multi-layers with unique optical properties such as negative phase velocity,

super-resolution imaging, canalization, and far-field imaging. We show that a thin metallic layer

can mimic a left-handed electromagnetic response at the surface plasmon resonance and that stacks

of metal and dielectric layers can do the same provided that the dielectric layer is sufficiently thin.

We also use EPCs to estimate resolution limits of both Pendry’s silver slab lens and the Veselago

lens and show that the image location and lateral image resolution of metal-dielectric layered flat

lenses can be described (and tailored) by the concavity and spectral reach of the dominant band in

their EPCs. Homogenization methods for describing the effective optical properties of various lay-

ered systems are validated by the extent to which they accurately mimic features in their EPCs.VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4900532]

I. INTRODUCTION

A metamaterial is a structure engineered on deep sub-

wavelength scales to elicit exotic electromagnetic properties.

A new breed of “plasmonic” metamaterials has emerged that

operate at the upper reaches of the visible spectrum by

exploiting a resonant optical mode called a surface plasmon

polariton (SPP). A spot-light was cast on this approach by

the pioneering work of Pendry,1 who showed that a thin sil-

ver layer at ultraviolet frequencies could mimic a Veselago

lens2 (a homogeneous slab possessing a left-handed electro-

magnetic response described by the constitutive parameters

�¼l¼�1) and perform super-resolution imaging by evan-

escent wave amplification. Following a series of experiments

demonstrating super-resolved images made by a thin silver

layer,3–6 more complex layered plasmonic metamaterial

structures were introduced with unique refractive properties.

Several multi-layered metal-dielectric systems were shown

to exhibit sub-diffractive image translation across an opti-

cally thick extent,7–10 a process called “canalization” in ref-

erence to their ability to support long, thin canals of light.

Just last year, a multi-layered metal-dielectric system

designed on the principle of coupled plasmonic waveguides

was experimentally shown to be capable of far-field imag-

ing,11 in a manner strikingly similar to what would be

expected of a Veselago lens.

There are many ways to describe and model the behav-

ior of light in layered plasmonic metamaterials. Pendry1

employed the electrostatic approximation to argue for the

analogy between a thin silver layer and a negative-index

layer. More complex metal-dielectric multi-layers have been

modeled as homogeneous slabs composed of anisotropic6,7,9

or negative-index11 media, enabling intuitive visualization of

light refraction and bulk propagation based on geometric

optics. Perhaps that most widespread approach is to use nu-

merical electromagnetic simulations based on Maxwell’s

equations to directly calculate spatial field and energy distri-

butions in and about the medium. Numerical simulations are

powerful because they can be adapted to model any system,

but are also limited because they do not offer general physi-

cal insights. As a complement to these methods, we explore

the use of band diagrams, or equipotential contours (EPCs),

as a graphical method to describe a wide range of refractive

properties associated with layered plasmonic metamaterials.

Band diagrams are indispensable for illustrating the re-

fractive properties of dielectric photonic crystals. The con-

ventional top-down method to obtain band diagrams is to

leverage the Floquet-Bloch theorem12,13 to isolate discrete

sets of real-valued wave vectors ~k and then solve for corre-

sponding frequencies xð~kÞ from a linear eigenvalue equation

derived from the Helmholtz equation.14 Surfaces of constant

frequency are then displayed as a function of real-valued

wave-vector coordinates, providing graphical information on

phase and group velocity in the medium along different

directions. Unlike photonic crystals, plasmonic metamateri-

als are typically finite-sized, lossy, and inherently dispersive

(due to operation near the surface plasmon resonance fre-

quency). Loss and finite extent restrict the validity of the

Floquet-Bloch theorem, and dispersion turns the Helmholtz

a)Author to whom correspondence should be addressed. Electronic mail:

kenneth.chau@ubc.ca

0021-8979/2014/116(17)/173101/9/$30.00 VC 2014 AIP Publishing LLC116, 173101-1

JOURNAL OF APPLIED PHYSICS 116, 173101 (2014)

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equation into a non-linear eigenvalue equation, which

requires iterative algorithms that are time-consuming and

sensitive to initial guesses. As a result, band diagrams are

rarely used to describe the properties of plasmonic metama-

terials, an observation previously noted in Ref. 15. Recent

methods to derive band diagrams for periodic plasmonic sys-

tems have been proposed using a finite-element method for-

mulation to solve for complex-valued wave vectors ~k as a

function of frequency xð~k Þ.15–18 The resulting surfaces of

constant frequency, however, must be displayed as a func-

tion of both real and imaginary wave-vector coordinates.

