Master’s Thesis in Electrical Engineering by Adnan Daud Khan & Muhammad...

Technical report, IDE0946, June 2009

PHOTONIC CRYSTAL DESIGNS

(PCD)

Master’s Thesis in Electrical Engineering

by

Adnan Daud Khan & Muhammad Noman

School of Information Science, Computer and Electrical Engineering

Halmstad University

Title

PHOTONIC CRYSTAL DESIGNS (PCD)

Master’s thesis in Electrical Engineering

School of Information Science, Computer and Electrical Engineering

Halmstad University

Box 823, S-301 18 Halmstad, Sweden

June 2009

Description of cover page picture: Triangular Photonic Crystal Cavity.

i

Preface

The field of wireless communications develops upon electromagnetism principles.

Advancement of the AM radio, Radio navigation, Commercial AM, FM radio, Microwave

communication, Radar, Communication satellites, cellular phones, AMPS, GSM, 3G mobile

systems etc are some of the offerings of electromagnetism.

The field of electromagnetism inspire by its applications in field of high-speed

communications, modelling of microwave devices, optics, wireless communications,

Electromagnetic Interference, filtering, Cellular communications, antenna, electromagnetic band

gap structure, imaging of human body etc.

Since the first conversation on the photonic crystals, scientists and engineers have

established that photonic crystals can be used for a huge number of applications in all areas of

engineering. Photonic crystals have exposed promising results in engineering applications

particularly in communication engineering. These photonic crystals have applications for

instance, guiding light through photonic crystal slab waveguides with fewer losses, optical

switching, optical filtering, optical cavities etc.

Electromagnetism earlier relied on mathematical simplifications. Engineering problems

involve getting solutions to partial differential equations. Although only easy problems can be

resolved by using analytical methods, the majority of the practical problems are complex enough

to solve analytically. For that reason, to solve these problems, numerical techniques are the only

existing tool. There are number of numerical techniques for time evolution of electromagnetic

fields.

So, the purpose of this project is to solve problems in electromagnetism, which are not

possible to solve analytically, by making use of numerical technique such as the finite difference

time domain (FDTD) method. We have simulated a waveguide bend, which is very useful in

photonic integrated circuits. We have worked on photonic crystal cavities and made monopole,

dipole and Quadra pole. We have also simulated a double heterostructure nano cavity to achieve

high Q-factor. All these simulations have been performed as a thesis at Halmstad University for

the degree of Master of Science with a major in Electrical Engineering.

Adnan Daud Khan & Muhammad Noman

Halmstad University, June 2009

ii

Abstract

Photonic Crystal (PC) devices are the most exciting advancement in the field of photonics.

The use of computational techniques has made considerable improvements in photonic crystals

design. We present here an ultrahigh quality factor (Q) photonic crystal slab nanocavity formed

by the local width modulation of a line defect. We show that only shifting two holes away from a

line defect is enough to attain an ultrahigh Q value. We simulated this double heterostructure

nano cavity by using Finite Difference Time Domain (FDTD) technique. We observed that

photonic crystal cavities are very sensitive to the frequency, size and position of the source. So

we must choose the right values for these parameters.

iii

CONTENTS

PREFACE --------------------------------------------------------------------------------------------------------------------------- I

ABSTRACT ------------------------------------------------------------------------------------------------------------------------ II

1. INTRODUCTION TO PHOTONIC CRYSTALS -------------------------------------------------------------------- 1

2. MAXWELL’S EQUATION FOR PHOTONIC CRYSTALS ------------------------------------------------------ 4

2.1 BLOCH WAVES AND BRILLOUIN ZONES IN PHOTONIC CRYSTALS: ----------------------------------------------------- 7

3. ONE DIMENSIONAL AND TWO DIMENSIONAL PHOTONIC CRYSTALS ----------------------------- 10

3.1 PHYSICAL ORIGIN OF PHOTONIC BAND GAP: ------------------------------------------------------------------------------ 10

3.1.1 THE SIZE OF THE BAND GAP: --------------------------------------------------------------------------------------------- 11

3.2 TWO DIMENSIONAL PHOTONIC CRYSTALS: ------------------------------------------------------------------------------- 13

3.3 COMPLETE BAND GAP FOR ALL POLARIZATIONS: ----------------------------------------------------------------------- 17

3.4 POINT DEFECTS: ------------------------------------------------------------------------------------------------------------- 18

4. PHOTONIC CRYSTAL CAVITIES ----------------------------------------------------------------------------------- 19

4.1 QUALITY FACTOR (Q): ------------------------------------------------------------------------------------------------------ 20

5. FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) ---------------------------------------------------- 22

5.1 INTRODUCTION: -------------------------------------------------------------------------------------------------------------- 22

5.2 FINITE-DIFFERENCE EXPRESSION OF MAXWELL’S EQUATIONS: ------------------------------------------------------- 23

6. ELECTROMAGNETIC SIMULATIONS TOOL ------------------------------------------------------------------- 28

6.1 TRANSMISSION AND REFLECTION SPECTRUM: --------------------------------------------------------------------------- 28

6.2 RESONANT MODES: --------------------------------------------------------------------------------------------------------- 29

6.3 HARMINV: -------------------------------------------------------------------------------------------------------------------- 29

6.4 BOUNDARY CONDITIONS IN MEEP: --------------------------------------------------------------------------------------- 30

6.5 UNITS IN MEEP: ------------------------------------------------------------------------------------------------------------- 31

7. SIMULATIONS OF PHOTONIC CRYSTAL DEVICES --------------------------------------------------------- 32

7.1 PHOTONIC CRYSTAL WAVEGUIDES: -------------------------------------------------------------------------------------- 32

7.2 CAVITY SIMULATIONS: ----------------------------------------------------------------------------------------------------- 35

7.3 ULTRA-HIGH LINE MODULATED CAVITY: ------------------------------------------------------------------------------- 37

7.3.1 RESULTS: ------------------------------------------------------------------------------------------------------------------- 38

7.3.2 DISCUSSION: --------------------------------------------------------------------------------------------------------------- 41

CONCLUSION -------------------------------------------------------------------------------------------------------------------- 48

REFERENCES -------------------------------------------------------------------------------------------------------------------- 49

iv

LIST OF FIGURES Figure 1: Photonic Crystal --------------------------------------------------------------------------------------------------------- 3

Figure 2: Multilayer film ---------------------------------------------------------------------------------------------------------- 12

Figure 3: A 2-D photonic crystal of dielectric rods in air --------------------------------------------------------------------- 13

Figure 4: Hexagonal structure of dielectric rods in air ------------------------------------------------------------------------ 14

Figure 5: Brillouin Zone for the triangular photonic crystal ------------------------------------------------------------------ 15

Figure 6: Band diagram of a triangular photonic crystal. The first one is for the TE and the second one is for the TM

modes --------------------------------------------------------------------------------------------------------------------------------- 16

Figure 7: Brillouin Zone of a square photonic crystal ------------------------------------------------------------------------- 17

Figure 8: Band structure of a square photonic crystal ------------------------------------------------------------------------- 17

Figure 9: A two dimensional Square lattice photonic crystal with a dielectric rod missing in the lattice --------------- 19

Figure 10: Square crystal cavity formed by making one of the rods of crystal bigger than the normal size ----------- 20

Figure 11: A waveguide bend ---------------------------------------------------------------------------------------------------- 32

Figure 12: Light propagates in a waveguide bend. The blue spot shows the negative cycle, the white shows zero and

the red shows the positive cycle. -------------------------------------------------------------------------------------------------- 33

Figure 13: Photonic Crystal waveguide bend ----------------------------------------------------------------------------------- 33

Figure 14: Light propagates in a photonic crystal waveguide bend --------------------------------------------------------- 34

Figure 15: Flux spectrum of a waveguide bend. The blue curve shows the input flux and the red curve shows the

bend flux. Bend flux is slightly higher due to back reflections --------------------------------------------------------------- 34

Figure 16: A two dimensional Square lattice photonic crystal with a dielectric rod missing in the lattice ------------- 35

Figure 17: Square crystal cavity formed by making one of the rods of crystal bigger than the normal size ----------- 35

Figure 18: Monopole field distribution in a cavity ---------------------------------------------------------------------------- 36

Figure 19: Dipole field distribution in a cavity --------------------------------------------------------------------------------- 36

Figure 20: Quadra-pole field distribution in a cavity -------------------------------------------------------------------------- 37

Figure 21: Triangular photonic crystal cavity ---------------------------------------------------------------------------------- 38

Figure 22: Proposed structure of the Triangular Photonic crystal cavity --------------------------------------------------- 39

Figure 23: Band diagram of Triangular Photonic Crystal --------------------------------------------------------------------- 40

Figure 24: Field view of a double heterostructure nano cavity --------------------------------------------------------------- 41

Figure 25: Irreducible brillouin zone -------------------------------------------------------------------------------------------- 42

Figure 26: Position of the plane wave source vs the quality factor ---------------------------------------------------------- 44

Figure 27: Size of the plane wave source vs the quality factor --------------------------------------------------------------- 45

Figure 28: Frequency of the light vs the quality factor ------------------------------------------------------------------------ 47

INTRODUCTION TO PHOTONIC CRYSTALS

1

1. INTRODUCTION TO PHOTONIC CRYSTALS

A crystal represents a periodic array of atoms or molecules. The arrangement with which

the atoms or molecules are repeated in space is called the crystal lattice. Photonic crystals are

artificial crystal structures that manipulate the propagation of light. Photonic crystals are also

called periodic materials of light or photonic band gap materials. These crystal structures

manipulate light in the same way that electronic and semi conducting materials manipulate the

electron waves due to periodic potential. We can also name them “Semiconductors of Light”

[17].

In electronic band gap materials, the electronic band gap is an array of discrete energies

that electron waves cannot occupy. All the fancy functionalities carried out by the improvements

in the field of electronic materials and devices made up of these materials are due to the control

of the availability of the electrons and holes above and below the electronic band gap. The

existence of the electronic band gap depends on the material, its crystal structure, the type of

atoms or group of atoms in the material. By varying the sorts of materials, and therefore using

different atoms called dopant atoms. One can dramatically vary the properties of the materials. In

electronic and semi conducting materials, atoms nearby to each other are separated by a quarter

of a nanometer. Photonic band gap materials, or photonic crystals, have comparable sorts of

structure, but at a larger scale than electronic materials. In photonic crystals, the spacing can be of

the order of 400 nanometers. The spacing works as analogous to atoms in semiconductors. In

general, however, the wavelengths of the wave to be controlled should be close to the spacing in

the photonic crystal. We can take an example of a material having a specific number of holes,

which are spaced a few hundred nanometers from each other. The light entering the material will

partially refract and reflect, creating complex patterns of light beams. These light beams will

strengthen, or cancel, at certain locations according to the light’s wavelength, the direction of the

light propagation, the dielectric constant of the material, and the arrangement of the crystal. If the

light is cancelled completely within a specific range of wavelengths, leading to a band gap for the

light, i.e. the light having that range of wavelengths, it will not propagate through the crystal.

Also if the properties of the photonic crystal are changed the same way, changing the doping


2

atoms in semiconductors, it will change the performance of the crystal for the light beam

propagating through it.

Metallic waveguides and cavities can be related to photonic crystals. They are

extensively used to manage the transmission of microwaves. The transmission of the

electromagnetic waves will be disallowed by the walls of the metallic cavity with frequencies

lower than a particular threshold frequency, and a metallic waveguide accept transmission just at

its axis. It would be good to have these similar potentials for electromagnetic waves with

frequencies beyond the microwave scheme, like the visible light. However visible light energy is

rapidly dissipated within metallic machinery, which renders this technique of optical control

impractical to generalize a wider choice of frequencies. We can build a millimetre dimension

photonic crystal of a known geometry for the control of microwave, or else for micrometer

dimension for handling the infrared.

Photonic crystals are considered to be an engagement of electromagnetism and solid-

state physics. Crystal arrangements are populace of solid state physics, except that the electrons

are replaced by electromagnetic waves in photonic crystals [1].

The periodic layers in the photonic crystal reflect light beams with a range of

wavelengths, producing a photonic band gap. Hence, we have a certain range of frequencies of

light which cannot propagate through the photonic crystal. Outside the band gap region, the shape

and speed of light beam can be very different and can be manipulated This permits us to control

the flow of light and guide it efficiently through the guiding structure, slowing it down to a

desired speed. Photonic crystals have so far been used to make efficient optical switches, optical

cavities, optical filters, and optical waveguides, and can be used to make optical circuits in future

[4][8][14].

The figure below shows the photonic crystals, which contains the high dielectric and low

dielectric regions.

INTRODUCTION TO PHOTONIC CRYSTALS

3

Figure 1: Photonic Crystal

High dielectric

Low dielectric


4

2. MAXWELL’S EQUATION FOR PHOTONIC

CRYSTALS

The investigation and study of wave propagation in periodic media was first made by

Felix Bloch [1][2]. He showed that the wave propagation in periodic media can be successfully

accomplished without scattering of the waves. He proved that the wave drifting in the periodic

medium is modulated by a periodic function. This periodic function is due to the periodicity of

the structure of the medium, identical to the travelling of the electronic waves in the crystal

structures. The identical analogue can be applied to the electromagnetic wave propagation but,

here, the periodicity is in the dielectric difference of the medium. We start the mathematical

analysis by Maxwell’s curl equations.

0=∂

∂+×∇

t

BE

Jt

DH =

∂

∂−×∇

( 2.1)

ρ=⋅∇

=⋅∇

D

B 0

The quantities E and H are the electric and magnetic fields, measured in units of [volts/m]

and [Ampere/m] respectively. The quantities D and B are the electric and magnetic flux densities

respectively. In linear, isotropic and non-dispersive materials, the E and H fields, relate to electric

and magnetic flux through the following equations:

EED ro ξξξ == (2.2)

HHB ro µµµ ==

MAXWELL’S EQUATION FOR PHOTONIC CRYSTALS

5

Now assume no current flow or charge density in the system, then:

0=∂

∂+×∇

t

BE (2.3)

0=∂

∂−×∇

t

DH

So

0.

0.

=∇

=∇

∂

∂=×∇

∂

∂−=×∇

E

H

t

EH

t

HE

ro

o

εε

µ

(2.4)

Generally, E and H are both complex functions of space and time, since the Maxwell

equations are linear, though, from the spatial dependence, we can break up the time dependence

by increasing the fields into a set of harmonic-modes. So the harmonic-mode as a spatial

prototype or “mode profile” times a complex exponential is set by:

ti

ti

erEtrE

erHtrH

ω

ω

−

−

=

=

)(),(

)(),( (2.5)

By putting equation (2.5) in equation (2.4), the two divergence equations are given by:

0)(.

