Chapter 1 An Overview: Photonic Band Gap...

Chapter 1 An Overview:

Photonic Band Gap Materials

Chapter 1

An Overview: Photonic Band Gap Materials

1.1 General Introduction Scientific and technological advances have paved the way for the

growth of our species and improvement in the overall standard of living,

throughout human history. This process has taken shape due to manipulation

and understanding of the environment around us. In the last century,

constituent parts of an atom have been discovered. By the turn of the twenty

first century, machines and devices based on sub-micron technologies have

been invented, making millions of computations per second possible. One

common theme that all these advances have is the electronic properties of

matter which make up a wealth of physical interactions that dominate the

modern world from household appliances to industries.

In 1864, James Clark Maxwell summarized the theory of

electromagnetic waves by a set of mathematical equations and explained

their nature of propagation when they travel through a medium. He

established close relationship between optics and electromagnetism [1]. In

1887, Lord Rayleigh investigated a purely periodic system extending to

infinity in one direction for the first time and found that such type of

structure exhibits a range of wavelengths that are forbidden to propagate

inside this periodic arrangement [2]. In the 1930s, condensed matter

physicists realized that electrons are confined to some intervals of permitted

energies surrounded by forbidden energy bands due to the periodic potential

of the atomic lattice. One of the main tasks in developing the new science of

Chapter 1: An Overview: Photonic Band Gap Materials

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electronics was to control these forbidden bands; and we have seen a lot of

development in the field of electronics, computer technology and

telecommunication.

In 1946, Brillouin found the discontinuities in the relation between the

frequency ( ) and the wave vector (K) for any wave propagating in a

periodic medium [3]. These discontinuities are the frequency gaps in the

dispersion relation = f(K) for electrons in crystals. Later in 1958, for the

periodic dielectric materials, Slater analyzed, in detail, the conditions behind

the appearance of forbidden gaps [4]. In the forbidden gaps, the wave

vectors are purely imaginary in periodic dielectric materials. An imaginary

wave vector corresponds to damping of the wave in the crystal. Thus, the

electromagnetic waves having the energies within the gap could not be

transmitted through the bulk of such crystals.

For the past 50 years, semiconductor physics has revolutionized the

electronic industry and has played a vital role in almost all aspects of

modern technology due to the invention of the Si-based devices like

transistors, diodes, etc. Thus, it is evident that the success of electronics lies

in the semiconductor materials. Now-a-days, electronic devices based on

semiconductor technology are one of the most common objects around us.

And the demand for faster and smaller electronic devices is still on the rise.

Semiconductor crystals have a periodic arrangement of atoms occurring

naturally in them. All these developments are due to intrinsic property of

having narrow forbidden band gaps or the manipulation done to exhibit

narrow forbidden band gaps of the semiconductor materials.

Today, the global telecommunication market is growing with an

extraordinary speed and is driven largely by the explosion of the �Internet�

which has fuelled the increase in information capacity and data bandwidth


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required for telecommunication and data communication applications. It

demands for broadband communication network. Optical communication

systems have a very large bandwidth (of the order of terahertz), high speed

and low loss in comparison to the electronic systems. Numerous optical and

optoelectronic devices have provided solutions to some of the technological

problems encountered in the electronic, computing and information

revolutions. Since the invention of Laser in the mid 1960s, an explosion of

academic and industrial interests in optoelectronics already happened. But

the main drawback of the optical system is that there is no multipurpose

optical device analogous to the transistor in electronics. So, it is necessary to

develop new materials and concepts with increased optical functionality for

a variety of applications. In the past ten years, many researchers have

suggested that we may now be able to accomplish similar things with light.

In order to realize the more advanced optical elements needed for networks,

new approaches for the manipulation of photons will have to be developed.

This goal is achieved by a new class of materials called photonic crystals

which is an optical analogue of the electronic semiconductors.

1.2 Photonic Band Gap Materials Originally photonic crystals were introduced with the goal to control

the optical properties of materials. Indeed, the last century has seen our

control of the electrical properties of materials using semiconductors (to

tailor the conducting properties of certain materials). Photonic crystals offer

the same control for the electromagnetic properties of the materials. Using

the formal analogy between Schroedinger�s and Helmholtz�s equations, Eli

Yablonovitch [5] had the idea in 1987 to built artificial periodic structures


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manipulating the permittivity in order to inhibit totally the propagation of the

light. Thus, the concept of a photonic band gap (PBG) material was born. To

test this idea, he realized a prototype with a three-dimensional diamond hole

lattice in Plexiglas. With this material he demonstrated the capability of the

PBG material to control the propagation of electromagnetic wave in such

structure. Almost simultaneously, John et al [6] proposed the concept of

strong localization of photons in disordered dielectric superlattices. One of

the most important properties of the photonic band-gap materials is the

emergence of localized defect modes in the gap frequency region when a

disorder is introduced to their periodic dielectric structure [7, 8]. In addition

to the purely scientific interest in these strongly localized eigenstates of

photons, several applications to optical devices are expected. For example,

as was pointed out by Yablonovitch, the single-mode light emitting diode

that utilizes spontaneous emission through a localized defect mode in a

photonic band gap may have such properties as good temporal and spatial

coherence, high efficiency, low noise, and high modulation rate. Another

example is the waveguides composed of defects introduced into regular

photonic band-gap materials, for which quite a high transmittance for the

guided modes through sharp bends was theoretically predicted. This feature

originates from the nonexistence of the electromagnetic modes outside the

waveguides, and is a striking contrast to a large loss at the sharp bends

observed for ordinary optical waveguides. This feature may be quite useful

for optical microcircuits.

First photonic band gap materials were realized with dielectric

materials. However, different research groups have progressively introduced

more complicated structures. For instance, in the microwave domain,

metallo-dielectric material was often used. Metallo-dielectric PBG materials


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are constituted of a periodic arrangement of metallic parts (rods for instance)

either in air or embedded in a more complicated dielectric structure. They

have some properties very different from purely dielectric PBG crystals.

They have a gap down to very low frequencies. These materials have many

advantages in this frequency domain: easy fabrication, robustness,

conformability and low cost. More recently, controllable PBG materials

were proposed at microwave and optical frequencies [8, 9]. Metallo-

dielectric materials also allow the insertion of electronic devices in the core

of the material leading to controllable structures. The reader could also find

in the literature the electromagnetic gap (EMG) material in place of metallo-

dielectric structures. However, we do not used this terminology here to avoid

confusion.

Sometimes, metallo-dielectric photonic band gap materials, which are

for instance reserved to the centimeter and millimeter wavelengths, are

called electromagnetic gap (EMG) materials. But some of these structures

may be used at higher frequencies, in the infrared or submillimetric domain

for example. Also, purely dielectric structures may be used at lower

frequencies. Therefore the distinction between PBG and EMG materials is

not evident. Another common name is photonic crystals (PC). This name

may assign any structures that interact with the light. We use indifferently

PBG or PC in the following sections. Many applications have been proposed

for these materials, in optics or in the microwave domain. In optics, several

authors have proposed high-Q microcavities and low threshold lasers, novel

types of filters; low-loss bent waveguides, novel LEDs, optical fibers [10�

19].

In the microwave domain, numerous applications of metallo-dielectric

PBG have been investigated, such as reflectors and substrates for antennas,


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high impedance surfaces, compact uniplanar slow-wave lines; broad band

filters [20�30]. The following is devoted to an introduction to photonic

crystals and their emerging applications. The reader is referred to [19] for

other reviews on many aspects of PBG materials.

PC like structure can be observed in the nature in the skins and furs of

small creatures. For example, the Butterfly wing of the mitouragrynea

produces a greeny blue iridescent reflection depending on the angle from

which it is viewed (Figure 1.1). A photonic 3D square lattice structure was

found to be the origin of the effect in regions of the wings that exhibits these

properties. Some more examples of the photonic crystals in the Nature are

like the wings of some Coleopterans built by stacking periodic layers of

organic materials or like pearls in which organic and inorganic layers are

alternate. Some Mexican and Australian opal gemstones (minerals), whose

surfaces present periodic stacks of the silica particles, are the other visible

examples. Some other materials like colloids or polymers also present

spontaneously organized structures.

Figure 1.1: Natural Photonic crystals.


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1.3 Important Crystal Parameters and Classifications The Parameters on which the optical features of a photonic crystal

will depend are listed below-

� The type of symmetry of the structure- The position of the building blocks

of the PCs will set the symmetry of the lattice. Its example are the sc (simple

cubic), bcc, fcc, sh, hcp and diamond structures.

� Topology- It can be varied by interpenetrating the building blocks (network

topology) or isolating them (cermet topology). Economou and Sigalas have

published a general discussion about topologies in PBG theory.

� Lattice Parameters- It is the distance of separation between scattering

building blocks. The working range of wavelength of the PC will be

proportional to the lattice parameter.

� Filling fraction- It is the ratio between the volumes occupied by each

dielectric with respect to the total volume of the composite.

� Refractive index contrast ( ) - It is defined as the ratio between the

refractive index of the high dielectric constant material and the low dielectric

constant material.

� The shape of the scattering centers.

� Scalability.

�Dimensionality- PCs are categorized as one-dimensional (1-D), two-

dimensional (2-D) and three-dimensional (3-D) crystals according to the

dimensionality of the building stacks (Figure 1.2). 1-D PC consists of

alternate layers of the materials having low and high indices of refraction

and the dielectric constant is modulated along only one direction. 1-D

photonic crystal can be fabricated on a needed wavelength scale easily and

cheaply. One dimensional photonic crystals can be used as omnidirectional


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totally reflecting mirrors, frequency filters, microwave antenna substrates

and enclosure coatings of waveguide etc. Its applications depend on the

chosen geometry and frequency regions [31-33].

Figure 1.2: Schematic depictions of photonic crystals periodic in one, two,

and three dimensions, where the periodicity is in the material structure of the

crystals.

In the 2-D photonic crystals, the dielectric constant is periodic in one

plane and extends to infinity in the third direction. These include quadratic,

hexagonal and honeycomb types of lattices. The 2-D PCs are less difficult to

fabricate in comparison to the three-dimensional dielectric arrays. The

observation of some fundamental phenomenon, such as Anderson

localization of light, may be easier in 2D structures [34]. In a 2D dielectric

array, the two orthogonally polarized waves, one with its E-field polarized in

the 2D-plane (TE-mode) and the other with its E-field polarized

perpendicular to the 2D plane (TM-mode) have very different dispersion

[35]. Because of this a �complete photonic band gap� i.e. a frequency region

in which the propagation of EM wave is completely forbidden for all-


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directions of propagation and polarization, is less likely to form. Reason for

this is the band gap for individual polarization is unlikely to overlap. Many

researchers have concluded that the hole array structures are more likely to

generate a complete gap than the usual rod array structures [35, 36] in a 2-D

photonic crystals. Also, thin semiconductor layers are recognized as very

attractive candidates in order to achieve light molding in a planar photonic

integrated circuit. These structures are obtained by drilling a triangular array

of holes in the layered structures [37, 38]. In the recent years, the uses of 2D

PCs have been widely explored in order to improve the overall performance

of optoelectronic devices [38-43]. For example, 2D photonic crystal micro-

lasers already rival the best available micro cavity lasers both in size and

performance.

In the 3-D PBG structures, refractive index modulation is periodic

along all the three directions. These types of materials facilitate complete

localization of light and provide complete inhibition of spontaneous

emission of light from atoms, molecules and other excitations. Such

feedback effects have important consequences on laser action from a

collection of atoms. The 3D-Photonic crystals, such as the inverse opals,

exhibit forbidden frequency ranges over which the ordinary linear

propagation is forbidden irrespective of the direction of propagation and

polarization mode [44, 45].

The existence of these complete photonic band gaps allows the

complete control over the radiative dynamics of active materials embedded

in photonic crystals such as the complete suppression of spontaneous

emission for atomic transition frequencies deep in the PBG. In 3D-PBG

structures a large number of self-assembling periodic structures already

exist. These include colloidal systems and artificial opals [46, 47]. A face


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centered cubic lattice consisting of low dielectric inclusions in a connected

high dielectric network, called inverse structure, can exhibit small photonic

band gap [48]. The �wood pile� structure also represents a 3D-PBG structure

that is built by the layer-by-layer fabrication technique [49]. Recently,

Kosaka et al. [50, 51] demonstrated highly dispersive photonic

microstructures in a 3D- photonic crystal, which was termed as optical super

prism. Superprism effect allows wide-angle deflection of the light beam in a

photonic crystal by a slight change of the wavelength or incident angle. A

recent study of super prism effect is done by T. Baba and M. Nakamura [52].

1.4 Origin of Photonic Band Gaps Photonic crystals, like the familiar crystals of atoms, do not have

continuous symmetry; instead, they have discrete translational symmetry.

Thus, these crystals are not invariant under translations of any distance, but

only under distances that are multiple of a fixed step length. This basic step

length is known as the lattice constant �a�, and the basic step vector is called

primitive lattice vector �a�. Because of this symmetry, (r) = (r + R). By

repeating this translation, we can see that (r) = (r + R) for any R that is

integral multiple of �a�. The dielectric unit that is repeated over and over is

known as the unit cell.

The discrete periodicity in a certain direction leads to a dependence of

H for that direction that is simply the combination of plane waves,

modulated by a periodic function because of the periodic lattice:

H(r)= exp(ikr.r) uk(r) (1.1)

where, uk(r) is periodic in the real space lattice. This result is commonly

known as Bloch�s theorem and the form of above equation is known as


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Bloch state. The wave vectors kr that differ by integral multiples m of 2 /a

are not different from a physical point of view. In fact, all the modes with

wave vector of the form kr+m(2 /a), where m is an integer, form a

degenerate set and leave the state unchanged. Thus the mode frequencies

must also be periodic in kr i.e. (kr)= (kr+m(2 /a)). In fact, we only need to

consider kr to exist in the range aka r // . This region of non-

redundant values of kr is called the Brillouin zone. Substituting the Bloch

state into the Master Equation, we can get a reduced form of Master

equation

)()()()()(

1)(2

ruckruik

rik kk (1.2)

The above equation can be solved numerically for all k in the first

Brillouin zone, resulting in an infinite set of modes with discretely spaced

frequencies labeled with the band index n.

Figure 1.3: Energy dispersion relations for free electron (left) and for

electron in a 1D solid (right), and for a free photon (left) and a photon in a

photonic crystal (right).


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Photonic bands n(k) of the crystal: They are a family of continuous

functions, indexed in order of increasing frequency by the band number. The

information contained in these functions is called the band structure of the

photonic crystal. The optical properties of the crystals can be predicted by

studying the band structure of a crystal.

Figure 1.3 shows the parallelism between electrons in crystalline solids and

photons in photonic crystals. The energy dispersion relation for an electron

in vacuum is parabolic with no gaps. When the electron is under influence of

a periodic potential, gaps are found and electrons with energies therein have

localized (non-propagating) wave functions as opposed to electrons in

allowed bands that have extended (propagating) wave functions. Similarly, a

periodic dielectric medium will present frequency regions where

propagating photons are not allowed and will find it impossible to travel

through the crystal. One important difference between electrons and photons

rests on the different nature of their associated waves. Electrons are

associated with scalar waves, while photons are associated with vectorial

ones. This implies that polarization must be taken into account while dealing

with photons.

1.5 Calculation of Band Structure of the PBGs The calculations on photonic band gap (PBGs) materials are similar to

the calculation on atomic crystals. In case of an atomic crystal, the

Schrödinger equation is fundamental, in which the atomic crystal is

described by periodicity of the atomic potential. The periodic nature of the

lattice allows the application of the Floquet-Bloch theorem which states that

eigen function of the wave functions for a periodic potential are the product


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of a plane wave (eik.r) times a function (uk) with the periodicity of the crystal

lattice vector. This implies that for any k-vector in reciprocal space the

dispersion relation can always be shifted back to the first Brillouin zone by

adding or subtracting an integral multiple of reciprocal lattice vectors. In

general the band structure is only plotted along the characteristics path of the

irreducible part of the Brillouin zone, i. e. a line following all edges of the

irreducible part. All maxima and minima of the band structure lie on the

characteristics path. Hence, the existence of the frequency range of the

photonic band gap can be deduced from a plot of the band structure along

the characteristics path.

1.6 Theoretical Formalism In the photonic crystals, the electromagnetic wave interacts at the

interfaces of the building blocks. Maxwell�s equations can be used to predict

the photonic behavior of light propagating in the structure in terms of Bloch

functions, band structures and band gaps [53-55],

D (1.3)

0B (1.4)

tBE (1.5)

tDJH (1.6)

where H and E are the magnetic and electric fields, B and D are the

magnetic and electric flux density, J is the current density and is the

electric charge density. These equations can be simplified for the case of

electromagnetic wave propagation in photonic band structures. These


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structures are multilayer of different homogeneous dielectric materials.

There are no free charges or currents therefore, J = =0.

It is assumed that the materials behave linearly and isotropic with

respect to light propagation hence the electric field and electric flux density;

and magnetic field and magnetic flux density obey the following relations.

ErD )(0 and HrB )(0 (1.7)

where )(r and )(r are the electric permittivity and magnetic permeability

respectively. But for dielectric materials, 1)(r ; hence HB 0 .

Applying these conditions Maxwell�s equations can be written as

0),()( trEr (1.8)

0),( trH (1.9)

ttrHrtrE ),()(),( 0 (1.10)

ttrErtrH ),(

)(),( 0 (1.11)

The time dependence of magnetic field and electric field can be

separated from the spatial dependence by expansion into a set of

harmonically oscillating modes of single frequency, which can be written as tierEtrE )(),( (1.12)

tierHtrH )(),( (1.13)

where represents the angular frequency. Substituting equations (1.9) and

(1.10) in the above equations, we get

0)(),( 0 rHitrE (1.14)

0)()(),( rEritrH o (1.15)

Now taking curl of equation (1.11), on both sides, we get

)()],([ 0 rHitrE (1.16)


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Now using equation (1.12) for eliminating )(rH from equation (1.13),

we get

0)()()( 2

22 rEr

crE (1.17)

Equation (1.17) is an eigen value problem; by solving this equation

one can calculate the band structure, dispersion relation and the propagation

characteristics of the photonic band gap materials. Such calculations are

done numerically and the effect of the periodicity of the lattice is considered

by imposing the periodic boundary conditions.

1.7 Numerical Methods of Simulation of PBG Materials There are six main methods generally employed to study properties of

photonic band gap materials numerically:

(1) The Plane Wave Method [56],

(2) The Finite Difference Time Domain (FDTD) method [57],

(3) The Finite Element method [58],

(4) The Transfer Matrix Method (TMM) [59],

(5) A method based on a rigorous theory of scattering by a set of rods (for a

two-dimensional crystal), [60] or a set of spheres (for a three-dimensional

crystal) [61],

(6) The study of diffraction gratings [62].

All of these methods calculate with high efficiency and accuracy and

are in good agreement with experimental results. These methods are chosen

according to the nature of the problem to be tackled. Some of these methods

(methods (1) to (4)) can simulate any doped or non-doped crystals [56-59] as

they are highly flexible. Method (5) is limited to certain types of PCs which

are made up of parallel cylinders (for 2D photonic crystals) and spheres (for


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3D) [60, 61]. Some of these methods as (1), (4) and (6) can deal only with

infinite crystals [56, 59, 62] and method (5) can deal with finite-sized

structures [60, 61]. Finally, methods (1), (4) and (6) use a super-cell to study

the defect structures. On the contrary, methods (2), (3) and (5) can deal with

a finite structure having a single defect. In the following sections, we outline

briefly the main numerical methods used to study photonic crystal

properties.

The Plane Wave Expansion method is very easy to implement and

obtain the band structure when the direction is specified. The codes give all

the propagating/evanescent energies for that direction. A defect in the

infinite photonic crystal will be treated using a super-cell. Many results have

been obtained with this method [20, 63, 64]. The limitation of the method is

linked to the memory storage that depends on the number of plane waves

used for the expansion of the field, and this number escalates when the

photonic crystal diverges from a periodic structure. The calculation of

sophisticated defects is not possible by this method.

The FDTD method analyses the Maxwell�s equations in time domain

and the results are in good agreement with experimental measurements as

found in many works on photonic crystals [65-67]. Many works on photonic

crystals have been reported using this method. As for the Finite Element

method, electromagnetic modes of a defect can be calculated as the

transmission ratio of the material. To obtain the transmission spectrum of the

crystal, an electromagnetic pulse is sent on the material and the output signal

is recorded. A fast Fourier transform is applied to both incident and

transmitted signals and the transmission spectrum is calculated. The Finite

Difference Time Domain method allows the simulation of finite or infinite

crystals with inner or outer electromagnetic sources. In some cases, this


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method permits the simulation of an entire experimental setup with a

photonic crystal. Results of this experiment are then analyzed. This is the

most common technique to simulate a photonic crystal. The limitation of this

method is the size of the memory to calculate a large crystal and the lack of

an accurate electromagnetic model for some particular objects like thin wires

for example. Another advantage of this method is the attractive capability to

simulate nonlinear materials [57].

The Finite Element method is well established in electrodynamics and

has the great advantage to be implemented in very efficient commercial

software�s as MAFIA, HFSS etc. It can simulate infinite and finite doped or

non-doped crystals with inner or outer source.

The Transfer Matrix Method (TMM) is a well-described method [59].

The TMM involves writing the Maxwell�s equations in the k-space and

rewriting them on a mesh. It is capable of handling PBG materials of finite

thickness with layer by layer calculations. Structures with defects can be

dealt only by considering a super-cell. The band structures, reflectivity and

transmission coefficients can be found by this method easily. Many

researchers have used this method [68-70]. It has also been proved to be

very useful and accurate when comparisons with experimental structures are

undertaken [69, 70]. The limitations of this method are the memory storage

but also it is difficult to deal with geometry different from the cubic

geometry.

Many working groups implement the method based on the rigorous

scattering of light by a set of finite sized cylinders/spheres [60, 61]. The

main advantage of this method is that cylinders/spheres can be located

anywhere in the space. Accordingly, a periodic arrangement is just a

particular case and it is possible to deal with a single defect without the need


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of a super-cell. Also, it is very simple to change the geometry of the

structure, although, limitations are linked to the size of the memory when a

large number of cylinders have been implemented (about one hundred).

The use of diffraction gratings theory [62] allows the calculation of

reflection and transmission coefficients of a photonic crystal constituted by a

stack of a finite number of infinite grating layers. The method can deal only

with an infinitely long cavity as a defect for the structure. But this method

cannot simulate new PBG materials that are sophisticatedly doped and active

structures.

We have adopted the TMM method for photonic band gap structure

calculations and optical properties of one-dimensional photonic crystals are

studied.

1.7.1 Transfer Matrix Method for 1-D PBG Materials The wave behavior in one-dimensional periodic lattice can be

described by using the Transfer Matrix Method (TMM) techniques. This

method is largely based on interfaces of the two layers [53-55].

Figure 1.4: Schematic diagram of bi-layers unit cell of refractive indices n1

and n2 with thicknesses d1 and d2 respectively.


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Let us consider a periodic arrangement of multilayer film (Figure 1.4),

with refractive indices n1 and n2 and each having thicknesses d1 and d2

respectively.

The solution for the master equation (1.17) will be the superposition

of plane waves traveling to the right and to the left. Say, for the layer with

index n1, the right going and left going plane waves have amplitudes A1 and

B1 respectively and the right going and left going plane waves have

amplitudes C1 and D1 for layer with index n2 respectively in the unit cell

considered. Hence for layer with index n1 the solution of equation (1.14) is, xikxik xx eBeAxE 11

11)( (1.18) )(

1)(

11212)( dxikdxik xx eDeCxE (1.19)

for the layer with index n2. The wave numbers k1x and k2x are defined as,

jjjx nc

k cos , j=1, 2 (1.20)

where 1 and 2 are the ray angles in the two mediums respectively.

At the interface between layers (x = d1), the solution and its derivative

should be continuous. This gives a relation between plane wave amplitudes:

1

112

1

1

BA

MDC

(1.21)

with, 1111

1111

2

1

2

1

2

1

2

1

12

1211

21

1211

21

dik

x

xdik

x

x

dik

x

xdik

x

x

xx

xx

ekke

kk

ekke

kk

M (1.22)

and, also at x = d, the continuity of the plane waves at the interface between

layers with indices n2 and n1 and its derivative gives -


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1

121

2

2

DC

MBA

(1.23)

where the matrix M21 is the same as (1.21) but with interchanging the

indices.

From the two matrix equations (1.21) and (1.23), we have,

1

11221

2

2

BA

MMBA

(1.24)

1

1,

2

2

BA

MBA

ji (1.25)

where, Mi,j = M21M12.

The matrix element of the matrix Mi,j are given by

)sin(12

)cos( 22221,11 dkidkeM xx

bik x (1.26)

)sin(12 222,1

11 dkieM xdik x (1.27)

2,11,2 MM and 1,12,2 MM (1.28)

x

x

kk

2

1 for TE mode and 2

12

221

..nknk

x

x for TM mode (1.29)

The matrix Mi,j is called as the transfer matrix of one unit of the

periodic lattice. The matrix Mi,j depends on the frequency , and it is

unimodular (it is a square matrix with determinant equal to unity). Hence,

for each , the matrix Mi,j defines a unique mapping for amplitudes of the

plane waves in layer n1 into the amplitude of the next layer with index n2.

For an infinite lattice extending on the whole x-axis, the solution of

the equation (1.17) can be written in terms of Bloch waves [53-55, 71, 72].


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xiKK exUKxE )().(),( (1.30)

where UK(x) is a complex valued periodic function with the period of the

lattice (d=d1+d2 ), UK(x) = UK(x+d). The parameter K( ) is called the Bloch

wave number for a periodic lattice with indices n1 and n2.

The expression for K( ) is as follows,

)(21cos1)( ,

1jiMTr

dK (1.31)

with Mi,j given in (1.25).

After simplifying (1.31), one can obtain as,

)sin()sin(121)cos()cos(cos1)( 22112211

1 dkdkdkdkd

K xxxx (1.32)

jjjx nc

k cos , with j=1,2. (1.33)

The equation (1.32) is known as the dispersion relation of the periodic

lattice with refractive indices n1 and n2 and thicknesses d1 and d2

respectively.

The behavior of Bloch waves is characterized by the dispersion

relation. The behavior of Bloch wave can be divided into three cases-

1. For real K( ), which lies in the first Brilluion zone [0, /d], E(x, K) is a

periodic and traveling wave function. In this case, it is said that is outside

the band gap.

2. For imaginary K( ), defined by K( ) = /d + i ( ) , E(x, K) is a standing

wave function, a product of two periodic functions with an exponentially

increasing and a decreasing function, depending on the sign of ( ) . In this

case, is inside the band gap.

3. For K( ) = /d, E(x, K) is a periodic function of period 2nd with special

properties that it is a d-shift skew symmetric, E(x + d, K) = -E(x, K).


- 22 -

1.7.2 Transmittance and Reflectance In this section, a brief discussion on how transmittance and

reflectance of periodic lattices can be calculated for a multilayerd structure

will be made. The transmission properties of photonic crystals show that

PCs resemble a typical device which functions as a filter or mirror for some

interval of wavelength or frequency that lies in the band gap [55, 73-77]. It

is well known from the theories in optics that a ray of light incidence on the

boundary of two materials of differing index of refractions will be partially

reflected and partially transmitted. However, in recent years, such

manufacturing techniques have been developed that led to the production of

devices, which can take full advantages of these effects. Photonic crystals

having finite thickness and made up of appropriate materials can be applied

in conjunction with optical surfaces to eliminate unwanted reflection.

On the other hand, a multiple layered structure can be used as anti-

reflection coating at a desired wavelength in application such as non-

absorbing beam splitters and dichroic mirror, transmitting the desired

wavelength and reflecting others. Multilayered narrow band pass filters can

be made to transmit light over a specific spectral range, and find a multitude

of practical applications [78, 79]. A periodic multilayered structure made up

of alternating layers of two materials, one of then fairly high index of

refraction than the other, is called one-dimensional photonic crystals. In such

structures, we consider a periodic multilayer film with refractive index

n1=( 1)1/2 and n2=( 2)1/2 with thickness d1 and d2 respectively and taken N

unit of these cells and stack than as depicted in figure 1.5.


- 23 -

Figure 1.5: Schematic diagram of bilayers unit cell of dielectric constant 1

and 2 with thickness d1 and d2 respectively.

Figure 1.6: The dispersion and transmittance of one dimensional photonic

crystal with n1(= 1) = 1.25, n2(= 2) = 2.5 and n1d1 n2d2 /4.


- 24 -

When light is incident on this type of a structure described above, the

transmission spectrum exhibits frequency region where energy is freely

transmitted or prohibited. This can be shown by the gaps in the plot of

dispersion and transmittance spectrum. A simplified plot of dispersion and

transmittance versus normalized frequency for such structure is shown in

figure 1.6. Because there are frequency ranges where the incident wave is

not transmitted, these structures have become known as photonic band gap

or photonic band gap materials.

The reflectance/transmittance properties of the structures can be tuned

by designing the layers at thickness associated with the frequency of light

with wavelength ( 0=2 c/ 0) desired to be reflected or transmitted. The band

gap is centered at the reference wavelength 0, and its width is a sensitive

function of the number of periods, the values of n1 and n2, and their relative

difference 21 nnn , sometimes called the index modulation depth. In

order to calculate the reflectance and transmittance coefficients, we have

taken periodic structure with left and right exterior (background), there is a

homogenous medium with index n0=1 for air. Light is incidence from the

left exterior, say with unit amplitude and frequency . The light will interact

within this structure, resulting into a right going plane wave with amplitude

(t) in the right exterior, and a reflected plane wave with amplitude (r) to the

left. Using the transfer matrix method, it can be shown easily that there is

relationship between plane wave amplitudes in the left and right of any

interface. The matrix can be related to the transmission coefficient (t) and

reflection coefficient (r) [55]. The reflection and transmission can be related

easily between the plane wave amplifications.

r

mt 10

(1.34)


- 25 -

and 2221

1211

mmmm

m with 211111 NN UUMm , 12121 NUMm ,

211212 NN UUMm , 212222 NN UUMm and ]).(sin[

]).().1sin[(dK

dKNU N

where Mi,j are same as section (1.7.1) and transmission and reflection

coefficients are given by

22

211211

.m

mmmt (1.35)

22

21

mmr (1.36)

The associated transmittance (T) and reflectance (R) are obtained by

taking the absolute square of t and r respectively

2tT and 2rR (1.37)

1.7.2.1 Omnidirectional Reflection A complete photonic band gap requires that there be no states in the

given frequency range for propagation in any direction in the structure [33,

80, 81]. For frequencies within the complete bandgap, the structure may

exhibit total reflectivity for all incident angles and for all polarizations. This

phenomenon is known as omnidirectional reflection (ODR). As an example,

the total ODR range for n1=1.5, n2=3.7 and d1=0.7d, d2=0.3d is shown in

Figure 1.7. For both TE and TM polarizations, the ODR bands coincide for

the angle of incidence 0° [80].


- 26 -

Figure 1.7: Total ODR range for n1=1.5, n2=3.7 and d1=0.7d, d2=0.3d.

1.7.3 Group Velocity and Effective Index of Refraction

in 1-D PBG Materials The group velocity and effective indexes of refractions (group &

phase index) are calculated inside the photonic crystals using the dispersion

relation given above. The calculation of the group velocity and effective

refractive indices are an essential task for the understanding of their optical

properties. The group velocity of the radiation modes has very important

role in light propagation and optical response in photonic crystals. The group

velocity is defined as 1)(

ddKvg , where K( ) is the dispersion relation of

one-dimensional photonic crystal [31, 82]. For electromagnetic pulse

propagation in a dispersive media the group velocity has an important role,


- 27 -

and it will represented as ddnn

cd

dKvg /)()( 1

, for the

propagation of an electromagnetic pulse in a linear dispersive but non-

absorbing medium. However, in regions of strong anomalous dispersion, the

group velocity can exceed speed of light in medium space or even become

negative. The common belief is that the meanings of group velocity break

down and the behavior of the pulse becomes much complicated [31, 82, 83].

The effective phase and group index is taken using p

eff vcpn )( and

geff v

cgn )( . For example, it can be seen that the band structure in the 1D

photonic crystal with parameters n1=1.5 and n2=2.5 and n1d1 = n2d2, a series

of band gaps occur, and the lowest band gap is at normalized frequency

around =0.27. From the dispersion curves, the phase velocity ()(K

v p )

and group velocity ()(dK

dv p ) both, can be obtained, and the results for the

two lowest frequency bands are presented in Figure 1.8. At frequencies far

away from the band edge, both the group velocity and phase velocity are

nearly constant. Near the band edge, vp exhibits a slight increase with

frequency, while vg shows abnormal behavior and significantly slows down.

In practice, the phase velocity (vp) is often expressed in terms of the

refractive index of the material vp=c/neff(p); here neff(p) is used to indicate

that this is an effective phase index for the PC and c is the vacuum light

velocity. As shown in Fig. 1.8(a), at frequencies, far away from the band

edge, neff(p) is only weakly frequency dependent. When approaching the

band edge, a large change of neff(p) with frequency (frequency dispersion)

occurs [82].


- 28 -

Figure 1.8 The phase velocity vp and effective refractive index (neff(p)) and

the group velocity vg and effective refractive index (neff(g)) derived from the

dispersion curves obtained n1=1.5, n2=2.5 and n1d1=n2d2.

Inside the band gap, neff(p) is actually complex and also exhibits

abnormal dispersion. Besides neff(p) for the refractive index of the phase

velocity in a PC, the refractive index of the group velocity, neff(g), was

introduced by Sakoda to describe the ratio of c/vg [31]. For the 1D PC,

neff(g) thus obtained is also presented in Fig. 1.8(b). It is apparent that only

for the lowest branch of the dispersion curve and at low frequencies (long

wavelength limit), neff(p) and neff(g) are the same. Near the band edges neff(g)

exhibits sharp increase and is much larger than neff(p), reflecting the marked

slowing down of vg. Also, Ojha et al. have also studied the group velocity

and effective group index using this concept. They have found a remarkable


- 29 -

results that the ultra-high refraction for the Yablonovite structure for high

refractive index contrast, larger than 2 [84].

1.7.3.1 Superluminal Propagation in PBG Structures Superluminal (also Faster-than-light or FTL) communications refer to

the propagation of information or matter faster than the speed of light [83].

Under the special theory of relativity, a particle (that has mass) with

subluminal velocity needs infinite energy to accelerate to the speed of light,

although special relativity does not forbid the existence of particles that

travel faster than light at all times. The group velocity is often thought of as

the velocity at which energy or information is conveyed along a wave. In

most cases, this is accurate and the group velocity can be thought of as the

signal velocity of the waveform. However, if the wave is travelling through

an absorptive medium, this does not always hold good. Since the 1980s,

various experiments have verified that it is possible for the group velocity of

laser light pulses sent through specially prepared materials, significantly to

exceed the speed of light in vacuum. However, superluminal communication

is not possible in this case, since the signal velocity remains less than the

speed of light. It is also possible to reduce the group velocity to zero,

stopping the pulse, or have negative group velocity, making the pulse appear

to propagate backwards. However, in all these cases, photons continue to

propagate at the expected speed of light in the medium.

Anomalous dispersion happens in areas of rapid spectral variation

with respect to the refractive index. Therefore, negative values of the group

velocity will occur in these areas. Anomalous dispersion plays a

fundamental role in achieving backward propagation and superluminal light.


- 30 -

The propagation of electromagnetic radiation in dispersive media was

extensively studied by Sommerfeld [85] and Brillouin [86, 87]. They

observed the amazing results stating that the group velocity in the region of

anomalous dispersion close to absorption line can exceed the speed of light

in vacuum and it becomes very large (infinite) at particular frequency, it can

attain negative values too. Though relativistic causality is not violated for

wave propagation in a Lorentzian medium, Brillouin considered

superluminal or negative group velocity as mathematical achievement not

physical reality. Garrett and McCumber [88] considered the propagation of

Gaussian pulse and concluded that superluminal or negative group velocity

could be obtained without significant distortion in pulse shape. The velocity

of the pulse which propagates at a velocity greater than the velocity of light

in vacuum does not violate special theory of relativity or causality relations.

According to Crisp [89], the effect is attributed to a pulse reshaping and

despite attenuation the shape and width of the pulse may remain intact even

after it emerged from the material.

For the first time Chu and Wong [90] measured the pulse velocity in a

sample of GaP:N. The pulse was seen to propagate through the material with

little distortion in shape, and with an envelope velocity given by the group

velocity even when the group velocity exceeds 3.0 108 meter/sec., equals

, or becomes negative confirming the predictions of Garret and

McCumber [88]. Based on the Kramers-Kroning relation, Bolda and Chiao

[91] proved general theorems stating that for any dispersive medium,

superluminal, infinite or negative group velocities must exist at some

frequency, and that at frequency at which the attenuation (or gain) is the

maximum, the group velocity must be abnormal.


- 31 -

1.8 Yablonovite Structure This structure was given by Yablonovitch [20, 74, 92] in which 85%

of lattice structure is taken as index n1 = 1 for air and 15% of the lattice

structure is taken as index n2 = 1.5 for glass (SiO2) or semiconductor

materials (GaAs) etc. Such a structure shows complete photonic band gap in

three-dimensional structure. Same structures parameters are taken to design

tunable band pass filter using one-dimensional nano-photonic structures is

done by Ojha et al [78, 79]. It is possible to get desired ranges of the

electromagnetic spectrum filtered with such structure by changing the

incidence angle of light and/or changing the value of the lattice parameters.

1.9 Quarter Wave Stack Structure A specific case of a periodic structure, called a quarter wave

structures, is a case where for each layer of films, the optical path length is

equal to the quarter of the wavelength [55, 77]. Taking periodic structure

with indices n1=1.25 and n2=2.5 the corresponding thickness of each layer

are d1= /4n1= /5 and d2= /4n2= /10. Performing the reflection for this

structure one can calculate the power of the reflected light on the periodic

lattice for any frequency . The power reflected is given by a quantity called

the reflectance defined as 2rR where22

21

mmr . As an example, the

reflectance curve as a function of frequency has been illustrated in Figure

1.9 for the parameters given above. There is a region of 100% reflectance.

The region is called band gap shown in Figure 1.9 by shaded region.


- 32 -

Figure 1.9: Reflectance versus normalized frequency of one-dimensional photonic crystal with n1=1.25, n2=2.5 and n1d1=n2d2 /4.

1.10 Fabrication of Photonic Crystals Several approaches have been followed to fabricate photonic crystals

according to the needed wavelength scale. Crystal periodicity varies from

several hundred nanometers, if needed for application in the visible regime,

to a few microns for those operating in the near infrared (NIR). Over the

years, the nanofabrication problem has proven to be a main research

direction for many research groups. Three dimensional photonic crystals

have been fabricated by at least four methods: Self assembled colloidal

crystal, GaAs based three axis dry etched crystal, layer by layer lithography

and wood-pile method [93].

By means of self assembly approach, large 3D colloidal crystals can

be grown but it is difficult to control crystallization process in such a way as

to make structures with different lattice symmetries. This process leads to

the incorporation of random defects in the crystal. In addition, colloids do

not have a high enough index contrast to obtain a complete photonic band

gap. The etching technique requires the fabrication lithography masks with

feature size less than a 100 nm. The mask is then used in an anisotropic


- 33 -

etching process in high index contrast materials. Most masks are fabricated

by electron beam lithography because it provides a high level of control over

the structure. A deep UV-optical lithography can also be used for this

purpose. This technique is most suited for the 2D structures. A smaller

minimum feature size can be obtained by this process which enables the

fabrication of the structures for visible and near infrared wavelengths but the

major challenge is the optimization of the processing conditions to fulfill the

requirements on the aspect ratio. Galli et al. [94] have studied 2D GaAs PCs

experimentally and have fabricated their structure by X-ray lithography

followed by the radiative ion-etching. 3D photonic crystals present a greater

fabrication challenge.

Yablonovitch demonstrated experimentally the existence of a band

gap in microwave frequencies using Yablonovite structure [8, 95]. But it

could not be readily scaled to optical wave lengths. The wood-pile structure

possesses a band gap near the optical telecommunication wavelength of 1.55

micrometer [93]. Recently, Toader and John proposed a square tetragonal

spiral structure having a large band gap at 1.55 micrometer [96]. Noda and

co-workers used a wafer fusion technique [97]. Lin et al. also developed a

five step process to fabricate a 3D photonic crystal [98]. Blanco et al.

fabricated the first inverse opal with a RI contrast high enough to show a

complete PBG [45]. The first structure made of touching air spheres with an

fcc symmetry was obtained by Velev et al. [99] in 1997.

Other fabrication methods are continuously developed providing

interesting results. These are Block-copolymers self assembly, focused

ionbeam milling, glancing angle deposition and nano-robotic manipulations.


- 34 -

1D and 2D photonic crystals are easy to fabricate in comparison to 3D

photonic crystals and may lead to the applications and devices that do not

require complete inhibition of spontaneous emission.

1.11 Applications of the PBG Materials Photonic Crystals promise to provide us with a range of exciting

applications:

(i) Photonic Waveguides: PCs can be used in the construction of waveguides

with very low absorption and/or loss over much longer distances than

conventional waveguides. A standard photonic structure with the required

band gap can be constructed. Light is confined within the waveguide. Also,

it has the ability to guide the light around sharp corners that is not possible

through the conventional methods. This effect can be used to form

waveguide splitters that can split a beam of light with the resultant beams

being transmitted in opposite directions to each other [100]. Photonic effects

are also used for guidance in the optical fibers.

(ii) Perfect Reflectors: A 3D photonic crystal can behave as an

omnidirectional reflector with little or no loss. Omnidirectional mirror can

be used as the walls of laser cavities. Metallic mirrors are used for the

frequencies in the optical regime. 1-D PC is easier to fabricate and it can

also be used as the omnidirectional reflector in the optical region [101].

(iii) Light Emitting Diodes: Photonic crystals can produce new high

efficiency light sources [102]. By using a photonic crystal as the active

material in the LED, one can forbid all modes of photons except those which

would normally escape the crystal. Since spontaneous emission in the other

modes is forbidden, so all the energy will then go into those modes which


- 35 -

can escape. Such type of LEDs can take advantage of the high internal

quantum efficiency.

(iv) Photonic Crystal Lasers: PCs can be used to produce lasers with an

extremely low lasing threshold [103, 104]. Photonic Crystals have the

property of suppressing spontaneous emission inside the band gap. It is

forbidden to emit photons with these energies for the atoms in the crystal. A

defect can produce a frequency inside the band gap at which photons can

propagate with desired directionality. So, the lasing action will occur

without any loss as the unwanted spontaneous emission is suppressed.

(v) Photonic Integrated circuits: Much research is going on in this field and

it shall be some time before integrated photonic circuitry is produced. After

that PCs are likely to play a central role. This is why photonic crystals are

considered as the new-age crystals that should lead to the entirely optical

computer.

(vi) Nonlinear effects: Use of the materials with non-linear properties for

construction of photonic crystal lattices open new possibilities for molding

the flow of light. In this case the dielectric constant is additionally depending

on intensity of incident electromagnetic radiation and any non-linear optics

phenomenon can appear.

(vii) Other Applications: Photonic crystals without a complete PBG can be

designed to obtain super collimators and super lenses [105, 106]. PCs can

also be used as antenna substrates, resonant cavities and filters at microwave

frequencies. These crystals can be used to design micro scale light circuits

[107], multiplexers or demultiplexers based on inhibition of spontaneous

emission [108], super-prism phenomenon [51, 109] etc. Studies of plasmon

frequencies occurring for metal photonic crystals have also shown that the

plasmon frequency can be controlled in the microwave region. Many


- 36 -

developments are concerned with the direct control of the electromagnetic

energy and its transmission: mirrors, electromagnetic windows, radiation

pattern control etc [24, 110]. Other applications include duplexers [111] and

controllable PBG materials. Recent works have investigated the capability to

fabricate and to experimentally test these materials. Industrial applications of

these crystals concern mainly aerospace and telecom domains and are under

development.

1.12 A Brief History of PBG Materials Researchers all around the world have shown much interest in the

field of photonic crystals since they were proposed in 1987. This can be seen

by the spectacular exponential growth of the published work and, at present,

there are no signs of saturation in this field. Here, I am presenting a brief

summary of the most important works related to photonic crystals.

Upto 1987:

The interaction of light with ordered dielectric structures had already

been observed and studied in the optical regime before terms such as

photonic band gap or photonic crystal were invented. The results of

structural analysis and light diffraction experiments of Colloidal particles

with diameters close to the optical wavelength (polystyrene micro spheres)

were attributed to Bragg reflection of visible light in the first half of the 20th

century [112]. Ohtaka [113] published a dynamical theory of the diffraction

of visible and ultraviolet light in 1979 in which he analyzed the interaction

of light with a dielectric system composed of identical spheres ordered in a

three-dimensional lattice. He used the tools widely used in energy band


- 37 -

calculations in semiconductors. Ohtaka took into account the full vector

character of photons and did not use a scalar approximation. He developed

his theory borrowing many aspects from semiconductors but he did not state

the possibility of having photonic band gaps and their potential applications.

For this reason, his work has remained unknown for many years. Besides,

the systems he modeled in that paper have also had a tremendous importance

as photonic crystals.

From 1987 to 1994:

On May 18th 1987, two independent papers appeared in the same

issue of the journal �Physical Review Letters�. The first one, published by

Yablonovitch [5], dealt with the possibility of inhibiting spontaneous

emission of electromagnetic radiation using a three dimensionally periodic

structure. This lattice has a region of forbidden energy states for photons

showing photonic band gap. Secondly, Sajeev John [6] discussed the strong

Anderson localization [114] of photons in disordered dielectric superlattices.

He suggested that defects present within a lattice should trap EM radiations

where certain energy states were forbidden for photons. These two works are

considered as the origin of the �photonic crystals or photonic band gap

materials�.

In 1989, John published another work in which he proposed that an

fcc structure could show a complete PBG between the second and third band

[115]. 1990 was a very critical and exciting year for photonic crystals. At the

beginning of this year, Satpathy et al. [116] and Leung and Liu [117]

published implementation of the plane wave method with the scalar

approximation (PWM) to photonic band calculations independently. This

time theoretical calculations and experimental data showed excellent


- 38 -

agreement. These events lead the editor of the well known journal �Nature�

to assure that “Photonic Crystals bite the dust” [118]. Ho et al. [119]

demonstrated that although fcc lattices with spherical atoms did not show the

�missing� gap, but a diamond structure can do it. Later, Yablonobitch et al.

submitted another work presenting a structure based on an fcc lattice with

non-spherical atoms and which presented a complete PBG, this structure was

called �Yablonovite� [8]. Sözüer et al [48] improved the plane wave method

to show the behavior of higher energy bands in 1992. They showed that a

complete PBG was formed for an fcc lattice of air holes in a semiconductor

between the 8th and 9th band. In 1994, a newly proposed woodpile or layer-

by-layer structures following the diamond symmetry presented a cPBG

between the 2nd and 3rd bands. At that time, it could be constructed in the

optical regime by means of photolithographic techniques.

From 1995 to 1999:

Yablonovite structure had been fabricated in the microwave regime at

the end of 1994. The band structures and optical properties had been

presented. At the fabrication part, two groups at Sandia Labs (USA) and

Kyoto University (Japan), independently presented four-layer crystals based

on the woodpile or layer-by-layer structures at the end of 1998. These

crystals showed the band gap effect at mid-infrared wavelengths [98, 120].

Artificial opals were a method that all research laboratories could afford and

soon attracted the interest of many other groups. In 1997, Velev et al. [99]

succeeded to obtain the first inverse structure (inverse opal). In 1996, Lin et

al. [121] observed that photons were strongly dispersed in 2D crystals when

their frequency was close to the band gap edges. Kosaka et al [51] showed

experimental evidences of novel anomalous dispersion phenomena


- 39 -

(including negative refraction) explained it on the basis of dispersion

surfaces and group velocity instead.

From 1999 to till date:

In 1999, Fleming and Lin [122] presented the first photonic crystal

working in the NIR. One year later, Noda et al [93] fabricated an eight layer

crystal by the wafer fusion method. In 1999, the first artificial opals with the

appropriated periodicity were obtained and an inverse opal of silicon was

presented by Blanco et al. [44] in 2000. The work on photonic crystal is now

shifting towards negative index metamaterials. In 1999, Pendry et al [123]

showed that how a negative material could be created using split ring

resonators (SRR). A boost to the field came when negative refraction was

experimentally verified by Shebly et al. [124] in their composite material

which was made of wire array and SRR. Furthermore J. B. Pendry in 2000

[125] proposed that a negative index medium could be used to make a

perfect lens. Negative refraction in photonic crystals has also been

demonstrated [126]. A new type of photonic gap obtained by stacking

alternating layers of ordinary (positive-n) and negative-n materials is

proposed in 2003 by J. Li et al. [73]. This type of gap, which is invariant

with respect to a (length) scale change and insensitive to randomness, arises

when the volume averaged effective refractive index (nav) equals zero.

Panoiu et al. (2006) [127] demonstrated that photonic superlattices

consisting of a periodic distribution of layers of materials with positive index

of refraction and photonic crystal slabs that, at the operating frequency, have

negative effective index of refraction present a photonic gap. Ricci et al.

[128] demonstrated experimentally the properties of low loss

superconducting materials and Peminov et al. [129] realized negative


- 40 -

refraction in ferromagnetic superconductor supperlattices at millimeter

waves experimentally.

In last three years some researchers have studied the optical properties

of graded dielectric multilayered structures [130, 131] and photonic crystals

with single negative index materials [132, 133]. Recently Ojha et al. [134,

135] calculated large forbidden bands in one dimensional exponentially

graded structure and also presented the design of broadband optical reflector

in simple graded structure. X. Xu et al [136] obtained complete photonic

band gaps for all polarizations in the structure made of a conventional

dielectric material and a negative-permittivity and a negative-permeability

meta-material. This result has revealed the origin of the complete PBGs

which lies in the existence of surface waves for all polarizations.

Recently for oblique propagation, the dispersion relation and

associated electric fields of one-dimensional photonic crystals composed of

alternating layers of right-handed and left-handed materials (RHM and

LHM) have been investigated. The dielectric permittivity and magnetic

permeability are constant in the RHM, whereas both parameters of the

plasma frequencies in the LHM are assumed to perform the calculation of

the dispersion curves and associated electric fields [137].

1.13 Outline of the Thesis In this thesis, the study of electromagnetic wave propagation through

different types of one-dimensional photonic structures is presented.

Chapter 1 serves as an introduction to the photonic crystals outlining

the key pieces of work which first created an interest in this area. We look at


- 41 -

the progress which has been made so for in this emerging field of research,

both experimentally and theoretically.

In chapter 2, the design of optical filter based on photonic band gap

material in the ultraviolet region of EM spectrum has been suggested. To

calculate the characteristics equation of the structure we have used the

analytical method for solving the boundary value problem. From the

analysis of the dispersion relation of the structure, it is found that the

structure behaves like a filter. Also, we have suggested by cascading such

filters one can achieve a composite structure that acts as a monochromator.

Chapter 3 deals with the optical properties of one dimensional

dielectric-plasma photonic band gap materials. There are two structures

considered for numerical computation. First we choose SiO2 and in secondly

TiO2 as the dielectric material layer. In this chapter, we have studied the

photonic band structure, reflectivity, group velocity and the effective group

index for both the structures. Also, we compared the results of both

structures. The band structure of the structures is obtained by solving a

Maxwell�s wave equation using transfer matrix method.

Chapter 4 deals with the frequency bands of negative refraction in

finite one dimensional photonic band gap material choosing a periodic

multilayer structure of Na3AlF6/Ge system. In this chapter, we have studied

the photonic band structure and group velocity and then frequency bands of

negative refraction are obtained. We have also studied the optical properties

like transmittance and phase velocity. It is interesting to note that it is

possible to enlarge the omnidirectional reflection by cascading three such

structures.

Chapter 5 is devoted to the enlargement of omnidirectional reflection

range by using photonic band gap materials. The condition for obtaining


- 42 -

large ODR range is described. Two types of structures are considered; the

first one consists of photonic quantum well structure and the second consists

of gradual stacked 1D PC. It is observed that the omnidirectional reflection

range increases in both the cases.


- 43 -

References: [1] J. C. Maxwell, Philos.Tran. Royel Society of London 155, 159 (1865).

[2] L. Rayleigh, Philos. Mag. 24, 145 (1887); L. Rayleigh, Proc. Royel

Society London 93, 556 (1917).

[3] L. Brilloun, �Wave propagation in periodic structure�, (McGraw-Hill,

Chapter 4, 1946).

[4] J. C. Slater, Rev. Mod. Phys. 30, 197 (1958).

[5] E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).

[6] S. John, Phys. Rev. Lett. 58, 2486 (1987).

[7] J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals,

Molding the Flow of Light, Princeton University Press, 1995.

[8] E. Yablonovitch, Phys. Rev. Lett. 67(17), 2295 (1991).

[9] A. de Lustrac, F. Gadot, S. Cabaret, J. M. Lourtioz, T. Brillat, A.

Priou and E. Akmansoy, Appl. Phys. Lett. 75(11), 1625 (1999).

[10] S. W. Leonard, J. P. Mondia, H. M. van Driel, O. Toader, S. John, K.

Busch, A. Birner, U. G¨osele and V. Lehmann, Phys. Rev. B 61,

R2389 (2000).

[11] O. Painter et al., Science 284, 1819 (1999).

[12] P. R. Villeneuve et al., Appl. Phys. Lett. 67(2), 167 (1995).

[13] D. Labilloy et al., Phys. Rev. Lett. 79(21), 4147 (1997).

[14] R. D. Meade et al., J. Appl. Phys. 75(9), 4753 (1994).

[15] A. Mekis, S. Fan and J. D. Joannopoulos, Phys. Rev. B 58(8), 4809

(1998).

[16] M. Boroditsky et al., 1999 IEEE LEOS Annual Meeting Conference

Proceeding, LEOS’99, 238 (1999).

[17] J. C. Knight, T. A. Birks, P. S. J. Russell and D. M. Atkin, Optics

Letters 21(19), 1547 (1996).


[18] A. Bjarklev et al., J. Opt. A 2(6), 584 (2000).

[19] Review articles in the special issue on Photonic Crystals of the J. Opt.

Soc. Am. B 10, 1993; J. Mod. Opt. 41, 1994; Photonic Band Gap

Materials, C. M. Soukoulis (Ed.), Kluwer Academic Publishers, 1996

and the special issue of 1999 of the J. Light. Technol.

[20] B. Temelkuran and E. Ozbay, Appl. Phys. Lett. 74, 486 (1999).

[21] E. R. Brown, C. D. Parker and E. Yablonovitch, J. Opt. Soc.Am. B 10,

404 (1993).

[22] E. R. Brown and O. B. McMahon, Appl. Phys. Lett. 68, 1300 (1996).

[23] M. M. Sigalas et al., Micro. Opt. Tech. Lett. 15, 153 (1997).

[24] B. Temelkuran et al., J. Appl. Phys. 87, 603 (2000).

[25] M. Thevenot, C. Cheype, A. Reineix and B. Jecko, IEEE: Trans.

Microwave Theory Tech. 47, 2115 (1999).

[26] D. Sievenpiper, L. Zhang, R. Broas, N. Alexopolous and E.

Yablonovitch, IEEE: Transactions on Microwave Theory and

Techniques 47, 2059 (1999).

[27] F. R. Yang, Y. Qian, R. Coccioli and T. Itoh, IEEE: Microwave and

Guided Wave Lett. 8, 372 (1998).

[28] F. R. Yang, K. P. Ma, Y. Qian and T. Itoh, IEEE: Trans. Microwave

Theory Tech. 47(8), 1509 (1999).

[29] K. P. Ma, K. Hirose, F. R. Yang, Y. Qian and T. Itoh, Electronic Lett.

34(21), 2041 (1998).

[30] F. R. Yang, K. P. Ma, Y. Qian and T. Itoh, IEEE: Trans. Microwave

Theory and Tech. 47(11), 2092 (2000).

[31] A. Yariv and P. Yeh, Optical waves in crystals, (Wiley, New York,

1984, chapter 6).

- 44 -


- 45 -

[32] D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky and S. V.

Gaponenko, Optics and Photonics News: Optics 10, 33 (1999).

[33] D. N. Chigrin and C. M. S. Torres, Optics and spectroscopy 91, 484

(2001).

[34] M. Plihal, A. Shambrook and A. Maradudin, Opt. Comm. 80, 199

(1991).

[35] P. Villeneue and P. Michel, Phys. Rev. B. 46, 4969 (1992).

[36] R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos,

Appl. Phys. Lett. 61, 495 (1992).

[37] O. Painter, A. Husain, A. Scherer, P. T. Lee, I. Kim, J. D. O'Brien and

P. D. Dakpus, IEEE: Photon. Technol. Lett. 12, 1126 (2000).

[38] P. Pottier, C. Seassal, X. Letartre, J. L. Leclercq. P. Viktorovitch, D.

Cassagne and C. Jounin, IEEE: J. Lightwave Technol. 17, 2058

(1999).

[39] J. D. Joannopoulos, S. Fan, A. Mekis and S. G. Johnson, Novelties of

light in Photonic Crystals and Light Localization in the 21st century;

edited by C. M Soukoulis (Kluwer Academic, Dordrecht, 1-24, 2001).

[40] S. Fan and J. D. Joannopoulos, Appl. Opt.: Photon. News 11, 28

(2000).

[41] R. Costa, A. Melloni and M. Martinelli, IEEE: Photonic Technol. Lett.

15, 401 (2003).

[42] L. Raffaele, R. M. De La Rue, J. S. Roberts and T. F. Krauss, IEEE:

Photon. Technol. Lett. 13, 176 (2001).

[43] M. Tokushima, T. H. Kosaka, A. Tomita and H. Yamada, Appl. Phys.

Lett. 76, 952 (2000).

[44] A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, Sajeev John,

Stephen W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P.


- 46 -

Mondia, G. A. Ozin, O. Toader and H. M. van Driel, Nature 405, 437

(2000).

[45] Y. A. Vlasov, Xiang-Zheng Bo, J. C. Sturm and D. J. Norris, Nature

414, 289 (2001).

[46] I. I. Tarhan and G. H. watson, Phys. Rev. Lett. 76, 315 (1996).

[47] Y. A. Vlasov, V. N. Astratov, O. Z. Karimov, and A. A. Kaplyanskii,

V. N. Bogomolov and A. V. Prokofiev, Phys. Rev. B. 55, R13357

(1997).

[48] H. S. Sozuer, J. W. Haus and R. Inguva, Phys. Rev. B. 45, 13962

(1992).

[49] S. Noda, A. Chutinan and M. Imada, Nature 407, 608 (2000).

[50] H. Kosaka, T. Kawahima, A. Tomita, M. Notomi, T. Tamamura, T.

Sato and S. Kawakami, J. Lightwave Technol. 17, 2032 (1999).

[51] H. Kosaka, T. Kawahima, A. Tomita, M. Notomi, T. Tamamura, T.

Sato and S. Kawakami, Phys. Rev. B. 58, R10096 (1998).

[52] T. Baba and M. Nakamura, IEEE: J. Quantum Electron. 38, 909

(2002).

[53] P. Yeh, Optics in Layered Media, (John Willey and Sons, New York,

1988).

[54] A. Sopaheluwakan, Thesis on Defect States and Defect modes in 1D

photonic crystals, (Univ. of Twente, Nitherland, December, 2003); A.

Rung, Thesis on Numerical studies of energy gaps in photonic

crystals, (Uppsala Univ., 2005).

[55] M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford,

1964, 2nd Edition).

[56] J. B. Pendry, J. Phys. Cond. Mat. 8, 1085�1108 (1996) and references

therein.


- 47 -

[57] A. Taflove, Computational electrodynamics: The Finite-Difference

Time-Domain Method, (Artech House, Boston, London, 1995).

[58] G. Pelosi, R. Coccioli, and S. Selleri (Eds.), Quick Finite Elements for

Electromagnetic Waves, (Artech House).

[59] J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 (1992).

[60] D. Felbacq, G. Tayeb and D. Maystre, J. Opt. Soc. Am. A 11, 2526

(1994).

[61] F. de Daran, V. Vigneras-Lefebvre and J. P. Parneix, IEEE: Trans.

Magnetics 31, 1598 (1995).

[62] D. Maystre, Pure Appl. Opt. 3, 975 (1994).

[63] R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos,

Phys. Rev. B. 44, 13772 (1991).

[64] Z. Y. Li, L.-L. Lin and Z.-Q. Zhang, Phys. Rev. Lett. 84, 4341 (2000).

[65] S. Fan, P. R. Villeneuve, J. D. Joannopoulos and E. F. Schubert, Phys.

Rev. Lett. 78, 3294 (1997).

[66] C. T. Chan, Q. L. Yu and K. M. Ho, Phys. Rev. B. 51, 16635 (1995).

[67] Taflove (Ed.), Advances in Computational Electrodynamics (Artech

House).

[68] J. B. Pendry and L. Martin-Moreno, Phys. Rev. B 50, 5062 (1994).

[69] D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho and C. M.

Soukoulis, Appl. Phys. Lett. 65, 645 (1994).

[70] F. Gadot, A. Chelnokov, A. De Lustrac, P. Crozat and J.-M. Lourtioz,

D. Cassagne and C. Jouanin, Appl. Phys. Lett. 71, 1780 (1997).

[71] S. G. Johnson, Photonic Crystal: From Theory to Practice (Chapter 1,

2001).

[72] K. Sakoda, Optical Properties of Photonic Crystal (Springer-Verlag

Berlin Heidelberg New York, 2001).


- 48 -

[73] J. Li, L. Zhou, C. T. Chan and P. Sheng, Phys. Rev. Lett. 90, 083901

(2003).

[74] I. V. Shadrivov, A. A. Sukhorukov and Y. S. Kivshar, Phys. Rev. Lett.

95, 193903 (2005).

[75] L. Wu, S. He and L. Shen, Phys. Rev. B 67 , 235103 (2003).

[76] J. Zi, J. Wan and C. Zhang, Appl. Phys. Lett. 73, 2086 (1998).

[77] E. van Groesen and A. Sopalheluwakan, J. Nonlinear Optical Phys. &

Materials 12, 135 (2003).

[78] Sanjeev K. Srivastava, S. P. Ojha and K. S. Ramesh, Microwave Opt.

Technol. Lett. 33, 308 (2002).

[79] A Banerji, S. K. Awasthi, U. Malviya and S. P. Ojha, J. Mod. Opt. 53,

1739 (2006).

[80] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos

and E. L. Thomas, Science 282, 1679 (1998).

[81] J. N. Winn, Y. Fink, S. Fan and J. D. Joannopoulos, Optics Lett. 23,

1573 (1998).

[82] D. Y. Jeong, Y. H. Ye and Q. M. Zhang, J. Appl. Phys. 92, 4194

(2002).

[83] John Marangos, Nature 406, 243 (2000).

[84] S. P. Ojha and Sanjeev K. Srivastava, Microwave Opt. Technol. Lett.

42, 82 (2003).

[85] A. Sommerfeld, Ein Einwand gegen die Relativ theorie der

Elektrodyanmik und sienc Beseitgung 8, 841 (1907).

[86] L. Brillouin, Wave propagation in periodic structure, McGraw-Hill,

New York, 1946.

[87] L. Brillouin, Wave propagation and group velocity, Academic, New

York, 1960.


- 49 -

[88] C. G. B. Garrett and D. E. Mc Cumber, Phys. Rev. A 1, 305 (1970).

[89] M. D. Crisp, Phys. Rev. A 4 , 2104 (1971).

[90] S. Chu and S. Wong, Phys. Rev. Lett. 48, 738 (1982).

[91] E.L. Bolda and R.Y. Chiao, Phys. Rev. A 48, 3890 (1993).

[92] J. P. Dowling and C. M. Bowden, J. Mod. Opt. 41, 345 (1994).

[93] S. Noda, K. Tomoda, N. Yamamoto and A. Chutinan, Science 289,

604 (2000).

[94] M. Galli, M. Agio, L. Andreani, L. Atzeni, D. Bajoni, G. Guizzetti, L.

Businaro, E. Di Fabrizio, F. Romanato and A. Passaseo, The

European Physics Journal B 27, 79 (2002).

[95] E. Yablonovitch, Scientific American 285, 47 (2001).

[96] O. Toader and S. John, Science 292, 1133 (2001).

[97] S. Noda, N. Yamamoto and A. Sasaki, Jpn. J. Appl. Phys 35, L909

(1996).

[98] S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas,

K. M. Ho, M. M. Sigalas, W. Zubrycki, S. R. Kurz and J. Bur, Nature

394, 251 (1998).

[99] O. D. Velev, T. A. Jede, R. F. Lobo and A. M. Lenhoff, Nature 389,

447 (1997).

[100] G. Parker and M. Charlton, Physics World 13, 29 (2000).

[101] D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky and S. V.

Gaponenko, J. Lightwave Technol. 17, 2018 (1999).

[102] T. F. Cross and Richard M. De La Rue, Progress in Quantum

Electronics 23, 51 (1999).

[103] C. Lopez, A. Blanco, H. Mignez and F. Mesquer, Optical Materials

13, 187 (1999).


- 50 -

[104] S. V. Frolov, Z. V. Vardeny, Anvar A. Zakhidov and R. H.

Baughman, Opt. Commun. 162, 241 (1999).

[105] H. Kosaka et al., Appl. Phys. Lett. 74, 1212 (1999).

[106] M. Notomi, Opt. Quantum Electron 34, 133 (2002).

[107] Y. S. Seo, K.B. Chung and H. K. Hoeji, South Korea 11, 198 (2000).

[108] S. Y. Zhu, Y. Yang, H. Chen, H. Zheng and M. S. Zubairy, Phys. Rev.

Lett. 84, 2136 (2000).

[109] H. Kosaka, H., T. Kawashima, A. Tomita, M. Notomi, T. Tamamura,

T. Sato and S. Kawakami, Appl. Phys. Lett. 74, 1370 (1999).

[110] M. Thevenot, A. Reineix and B. Jecko, Microwave Opt. Technol. Lett.

21, 411 (1999).

[111] S. S. Oh, C. S. Kee, Jae-Eun Kim, Hae Yong Park ,Tae I. Kim, Ikmo

Park, and H. Lim, Appl. Phys. Lett. 76, 2301 (2000).

[112] P. A. Hiltner, and I. M Krieger, J. Phys. Chem. 73, 2386 (1969).

[113] K. Ohtaka, Phys. Rev. B. 19, 5057 (1979).

[114] P. W. Anderson, Phys. Rev. 109, 1492 (1958).

[115] S. John and R. Rangarajan, Phys. Rev. B 38, 10101 (1988).

[116] S. Satpathy, Z. Zhang and M. R. Salehpour, Phys. Rev. Lett. 64, 1239

(1990).

[117] K. M. Leung and Y. F Liu, Phys. Rev. B 41, 10188 (1990).

[118] J. Maddox, Nature 348, 481 (1990).

[119] K. M. Ho, C. T. Chan and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152

(1990).

[120] N. Yamamoto, S. Noda and A. Chutinan, Jpn. J. Appl. Phys. 37,

L1052 (1998).

[121] S. Y. Lin, V. M. Hietala, L. Wang and E. D. Jones, Opt. Lett. 21, 1771

(1996).


- 51 -

[122] J. G. Fleming and S. Y. Lin, Opt. Lett. 24, 49 (1999).

[123] J. B. Pendry, A. J. Holden, D. J. Robbins and W. J. Stewart, IEEE

Trans. Microwave Theory Tech. 47, 2075 (1999).

[124] R. A. Shelby, D. R. Smith and S. Schultz, Science 292, 77 (2001).

[125] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).

[126] P. V. Parimi and W. T. Lu et al. Phys. Rev. Lett. 92, 127401 (2004).

[127] N. C. Panoiu, R. M. Osgood Jr., Shuang Zhang and Steven R. J.

Brueck, J. Opt. Soc. Am. B 23, 506 (2006).

[128] M. Ricci, N. Orloff and C. M. Anlage, App. Phys. Lett. 87, 034102

(2005).

[129] A. Peminov, A. Loidl, P. Przyslupski and B. Dabrowski, Phys. Rev.

Lett. 95, 247009 (2005).

[130] D. M. Sedrakian, A. H. Gevorgyan, A. Zh. Khachatrian and V. D.

Vadalian, Opt. Commun. 271, 451 (2007).

[131] Z. F. sang and Z. F. Li, Opt. Commun. 273, 162 (2007).

[132] L. Zhang, Y. Zhang, L. He, Z. Wang, H. Li and H. Chen, J. Phys. D:

Appl. Phys. 40, 2579 (2007).

[133] T. Shang, H. Zhang, Y. Zhang, P. Wang and J. Yao, Int. J. Infrared

Milli. Waves 10762, 9236 (2007).

[134] P. C. Pandey, K. B. Thapa and S. P. Ojha, Opt. Commun. 281, 1607

(2007).

[135] S. K. Srivastava and S. P. Ojha, J. Mod. Opt. 56(1), (2009) 33.

[136] X. Xu, F. Wu, D. