alok project

ISP, CUSAT

Analysis of point and line defects of 2D PBG

structures:

For applications and designs M.Tech thesis by

Alok Kumar Jha

M.Tech,

Optoelectronics and Laser Technology (2009 – 2011batch) International School of Photonics,

Cochin University of Science and Technology

Under the guidance of PROF. T.SRINIVAS

Applied Photonics Laboratory Electrical Communication Engineering Department

INDIAN INSTITUTE OF SCIENCE Bangalore – 560012

Analysis of point and line defects of 2D PBG structures:

For applications and designs

A Thesis submitted for the partial fulfillment for the Degree of MASTER OF TECHNOLOGY in OPTOELECTRONICS AND LASER TECHNOLOGY

By

Alok Kumar Jha

M.Tech, Optoelectronics and Laser Technology (2009 – 2011batch)

International School of Photonics Cochin University of Science and Technology

Under the guidance of PROF. T.SRINIVAS Applied Photonics Laboratory

Electrical Communication Engineering Department INDIAN INSTITUTE OF SCIENCE

Bangalore – 560012

Indian Institute of Science, Bangalore

Dr T Srinivas

Associate Professor

ECE Dept

[email protected]

PROJECT CERTIFICATE

This is to certify that the thesis work entitled “Analysis of point and line defects of 2D

PBG structures- for applications and designs” being submitted by Mr. Alok Kumar Jha, in

partial fulfilment for the Degree of Master of Technology in Optoelectronics and Laser

Technology of Cochin University of Science And Technology is a record of the bonafide

work done by him during the period July 2010 – June 2011 at Applied Photonics Laboratory,

Electrical Communication Engineering Department, Indian Institute of Science, Bangalore-

560012, under my guidance and supervision.

Project supervisor,

( T Srinvias)

11.07.2011

International School of Photonics

Cochin University of Science and Technology

Cochin - 682022, Kerala

PROJECT CERTIFICATE

This is to certify that the thesis entitled “Analysis of point and line defects of 2D

PBG structures --For applications and designs” is a bonafide record of work done by Mr. Alok Kumar Jha, under the supervision and guidance of Prof. T.Srinivas at Applied Photonics Lab, Electrical Communication Engineering Department, Indian Institute of Science, Bangalore in partial fullfilment of the requirements for the award of the degree of Master of Technology in Optoelectronics and Laser Technology from July 2010 to July 2011during academic year 2009-2011.

Prof. P. Radhakrishnan Project Coordinator and Director International School of Photonics CUSAT, Kochi-682 022 Kerala

Dedicated to my late maternal grandfather and my family members.

Acknowledgements

At the completion of my two semester project work for the M.Tech course I find myself

overwhelmed by the support, encouragement and affection of various persons involved with

me. The list being larger one so I may miss out some names. I beforehand apologise for that.

But some names stand too significant to be forgotten.

I express my gratitude to my project supervisor Dr. T Srinivas for his effective guidance and

mentorship with constant encouragement. I also thank all the people working in his lab for

sharing resources and suggestions and numerous help. Dr. Badrinarayna for helping me the

initial days when it was appearing a little difficult to set my work here. I express my

gratitude to Mr. Sandeep U for his ever-ready helping hand and trouble shooting approach. I

also thankMr. Basavraj Talwar, Mrs. Rajini V Honnungar, Mr. Nivesh Mangal, Ms. Akshata

Shenoy, Mr. Yadunath T R, Mr. Kalol Rai, Mrs Sathish Malathi and Mr. Narayan and Mr. G

S Hegde for there support and affection. I thank every person in IISc for their support

including non-teaching staff.

Coming back to ISP,CUSAT I am greatful to my professors and staff.I express my gratitude

to Prof. Prof. P.Radhakrishnan (Director,ISP) and Dr. V P N Nampoori (emeritus scientist)

and also Dr. Kailasnath M (asst. professor) all other faculties. I thank my batch mates for

their camaraderie.

I also thank my friends, working professionals in banglore viz. Mr. Pankaj Kumar, Mr.

kundan Pandit and Mr. Hari Sai Gangadhar.

Finally I thank my family members who are always with me in the thicks and thins of life.

Conference papers based on the project work proposed

The following three papers are being worked upon at applied photonics lab, ECE Dept, IISc Banglore under

supervision of Dr. T Srinivas for the upcoming photonic conferences at the time of project thesis submission.

1. Defect mode analysis involving point and line defects in 2D PBG structures for communication applications – Alok Kumar Jha, Sandeep U,Dr. T Srinivas;

(Indicon 2011 (IEEE),Hyderabad) – underprocess.

2. Analysis of energy profile or energy distribution for various point defect cases in 2D PC lattice structures -- – Alok Kumar Jha, Sandeep U,Dr. T Srinivas;

– underprocess.

3. Design of different types of symmetric and asymmetric binary splitters on 2D

lattice structures using multiple line defects – underprocess.

Abstract

Photonic crystal provides numerous applications of technology based on photon in two

unique ways. First it improves many applications of light and laser which are provided by

other means to far greater precision and efficiency. Second various applications which seem

to be impossible otherwise or by other methods, it makes them possible and realised.

This thesis implores and explores certain important physical aspects of the photonic crystal

mediums and the phenomena happening due to that by simulation of the prototype cases and

analysis of the factors involved. Although this field of study is too vast and broad that it

cannot be covered fully in this thesis. Nevertheless important and significant issues and

features have been covered which again can be utilised in many ways. This thesis begins with

the basic features and gradually approaches complex cases. The features of phtonic crystal

have been observed studied from multiple perspective viz. physical properties involving

energy distribution in the medium and band-gap features as well as engineering features

related to PBG structures based design possibilities. Explanations and conclusions have

arrived at for different modifications and manipulations of the photonic crystals on case to

case basis.

Contents

Topic Page number

1. Introduction and overview 1

Theory of Photonic Band Gap (PBG) structures 2 Theoretical modelling of photonic crystals 3

2. Description of topics and their features undertaken 7 for the project work 3. PART – 1: Simulation of PBG structures based 8 prototype cases Case 1: Simulation of band structure and associated features of a 9 5×5 2D rectangular lattice of circular dielectric rods in air. Case 2: Simulation of bandstructure and associated features of a 10 5×5 2D hexagonal lattice of circular air holes in dielectric. Case 3: Study of defect modes in 2D rectangular and hexagonal lattices. 11 Case 4: Study of output pattern variation for variation of physical 17 and numerical parameters over certain range. Case 5: Full Brillouin zone analysis for band structure. 22 Case 6: Simulation of bandstructure and associated features of 2D Photonic 24 crystal slabs (a particular case of 3D structures). Case 7: Channel waveguide implementation in 2D Photonic crystal slab 27 of finite thickness. Case 8: FDTD based simulation of prototypes. 30 Case 9: Simulation of real design photonic crystal on the lines of the 33 prototype cases discussed above.

4. PART – 2: Analysis of point defects and line defects. 37 Analysis of waveguide modes for square –lattice of dielectric 38 cylindrical rods in air.

Analysis of waveguide modes for triangular–lattice of dielectric 42 cylindrical rods in air. Analysis of waveguide modes for triangular–lattice of air-holes 47 in dielectric slab. Analysis of point defect modes for the three lattice structures by 51 removing central rod or hole. Analysis of D and H energy distribution in point defect region for 55 square lattice structure. Analysis of D and H energy distribution in point defect region for 58 triangular lattice structure. Analysis of point defect modes for the square lattice structure 61 with varying point defect. Analysis of D energy concentration in point defect dielectric rod for 64 square lattice structure. Analysis of D energy concentration in point defect dielectric rod for 66 triangular lattice structure.

5. Reference1-softwares and simulation tools. 68 6. Reference2-literature survey. 69

1

Introduction and overview

Photonic crystals represent a class of nano-structured materials in which alternating domains

of higher and lower refractive indices produce an ordered structure with periodicity on the

order of wavelength of light ―λ‖.

The periodic modulation of the optical index in the medium with a lattice constant on the

order of wavelength of light gives rise to ―λ photonics‖ i.e. control of spatio-temporal

trajectory of photons at the scale of their wavelength and of their periodic oscillation

duration.

In the case of a large refractive index contrast (defined by the ratio n1/n2) photonic crystal

with a proper shape of building blocks (domains) and proper crystal symmetry, a complete

bandgap develops. In this case, the bandgap is not dependent on the direction of wavevector,

which defines the light propagation; also, the density of photon states goes to zero in the

bandgap region.

These materials with a complete gap are often called photonic bandgap materials.

A graphical comparison of photons dispersion in vaccum to that in the photonic crystals.

Photonic crystals are thus arrangements of two or more dielectric media, periodic along one,

two, or three dimensions.

This is depicted schematically here by "red" and "yellow" materials arranged in simple 1d/2d/3d lattices

from left to right.

2

Theory of Photonic Band Gap (PBG) structures

Photonic crystals are classified into one dimensional, two-dimensional and three dimensional

according to the periodicity of dielectric as shown in above picture. The work described in

this thesis is based on 2D PBG structures or 2D PBG structures with finite height also called

Photonic crystal slabs.

Terms and parameters associated with PBG structures

(1) Lattice vectors:

A periodic function f (R) can be expressed in terms of a lattice for all periodic systems. Let

a1,a2, a3 be fixed vectors such that for all points R in the lattice,

R = la1 + ma2 + na3, for some integers l, m, and n.

The points R are known as lattice vectors, while the basis vectors a1, a2, a3 are termed

primitive lattice vectors. (In one or two-dimensional problems, only one or two primitive

lattice vectors are needed).

(2) Unit Cell:

A unit cell is any region of space which, when translated by every lattice vector in the lattice,

maps out the entire function. The unit cell is also not unique. A primitive unit cell is any unit

cell that has the minimum possible volume.

Following figure illustrates unit cell determination.

3

(3) Reciprocal lattice:

Associated with every lattice is a second lattice termed the reciprocal lattice. The vectors of

the reciprocal lattice are denoted by the vector equations:

The lattice and reciprocal lattice vectors are related by the equation: G・ R = n2πδij,

where, G = lb1 + mb2 + nb3 and δij = 1 if i = j otherwise δij = 0 .

The lattice and reciprocal lattice are thus essentially inverses of each other. Since the lattice

vectors have the dimensions of length, the reciprocal lattice vectors have dimensions of

inverse length and span reciprocal space.

(4) First Brillouin Zone:

It constitutes the unit cell of the inverse space such the solution of the dispersion relation

k) is the distinct set of solutions. Any other solution beyond this region is periodic

repetition as described by the equation G).

Following figure illustrates determination of Brillouin zone and Dispersion curves (band structures) for

modes of a 1 D lattice.

(5) Master equation:

The master equation is the eigen equation for the periodic medium and is derived from

Maxwell‘s equations for dielectric medium.

where, and ε(r) = ε(r + R)

4

Using Bloch’s theorem, it can be written that H(r) = exp(-ik.r)uk(k) where uk(k) is a

function with the periodicity of the lattice. Inserting this expression in the Helmholtz

equation, the relation between various mode frequencies and the allowed wave-vector K of

the lattice medium constitutes the Band-structure as described in Part-1 of the project work.

Theoretical modelling of photonic crystals

The band structure of a given photonic crystal defines its optical properties, such as

transmission, reflection, and angular dependence.

Many methods for photonic band structure determination—theoretical as well as numerical—

have been proposed. These methods fall in two broad categories:

Frequency-domain techniques and Time-domain techniques.

In frequency-domain techniques the photon eigen-value equation, is solved to obtain the

allowed photon states and their energies. The advantage provided by this method is that it

directly provides the band structure. Examples are the Plane Wave Expansion Method,

PWEM and the Transfer Matrix Methods, TMM.

In time-domain techniques, the temporal evolution of the input electromagnetic field

propagating through the crystal is calculated. Then, the band structure is calculated by the

Fourier transform of the time-dependent field to the frequency domain. A widely used time-

domain method is Finite-Difference Time Domain (FDTD) which calculates time evolution

of the electromagnetic waves by a direct discretization of Maxwell‘s equations. In this

method, the differentials in Maxwell‘s equations are replaced by finite differences to connect

the electromagnetic fields in one time interval to the ones in the next interval.

Modelling techniques used for the project work

The project work is based on PWEM and FDTD methods using the software tools as

mentioned in Reference – 1 of this thesis.

These two modelling techniques are concisely described as follows:

(1)PWEM (Plane Wave Expansion Method):

This is a frequency domain technique and involves expanding both the periodic dielectric

function and field solution in infinite series of uniform plane waves (as shown) and thus

reducing BVP (Boundary value problems) to a set of Eigen value equations as described

next.

5

and,

Hence, it depends on the expansion of the electromagnetic field with a plane wave basis set

as per the equation.

Here, k is a wave-vector in the Brillouin zone, G is a reciprocal lattice vector, and eλ are unit

vectors perpendicular to k + G.

These descriptions for the reciprocal lattice utilize the terminologies of solid-state physics.

The solution of the eigen-value equation (Helmholtz equation),

provides the band structure of the crystal by numerical means.

This is a frequency domain technique and involves expanding both the periodic dielectric

function and field solution in infinite series of uniform plane waves (as shown) and thus

reducing BVP (Boundary value problems) to a set of Eigen value equations.

(2) FDTD (Finite-Difference Time Domain) method

Imagine a region of space which contains no flowing currents or isolated charges. Maxwell's

curl equations in can be written in Cartesian coordinates as six simple scalar equations. Two

examples are:

The other four are symmetric equivalents of the above and are obtained by cyclically

exchanging the x, y, and z subscripts and derivatives. Maxwell‘s equations describe a

situation in which the temporal change in the E field is dependent upon the spatial variation

of the H field, and vice versa. The FDTD method solves Maxwell's equations by first

6

discretizing the equations via central differences in time and space and then numerically

solving these equations in software.

The most common method to solve these equations is based on Yee's mesh and computes the

E and H field components at points on a grid with grid points spaced Δx, Δy, and Δz apart.

The E and the H field components are then interlaced in all three spatial dimensions as shown

in Fig. 2-1. Furthermore, time is broken up into discrete steps of Δt. The E field components

are then computed at times t = nΔt and the H fields at times t = (n+1/2)Δt, where n is an

integer representing the compute step. For example, the E field at a time t = nt is equal to the

E field at t = (n-1) Δt plus an additional term computed from the spatial variation, or curl, of

the H field at time t.

This method results in six equations that can be used to compute the field at a given mesh

point, denoted by integers i, j, k. For example, two of the six are:

These equations are iteratively solved in a leapfrog manner, alternating between computing

the E and H fields at subsequent Δt/2 intervals.

7

Description of topics and their features undertaken for the project work

The contents or topics studied and worked upon of this project thesis are divided into two

parts. Part-1 comprises of the simulation of some standard prototypes of the PBG structures

and there by illustration of their important features and characteristics with relevant

comments. Part-2 comprises of the analysis of point and line defects across range of their

parameters to get insight into their behaviour pattern and some relevant physical aspects.

Three popular 2D lattice structures (out of numerous types of Photonic crystals) have been

studied and worked upon viz. square lattice of dielectric rods in air, triangular lattice of

dielectric rods in air and triangular lattice of air-holes in dielectric slab.

The period for any lattice analysis has chosen as 1 micron in most cases although not always.

This is because the parameters and solutions of the PBG structures are scalable i.e making the

period of lattice half doubles the corresponding frequency values. The frequency in the PBG

structure has been normalised ofr this reason and as such they are expressed in the units of

inverse of wavelength i.e per micron following the common and easy to work conventions of

expressing the frequency in the PBG structures.

Further part-1 is based on the work done on Rsoft CAD tools so the conventional definition

of TE and TM modes or EVEN and modes have been altered. While this aspect has been

acknowledged and warned at the end of every simulation case of part-1,it was a bit not

practical to eliminate it as various CAD tool‘s parameters, features and aspects are directly

based on it. So, the final resul and conclusion only needs to be changed as per common

conventions.

This issue is not present in part-2 which is based on work done on MIT ab initio group

software which completely agrees with the standard conventions.