We introduce an elegant bottom-up approach to derive

band diagrams for any layered plasmonic metamaterial as a

function of real wave-vector coordinates. The method relies

on spatial-frequency representation of the electric and mag-

netic fields in an arbitrary lossy layered medium, from which

a spectral time-average Poynting vector and its associated

equipotential contour can be directly calculated. The EPCs

of layered plasmonic metamaterials—ranging from the sim-

plest configuration of a thin silver layer to complex periodic

systems composed of multi-layered unit cells—are studied to

shed further light on interesting reported effects such as neg-

ative phase velocity, super-resolution, canalization, and far-

field imaging. We show that the SPP mode in a thin silver

layer possess anti-parallel phase and group velocities closely

mimicking a left-handed electromagnetic response, aniso-

tropic metamaterials capable of canalization have a flat

band-diagram extending far beyond the free-space cutoff,

and far-field flat lens imaging with a layered structure is pos-

sible when a dominant band in the EPC has an upward con-

cavity. We also use EPC engineering to design a highly

transmissive layered flat lens that is capable of far-field

imaging while incorporating a minimal amount of metal.

Throughout these explorations, various homogenization

methods commonly used to describe layered plasmonic

metamaterials are validated by their accuracy in capturing

the salient features of their EPCs.

II. EQUIPOTENTIAL CONTOURS

We present a recipe to obtain EPCs for any layered plas-

monic metamaterial. Consider a generic layered medium

bounded by semi-infinite half spaces where the layers are

parallel to the xy plane. The medium is excited by a plane

wave incident from one of the half spaces at an angle h in

the xz plane. Field continuity across the interfaces naturally

parameterizes the solutions in terms of the wave-vector com-

ponent parallel to the plane of the layers, kx. Time-harmonic

solutions for the electric ~Eðx; zÞ and magnetic ~Hðx; zÞ fields

in the medium can be obtained by standard techniques such

as the transfer matrix method.19 Spatial Fourier transforma-

tion of the solutions yields spectral electric and magnetic

fields defined as a function of kz, the spatial frequency along

z, which exist for a fixed kx. Sweeping kx over a range (by

varying h) yields two-dimensional k-space distributions of

the electric ~Eð~kÞ and magnetic ~Hð~kÞ fields as a function of

the spatial-frequency variable ~k ¼ kxx þ kzz. Together, the

electric and magnetic field distributions define a plane-wave

spectrum in which each plane-wave component is

completely described by the triad ~k-~Eð~kÞ-~Hð~kÞ.20–22 Time-

averaged power flow of each plane-wave component is given

by the spectral Poynting vector20

h~S ~kð Þi ¼ 1

2< ~E ~kð Þ � ~H� ~kð Þ� �

; (1)

whose scalar potential is defined by

U ~kð Þ ¼ 1

4p

ðr~k 0 � h~S ~k 0ð Þid~k 0

j~k � ~k 0j: (2)

The gradient of the scalar potential yields an approximation

of h~Sð~kÞi that becomes exact when the spectral Poynting

vector is ir-rotational. The scalar potential can be described

by contour lines that form a diagram of bands in k-space.

Given the fundamental relationship23,24

~vg ¼h~S ~kð ÞihU ~kð Þi

; (3)

where hUð~kÞi is the time-averaged energy density, it can be

seen that EPCs defined by Uð~kÞ are similar to conventional

band diagrams defined by frequency xð~kÞ because both have

gradients along the group velocity. EPCs are also unique in

several ways. First, the EPC density conveys additional in-

formation on the k-space distribution of energy density,

where a high contour density indicates high energy density.

Second, EPCs can be derived without invoking the Floquet-

Bloch theorem, which enables general treatment of periodic

or non-periodic systems. Third, the effect of material loss is

captured by a continuum of plane-wave modes with real

wave vectors as opposed to discrete modes with complex

wave vectors, allowing EPCs to be plotted as a function of

real wave-vector coordinates. In many respects, this

approach builds on past efforts to examine phase and group

velocity in infinite, lossless photonic crystals displaying

anomalous refraction based on spatial-frequency decomposi-

tion of electric and magnetic fields.20–22 We have extended

this philosophy to incorporate finite, lossy media and intro-

duced a new method to extract EPCs from this spatial-

frequency decomposition.

We first demonstrate how EPCs derived from the pro-

posed technique provide a picture of refraction at a planar

air-slab interface consistent with geometric optics. We exam-

ine two cases where a homogeneous, isotropic slab, whose

interface is aligned along the z plane, has either a positive or

negative index. As shown in Figs. 1(a) and 1(b), the EPCs

for the homogeneous slabs are each dominated by a semi-

circular band. In the limit where the slabs are lossless and in-

finite, this band collapses to a semi-circular line describing

the plane-wave mode in a homogeneous medium (x¼ ck/n).

A plane wave incident onto the slab from free space at an

angle h can be described in k-space by a wave-vector~k0 ¼ kx;0x þ kz;0z. Tangential field continuity imposes con-

servation of the wave vector component parallel to the inter-

face (kx,0), which is visualized by the vertical lines in Figs.

1(a) and 1(b). For the positive-index slab, the incident wave

excites planes waves with parallel phase and group velocities

173101-2 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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along a direction consistent with Snell’s law. For the

negative-index slab, the excited plane waves have anti-

parallel phase and group velocities, where the former is

directed towards the interface and the latter is directed away

from the interface. Plane waves with anti-parallel phase and

group velocities are hallmarks of a left-handed electromag-

netic response and together impose negative refraction to the

same side of the normal, as first presented by Veselago.2

III. METHODOLOGY

In the analysis that follows, we will derive EPCs for a

range of layered plasmonic metamaterials, including a thin

silver layer sustaining a SPP mode, Pendry’s silver slab lens

and the Veselago lens, and metal-dielectric multi-layered

systems capable of canalization and far-field imaging. EPCs

are calculated based on analytical, spatial-frequency repre-

sentations, parameterized in terms of kx, for the electric and

magnetic fields in an arbitrary lossy layered medium of finite

extent under plane-wave illumination.25 We employ analyti-

cal expressions because they enable rapid, efficient, and sta-

ble computation of high-resolution EPCs (composed of up to

500 unique solutions versus kx), but could be replaced by any

of the many methods available to obtain electric and mag-

netic field solutions in lossy, finite, layered media. For each

case, we check that the solenoidal component of the spectral

Poynting vector is small compared to its ir-rotational compo-

nent. To probe the plane-wave response of layered plasmonic

metamaterials at high lateral spatial frequencies (where kx� k0),

we adopt the standard prism configuration where the layered

systems are exposed to plane-wave illumination from a high-

index dielectric medium. For the case of the Pendry silver

slab lens and the Veselago lens, a thin planar air region is

added between the prism and the layers to provide evanes-

cent illumination at spatial frequencies beyond the free-space

cutoff (matching the condition used to predict their super-

resolution capabilities). Predictions made from derived EPCs

are supported by full-wave simulations in which Maxwell’s

equations are solved over a discrete two-dimensional spatial

grid (with a spatial resolution of at least 1 nm) using the

finite-difference frequency-domain (FDFD) technique. We

use these simulations because they provide detailed visualiza-

tion of the spatial distribution of fields and energy density

in and about the medium and can accommodate two-

dimensional geometries that are not symmetric in the lateral

direction. The FDFD technique is selected because it allows

realistic modeling of lossy materials using tabulated complex

permittivity values, which are generally available in Refs. 26

and 27. Simulations intended to quantify the imaging resolu-

tion of layered plasmonic systems use a conventional test

object consisting of two k0/15-wide openings in an optically

thick chromium mask spaced a variable distance T apart.

The minimum resolvable feature is quantified by the mini-

mum distance T at which the two openings can be uniquely

identified in the energy density distribution in the image

region.

FIG. 1. Light refraction at an interface

described by EPCs. EPCs for finite,

lossy slabs of thickness 2k0 having ei-

ther (a) a positive refractive index n ¼1:5þ 0:1i or (b) a negative refractive

index n ¼ �1þ 0:1i. The bottom pan-

els depict the EPCs of an incident

plane wave (k0¼ 400 nm) impinging

on the slab. The resulting phase (blue

arrow) and time-averaged Poynting

vector (red arrow) are graphically

derived in k-space from the EPCs and

then depicted in real-space in the

insets.

173101-3 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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IV. SPP MODE AS A WINDOW INTO THENEGATIVE-INDEX WORLD

The simplest plasmonic system is a thin metallic layer

sustaining an SPP mode. We derive the EPC for a thin silver

layer under the Kretschmann configuration for SPP excita-

tion to demonstrate why this mode offers a pathway to realiz-

ing left-handed electromagnetic behavior. A 40-nm-thick

silver layer is sandwiched between a dielectric (n¼ 2) me-

dium and air and illuminated by a plane wave (free-space

wavelength k0¼ 400 nm, transverse-magnetic (TM) polar-

ization) incident from the dielectric medium. The silver is

characterized by a permittivity � ¼ �4:4þ 0:2i.26 The most

prominent feature of the EPC, shown in Fig. 2(a), is a high

density of vertical lines localized at a spatial frequency just

beyond the free-space cutoff k0. This feature corresponds to

a sub-diffractive guided mode propagating along the surface,

i.e., the SPP mode. Plane-wave excitation of the SPP mode

at kx ’ 1:1k0 can be visualized by drawing a vertical line in

the EPC at that wave vector coordinate and then forming a

wave vector ~k from the origin to the point of highest line

density along the line. The phase velocity along the ~k vector

points away from the origin, whereas the group velocity

along the frequency gradient points towards the origin. A sil-

ver layer, therefore, can mimic the defining characteristic of

a left-handed medium (i.e., anti-parallel phase and group

velocities), albeit under highly restrictive conditions. The

left-handed response is achieved solely within the silver

layer, in a direction along the plane of the layer, and at a

spatial-frequency that is inaccessible to free-space plane-

wave illumination. Moreover, it is inherently dispersive due

to operation near a resonance and requires TM polarization.

If, for example, the illumination switched to transverse-

electric (TE) polarization, the resulting EPC [Fig. 2(b)] has

an inverted potential gradient so that group velocity is gener-

ally directed away from the origin.

The SPP mode mimics the behavior of a negative-index

medium due to an effective magnetic response established

by microscopic circulation of its electric field in the silver

layer. To illustrate this phenomenon, Fig. 2(c) shows a

FDFD simulation of the electric field (blue arrows) and mag-

netic field (contour lines) in the 40-nm-thick silver layer

excited at the surface plasmon resonance. The electric field

circulation is synchronized with the background magnetic

field and is most pronounced at the silver-air interface where

the SPP mode resides. Contributions from the in-plane elec-

tric field to the magnetic response can be described by a neg-

ative permeability using homogenization methods in which

the electric field circulation is folded into an effective mag-

netic flux density.28,29 In contrast to shaped resonators used

FIG. 2. Band diagrams reveal interesting propagation characteristics of the SPP mode. We first examine EPCs for a single 40-nm-thick silver layer illuminated

by a plane wave (k0¼ 400 nm) incident from a dielectric (n¼ 2) prism for either (a) TM or (b) TE polarization. (c) The electric (blue arrows) and magnetic

(contour lines) fields in the 40-nm-thick silver layer at the surface plasmon resonance. The x and z scales are the same. Under TM-polarized illumination at the

same wavelength, we next examine EPCs for a 5-layer stack of silver and silicon nitride (n¼ 2) layers, where the silver layer thickness is fixed at 40 nm and

the thickness of the silicon nitride is either (d) 10 nm or (e) 40 nm. (f) The electric (blue arrows) and magnetic (contour lines) fields in the multi-layer structure

at the surface plasmon resonance for the case of dielectric thickness 10 nm. The insets in the EPCs depict the geometrical configuration for each case. The blue

and red arrows describe the phase and time-averaged Poynting vector, respectively, of the most dominant mode excited at the spatial frequency kx ’ 1:1k0.

The location of the dominant mode corresponds to the highest density of contour lines intersected by the vertical line describing the conserved wave-vector

component of an incident plane wave.

173101-4 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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at microwave frequencies to achieve artificial magnetism,

the silver layer exploits a natural material resonance at a

smooth surface to enable electric circulation over the small-

est possible size scales. This represents the ultimate minia-

turization of the principles behind the original split-ring

resonators and offers what is now the only pathway to

achieve artificial magnetism (and left-handed behavior) at

visible and UV frequencies.

The left-handed behavior of the SPP mode can be made

to extend across metallic and dielectric regions provided that

the dielectric fill fraction is sufficiently low. We consider a

metal-clad five-layer waveguide made of 40-nm-thick silver

claddings and two 10-nm-thick dielectric (n¼ 2) cores sepa-

rated by a 40-nm thick silver layer, resulting in a dielectric

fill fraction of 0.14. The EPC for plane-wave illumination

(k0¼ 400 nm, TM-polarization) from a semi-infinite dielec-

tric (n¼ 2) medium is shown in Fig. 2(d). The EPC consists

of concentric elliptical contours of increasing potential closer

to the origin. Plane-wave illumination at kx¼ 1.1k0 [the sur-

face plasmon resonance condition in Fig. 2(a)] yields an

excited wave with phase and group velocities that are nearly

anti-parallel. Left-handed behavior predicted from the EPC is

consistent with both theoretical investigations of backward-

propagating SPP modes28,29 and experiments showing in-

plane negative refraction of visible light in metal-clad wave-

guides with ultra-thin dielectric cores.30 Similar to the case for

the single silver layer, the multi-layer system has a left-

handed response arising from microscopic electric field circu-

lation at the surface plasmon resonance [Fig. 2(f)]. This

left-handed response is highly sensitive to the dielectric fill

fraction. When the dielectric fill fraction increases to 0.40 (by

increasing the thickness of the dielectric cores to 40 nm), for

example, the EPC changes dramatically and flattens out into a

series of horizontal lines with an upward potential gradient

[Fig. 2(e)]. The EPC indicates that the system now supports

bulk modes propagating through the medium as opposed to

guided modes propagating along the surface.

V. SUPER-RESOLVING SILVER SLAB LENS AND THEVESELAGO LENS

Theory, simulations, and experiments indicate that a

thin silver layer is capable of imaging beyond the diffraction

limit. It is believed that this capability arises from a funda-

mental likeness to a Veselago lens,1 which has the unique

potential for imaging with perfect resolution due to ampli-

tude amplification of evanescent waves. Here, we derive the

EPCs for both a silver slab lens and the Veselago lens under

evanescent wave illumination conditions to understand the

origins of their resolving power. This approach offers two

novelties beyond the considerable amount of work already

devoted to this topic. First, evanescent waves are coupled

into the lenses using a realistic prism-coupling configuration

that can be implemented in a laboratory. Second, EPCs pro-

vide an intuitive picture of imaging based on phase and

power flow in the lens, similar to how classical imaging sys-

tems are analyzed.

The EPCs for a 40-nm-thick silver slab lens and a

Veselago lens of equivalent thickness are shown in Figs. 3(a)

and 3(b). The object plane is located 20 nm from the entrance

of the lens and is defined by the interface between a dielec-

tric prism and air. A prism with an inordinately high refrac-

tive index (n¼ 5) is used to allow plane-wave excitation of

high-k modes with large lateral spatial frequencies. The

EPCs of the two lenses are strikingly similar, which justifies

their analogy under near-field conditions. The behavior of

FIG. 3. Band diagram analysis of thin, super-resolving layers. EPCs for (a) Pendry’s 40-nm-thick silver slab lens and (b) a Veselago lens (�¼l¼�1) of

equivalent thickness for TM-polarized illumination at the wavelength of k0¼ 357 nm. The layers are illuminated from a high-index dielectric prism (n¼ 5)

with a 20-nm-thick air region separating the prism face from the entrance of the layers. This prism configuration is a realistic emulation of the evanescent

wave illumination conditions used in Ref. 1 to first propose super-resolution imaging in these systems. The insets describe the geometrical configurations for

each case. The red arrows trace out the time-averaged Poynting vector along a frequency contour, revealing a “funneling” effect at high lateral spatial frequen-

cies, a possible culprit for their super-resolution capabilities. Simulations of a test object imaged by Pendry’s silver slab lens when the two point-like features

of the object are spaced by (c) k0/2.5, (d) k0/3.0, and (e) k0/3.5. Note the gradual disappearance of discrete lobes in the image region (above the layer) suggest-

ing that the two objects become indiscernible. Simulations of a test object imaged by the Veselago lens when the two point-like features of the object are

spaced by (f) k0/4, (g) k0/5, and (h) k0/6.

173101-5 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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both lenses at large spatial frequencies is dominated by sur-

face modes (indicated by the dense vertical lines outside the

free-space cutoff) possessing a left-handed response (indi-

cated by potential gradients directed towards the origin).

Several additional observations can be made from the EPCs.

First, the layers support more modes beyond the free-space

cutoff than they do below. These high-k modes are the origin

of their super-resolution capabilities. Second, the high-kmodes have group velocities that are generally directed

towards the normal, indicated by the kx¼ 0 line. This implies

that the lenses are capable of collecting and concentrating

light across their extent, a feature that is emphasized in the

EPCs by traces of the time-averaged Poynting vector along a

potential contour. Third, the density of modes is non-

uniform and gradually diminishes for increasing k, indicating

that, even for the Veselago lens, there is a finite limit to their

resolving capabilities (for the realistic prism configuration

studied here).

Assuming that left-handed guided modes—the dominant

and common feature in the EPCs for both the silver slab lens

and the Veselago lens—are responsible for sub-diffractive

imaging, resolution limits can now be estimated by the

spatial-frequency bandwidth over which these modes exist.

Dense vertical lines in the EPC for the silver slab, for exam-

ple, are concentrated just outside the free-space cutoff, up to

a spatial frequency between 1.5k0 and 2.0k0. Based on this

bandwidth, the silver slab should have a minimum resolution

that is just less than double the free-space diffraction limit

(k0/2). This estimate is backed by FDFD simulations, which

show that the minimum resolvable feature by the silver slab

is T¼ k0/3 [Figs. 3(c)–3(e)]. The guided modes in the

Veselago lens, on the other hand, extend up to a spatial fre-

quency between 3k0 and 4k0, which is about double the

bandwidth of the silver layer. FDFD simulations concur with

this estimate, revealing that the Veselago lens has a mini-

mum resolvable feature of T¼ k0/6 [Figs. 3(f)–3(h)], about

twice as small as that for the silver layer. Finite resolution of

the Veselago lens may be initially surprising due to their the-

oretical potential for perfect resolution, but can be explained

by the use of a lossy, three-dimensional object in the simula-

tion, which is sufficient to perturb the delicate conditions

required for perfect imaging.

VI. CANALIZATION IN MULTI-LAYERED STRUCTURES

Simulations have established that metal-dielectric multi-

layers can guide light across its extent with little to no dif-

fraction. Given the complexity of these systems—which can

incorporate up to 40 alternating layers—simplifying models

of their refractive properties have been built upon effective

medium theory (EMT). As a case study, we adopt the config-

uration proposed in Ref. 7 consisting of 20 repetitions of a

bi-layer unit cell composed of a 7.8-nm-thick silver layer

(�¼�15) and a 7.2-nm-thick silicon layer (�¼ 14), at the

wavelength k0¼ 600 nm. Volumetric averaging of the local

permittivity by Maxwell-Garnett theory yields the basic fre-

quency contour shown in Fig. 4(a) consisting of a single hor-

izontal line located at kz ’ 0. Comparison to the true EPC

derived from the internal electric and magnetic fields reveals

some limitations of EMT. While EMT accurately predicts

the flat horizontal contours that dominate the EPC especially

below the free-space cutoff, it fails to predict their locations

in k-space. As seen in Fig. 4(a), the EPC consists of flat ellip-

tical contours roughly symmetric about kz ’ 0, with a promi-

nent band of horizontal lines centered at kz ’ k0 and another

centered at kz ’ –k0. These bands describe forward- and

FIG. 4. Band diagram for the metal-

dielectric multi-layer systems studied

in Ref. 7 for the case of TM-polarized

plane-wave (k0¼ 600 nm) illumination

from a high-index dielectric (n¼ 5)

prism. The blue line depicts the simpli-

fied band diagram predicted from

effective medium theory (namely,

Maxwell-Garnett theory). FDFD simu-

lations of the multi-layers imaging

a test object when the two point-like

features of the object are spaced by

(b) k0/8, (c) k0/9, and (d) k0/10.

173101-6 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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backward-propagating modes. Remarkably, because the con-

tours remain flat beyond the free-space cutoff, an incident

plane wave will excite modes that carry energy strictly in the

forward or backward direction, independent of the incidence

angle. A broad and flat EPC leads to an ability to channel

light straight through the medium, which is visualized by

FDFD simulations of thin light streams—spaced as little as

k/10 apart—flowing across the extent of the layers [Figs.

4(b)–4(d)]. It should be noted that these simulations have

employed the ideal lossless material parameters originally

used in Ref. 7. Incorporating realistic material losses will

likely reduce their resolving power.

VII. FAR-FIELD FLAT LENS IMAGING

EPCs can also be used to better understand the princi-

ples behind a planar layered structure recently shown to ex-

hibit far-field imaging (i.e., a flat lens). We consider the

structure reported in Ref. 11 consisting of three repetitions of

a five-layer unit cell with the layer sequence Ag (33 nm)—

TiO2 (28 nm)—Ag (30 nm)—TiO2 (28 nm)—Ag (33 nm),

which is then immersed in free space. This structure has

been modeled as a homogeneous, isotropic, negative-index

medium, a theoretical construct that enables its external opti-

cal properties to be understood by simple geometrical optics,

but lacks insight into its internal field dynamics. EPCs can

be used to provide clearer links between internal and external

light behavior. Fig. 5 illustrates the EPC of the structure, in

addition to simplified band diagrams derived from two ho-

mogenization methods (S-parameter method and Floquet-

Bloch modes). The most prominent feature of the EPC is a

curved band with an upward concavity. This upward concav-

ity is similar to that observed in the EPC of the negative-

index slab in Fig. 1(b) and ensures that the system satisfies

two important conditions—one related to power flow and

another related to phase—needed to achieve flat lens imag-

ing. First, group velocity in the medium is directed towards

the normal, indicated by the kx ¼ 0 line. This feature is

necessary to collect and concentrate light into a real image.

Second, the wavefront curvature in the medium, described

by the upward curvature in the EPC, opposes the wavefront

curvature in free-space, which has a downward curvature,

such that a wavefront exiting the medium can re-form into a

real image in the transmitted region. The curved band in the

EPC exists up to the free-space cutoff, beyond which it

breaks up and diffuses. This fundamentally limits the images

created by the structure to the free-space diffraction limit,

which is consistent with past experiments. Finally, EPCs can

be used to determine the best effective medium model for

the structure31 by the one that most accurately describes EPC

features. As shown in Fig. 5(b) the S-parameter method and

the Floquet-Bloch theorem—both widely used to derive ho-

mogeneous models for metamaterials—yield an infinite

number of potential solutions in k space. However, only one

of the solutions accurately describes the concavity and posi-

tion of the dominant band of the EPC. In this case, the best

solutions are the m¼ 2 branch from the S-parameter method

and the m¼ 1 branch from Floquet-Bloch theorem.

VIII. FAR-FIELD FLAT LENS WITH LESS METAL

We finally apply EPC engineering to design a metal-

dielectric layered flat lens that achieves far-field imaging with

minimal use of metal. We start with the basic template of a

tri-layer unit cell composed of a TiO2 layer sandwiched by

two silver layers of identical thickness. The structure operates

at the UV wavelength of k0¼ 330 nm. The design objective is

to achieve an EPC consisting of a single dominant band with

an upward concavity—the key feature that leads to phase and

power flow essential for far-field imaging—while using the

thinnest silver layers possible. Optimization yields a final

design consisting of 8 repetitions of a unit cell with the layer

sequence Ag (7.5 nm)—TiO2 (25 nm)—Ag (7.5 nm). The pro-

posed design has a metal fill fraction of 0.38, which is about

half the metal fill fraction of the more complex structure dis-

cussed in Sec. VII (0.63). As shown in Fig. 6(a), the flat lens

FIG. 5. (a) EPC for the flat lens struc-

ture shown in Ref. 11 to be capable of

far-field imaging in the UV. The EPC

is derived for TM-polarized plane-

wave (k0¼ 365 nm) illumination from

a dielectric (n¼ 2) prism. The inset

depicts the geometrical configuration

of the flat lens. (b) The simplified

EPCs derived using two common ho-

mogenization techniques: S-parameter

method (blue solid) and Floquet-Bloch

modes (red dashed).

173101-7 Al Shakhs, Ott, and Chau J. Appl. Phys. 116, 173101 (2014)

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design possesses the desired EPC feature of a band with

upward concavity. Its far-field imaging performance is con-

firmed by FDFD simulations [Fig. 6(b)], which reveal that it

is indeed capable of forming a real image of an object spaced

about a wavelength from the exit surface. Reduction in the

metallic content of the flat lens translates into reduced internal

losses and boosted transmission, a desirable feature if these

structures are to eventually be used in real-world UV micros-

copy applications.

IX. CONCLUSION

We have proposed a method to draw band diagrams, or

EPCs, for any layered plasmonic metamaterial through a

generic expression of the spectral Poynting vector in a lossy,

layered medium of finite extent. A unique feature of the EPCs

proposed here is that they display potential contours as a func-

tion of real wave vector coordinates, enabling visualization of

phase and group velocity on a single graph. The evolution of

layered plasmonic metamaterials—starting with a basic metal

layer and ending with complex metal-dielectric multi-layers—

has been described, interpreted, and analyzed through their

EPCs, leading to a consolidated picture of interesting effects

such as negative phase velocity, super-resolution imaging, ca-

nalization, and far-field flat-lens imaging.

The EPCs introduced here are a starting point to under-

standing how plasmonic metamaterials work and unlocking

their potential for creating new optical devices. Although a

major limitation is that they do not provide information on

the phase and amplitude of waves reflecting from and trans-

mitting through a medium, they can play a useful role in pro-

viding intuitive understanding and a means to communicate

the intricacies of light flow in plasmonic media. These

insights can be combined with existing techniques, such as

numerical simulations, to explore larger parameter spaces

and to develop a framework for organizing and categorizing

the growing number of plasmonic systems. What we have

attempted in this work is to initiate this process by a thor-

ough treatment of plasmonic systems based on the simplest

configuration of planar layers. Our treatment can and should

be extended to more complex geometries.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge helpful discussion with

Thomas Johnson and L€oic Markley from The University of

British Columbia and Henri Lezec from the National

Institute of Standards and Technology. This work was

supported by Natural Sciences and Engineering Research

Council of Canada (NSERC) Discovery Grant No. 366136.

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FIG. 6. (a) EPC of a new layered flat

lens structure that is capable of far-

field imaging yet possesses about half

the metallic fill fraction of the flat lens

presented in Ref. 11. The proposed

structure consists of 8 repetitions of a

unit cell with the layer sequence Ag

(7.5 nm)—TiO2 (25 nm)—Ag (7.5 nm).

The EPC is derived assuming TM-

polarized plane-wave (k0¼ 330 nm)

illumination from a dielectric (n¼ 2)

prism. (b) A FDFD simulation of the

proposed flat lens imaging a test object

placed on its surface. Tapering of the

magnetic energy density in the image

region at a location spaced about a

wavelength from the exit surface con-

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