0)(.

=∇

=∇

rE

rH (2.6)

The two curl equations relate E(r) to H(r) is:

)()(

)()(

rEirH

rHirE

ro

o

εωε

ωµ

=×∇

−=×∇ (2.7)


6

If we divide the magnetic curl equation in equation (2.7) by rε and take the curl, we obtain the

following equation, called the master equation [1][2].

)()(1

2

rHc

rHr

=

×∇×∇

ω

ε (2.8)

Where )(rε , is the dielectric constant and is function of the position. Equation (2.8) is an Eigen

value equation, with Eigen value (ω/c)2 (ω are real (lossless)) and an Eigen operator ( ×∇×∇

ε

1 )

which is called Hermetian operator. H(r) is the Eigen state, and the Eigen states are orthogonal.

The two curls gives “kinetic energy” and ε

1 gives the “potential energy”, compared to the

Hamiltonian in Schrodinger equation.

The equation given above is a slightly different form of Maxwell’s equation. For the

derivation of any standard wave equation, we do the similar procedure. We can also express the

master equation (ME) is terms of electric field, but expressing it in terms of magnetic field H,

because it has a series of properties that has got very important physical consequences and these

properties are relatively easily to derive. By expressing the master equation in the Eigen value

equation as:

)()()( 2

rHc

rHω

=Θ (2.9)

Where

×∇×∇=Θ HH

ε

1 (2.10)

The condition for the E-field is:

)()()()( 2 rErc

rE ξω

=×∇×∇ (2.11)


7

It is referred to as a generalized Eigen problem, since there are operators on both sides of this

equation. It is an easy matter to change this into a normal, Eigen problem by dividing equation

(2.11) byξ , but then operator is not a Hermitian.

We can reinstate a simpler transversality constraint by using D in place of E, since 0. =∇ D .

Putting oE

Dξ for E in equation (2.11) and, to keep the operator Hermitian, divide both sides by

ξ , which yields:

)()(

1)()(

)(

1

)(

1 2rD

rcrD

rr ξ

ω

ξξ=×∇×∇ (2.12)

This equation looks like being much more complex, so we prefer the “H” for numerical

calculations.

2.1 Bloch Waves and Brillouin Zones in Photonic Crystals:

Similar to traditional crystals of atoms or molecules, photonic crystals do not have

continuous translational symmetry. They have (photonic crystal) discrete translational symmetry

i.e. under translations of any distance they are not invariant but, instead, just distances, which are

the multiples of certain fixed step lengths [1][2].

In photonic crystals, the wave propagates according to the Bloch’s theorem [1][2]. The

propagation is a function of a periodicity in the medium. In optical periodic media (photonic

crystals), there is a periodic variation in dielectric constants. The refractive indices are functions

of positions.

)()(

)()(

axx

axx

+=

+=

µµ

εε (2.13)

Where, ‘a’ is any arbitrary lattice vector. The above equations state that the medium repeats its

properties after position ‘x+a’.

The Bloch’s theorem is given by


8

( ) )(),( xHetxHk

txki rrrrr

rrω−⋅= (2.14)

Where ( )txkie

ω−⋅rr

is a plane wave, and )( xHk

rrr is periodic “envelop”. K is conserved, i.e.

no scattering of Bloch wave occurs. ω are discrete )(knω .

It is important to know that Bloch state having wave vector ‘k’, and the Bloch state having wave

vector ‘k+mc’, are the same. So the k’s that are different by integral multiples of c = 2π/a, are

not distinct from the physical point of view. Therefore, the frequencies of mode should also be

periodic in k. K exists in the range -π/a < k < π/a. This area of imperative, non-redundant values

of k is known as the Brillouin zone. The shortest area inside the Brillouin zone, for which the

)(knω (frequency bands) are not linked by symmetry, is known as irreducible Brillouin zone.

The irreducible Brillouin zone consists of a triangular block with 1/8 the area of the complete

Brillouin zone, and the remainder of the zone contains redundant versions of the irreducible zone.

When the dielectric is periodic in three dimensions, then dielectric is invariant in that

case under translations through a large number of three dimensional lattice vectors R. These

vectors can be inscribed as a specific arrangement of the three primitive lattice vectors, (a1, a2,

a3), that are called to “span” the space of lattice vectors, or each namalaR 321++= for

various integers l, m, and n, the vectors (a1, a2, a3) provide three primitive reciprocal lattice

vectors (c1, c2, c3)

So that

jiji ca ,2. πδ= (2.15)

These reciprocal vectors span a reciprocal lattice of their individual that is occupied by the wave

vectors. The modes of a 3D periodic system are Bloch states that can be marked by a Bloch wave

vector

332211 ckckckK ++= (2.16)

Where, “K” is in the Brillouin zone. Every value of the wave vector K within the Brillouin zone

recognizing an Eigen state of ×∇×∇ε

1 with frequency )(kω and an eigenvector kH of the type


9

)()( .

ruerH k

rik

k = (2.17)

Where )(ruk , a periodic function of the lattice is: )()( Rruru kk += for each lattice vectors R.


10

3. ONE DIMENSIONAL AND TWO DIMENSIONAL

PHOTONIC CRYSTALS

3.1 Physical origin of Photonic band gap:

The electronic band gap in crystal structure occurs because of the relations of electronic

waves with the periodic potential due to ion cores, and this gap is the states of energies that

electrons cannot take up. In the same way, in electrodynamics, the photonic band gap is an array

of frequencies for the light wave, where there are no propagating explanations to Maxwell’s

equation. There may be band gaps which are incomplete, but can survive for several set of wave

vectors, polarization and/or symmetries. The basis of both the band gaps is alike. We will look at

the band gap of a photonic crystal with the support of the analysis of one dimensional photonic

crystal.

Assume the situation of one dimensional photonic crystal, with the periodicity in the

crystal is “a”. The one dimensional photonic crystal contains alternating layers of dielectric

constants. The result of this assembly to the incident waves is comparable to the concept of

interference in thin films. When a wave inserts into the crystal, partial reflections and

transmissions occur from every boundary. The waves which are reflected and transmitted either

interfere constructively or destructively. For a specific series of frequencies, the wave moving in

the forward direction and the waves reflecting off the interfaces, interfere destructively, with the

intention that there are no travelling waves in the forward direction for that series (range) of

frequencies. This range of frequencies is known as photonic band gap, where the wave is trapped.

For the one dimensional photonic crystal, the wave vectors k = -π/a and π/a results in standing

waves with wavelengths 2a. Because of the symmetry of the crystal structure, we can place its

nodes either in low refractive index layer (or low dielectric region) or high refractive index layer

(or high dielectric region). Low frequency wave vectors will focus their energy in the high index

region, and high frequency wave vectors will focus their energy in the low index region. This

focusing of energies of high and low frequencies creates a band gap of frequencies.

The band over and under the gap can be notable where the energy of their modes is

focusing i.e. inside the high or low dielectric area. Normally, in the two and three dimensional

ONE DIMENSIONAL AND TWO DIMENSIONAL PHOTONIC CRYSTALS

11

crystals, the low dielectric regions are air regions. Therefore, it is reasonable to describe the band

over a photonic band gap as the air band and the band under a gap as the dielectric band. The

condition is analogous to the electronic band structure of the semiconductors, where the

conduction band and the valence band set the elementary gap.

3.1.1 The size of the band gap:

The degree of a photonic band gap can be considered by its frequency width ω∆ .

However, this is not a very fine calculation. When the crystal was expanded by some factor e,

then the equivalent band gap would have a width e

ω∆ . Here, we will introduce a gap-midgap

ratio, which is very useful, and is independent of the scale of the crystal [1]. Suppose mω is the

frequency at the middle of the gap, so the gap-midgap ratio is given by

mωω∆

Where ω∆ is the frequency width of the photonic band gap. It is normally expressed as a

percentage. So if the system is scaled up or down, every frequency scale consequently, other than

the gap-midgap ratio because it is independent of the scale of the crystal. Therefore, whenever we

refer to the size or dimensions of a gap, we mean the gap-midgap ratio. The frequency and the

wave vector are always plotted in dimensionless units, and is given by:

ca

πω

2 and

π2ka

The dimensionless frequency is equal to λ

a , where λ is the vacuum wavelength that is given

byω

πλ c2= .

Now consider an example of a multilayer film, as shown in figure 2, with feeble

periodicity. Here, we can obtain an easy formula for the size of the band gap. If we suppose that,

the two materials have dielectric constants “ε ” and “ εε ∆+ ”, and thickness “a-d” and “d” in a

multilayer film. Here, we will see that if the dielectric difference is fragile i.e., 1<<∆ε

ε or the

thickness “d/a” is minute. After that the gap-midgap ratio among the first two bands is given by:


12

π

π

ε

ε

ω

ω )sin(. a

d

m

∆≈

∆ (3.1)

The above equation is suitable only for the smallε

ε∆ . Further, we can get a number of analytical

results for arbitraryε

ε∆ , which we sum up here. For two materials with refractive indices “ 1n ”

and “ 2n ”, and the thicknesses “ 1d ” and “ 12 dad −= ”, respectively, the normal-incidence gap is

maximized when 2211 ndnd = or21

21 nn

and

+= . In this situation, the midgap frequency mω is

given by:

a

c

nn

nnm

πω

2.

4 21

21 += (3.2)

Figure 2: Multilayer film

ε1

ε2

ε1

ε2

ε1

ε2

ε1

ε2

ε1

ε2

ε1

ε2

Adding a periodicity ε2= ε1+ ∆ε


13

3.2 Two dimensional photonic crystals:

Two-dimensional photonic crystals have variation in the dielectric constant, or refractive

index, of the material in space in two dimensions, while it is homogenous along the third

dimension. Typical example of a two dimensional photonic crystal is a square lattice of dielectric

rods in air as shown in the figure.

Figure 3: A 2-D photonic crystal of dielectric rods in air

For some values of the spacing between the columns of the dielectric rods, the crystal can

have a photonic band gap in the xy-plane. The incident light in this plane does not propagate

through the crystal at any angle, contrary to multilayer thin film (1-D photonic crystal) which

reflected light at normal incident of light.

Symmetries of the crystal play a very significant role in description of the electromagnetic

modes in the crystal. Two-dimensional crystal has discrete translational symmetry in xy-plane.

This means that the dielectric constant is periodic only if it follows the position of the lattice point

is linear combination of the primitive lattice vectors. The main fact about the photonic crystals in

two dimensions is to understand the fields in 2-D to be sub-divided into two polarizations. The

one is the Transverse magnetic (TM) mode and the other is the Transverse electric (TE) mode. In

the earlier case, the magnetic field is in the xy-plane and the electric field is perpendicular, i.e. in


14

the z-direction, and in the latter case, the electric field is in the crystal plane and the magnetic

field is perpendicular. Corresponding to the polarizations, there are two fundamental topologies

for two-dimensional photonic crystals, as shown in figure 4. High index rods are enclosed by low

index and low-index holes in high index, in other words, dielectric rods in air or air holes in a

dielectric substrate.

There are two ways to line up the dielectric rods in air or air holes in dielectric substrate.

One category of arrangement is called square lattice arrangement while the other one is called the

triangular, or hexagonal, arrangement. Figure 3 shows the square lattice arrangements. Likewise,

the following figure shows the hexagonal or triangular arrangements of two-dimensional photonic

crystals.

Figure 4: Hexagonal structure of dielectric rods in air

Square lattice arrangement of the dielectric rods in air or air holes in dielectric substrate

guides us to different sorts of band gaps of TM and TE polarizations. In a square lattice, TM

polarized mode band gaps are favored, for the reason that isolated high index regions compel TM

modes to have dissimilar fill factors, due to the appearance of the node in the higher mode that, in

turn, leads to higher TM gaps and lower TE gaps. The TE modes in square lattice were forced to

go through the low index regions for the field line to be continuous, so the fill factors for the

consecutive modes are low and not very far away. The point to understand is that TM band gaps

are favored in a lattice of high index region while TE modes are supported in a linked region.


15

To achieve better and larger band gaps, it is necessary to arrange photonic crystals with

both isolated and connected regions of high index region. This is accomplished by the triangular

or hexagonal arrangement of air holes or dielectric rods. The triangular or hexagonal lattice gives

better band gaps for both the polarizations. The band gaps for both the arrangements of two-

dimensional photonic crystals are given below. Figure 6 shows the band structure of the

triangular photonic crystal and figure 8 shows the band structure of the square photonic crystal.

We draw the band structure for the triangular photonic crystal by using brillouin zone as shown in

the figure below.

Kr

Figure 5: Brillouin Zone for the triangular photonic crystal

We get this brillouin zone by drawing perpendiculars from each lattice point in figure 4. This

brillouin zone is large enough, we do not need to go inside of it, we will just travel at its edges.

So for that we will draw a triangle inside of it. This triangle is called the irreducible brillouin

zone. If we rotate this triangle, than we can construct the whole brillouin zone. When we move

from Γ to K, K to M and from M to Γ than the first band will be formed, than the second band

and so on as shown in figure 6.


16

Figure 6: Band diagram of a triangular photonic crystal. The first one is for the TE and the

second one is for the TM modes

ω

ω


17

For the square photonic crystal we get the following brillouin zone by drawing perpendiculars

from each lattice point in figure 3.

Figure 7: Brillouin Zone of a square photonic crystal

So here we draw the band structure by moving from Γ to X, X to M and from M to Γ.

Figure 8: Band structure of a square photonic crystal

3.3 Complete Band Gap for All Polarizations:

A photonic crystal can be designed that has a band gaps for both TM and TE polarizations.

By manipulating the lattice dimensions, we can even have the complete band gap for all

polarizations.

ω


18

Due to the appearance of a node in the higher frequency mode, the isolated high dielectric

constant spots of the square lattice of dielectric columns forced successive TM modes in order to

have different concentration factors. This results in the large TM photonic band gap.

In the square lattice of dielectric rods, the field lines had to cross dielectric boundaries so

the TE modes were forced to go through the low dielectric constant regions. As a consequence of

this, for successive modes, the concentration factors were both near and low.

In short, we can say that, TM band gaps are privileged in a lattice of isolated high dielectric

constant regions, and also the TE band gaps are privileged in a connected lattice.

It looks impractical to arrange a photonic crystal together with connected regions of

dielectric material and isolated spots. The solution is a sort of compromise; we can put a

triangular lattice of low dielectric constant columns inside a medium with high dielectric

constant. If the radius of the column is very large, then the spots between the columns seem to be

localized regions of high dielectric constant material, which are connected to the adjacent spots.

3.4 Point Defects:

In 2-D, we can replace a column with another column whose shape, size, or dielectric

constant is different, or remove a single column from the crystal. Perturbing a single lattice spot

causes a defect, and this perturbation is localized to a specific point in the plane; we called this

perturbation as a point defect.

PHOTONIC CRYSTAL CAVITIES

19

4. PHOTONIC CRYSTAL CAVITIES

One of the important applications of photonic crystals is cavities. They behave like optical

micro-cavities for light confinement to small dimensions and volumes comparable to the

wavelength of the light. These devices work on the basis of resonant circulation, which means

that light can be trap inside the cavity when resonant mode is formed. Photonic crystal cavities

can be used in a variety of applications, such as micro cavities, which coax quantum dots to emit

spontaneous emissions in the required direction; they can also be used to control the laser

emission for long distance optical communication through optical fiber etc [14][18].

An ideal cavity has a very sharp resonant frequency and confines the light for an indefinite

period. When cavity deviates from its ideal behavior, it leads to a characteristic parameter, known

as Q-factor. The Q-factor depends directly on the time of the light confinement in the cavity, in

terms of optical period.

Photonic crystals are the analog of the semiconductors in electronic devices. Single or

multiple defects in the photonic crystal give rise to the micro-cavities. These cavities are better

than the conventional cavities because they are spectrally more efficient and pure. The

conventional cavities, which are made by etching or by the oxide confinement, usually cause non-

radiative defects. A better way is to employ a cavity by periodic variation in the dielectric

constant of substrate. By etching circular air holes in dielectric substrate we can form photonic

crystal and, by eliminating one air hole, we can form cavity.

Figure 9: A two dimensional Square lattice photonic crystal with a dielectric rod missing in the

lattice


20

Figure 10: Square crystal cavity formed by making one of the rods of crystal bigger than the

normal size

The frequencies in the crystal’s band gap will not propagate through it, but light will be

trapped by the defect, having the frequency equal to the resonant mode frequency formed due to

the existence of the defect. The micro-cavity that is formed has a feedback of the dominant modes

in all directions.

4.1 Quality Factor (Q):

The quality factor ‘Q’ is a measure of the losses in the cavity. Since the reflectivity of the

crystal enclosing the defect rises with the number of rods, we expect that Q will also increase

with the size of the crystal. To calculate Q, we decide to use a method which first includes

pumping energy into the cavity, then observing its decay [18]. We evoke that the quality factor is

defined as

Where E is the stored energy, ω0 is the resonant frequency, and P = -dE/dt is the dissipated

power. A resonator can consequently maintain Q oscillations before its energy decays by a factor

of π2−e (or about 0.2%) of its original value. After stimulating the resonant mode, the total energy

can be observed as a function of time, and Q can be calculated from the number or optical cycles

required for the energy to decay.

dtdE

E

P

EQ

oo

/

ωω−==

PHOTONIC CRYSTAL CAVITIES

21

The Q factor could also have been calculated using another technique. We recall that Q can

be defined as ωω

∆o , where ω∆ is the full width at half-power of the resonator’s Lorentzian

response. By calculating ω∆ from transmission calculation, we could have anticipated the value

of Q. This technique, however, would have led to larger uncertainties, especially for large values

of Q [18].

In order to compute Q, we will excite the resonance efficiently. The initial conditions are

chosen such that the pump mode and the resonant mode have a large overlap. Suppose that the

resonant mode is a monopole, we chose to initialize the system with a Gaussian field profile

centered around the defect. The energy inside the cavity was then calculated over time. Through

the early stages of the decay, every mode except the high-Q one rapidly emitted away, leaving

only the energy related with the resonant mode inside the cavity. The mode continued its gradual

exponential decay. From the rate of decay, we calculated Q.

It has been observed that Q increases exponentially with the number of rods. It reaches a

value near to 4

10 with as little as four lattices on either side of the defect [18]. Since the only

energy loss in the structure take place by tunneling through the edges of the crystal, Q does not

saturate yet for a very huge number of rods.


22

5. FINITE DIFFERENCE TIME DOMAIN METHOD

(FDTD)

5.1 Introduction:

The finite difference time- domain (FDTD) is a numerical method, which solves the

Maxwell’s equations. [5]. It solves electromagnetic wave propagation problems for a wide range

of applications. The FDTD technique is an increasingly admired method among scientists and

engineers in the area of computational electromagnetic.

The FDTD method can solve complex problems, but it requires large amounts of memory

and computational power. This is the reason why FDTD previously could not get the attention of

scientists and engineers because of lack of computational resources. Gradually, with the increase

in computational power, this method became popular within the computational electromagnetic

community.

Simplicity, and ease of implementation, was the main advantages of FDTD over other

computational methods, like Method of Moments (MoM) and Finite Element Method (FEM) [6].

The growing number of enhancements to, and expansions of, the method are being developed,

which is making FDTD the most preferred amongst other numerical methods.

The FDTD method is a simple and elegant way to discretize Maxwell’s equations. An

explicit finite difference approximation, both in spatial and temporal derivatives, appears in

Maxwell's equations. The method is based on central difference approximations on a staggered

Cartesian grid in space and time and, therefore, it is second order accurate in space and time

[5][6]. It calculates the E and H field everywhere in computational space, as they evolve with

time.

The finite difference time domain equations are solved for the future unknown fields, in

terms of known past fields. The two couple equations solve iteratively to advance the field in

time. This type of modelling is to test and measure fields inside a device.

FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

23

5.2 Finite-Difference Expression of Maxwell’s Equations:

It is a numerical method to solve Maxwell’s curl equations, i.e. Faraday’s and Ampere’s

Law.

(Faraday’s Law) (4.1)

(Ampere’s Law) (4.2)

5.2.1 FDTD uses central difference approximation:

5.2.1.1 FDTD for 1D wave:

1-D case of FDTD and simplified result of the curl equations in Ampere’s & Faraday’s Law

is

(4.3)

Et

H×∇−=

∂

∂

µ

1

Ht

E×∇=

∂

∂

ε

1

x

E

Hx

kji

HHH

zyx

kji

H

x

E

Ex

kji

EEE

zyx

kji

E

z

yzyx

z

zzyx

∂

∂−=

∂

∂=

∂

∂

∂

∂

∂

∂=×∇

∂

∂−=

∂

∂=

∂

∂

∂

∂

∂

∂=×∇

00

00

ˆˆˆˆˆˆ

00

00

ˆˆˆˆˆˆ


24

(4.4)

Now consider an Electric field:

(4.5)

Here, in the above equation, the superscript ‘q’ denotes the temporal step and ‘m’ denotes the

spatial step in the discrete 1-D grid. and are the spatial and temporal unit step sizes

respectively.

Now discretize Faraday’s law using central difference around the space-time point

(4.6)

(4.7)

Update equation for Ez field,

(4.8)

Similarly, update equation for Hy field

x

H

t

E

x

E

t

H

yz

zy

∂

∂=

∂

∂⇒

∂

∂=

∂

∂⇒

ε

µ

1

1

( ) ( ) ][,, mq

EtqxmEtxEzzz

=∆∆=

x∆ t∆

))2

1(, tqxm ∆+∆

))2

1(,))

2

1(, tqxmtqxm x

yH

t

zE

∆+∆∆+∆ ∂

∂

∂

∂=ε

x

m

q

yHm

q

yH

t

mqzEm

qzE

∆

−

+

−+

+

=∆

−+ ]

2

1[2

1

]2

1[2

1

][][1

ε

−−+

∆

∆+=

+++ ]

2

1[]

2

1[][][ 2

1

2

1

1mHmH

x

tmEmE

q

y

q

y

q

z

q

zε


25

(4.9)

5.2.1.2 FDTD for 3D wave:

Consider function n

kjiu ,, of space and time, evaluated at discrete point in grid and discrete

points in time, where n is the temporal step and i, j, k are the spatial steps in x, y, z coordinates

respectively:

Consider first partial derivative of u in x- direction, at point of time tntn ∆= .

x

uu

x

un

kji

n

kji

∆

−=

∂

∂ −+ ,,2/1,,2/1 (4.10)

Notice the half-spatial increment/ decrement in the i subscript, denoting the spatial finite

difference of x∆±2

1 . This would interleave the two adjacent E and H field components, so we

use that two consecutive spatial E field components to calculate H field component in between

them.

Similarly, we write central difference for y

u∂

∂ and z

u∂

∂ as follows:

y

uu

y

un

kji

n

kji

∆

−=

∂

∂ −+ ,2/1,,2/1, (4.11)

z

uu

z

un

kji

n

kji

∆

−=

∂

∂ −+ 2/1,,2/1,, (4.12)

Similarly, writing the central difference approximation for the function u as:

t

uu

t

un

kji

n

kji

∆

−=

∂

∂−+ 2/1

,,

2/1

,, (4.13)

Notice the half-spatial increment/ decrement in the n subscript, denoting the temporal

finite difference over t∆±2

1 . This would interleave the calculation of two adjacent E and H field

components in time intervals of t∆±2

1 .

Applying this finite difference approximation to vector components of curl operations of

Maxwell’s equation, we obtain [5]:

( )][]1[]2

1[]

2

1[ 2

1

2

1

mEmEx

tmHmH

q

z

q

z

q

y

q

y −+∆

∆++=+

−+

µ


26

∆

−−

∆

−

∆

+

+++

+++

++

−

++

+

++=

z

HyHy

y

HzHz

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiExEx

,2/1,1,2/1,

2/1,,2/1,1,

2/1,2/1,

2/1

2/1,2/1,

2/1

2/1,2/1,

ξ (4.14)

∆

−−

∆

−

∆

+

++−++

+−++−

++−

−

++−

+

++−=

x

HzHzz

HxHx

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiEyEy

2/1,1,12/1,1,

,1,2/11,1,2/1

2/1,1,2/1

2/1

2/1,1,2/1

2/1

2/1,2/1,2/1

ξ (4.15)

∆

−−

∆

−

∆

+

+−++−

++−++

++−

−

++−

+

++−=

y

HxHx

x

HyHy

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiEzEz

1,,2/11,1,2/1

1,2/1,11,2/1,

1,2/1,2/1

2/1

1,1,2/1

2/1

1,2/1,2/1

ξ (4.16)

∆

−−

∆

−

∆

+

+

++−

+

++−

+

++−

+

++−

++−

++−

+

++−=

y

EzEz

z

EyEy

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiHxHx

2/1

1,2/1,2/1

2/1

1,2/3,2/1

2/1

2/1,1,2/1

2/1

2/3,1,2/1

1,1,2/1

1,1,2/1

1

1,1,2/1

µ (4.17)


27

∆

−−

∆

−

∆

+

+

++

+

++

+

++−

+

+++

++

++

+

++=

z

ExEx

x

EzEz

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiHyHy

2/1

2/1,2/1,

2/1

2/3,2/1,

2/1

1,2/1,2/1

2/1

1,2/1,2/1

1,2/1,

1,2/1,

1

1,2/1,

µ (4.18)

∆

−

∆

−

∆

+

+

++−

+

+++

+

++

+

++

++

++

+

++=

x

EyEy

y

ExEx

tn

kji

n

kji

n

kji

n

kji

kji

n

kji

n

kjiHzHz

2/1

2/1,1,2/1

2/1

2/1,1,2/1

2/1

2/1,2/1,

2/1

2/1,2/3,

2/1,1,

2/1,1,

1

2/1,1,

µ (4.19)


28

6. ELECTROMAGNETIC SIMULATIONS TOOL

For the computational solution of arrangements using FDTD, MEEP (MIT

Electromagnetic Equation propagation) is used [3][16]. MEEP is a finite difference time domain

(FDTD) simulation open source software package, to model electromagnetic systems [16]. It is

developed at MIT. It has many features like:

1. Solving for finite difference time- domain solution, which gives a field pattern of the

system at every time step, stimulated by a source

2. FDTD simulation in 1D, 2D, 3D and cylindrical coordinates.

3. Provides distributed memory parallelism support using MPI standard.

4. Provides perfectly matched layer absorbing boundary conditions, Perfectly Electric

Conductor (PEC) and Bloch-periodic boundary conditions.

5. Exploitation of symmetries, to reduce the computational size.

6. Provides arbitrary material and source conditions.

6.1 Transmission and Reflection Spectrum:

Different finite structures are simulated, and transmission and reflection spectra are

drawn. At each frequency ω, we can separately calculate the fields and, ultimately, the

transmitted flux. However, it can calculate a response of broad-spectrum through a single

calculation much more efficiently by taking the Fourier transform of the response. This will give

the solutions for the system at different frequencies. It is much trickier if somebody wants to

calculate the transmission as well as reflection spectrum. In this case, we cannot calculate the flux

simply in the backward direction, since it would give the sum of the incident and reflected power.

Similarly, we cannot get the transmitted power simply by subtracting the incident power from

backward flux, since there will be interference between reflected and incident waves.

The solution to this is to simulate the system twice firstly without scattered and secondly

with the scattered. After that, subtract the Fourier transforms, before calculating the flux and after

ELECTROMAGNETIC SIMULATIONS TOOL

29

calculating the reflected power in the reflected plane, we will normalize to get the reflection

spectrum by the incident power.

6.2 Resonant Modes:

One of the most important features of MEEP is to calculate the resonant modes of a

structure. Consider a photonic crystal or a waveguide. MEEP can solve its harmonic (definite-ω)

modes at a given wave vector k. Similarly, consider a resonant cavity, which captures the light for

a long time in a small region, and we want to calculate the quality, factor Q and the resonant

frequency, ω.

In order to get the lifetime and frequencies with FDTD, the structure has been setup,

depending on whether it is a periodic or not, with Bloch periodic or/and absorbing boundaries.

Then the modes with a current has been excited which is placed in the system, with a short pulse.

After turning the current sources off, we get some fields in the system, which has been analyzed

to get the decay rates and frequencies.

The easiest way of harmonic analysis is to calculate the Fourier transform of the fields,

and the sharp peaks in the spectrum will be yielded by harmonic modes. However, this method

has some disadvantages; a very long running time is required by the higher frequency resolution,

and the problem of getting the decay rates followed to a poorly conditioned non-linear fitting-

problem. See the next section on Harminv.

Once the frequencies of modes have been found, than for the field patterns, the simulation

will be run once more, with a narrow bandwidth pulse only, to excite the mode in the question.

Given the field patterns, we can then perform other analyses. If we want the longest-lifetime

mode, we can use more time-steps after the source, enough for the other modes to decay away.

Computing modes in time domain needs care. For instance, if the source is almost orthogonal to

it, or if it is very near to the other mode, than a mode will be missed. Similarly, sometimes the

spurious frequencies peak will be indicated accidently by the signal processing. In addition,

analyzing periodic systems having non rectangular unit cells is very difficult in MEEP.

6.3 Harminv:

Instead, MEEP allows us to perform a more sophisticated signal-processing algorithm,


30

known as filter-diagonalization, and is implemented by Harminv-package. In a short time

Harminv gets all of the frequencies and their decay rates to a high level of accuracy.

Harminv is a freeware program that gives the solution to the harmonic inversion problem,

provided a discrete time, and finite signal length, it determines the decay constants, frequencies,

phases, and amplitudes of the sinusoids.

It is much more accurate when compared to the FFT peaks, extracting straight-forwardly

because for the signal takes a specific form. For finding the sinusoids, Harminv uses a low-

storage, "filter diagonalization method" (FDM), near a given frequency interval. This type of

spectrum analysis has vast applications in many areas of engineering and physics. For instance, it

can be used to extract the Eigen modes of a system from stimulus response, and their decay rates

in dissipative systems.

6.4 Boundary Conditions in MEEP:

As discussed in previous chapters, we can simulate only a finite region, which means that

we should finish the simulation with some kind of boundary conditions. In order to simulate for a

practical scenario, we need to cater for different types of boundary conditions. There are three

basic types of grid terminations supported with MEEP, and these are:

1. Bloch-Periodic Boundaries

2. Metallic walls

3. PML absorbing layers

As explained in previous chapters, for given periodic boundaries in a cell that has a size L,

the components of the field satisfy f(x + L) = f(x). Bloch Periodicity is a generalization where

)()( xfeLxf Likx=+ for Bloch wave-vector k.

Boundary conditions with a metallic wall at which, the fields on the boundaries are forced

to be zero. It is relatively simple; a perfect metal is around the cell (no absorption and no skin

depth). In addition, perfect metal materials can be placed anywhere in the computational cell.

We put the boundaries to simulate open boundary conditions in order to absorb all those

waves that incident on them, with zero reflections. This can be done with a phenomenon known

ELECTROMAGNETIC SIMULATIONS TOOL

31

as perfectly matched layers (PML). Strictly speaking, PML is not a boundary condition, but a

special absorbing material that is placed adjacent to the boundaries.

6.5 Units in MEEP:

MEEP has dimensionless units, where all these units are unified. Computation expresses

in ratio, therefore, these units end up canceling. Specifically, the Maxwell's equations are scale

invariant, and it is convenient to choose units which are dimensionless. Therefore, we can notice

the lack of annoying constants, like ε0, µ0, c, and 4π, where all these constants are in unity. In

practical terms, almost everything we want to calculate, i.e. transmission spectra, frequencies, are

ratios anyway, so the units end up canceling.

For instance, consider that, at infrared frequencies, we are defining some nano photonic

structure, where it is easy to describe the distances in micro-meter. Thus, let a = 1µm then, if we

want to define a source that corresponds to λ = 1.55µm, we define the frequency ω as 1/1.55 =

0.6452. If we run the simulation for 100 periods, we then run the simulation for 155 time units (=

100 / ω).


32

7. SIMULATIONS OF PHOTONIC CRYSTAL DEVICES

Photonic crystals have inspired great interest because of their potential ability to control the

propagation of light. Photonic crystals can modify, and even eliminate, the density of

electromagnetic states inside the crystal. Such periodic dielectric structures with complete band

gaps have many applications, including the fabrication of waveguides, cavities, bends, splitters

and filters etc for optical light.

7.1 Photonic Crystal Waveguides:

Waveguides are line defects rather than point defects. Linear line defects are created in the

crystal, which supports a guided mode that is in the band gap. The major advantage of using the

photonic crystal waveguides over other conventional waveguides like optical fibres is the

availability of waveguide bends. More examples can be found in [4][7][13][14][15].

We have done our simulations in MEEP (MIT Electromagnetic Equation Propagation)

software. It solves the Maxwell’s equations at different time and space steps and finds its solution

and then we use that solution to design different structures like waveguides, antenna, radar etc.

The figure below shows a waveguide bend, which we have simulated in MEEP. The

dielectric constant of the waveguide is 12 and it is place in low dielectric medium of dielectric

constant 1 i.e. air.

Figure 11: A waveguide bend

SIMULATIONS OF PHOTONIC CRYSTAL DEVICES

33

Figure 12: Light propagates in a waveguide bend. The blue spot shows the negative cycle, the

white shows zero and the red shows the positive cycle.

Here, in the waveguide, we bend the light at 90 degrees. It can be clearly seen that most of

the light is refracted and reflected, so the flux is very minimum at the output end. This kind of

waveguide will create a problem in an integrated circuit, where routing has been done. In order to

overcome this problem, we will design a photonic crystal waveguide bend.

As mentioned in the previous chapters, we can design a photonic crystal in two ways: we

can either put a high index rod in a low index region or make air holes in a high index region.

Here, in this crystal, we have made air holes in a high index or dielectric constant region. Its

dielectric constant is 12. The radius of the air holes is 0.2a.

Figure 13: Photonic Crystal waveguide bend


34

Figure 14: Light propagates in a photonic crystal waveguide bend

We have bent the light from a waveguide 1 to a waveguide 2 at 90 degrees. We have

designed a line defect, and a point defect, to overcome back reflection. As you can see in the

above figure, reflection is almost negligible. Here we have chosen the frequency of light which is

0.35. If we choose some other frequency like for example 0.335, than the light will not be guided,

it will pass through the whole crystal. This kind of structure is very useful in photonic integrated

circuits.

Figure 15: Flux spectrum of a waveguide bend. The blue curve shows the input flux and the red

curve shows the bend flux. Bend flux is slightly higher due to back reflections


35

7.2 Cavity Simulations:

Any defect bound by a photonic crystal with a band gap defines a cavity. A defect can

have any shape or size. It can be made by changing the refractive index, or dielectric, of a rod,

changing its radius, or removing a rod from a photonic crystal. The defect could also be made by

changing the refractive index, or the radius of several rods, like the following figures:

Figure 16: A two dimensional Square lattice photonic crystal with a dielectric rod missing in the

lattice

Figure 17: Square crystal cavity formed by making one of the rods of crystal bigger than the

normal size

Here, we have done some simulations of the cavities by changing the radius of a single

rod. Here every rod in a photonic crystal has a radius of 0.2a. The frequency of the mode can be

tuned by changing the size of the rod.

When we make the radius of the central rod in a photonic crystal equal to zero, we get a

monopole. When we make the radius equal to 0.33a, we get double degenerate dipole modes and,


36

by making the radius equal to 0.6a, we get non degenerate Quadra pole modes in the cavity. The

figures below show the images of the mentioned field distributions.

Figure 18: Monopole field distribution in a cavity

Figure 19: Dipole field distribution in a cavity


37

Figure 20: Quadra-pole field distribution in a cavity

7.3 Ultra-High Line Modulated Cavity:

We present an ultrahigh quality factor (Q) photonic crystal slab nanocavity formed by the

local width modulation of a line defect, which is given in [8][9]. We show that only shifting two

holes away from a line defect is enough to attain an ultrahigh Q value.

The authors in their paper named “Ultrahigh-Q photonic crystal nanocavities realized by the

local width modulation of a line defect” have designed a triangular photonic crystal to achieve a

high Q-factor. They use a double hetero structure approach here. They design a cavity by putting

a line defect in the crystal, and then they shift the lattice points through different distances as

shown in the figure.

The lattice constant of the crystal is 420nm, Slab index is 3.46 (Si), the radius of the air

holes is 108nm and the holes are shifted by a distance x, 2x/3, x/3 respectively. The width of the

line defect is 0.98W.


38

Figure 21: Triangular photonic crystal cavity

Through this structure they get the high value of Q-factor, which is 7 x 107, at a lattice shift of

9nm.

7.3.1 Results:

We design exactly the same structure given below by making air holes in a high index

region of 3.46 in a triangular form.


39

Figure 22: Proposed structure of the Triangular Photonic crystal cavity

In order to get the frequency of the light we make a band structure of this photonic crystal

because we have given the radius of the air holes so we have made its band diagram.


40

Figure 23: Band diagram of Triangular Photonic Crystal

In the band diagram, we found two band gaps, one from 0.2 to 0.26 and the other is from

0.41 to 0.46. So from here we get the center frequency about 0.215.

As this is a triangular photonic crystal, so the band gap is formed for the TE modes. A very

small or no band gap will be formed for the TM modes. So here we have assumed it for the TE

modes in which the E-field is in the crystal plan and the H-field is in the perpendicular direction.

From the above calculations we get the frequency of the light = 0.215, the size of the source

(2 0) and the position of the plane wave source (0 4). The crystal is placed in the xy plane and its

dimensions are 12 x 16. So by using these values we get the Q-factor = 4108.4 × . The figure

below shows the formation of the double heterostructure nano cavity in the crystal.


41

Figure 24: Field view of a double heterostructure nano cavity

7.3.2 Discussion:

The Q-factor we got over here is 4108.4 × which is less than 7 x 107. The reason for this is

that, that we do not know exactly some of the important parameters given below.

1) We do not know the position of the plane wave source.

2) We do not know the size of the plane wave source.

3) We do not know the frequency of the light.

4) We do not know about the polarization of light.

The crystal which we use over here to design a double heterostructure nano cavity is a triangular

photonic crystal. We draw the band structure for this crystal by using brillouin zone as shown in

the figure below. This brillouin zone is large enough, we do not need to go inside of it, we will

just travel at its edges. So for that we will draw a triangle inside of it. This triangle is called the

irreducible brillouin zone. If we rotate this triangle, than we can construct the whole brillouin

zone. When we move from Γ to K, K to M and from M to Γ than the first band will be formed,

than the second band and so on as shown in the band structure diagram given above. The bands


42

which are formed are discrete and these bands are the function of the wave vector Kr

, which

means that these bands depend upon the wave vector. We also know that when a resonant mode

is formed than that means that light is trapped inside the cavity. That mode is also in the band gap

because a mode in the cavity is also a mode in the band gap. So that mode also depends upon the

wave vector. Therefore, if we do not know the exact direction, position and size etc of the source,

than we cannot reproduce the same cavity or the Q-factor, which is given in the paper.

Kr

Figure 25: Irreducible brillouin zone

The frequency which we find here is by drawing the band structure for this photonic

crystal but this band structure is not the complete representation of the bands and the band gaps

because it is the band structure of the crystal not of the crystal with a defect but anyways we can

get a fair idea of the bands and the band gaps from it. So the frequency we find here is not the

exact frequency. For the polarization we have consider the TE modes in which the E-field is in

the crystal plan and the H-field is in the perpendicular direction because in triangular photonic

crystal the band gaps are formed for the TE modes and very small band gaps are formed for the

TM modes. So here we do not know exactly about the polarization. For the position and size of

the plane wave source we did some tests which are given below.

At the beginning we have made fix the frequency of light which is 0.215 and the size of the

plane wave source which is (2 0). Here we have only changed the position of the plane wave

source. The table below shows that how the Q-factor changes when the position of the plane

wave source changes.


43

Position of the Plane Wave Source Quality Factor

0 297.194

1 85.66

1.5 113

2 1171

2.5 221

3 110

3.5 411

3.999 123

4 48000

4.001 395

4.5 168

5 126

5.5 82

6 77

Table 7.1: Position of the plane wave source and the quality factor

The graph below shows the complete picture of the position of the plane wave source

verses the quality factor. So according to the graph we get the high Q-factor when the source is at

the position (0 4).


44

Figure 26: Position of the plane wave source vs the quality factor

We then fixed the position of the source which is (0 4) and changed the size of the plane

wave source. The table below shows that how the Q-factor changes when the size of the plane

wave has been changed.

Size of the Plane Wave Source Quality Factor

1 257

1.5 126.32

1.999 1583

2 48000

2.001 218

2.5 265

3 387

Table 7.2: Size of the plane wave source and the quality factor

Log

(Q)


45

The graph below shows the complete picture of the size of the plane wave source verses

the quality factor. So according to the graph we get the high quality factor when the source size is

(2 0).

Figure 27: Size of the plane wave source vs the quality factor

We have also tried different light frequencies. We have made fix the size (2 0) and

position (0 4) of the plane wave source and change frequencies. The table below shows how the

Q-factor changes when the frequency of the light changes.

Log

(Q)


46

Frequency of Plane wave source Quality Factor

0.2 69

0.212 3

0.213 92

0.214 166

0.215 48000

0.216 6

0.217 201

0.218 112

0.219 4

0.22 23

0.221 3

0.222 53

0.223 1095

0.224 2

0.225 255

0.226 59

0.227 145

0.228 74

0.229 61

Table 7.3: Frequency of the plane wave source and the quality factor

The graph below shows the complete picture of the frequency of light verses the Q-factor.

47

Figure 28: Frequency of the light vs the quality factor

Log

(Q)


48

CONCLUSION

The explanation of the Maxwell’s curl equations offers classical electromagnetic

phenomenon, for time varying fields. Maxwell originated the four of the most basic equations of

electromagnetic theory, which engineers and scientists exploit worldwide.

FDTD (Finite Difference Time Domain) is a numerical technique requiring very high

computational power, but with the development of fast computers has made this method growing

famous in the electromagnetic area.

We have shown that photonic crystals can be used for the production of high-Q micro

cavities. By set up a defect in a photonic crystal, intense resonant states can be formed in the area

of the defect. The properties of these modes like the frequency, polarization, symmetry, and field

distribution, can be controlled by varying the nature and the size of the defect. Furthermore, we

offer an ultrahigh quality factor (Q) photonic crystal slab nanocavity formed by the local width

modulation of a line defect. We show that only shifting two holes away from a line defect is

enough to attain an ultrahigh Q value. The Q-factor we got is 4108.4 × , which is less than

7107 × . The reason for this is that, that we do not know the exact frequency, polarization, size,

direction and position of the plane wave source. If we know exactly these parameters, than we

can get the high Q-factor. Nobody has designed the double heterostructure nano cavity in MIT

Electromagnetic Equation Propagation (MEEP) software. This software has additional features

like the sub pixel averaging, boundary conditions and symmetries etc.

Scientists and engineers will work on these double heterostructure nano cavities to get the

higher Q-value, which will be used to design the micro and nano lasers. The only problem in the

photonic crystal cavities is that, that they are very sensitive to frequency, polarization, size and

position of the source. So we should choose the right values for these parameters and should be

very careful to design such devices.

REFERENCES

49

REFERENCES

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50

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Master’s Thesis in Electrical Engineering by Adnan Daud Khan & Muhammad...

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