Quite few of the simulation cases of part-1 and all the analysis cases of part-2 involve 2D

photonic crystal lattice which is mathematical structures of infinite height. But conclusions

drawn there come in very close approximation of the various photonic crystal slabs of finite

height. However some of the real structure examples also have been worked upon in part-1.

8

PART – 1

Simulation of PBG structures based prototype cases.

9

Case1: A 5×5 rectangular array of dielectric cylindrical rods (2D lattice) in air is considered

whose thickness is assumed to be infinite and assumes no significance in band structure

analysis. The various parameters are mentioned. The lattice has uniform periodicity in XZ

plane.

A 5×5 2D rectangular lattice, its 2D dielectric constant profile and irreducible Brillouin Zone

with wave vector sampled along the sides of the triangle ΓXM.The reason for opting for this

line of symmetry is that various maxima and minima of a particular mode or band Occur

along this line only and hence band gap along this line gives the band gap for entire BZ.

The important parameters for the design of this prototype are mentioned below:

The radius is the radius of the dielectric rod in the air with refractive

index of 1 and the refractive index of the dielectric rod is√8.9 and ∆ is the index difference.

The band structures for TE, TM and both together are shown as follows and are calculated on the basis

of PWEM algorithm for first 8 bands.

Conclusion: Bands gap exists for TE case only when the electric field is solely confined in

the lattice plane and the magnetic field is normal to it. Further the frequency values have been

expressed in length units of microns with a common conversion factor of a/2 λ. The free

space wavelength chosen is 2.5 times lattice period but has no role in determining

bandstructure. The bandgap for TE structure comes around 0.4 micron as shown.

NOTE:- The general convention of describing the modes as TE or TM as whether the electric field or magnetic

field is in the plane of periodicity has been altered here as per the R-soft CAD tools scheme and hence from

conventional perspective TE modes illustrated here are actually TM modes and vice versa. This feature has

been retained throughout in this section.

10

Case2: A 5×5hexagonal array of cylindrical air holes in dielectric substrate (2D lattice) is

considered whose thickness is assumed to be infinite and assumes no significance in band

structure analysis. The various parameters are mentioned. The lattice has uniform periodicity

in XZ plane.

A 5×5 2D hexagonal lattice, irreducible Brillouin Zone and its 2D dielectric constant profile with wave

vector sampled along the sides of the triangle ΓMK.

The reason for opting for this line of symmetry is that various maxima and minima of a

particular mode or band Occur along this line only and hence band gap along this line gives

the band gap for entire BZ.


The radius is the radius of the the air hole in the dielectric

substrate with refractive index of 1 and the refractive index of the dielectric substrate is√13

which serves as the background index and ∆ is the index difference.

The band structures for TE/TM are shown as follows and are calculated on the basis of PWEM algorithm

for first 6 bands.

Conclusion: Bands gap exists for TM case when the magnetic field is solely confined in the

lattice plane and the electric field is normal to it and also for TE case when the electric field

is solely confined in the lattice plane and magnetic field is normal to it Further the frequency

values have been expressed in length units of microns with a common conversion factor of

a/2 λ. The freespace wavelength chosen is 2.5 times lattice period but has no role in

determining band structure. The band gap for TM structure comes around 0.5 microns as

shown. The band gap for TE structure comes around 0.5 microns as shown.

NOTE:- The general convention of describing the modes as TE or TM as whether the electric field or magnetic field is in the plane of

periodicity has been altered here as per the R-soft CAD tools scheme and hence from conventional perspective TE modes illustrated here are

actually TM modes and vice versa. This feature has been retained throughout in this section.

11

Case3: Defect modes in 2D rectangular and hexagonal lattices are being considered. The

defects considered are localised point defect and line defect. Various interesting applications

of photonic crystals are associated with imperfections or defects in a periodic structure

because defects often support localised optical states with completely different properties to

the extended states within the band.

(1) The point defect in rectangular lattice is created by removing the central dielectric rod and

leaving it void as shown.

An 11 × 11 2D Rectangular array of circular dielectric rods in air with a missing rod in the centre (above

arrow mark)and its 2D dielectric constant profile considering a supercell dimension of 5×5 lattice.


The band structure is plotted with wave vector sampled along the sides of the triangle ΓXM

of the irreducible BZ (shown below) as in case of standard rectangular lattice with no defect.

The band structure shows band gap for TE case but not for the TM case just as rectangular lattice

without defects do. So TE band structure is exclusively analysed.

TE band structure shows one defect state in the band gap represented by constant frequency value all

along (flat straight line) otherwise the band structure is typical of the standard rectangular lattice.

So, the next thing done is to get the band structure at single point say kz= 0 or Γ as shown

next for 6×6 super cell dimension for better resolution (initially 5×5 super cell dimension was

considered).

12

The defect state comes for mode no.= 35 for 6×6 super cell which would have been as mode no.=2 for 1×1

super cell dimension.

By the help of software electric and magnetic field intensities at the defect site can be

obtained where electric field for the defect is normal to lattice plane and magnetic field for

the defect is in the lattice plane contrary to the normal state where electric field is in the

lattice plane and magnetic field is normal to it.

Electric field and magnetic field intensity at defect site; Electric field is normal to lattice plane XZ.

Magnetic field intensity is in XZ plane.

(2) The point defect in hexagonal lattice is created by decreasing the central air hole in the

dielectric slab as shown.

An 11 × 11 2D Hexagonal array of circular air holes (red circles) in dielectric slab with a smaller hole in

the centre (above arrow mark)and its 2D dielectric constant profile considering a supercell dimension of

5×5 lattice.


13

The band structure is plotted with wave vector sampled along the sides of the triangle ΓMK

of the irreducible BZ (shown below) as in case of standard hexagonal lattice with no defect.

The following TE/TM band structure is obtained. The band structure shows band gap for TM case but

not for the TE case just as hexagonal lattice without defects do. So TM band structure is exclusively

analysed.

TE band structure shows four defect states in the band gap represented by constant frequency

values all along (flat straight lines) otherwise the band structure is typical of the standard

hexagonal lattice. So, the next thing done is to get the band structure at single point say kz= 0

or Γ as shown for 6×6 super cell dimension for better resolution (initially 5×5 super cell

dimension was considered).

The defect states come for mode nos. 27, 28, 29 and 30 for 6×6 super cell. Modes 27 and 28

are very close (almost overlapping).

By the help of software electric and magnetic field intensities at the defect site can be

obtained where magnetic field for the defect is normal to lattice plane and electric field for

the defect is in the lattice plane contrary to the normal state where magnetic field is in the

lattice plane and electric field is normal to it.

Magnetic field and intensity at defect site for m=27; Magnetic field is normal to lattice plane XZ.

14




15

(3) The line defect in rectangular lattice is created by removing the central column dielectric

rods and leaving it void as shown.

A 7× 11 2D Rectangular array of circular dielectric rods in air with a missing column in the centre

created from the standard rectangular array as shown along with its dielectric constant profile.

Array without defect:


The band structure is plotted with wave vector sampled along the sides of the triangle ΓXM

of the irreducible BZ (shown below) as in case of standard rectangular lattice with no defect.

The following TE/TM band structure is obtained.

The points marked express mode no.=8 and successive nth point for k vector. In this case the

super cell dimension comprised 1×9 lattice point so band gap which appears for 1×1 cell as

first mode appears here as 8th

mode but the frequency value does not vary much which can be

compared from the standard lattice band structure shown on right side.

Unlike as in the case of localised defect the shape of the defect band is not flat. This

difference arises because in the present case the defect state is a propagating mode that

travels along the line defect.

16

By the help of software electric and magnetic field intensities along the defect line can be

obtained where electric field for the defect is normal to lattice plane and magnetic field for

the defect is in the lattice plane contrary to the normal state where electric field is in the

lattice plane and magnetic field is normal to it.

Further some of the electric and magnetic field intensities for different k vectors as marked on the band

structure for 8th

mode are illustrated.

Conclusion: For localised defects defect modes exist as flat bands. For line defects the defect

state is a propagating mode that travels along the line defect and this can be utilised in

making waveguides. In both these cases the direction of electric and magnetic fields is

exchanged compared to the non-defect parts of the lattice. The electric and magnetic field

profile assumes different and complex features for different modes or different k vectors

point.




17

Case 4: Scan of lattice physical and numerical parameters for design (input) over certain

range of values taking one or more together is considered.

This serves the purpose that for most real applications performing a single band structure

calculation is insufficient. In order to create a workable structure it requires to identify the

important parameters and optimise their values to achieve the desired result.

(1) A 7× 11 2D Hexagonal dielectric slab with circular air holes is scanned for a range of radius values of

air holes

Radius is scanned for output pattern from lower value of 0.2µm and to beyond 0.5µm in the

fixed 140 incremental steps.The TE/TM band structure shown is for the lower value of radius

of 0.2µm.

Next the various output parameters for range of inputs illustrating the lattice behaviour

pattern are illustrated.

The following plots band map features for TE and TM band structures varying with radius. Band map as

measurement represents all the eigen values from a complete band structure as a single array.

18

The hollow regions in the band map plots represent band gaps which are illustrated in the plot of gap

map as shown.

Blue region represents band gap for TE band structure and red one for TM band structure

exclusively in both the cases. In case both the band gaps exist the region is shown in green

colour.

Likewise band gap center for TE and TM band structures exclusively and TE/TM band structure

combined are also illustrated as follows.

Also Gap ratio = Gap map / Gap centrefor the three cases are illustrated as follows.

(2) The same hexagonal dielectric slab with air holes is now scanned for a range of radius

values of air holes and range of delta values together.

Radius is scanned from a lower value of 0.2µm to beyond 0.5µm in the fixed 70 incremental

steps and Delta is scanned from a lower value of -2 to upper value of -1 in 10 incremental

steps for output pattern.

19

In the previous case the output parameter plot was 2D plane for single input parameter scan;

but in the present case with two input parameters scanned together the output parameter plot

is 3D volume and for more number of input parameters scanned together a single output plot

can‘t be obtained. So, 2D plane sliced output patterns are provided where principle input

parameter is taken on x axis (basis) and output is plotted for different values of other input

parameter across its range taking one at a time. In the present case the principal input

parameter is radius (taken on x axis) sliced output plot is taken for different values of delta

taking one at a time.

Some these sliced plots for different categories as in 1D scan case is shown on the as follows.

Sliced TE band map structures:

Sliced TM band map structures:

20

Sliced TE/TM gap map structures:

21

Sliced gap enter, gap width and gap ratio for TM bands:

`

Conclusion: Scanning technique is utilised in optimising the design of photonic crystal (or

any device based on it).

NOTE:- The general convention of describing the modes as TE or TM as whether the electric field or magnetic field is in the plane of periodicity has been altered here as per the R-soft CAD tools scheme and hence from conventional perspective TE modes illustrated here are


22

Case 5: Full Brillouin zone analysis for band structure calculation where every single k-

vector point on the Brillouin zone is considered and not just points along the line of

symmetry.

A 7×11 2D Hexagonal dielectric slab with air holes, its BZ (sampled along the line shown) and

corresponding TM band structure.

The lattice parameters for the above prototype are:

Next the entire band structure throughout the Brillouin zone is determined instead of the band

structure along a path between the symmetry points. The result is the creation of band

surfaces and equi-frequency contours and these plots are called whole zone analysis.

These features are shown on the as follows.

The many oscillations in the diagram are simply a result of the raster scan and have no particular

physical significance. As expected the band gap is preserved.

23

Following are the equi-frequency contours for different bands of TM band structure. Equi-frequency

contours for a particular band exhibit k-vector points on BZ with same frequency connected together.

Conclusion: Full zone BZ analysis is used to get the band surfaces and equi-freqency

contours which are helpful in identifying precise band features to improve the photonic

crystal design.

NOTE:- The general convention of describing the modes as TE or TM as whether the electric field or magnetic field is in the plane of periodicity has been altered here as per the R-soft CAD tools scheme and hence from conventional perspective TE modes illustrated here are


24

Case 6: 2D Photonic crystal slabs of finite thickness are considered here.

These are 3D photonic crystal slabs with periodicity in 2D (plane) and a finite thickness. Here

light is confined in the third dimension by traditional index guiding. Photonic crystals with

periodicity in 3D are difficult to fabricate. Hence, these structures attract considerable interest

as they posses many features of full 3D photonic crystals, but are substantially easier to

manufacture.

Concept of light line or light cone filter

By increasing the size of the super cell (number of unit cells considered together from three

different directions as a unit for simulation related calculations) a simulation domain is set up

that approximates a system where the structure is non-periodic in the vertical direction.

However, there are still leaky waveguide and radiation modes that exist for this system.

These modes would eventually decay into the background material and are not guided. All

these modes have an effective index lower than that of the cladding region, which in this case

is air. To remove these from the band diagram, we employ a light line or light cone (full BZ

analysis) filter. This filter removes all of the radiation states from the band calculation.

Concept of even and odd parity

As discussed in 2D crystal structures TE and TM modes cannot be strictly applied here. But

if a structure possesses one or more planes of reflection symmetry, the modes can be

classified as either even or odd.

These modes parallel the meaning of the TE and TM modes that are found in 1D and

2Dstructures. As a result, it is possible that a band gap exists between the even and odd bands

of a PCS bases upon their reflection symmetry about the 2D periodic plane being even and

odd respectively. Grossly speaking even mode corresponds to TM mode of 2D lattice and

odd mode corresponds to TE mode there.

Next the case of a7× 11 2D Rectangular array of circular dielectric rods in air with a finite height is

considered as shown below. Its parameters are as follows:

25

Various band structures along with the light line are shown below where k-vectors are being sampled

along the line of symmetry of BZ shown right:

Resolved and combined hybrid band structures respectively:

Similarly band structures and equi-frequency contours based on full BZ analysis are also shown:

26

Next the case of a 7×11 2D Hexagonal dielectric slab with air holes with a finite height is considered as

shown below. Its parameters are as follows:

Top view Structural view Hybrid Band Structure

Next the case of a 7×11 2D Hexagonal dielectric slab with air holes with a finite height and periodic

cladding is considered as shown below. Its parameters are as follows:

Concept of light line revisited

In this case, the light line cannot be defined by the cladding index because the cladding is

periodic and not solid. The correct light line to use in this case is defined by the effective

index values of the lowest band from a 2D band calculation performed for the cladding

structure. To calculate this band, we perform a 2D band calculation in the cladding to find the

required effective index curve. This can easily be done with this file by changing the

dimensions to 2D (X-Z), and then setting the Y cut plane to lie within the cladding material.

Band structure for light line is calculated as shown. The TE band is the lowest, and this band is used to

define the filter; then the following band structure is obtained:


periodicity has been altered here as per the R-soft CAD tools scheme and hence from conventional perspective TE modes illustrated here are actually TM modes and vice versa. This feature has been retained throughout in this section.

27

Case 7: Waveguide implemented in PCS. Here the case of PC slabs with embedded defect

waveguides is considered.

The PC slab considered is of a 6×7 2D Hexagonal dielectric slab with air holes with a finite height and

periodic claddingand the central column with holes of reduced radius (defect line) forms the waveguide as

shown:

Top view Side view 3D view: central column rods are thinner

Red segment represents dielectric slab, blue segment represents periodic cladding and green region represents air. Further, purple

rectangle encircling red segments in side view represents super cell dimension for simulation domain.

Its parameters are as follows:

In this case, we are interested in the dispersion relations of the guided modes which are

confined within the defect lines. In the dispersion relation, these modes lie between the

projected band structures of the uniform slab modes and below the projected light line of the

cladding. The new elements discussed here involve the creation of these projected light lines.

The first stage of the analysis is to understand the properties of the slab without the defect

line waveguide. This procedure is identical to case 6.

In brief, we find the light-line for the periodic cladding by finding the lowest band of the 2D

lattice in the cladding region, then find the 3D modes of even and odd parity that fall below

this light-line as shown next.

28

Hybrid Band Structure Band structure for light line

Defect mode properties: The essential issue for calculation of defect mode properties is to

find projected band structures of the light-line (or more accurately light-cone) and of the

regular lattice of the slab. It needs to calculate a Full zone band structure over the slab‘s

entire 2D Brillouin zone, and then project that band structure onto the 1BZ of the linear

waveguide.

Projected light line is obtained now over the entire 2D 1BZ of the periodic cladding, rather than just

along its symmetry points as shown:

The next step is to determine the correct 1D 1BZ on to which the light-cone should be

projected. The defect line is oriented along the Z-axis and that the period along that direction

is simply the period of the crystal (say ‗a‘).

Thus projected light cone along the waveguide (defect line) is obtained as shown:

Blue region represents light line for TE modes and red region represents light line for TM modes. TE modes lie below the TM, as

such TE mode is selected for light line.

29

Projected slab modes

As the next step an analogous procedure is performed for the slab modes. Modes confined in

the 0-1 gap of the odd mode band structure need to be searched.

Full BZ slab modes and there projected bands as the function of kz is shown next:

Guided waveguide modes

Finally, the mode guided in the defect guide is found. Band structure for BZ along the defect line region is

calculated first and then the three different modes along the waveguide are combined on a single graph to

present the complete picture of analysis as shown.( ΓX2 is the BZ along the waveguide).

The range of kz along the defect line has been chosen to illustrate the area of interest.



30

Case 8: Study of FDTD based prototype structures (electromagnetic wave propagation in

photonic crystals).

(1) Transmission spectra of semi-infinite square PBG lattice.

This prototype simulation discusses the computation of a transmission spectrum for a semi-

infinite photonic crystal lattice. Several band-gaps will be found for one propagation

direction with in the crystal.

Crystal layout for simulation is as shown. Its parameters are as follows:

The computed spectra can be viewed which contain the frequency and wavelength spectra

respectively. In both cases several band gaps are clearly visible. Computation of a

transmission spectrum for a semi-infinite photonic crystal lattice gives several band-gaps for

one propagation direction with in the crystal.

31

(2) PBG Y-Branch Power Splitter

Crystal layout for simulation is as shown. Its parameters are as follows:

32




33

Case 9: Simulation of real design photonic crystal on the lines of the prototype cases

discussed previously.

A 25×25 hexagonal PCS with air holes and uniform air cladding is used to make waveguide

by creating a defect line along the central row.

The simulation is performed on the lines of the simulation of prototype of case 7.When the

holes along the central row are simply removed (effectively filled with the same dielectric)

the waveguide strip has the width of W = (2×Period×sin600 - 2×Radius).Further two other

waveguides with widths 0.85W and 0.75W respectively are simulated.

Lattice layout (red circles represent air holes) is as shown. The common parameters for the waveguides

are as follows:

Structural view

The central dielectric strip represents the waveguide. Hybrid Band structure of the slab without

considering the defect is shown as follows:

Defect mode properties

The essential issue for calculation of defect mode properties is to find projected band

structures of the light-line (or more accurately light-cone) and of the regular lattice of the

slab. It needs to calculate a Full zone band structure over the slab‘s entire 2D Brillouin zone,

and then project that band structure onto the 1BZ of the linear waveguide.

As the next step an analogous procedure is performed for the projected slab modes. Modes

confined in the 0-1 gap of the odd mode band structure need to be searched.

34

Full BZ slab modes (and their projected bands as the function of kz ) and projected light cone along the

waveguide (defect line) is obtained as shown:

As the next step an analogous procedure is performed for the projected slab modes. Modes

confined in the 0-1 gap of the odd mode band structure need to be searched.

Full BZ slab modes and there projected bands as the functions of kz are shown below:

Guided waveguide modes

Finally, the mode guided in the defect guide is found. Band structure for BZ along the defect

line region is calculated first and then the three different modes along the waveguide are

combined on a single graph to present the complete picture of analysis as shown.

(1) Modes when strip width is W (defined initially):

Xlo-X is the BZ along the waveguide. Guided modes: The range of kz along the defect line has

been chosen to illustrate the area of interest.

35

(2) Modes when strip width is 0.85W:

(3) Modes when strip width is 0.75W:

Some FDTD simulation features for waveguide of width W

(a) (b)

(c) (d)

The computed a) time b) frequency c) wavelength spectra for waveguide involving TM CW input and d)

electric field strength profile.

36

(e) (f)

(g)

The computed e) frequency f) wavelength spectra for waveguide involving TM pulsed input and g)

electric field strength profile.



37

PART – 2

ANALYSIS OF POINT AND LINE DEFECTS

38

ANALYSIS – 1:

Analysis of waveguide modes for square –lattice of cylindrical dielectric

rods in air.

Description:

Line defect is created in the square lattice structure of cylindrical dielectric rods of infinite

height in air with periodicity plane profile as shown below. Three types of defects have been

analysed with chosen values of the defect-rod epsilon as mentioned in the regular lattice

parameters and the line defect parameters.

This lattice structure is analysed for TM band structure. Band-structure and the line-defect or

wave-guide modes for the line-defect are explained.

(a) (b) (c)

Regular lattice parameters:

Here the geometric and physical parameters of the non-defect dielectric rods are described.

Period = 1µm, radius of rods = 0.19µm, height of rods = infinity, epsilon of rods = 12.

Line-defect parameters:

Here the geometric and physical parameters of the defect dielectric rods are described.

A defect row is created in three possible ways as shown above and described below.

(a) Line defect created by removing a horizontal row of dielectric rods.

(b) Line defect created with the horizontal row of dielectric rods of half the radius of the

regular rods and their epsilon values as 4,8,12 and 16 analysed as four different cases.

(c) Line defect created with the horizontal row of dielectric rods of twice the radius of the


39

PLOT1:- TM band-structure for the line-def on square-lattice of dielectric rods in air with a horizontal

row of rods removed. A distinct waveguide mode is present between upper and lower PC modes.

K-points as indicated by k-index are sampled along the ΓX direction of Brillouin zone as shown as 18

equi-distant points between Γ (k = 0) and X (k = 0.5) including the both .

Subsequently, the other cases are also analysed along the same k-points as shown next.

40

PLOT 2:- TM band-structure for the line-def on squarer-lattice of dielectric rods in air with the

horizontal row of dielectric rods of half the radius of the regular rods and their epsilon values as 4,8,12

and 16 as different cases.

PLOT 3:- TM band-structure for the line-def on square-lattice of dielectric rods in air with the horizontal

row of dielectric rods of twice the radius of the regular rods and their epsilon values as 4,8,12 and 16 as

different cases.

41

Conclusion:

1. A distinct wave-guide or line-defect mode is observed for the cases (a) and (b).

2. The normalised eigen-frequencies in these cases are centred around 0.3 to 0.35 etc.

and are expressed in inverse of wavelength units (µm) for 1 µm period. So,the

corresponding free-space wavelengths come around 3.3 µm. As PBG structures are

scalable, so for a lattice period of about 0.5µm, the frequency becomes twice and

wavelength half i.e around 1.55µm which can be utilised for single mode optical

communication applications with proper fine tuning of the fill ratio of the lattice,

finite height of lattice of epsilon of regular of defect rods.

3. Also as observed in the case (b) with increasing defect-rod epsilon the line-defect

modes approach the lower photonic crystal modes.

4. No distinct wave-guide mode is observed for the case (c).

5. But in the case of defect-rod epsilon being 8 as shown in plot 3, a distinct pair of

linedefect or waveguide modes with positive and negative group velocity respectively

is present and can be suitably used for meta-material applications.

6. Further the distinct waveguide modes from the cases (a) and (b) are shown below for

comparision.

42

ANALYSIS – 2:

Analysis of waveguide modes for triangular –lattice of cylindrical dielectric

rods in air.

Description:

Line defect is created in the triangular lattice structure of cylindrical dielectric rods of infinite

height in air with periodicity plane profile as shown below. Three types of defects have been

analysed with chosen values of the defect-rod epsilon as mentioned in the regular lattice

parameters and the line defect parameters.

This lattice structure is analysed for TM band structure. Band-structure and the line-defect or

wave-guide modes for the line-defect are explained.

(a) (b) (c)


Here the geometric and physical parameters of the non-defect dielectric rods are described.

Period = 1µm, radius of rods = 0.19µm, height of rods = infinity, epsilon of rods = 12.


Here the geometric and physical parameters of the defect dielectric rods are described.

A defect row is created in three possible ways as shown above and described below.

(a) Line defect created by removing a horizontal row of dielectric rods.

(b) Line defect created with the horizontal row of dielectric rods of half the radius of the


(c) Line defect created with the horizontal row of dielectric rods of twice the radius of the


43

PLOT1:- TM band-structure for the line-def on triangular-lattice of dielectric rods in air with a

horizontal row of rods removed. A distinct waveguide mode is present between upper and lower PC

modes.

K-points as indicated by k-index are sampled along the ΓM direction of Brillouin zone as shown as 18

equi-distant points between Γ (k = 0) and X (k = 0.5) including the both .

Subsequently, the other cases are also analysed along the same k-points as shown next.

44

PLOT 2:- TM band-structure for the line-def on triangular-lattice of dielectric rods in air with the

horizontal row of dielectric rods of half the radius of the regular rods and their epsilon values as 4,8,12


PLOT 3:- TM band-structure for the line-def on triangular-lattice of dielectric rods in air with the

horizontal row of dielectric rods of twice the radius of the regular rods and their epsilon values as 4,8,12


45

Conclusion:

1. A distinct wave-guide or line-defect mode is observed for the cases (a) and (b).

2. The normalised eigen-frequencies in these cases are centred around 0.35 to 0.4 etc.

and are expressed in inverse of wavelength units (µm) for 1 µm period. So,the

corresponding free-space wavelengths come around 2.7µm. As PBG structures are

scalable, so for a lattice period of about 0.5µm, the frequency becomes twice and

wavelength half i.e around 1.31µm which can be utilised for single mode optical

communication applications with proper fine tuning of the fill ratio of the lattice,

finite height of lattice of epsilon of regular of defect rods.

3. Also as observed in the case (b) with increasing defect-rod epsilon the line-defect

modes approach the lower photonic crystal modes.

4. No distinct wave-guide mode is observed for the case (c).

5. But in the case of defect-rod epsilon being 8 as shown in plot 3, a distinct pair of line-

defect or waveguide modes with negative group velocity each is present and can be

suitably used for meta-material applications.

6. Further the distinct waveguide modes from the cases (a) and (b) are shown below for

comparison.

46

The distinct waveguide modes from the cases (a) and (b) of the square and the

triangular lattice structures each of the cylindrical dielectric rods in air is shown below

for comparison.

―+‖ indicates curves corresponding to sqr-lattice and ―*‖ indicates corresponding to trig-

lattice.

47

ANALYSIS – 3:

Analysis of waveguide modes for triangular –lattice of cylindrical air holes

in dielectric slab.

Description:

Line defect is created in the triangular lattice structure of cylindrical air-holes of infinite

height in the dielectric slab of infinite thickness with periodicity plane profile as shown

below. Four types of defects have been analysed with chosen values of the defect-hole radius

as mentioned in the regular lattice parameters and the line defect parameters.

This lattice structure is analysed for TE as well as TM band structure of which the TE case is

prominent. Band-structure and the line-defect or wave-guide modes for the line-defect are

explained.

(a) (b)

(c) (d)


Here the geometric and physical parameters of the non-defect air-holes and the dielectric slab

are described.

Period = 1µm, radius of air-holes = 0.42µm, height of air-holes = infinity, epsilon of

dielectric-slab = 12.

48


(a) Line defect created by replacing a horizontal row of air-holes with dielectric or in other

words radius of air-holes becomes zero.

(b) Line defect due to the horizontal row of smaller air holes of radius 0.1µm.

(c) Line defect due to the horizontal row of smaller air holes of radius 0.2µm.

(d) Line defect due to the horizontal row of smaller air holes of radius 0.3µm.

K-points as indicated by k-index on the following plots for TE and TM line-defect modes are sampled

along the ΓM direction of Brillouin zone as shown above as 18 equi-distant points between Γ (k = 0) and

M (k = 0.5) including the both.

Subsequently, TE and TM band structures are shown on next page for the four cases of the

line-defects each and 0.1r, 0.2r and 0.3r refer to defect-hole radius of 0.1,0.2 and 0.3 microns

respectively.

49

PLOT 1:- TE band-structure for the line-def on triangular-lattice of air-holes in the dielectric slab with a

horizontal row of air-holes of zero radius or smaller radius than the regular air-holes as described above.

PLOT 2:- TM band-structure for the line-def on triangular-lattice of air-holes in the dielectric slab with a

horizontal row of air-holes of zero radius or smaller radius than the regular air-holes as described above.

50

Conclusion:

1. No single distinct waveguide mode is present between upper and lower PC modes in

either case TE or TM.

2. In both these cases the waveguide modes are very narrowly spaced with the PC

modes.

3. In the TE band structure a near horizontal line-defect or waveguide mode is present

for each defect case and it will have a very small group velocity can be utilised for

very slow propagation of light.

4. In TM band structure the various modes diverge at Γ point and converge at M point.

This feature can be utilised in making the super-prism.

51

ANALYSIS – 4:

Analysis of point defect modes for the three lattice structures by removing

central rod or hole.

Description:

Point defect in the square and the triangular lattice structure of cylindrical dielectric rods of

infinite height in the air is created by removing one of the rods (shown in the centre of the

lattice structures). Point defect in the triangular lattice of air-holes in the dielectric slab of

infinite thickness is created by replacing one of the air-holes (shown in the centre of the

lattice structure).

The point defect modes (expressed in terms of eigen-frequency and Q-factor of the localised

mode in the defect region) for both the dielectric rods in air lattice structure are TM where as

the point defect mode for the air-holes in dielectric slab is TE considering their band-gap

features.

The point defect modes for the three lattice structure are analysed for the epsilon range of the

rods or slab from 4 to 20 each with radius of the rods or the holes in each structure being

fixed at an optimal value for which average TM (for dielectric rods in air) or TE (for air-holes

in dielectric slab) bandgap of regular lattice structure is maximum across the range of epsilon.

(a) (b) (c)


(a) Square-lattice of dielectric rods in air with period = 1µm, radius = 0.20µm and epsilon of

the rods = 4 to 20.

(b) Triangular-lattice of dielectric rods in air with period = 1µm, radius = 0.18µm and epsilon

of the rods = 4 to 20.

(c) Triangular-lattice of air-holes in dielectric slab with period = 1µm, radius = 0.42 and

epsilon of the dielectric slab = 4 to 20.

Height of the dielectric-rods or air-holes = infinity.

52

Band –structures for point-defect cases and confined D-field profile

Typical band structure for the square and the triangular lattices of dielectric rods in air with

the point defects and their D-field profile in the defect region are illustated respectively. The

resonant mode in each case is represented by constant value of frequency (shown as straight

horizontal line) across the band structure. Band structure is taken along the conventional

contour of BZ.

PLOT1:- Band structure and energy profile for square lattice structure of dielectric rods in air.

colour scale on the energy profile picture has red for positive, white for zero and blue for negative.

PLOT2:- Band structure and energy profile for triangular lattice structure of dielectric rods in air.

colour scale on the energy profile picture has red for positive, white for zero and blue for negative.

53

PLOT3:- Resonant frequency (point-defect-mode) variation for three lattice cases with variation in

epsilon of dielectric rods (square and triangular lattice of dielectric rods in air) or that of dielectric slab

(triangular lattice of air-holes in dielectric).

PLOT4:- Q-factor variation (on logarithmic scale) corresponds to the resonant frequency (point-defect-

mode) variation for three lattice cases with variation in epsilon of dielectric rods (square and triangular

lattice of dielectric rods in air) or that of dielectric slab (triangular lattice of air-holes in dielectric).

54

Conclusion:

1. As observed from plots 1 and 2 the point-defect mode has a constant eigen frequency

across the band-structure and the energy is confined in the region of the point defect

which confirms the standing wave pattern for the point-defect mode.

2. As observed from plot 3, the eigen frequency continuously decreases with increasing

epsilon of the dielectric rod or the dielectric slab for all the three lattice structures.

3. The nature of decrease in defect mode frequency indicated by slope of the frequency

curves on plot 3 reveals the decrease to be steepest in the case of air-holes in dielectric

slab structure. For the two dielectric rods in air structure the relative decrease is

almost identical indicated by the almost parallel frequency curves.

4. As observed from plot 4, the Q-factor more or less continuously increases with

increasing epsilon (upto the epsilon value of 12 and then the change is very less) of

the dielectric rod or the dielectric slab for all the three lattice structures which

indicates stronger confinement of the mode-energy (greater lifetime) for higher

epsilon of the dielectric rod or the dielectric slab for all the three lattice structures.

5. These features along with non-linear photonics and electro-optics can be utilized in

the design and implementation of digital photonics and bi-state mechanisms utilising

the epsilon values with large difference in Q-factors (eg: - epsilon values 4 and 12).

6. The epsilon values between 12 and 16 which offers somewhat constant Q-factor and

constant decrease in mode frequency can used for optical mux-demux design.

55

ANALYSIS – 5:

Analysis of D and H energy distribution in point defect region for square

lattice structure.

Description:

Point defect in the square lattice structure of cylindrical dielectric rods of infinite height in

the air is created by removing one of the rods (shown in the centre of the lattice structures).

The energy distribution (both D and H energy) in the point defect region (corresponding to

the point defect modes) for the dielectric rods in square lattice structure is considered for TM

modes considering its band-gap feature.

The energy distribution of point defect modes expressed in fraction for the square lattice

structure are analysed for the epsilon range of the rods from 4 to 20 each with radius of the

dielectric rods being fixed at an optimal value for which average TM band-gap of regular

lattice structure is maximum across the range of epsilon.

Further to explain the energy distribution profile four zones of increasing area with common

centre (at the centre of defect) have been considered as shown and explained below. The

spots at the centre of lattice structure are geometric regions and not any substance.

(a) (b) (c) (d)


Period = 1µm, radius of the rods= 0.20µm, height of the rods = infinity and epsilon of the

rods = 4 to 20.

Point-defect region parameters:

(a) area1: circular region with radius = 0.2 µm.

(b) area2: circular region with radius = 0.5 µm.

(c) area3: circular region with radius = 1/√2 µm..

(d) area4: circular region with radius = 1 µm.

56

PLOT1:- Fractions of D and H energies concentrated in the identified area regions of the point defect for

TM mode. D energy is densely concentrated between area1 and area2. H energy is densely concentrated

at the periphery of defect zone outside area3.

PLOT2:- (a) Ratios of D by H energies in each of the identified area regions for TM mode. D energy is

more concentrated towards the center and H energy is more concentrated towards the periphery of the

point defect region. (b) Ratios of the D and H energies separately between one area region and the other

area region. D energy concentration increases rapidly outside area1 and within area2. H energy is mostly

confined at periphery outside area3. (“*” is for D-energy curve and “+” is for H-energy curve).

57

Conclusion:

1. As observed from plots 1 (a), fraction of D-energy present in different area zones

continuously increase with increase in area but this increase is very much from area1

to area 2 and then gradual increase to the subsequent areas for a fair range of epsilon

(i.e. from 6 to 18). Hence, D energy is densely concentrated between area1 and area2.

2. As observed from plots 1 (b), fraction of H-energy present in different area zones

continuously increase with increase in area but this increase is very much from area 3


(i.e. from 6 to 20). Hence, H-energy is densely concentrated at the periphery of defect

zone outside area3.

3. As observed from plots 2 (a), the ratio of D by H energy is very large for area 1 and

very small for other areas, the smallest being for area 4. Hence, D-energy is more

concentrated towards the centre of the defect zone and H-energy is more concentrated

towards the periphery of the defect zone.

4. As observed from plots 2 (b), ratio of D-energy in area 1 by D-energy in area 2 is very

less compared to those of area 3 by area 4.Hence, D-energy is confined outside area 1.

Similarly, H-energy is very less present in defect zone up to area 2.

58

ANALYSIS – 6:

Analysis of D and H energy distribution in point defect region for

triangular lattice structure.

Description:

Point defect in the triangular lattice structure of cylindrical dielectric rods of infinite height in

the air is created by removing one of the rods (shown in the centre of the lattice structures).

The energy distribution (both D and H energy) in the point defect region (corresponding to

the point defect modes) for the dielectric rods in triangular lattice structure is considered for

TM modes considering its band-gap feature.

The energy distribution of point defect modes expressed in fraction for the triangular lattice

structure are analysed for the epsilon range of the rods from 4 to 20 each with radius of the

dielectric rods being fixed at an optimal value for which average TM band-gap of regular

lattice structure is maximum across the range of epsilon.

Further to explain the energy distribution profile four zones of increasing area with common

centre (at the centre of defect) have been considered as shown and explained below. The

spots at the centre of lattice structure are geometric regions and not any substance.

(a) (b) (c) (d)


Period = 1µm, radius of the rods= 0.18µm, height of the rods = infinity and epsilon of the

rods = 4 to 20.

Point-defect region parameters:

(a) area1: circular region with radius = 0.2 µm.

(b) area2: circular region with radius = 0.5 µm.

(c) area3: circular region with radius = 1/√2 µm..

(d) area4: circular region with radius = 1 µm.

59

PLOT1:- Fractions of D and H energies concentrated in the identified area regions of the point defect for

TM mode. D energy is densely concentrated between area1 and area2. H energy is densely concentrated

at the periphery of defect zone outside area3.

PLOT2:- (a) Ratios of D by H energies in each of the identified area regions for TM mode. D energy is

more concentrated towards the center and H energy is more concentrated towards the periphery of the

point defect region. (b) Ratios of the D and H energies separately between one area region and the other

area region. D energy concentration increases rapidly outside area1 and within area2. H energy is mostly

confined at periphery outside area3. (“*” is for D-energy curve and “+” is for H-energy curve).

60

Conclusion:

1. As observed from plots 1 (a), fraction of D-energy present in different area zones

continuously increase with increase in area but this increase is very much from area1


(i.e. from 6 to 18). Hence, D energy is densely concentrated between area1 and area2.

2. As observed from plots 1 (b), fraction of H-energy present in different area zones

continuously increase with increase in area but this increase is very much from area 3


(i.e. from 6 to 20). Hence, H-energy is densely concentrated at the periphery of defect

zone outside area3.

3. As observed from plots 2 (a), the ratio of D by H energy is very large for area 1 and

very small for other areas, the smallest being for area 4. Hence, D-energy is more

concentrated towards the centre of the defect zone and H-energy is more concentrated

towards the periphery of the defect zone.

4. As observed from plots 2 (b), ratio of D-energy in area 1 by D-energy in area 2 is very

less compared to those of area 3 by area 4.Hence, D-energy is confined outside area 1.

Similarly, H-energy is very less present in defect zone up to area 2.

61

ANALYSIS – 7:

Analysis of point defect modes for the square lattice structure with varying

point defect.

Description:


the air is created by changing the radius and epsilon of one of the dielectric rods (shown in

the centre of the lattice structure) for a range of values. For regular dielectric rods their

epsilon remain constant.

The point defect modes (expressed in terms of eigen-frequency and Q-factor of the localised

mode in the defect region) for this lattice structure are TM considering their band-gap

features.

The point defect modes for this lattice structure are analysed for the six different cases of the

defect-rod radius from 0.1 to 0.6 and the epsilon range of the defect-rod from 2 to 16 for each

defect-radius case. The figures below represent some cases of the point defects with the

relative radius and the relative epsilon of the defect-rod at the centre exhibited.

(a) (b) (c) (d) (e)


Period = 1µm, radius-rods = 0.19 µm, height-rods = infinity, epsilon-rods = 12.

Point defect parameters:

Radius of defect of rods = 0.1 to 0.6 µm, height of defect rods = infinity, epsilon of defect

rods = 2 to 16. The point defect cases are considered for all the possible combinations of

radius and epsilon values for the respective ranges mentioned.

Figures shown above:

(a) Point defect dielectric rod with radius = 0.1 and epsilon = 6.

(b) Point defect dielectric rod with radius = 0.3 and epsilon = 4.

(c) Point defect dielectric rod with radius = 0.4 and epsilon = 15.

(d) Point defect dielectric rod with radius = 0.5 and epsilon = 5.

(e) Point defect dielectric rod with radius = 0.5 and epsilon = 8.

62

PLOT1:- Resonant frequency (point-defect-mode) variation for the six defect-rod radius cases with

variation in epsilon of defect-rods. For defect radii r4, r5 and r6 there is steep shift in resonant frequency

at a particular value of defect-rod epsilon.

PLOT2:- Q-factor variation (on logarithmic scale) corresponds to the resonant frequency (point-defect-

mode) variation for the square-lattice structure with variation in epsilon of defect-rod. For the cases of

defect-radii having gradual convergence of resonant frequency there is gradual convergence of Q-factor

value also. But for the cases of steep change in resonant frequency the large although not very steep

change in Q-factor value.

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Conclusion:

1. As observed from plots 1, there is a steep shift in frequency at a particular value of

defect-rod epsilon for the three cases where the defect-rod radius is significantly

larger than the regular lattice rod. The greater is the defect-rod radius the lesser is its

epsilon value at which this shift occurs.

2. There is large increase in Q-factor for this resonant frequency shift.

3. For the other three cases where the resonant frequency converges or constantly

decreases, corresponding Q-factor also converges.

4. For the case of defect-rod radius r1=0.1micron the resonant frequency decreases

linearly and Q-factor remains almost the same.

5. For defect radii cases of r4=0.4, r5=0.5 and r6=0.6 microns the steep shift in resonant

frequency can be used for digital photonics (binary system logic) designs using non-

linear photonics.

6. The linear variation in resonant frequency with very small change in Q-factor for the

case of the defect radius r1 can be suitably used for stable analog system designs.

This can also be utilised for optical multiplexer design.

64

ANALYSIS – 8:

Analysis of D energy concentration in point defect dielectric rod for square

lattice structure.

Description:





The D-energy fraction in the point defect region (corresponding to the point defect modes)

for the dielectric rods in square lattice structure is considered for TM modes considering its

band-gap feature.

The energy distribution for this lattice structure are analysed for the six different cases of the




(a) (b) (c) (d) (e)











(d) Point defect dielectric rod with radius = 0.5 and epsilon = 5.

(e) Point defect dielectric rod with radius = 0.5 and epsilon = 8.

65

PLOT1:- Variation of D-energy (expressed in fraction of the total in lattice) with defect-rod epsilon for six

radii cases.

Conclusion:

1. For larger defect-radius cases r4, r5 and r6 energy in defect-rod drops quickly to

almost zero with at defect-epsilon = 6 and this feature can be utilized for sensor

application, switching and digital photonics in conjugation with non-linear photonics.

2. For defect-radius case r1=0.1, energy in the defect-rod increase almost linearly for the

given range of epsilon and this feature can be suitably utilized for analog system

designs.

66

ANALYSIS – 9:

Analysis of D energy concentration in point defect dielectric rod for

triangular lattice structure.

Description:

Point defect in the triangular lattice structure of cylindrical dielectric rods of infinite height in




The D-energy fraction in the point defect region (corresponding to the point defect modes)

for the dielectric rods in triangular lattice structure is considered for TM modes considering

its band-gap feature.

The energy distribution for this lattice structure are analysed for the six different cases of the




(a) (b) (c)











67

PLOT1:- Variation of D-energy (expressed in fraction of the total in lattice) with defect-rod epsilon for six

radii cases.

Conclusion:

1. For larger defect-radius cases r4, r5 and r6 energy in defect-rod drops quickly to

almost zero with at defect-epsilon = 6 and this feature can be utilized for sensor

application, switching and digital photonics in conjugation with non-linear photonics.

2. For defect-radius case r1=0.1, energy in the defect-rod increase almost linearly for the

given range of epsilon and this feature can be suitably utilized for analog system

designs.

68

Reference 1 -softwares and simulation tools

The following software and CAD tools have been used to work out the contents of this

project work.

(i) RSOFT Design Group, INC, USA

(1) Bandsolve – PWEM simulations.

(2) Fullwave – FDTD simulations.

(ii) MIT Ab Initio Physics Group, MIT, USA

(1) MPB – PWEM simulations.

(2) MEEP –FDTD simulations.

(iii) MATLAB – graphs and plots.

69

Reference 2 – literature survey

(i) Books

(1) Photonic Crystals

Molding the flow of light -- J D Jonnopoulos, S G Jhonson, J N Winn, R D Meade.


Advances in Design, Fabrication, and Characterization – K Busch, S lolkes, H Foll.

(3) Photonic Crystal - Physics & Practical Modeling – Springer Series in optical

sciences.

(ii) Phd thesis


From theory to practice – S G Johnson at Dept of Physics, MIT, USA-- June

2001.

(2) Analysis and Simulation of Photonic Crystal Components for Optical

Communication -- Dinesh Kumar V at Dept of ECE, IISc Bangalore, India--

December 2003.

(iii) Papers

(1) Design and fabrication of Silicon Photonic Crystal Optical waveguide -- Marko

Loncar, Theodor Doll,J Vuckovic and A Scherer - JOURNAL OF LIGHTWAVE

TECHNOLOGY, VOL. 18, NO. 10, OCTOBER 2000.

(2) Photonic Bandgap Materials: A semiconductor for light – Sajeev John,

O Toader, K Busch.

(3) Structural Tuning of Guiding Modes of Line-Defect Waveguides of Silicon-on-

Insulator Photonic Crystal Slabs -- Masaya Notomi, A Shinya, K Yamada,

J Takahashi, C Takahashi, and I Yokohama-- IEEE JOURNAL OF QUANTUM

ELECTRONICS, VOL. 38, NO. 7, JULY 2002.

(4) Light Propagation Characteristics of Straight Single-Line-Defect Waveguides in

Photonic Crystal Slabs Fabricated Into a Silicon-on-Insulator Substrate –

Toshihiko Baba, Member, IEEE, A Motegi, T Iwai, N Fukaya, Y Watanabe, and

A Sakai -- IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 38, NO. 2, JULY 2002.

(5) Guided modes in photonic crystal slabs -- Steven G. Johnson, Shanhui Fan, Pierre

R. Villeneuve, and J. D. Joannopoulos, Department of Physics, Massachusetts

Institute of Technology, Cambridge, Massachusetts 02139 -- PHYSICAL REVIEW B –

AUGUST 1999

70

(6) Linear waveguides in photonic-crystal slabs --Steven G. Johnson, Pierre R.

Villeneuve, Shanhui Fan, and J. D. Joannopoulos--Department of Physics,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 -- PHYSICAL REVIEW B – SEPTEMBER 2000.

(7) Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in

Photonic Crystal Slabs--M. Notomi, K. Yamada. A. Shinya, J. Takahashi,

and I. Yokohama --NTT Basic Research Labs., NTT Corporation, Atsugi, 243-

0198 Japan-- PHYSICAL REVIEW B – DECEMBER 2001.

(8) Nature of the photonic band gap: some insights from a field analysis-- R. D.

Meade, A. M. Rappe, K. D. Brommer and J. D. Joannopoulos -- J. Opt. Soc. Am B 10,

328 (1993).

(9) Existence of a photonic band gap in two dimensions-- R. D.Meade, A. M. Rappe,

K. D. Brommer and J. D. Joannopoulos -- Appl. Phys. Lett. 61, 495 (1992).

(10) Existence of photonic gaps in periodic dielectric structures-- K. M. Ho, C. T.

Chan and C. M. Soukoulis -- Phys. Rev.Lett. 65, 3152 (1990).

(11) Photonic bandgap formation and tunability in certain self-organizing systems--

S. John and K. Busch -- J. Lightwave Technol. 17, 1931 (1999)

(12) All-optical ultracompact photonic crystal AND gate based on nonlinear ring

Resonators -- Parisa Andalib and Nosrat Granpayeh-- J. Opt. Soc. Am. B/Vol. 26, No.

1/January 2009.

(13) Design of Optical Logic Gates Using Photonic Crystal --A.P. Kabilan, X. Susan

Christina, P. Elizabeth Caroline -- 978-1-4244-4570-7/09/$25.00 ©2009 IEEE.

(14) Full 2-D photonic bandgaps in silica/air structures -- T. A. Birks, P. J. Roberts, P.

St. J. Russell, D. M. Atkin and T. J. Shepherd-- Electron. Lett. 31, 1941 (1995).

(15) Analysis of air-guiding photonic bandgap fibers -- J. Broeng, S. E. Barkou, T.

Søndergaard and A. Bjarklev -- Opt. Lett. 25, 96 (2000).

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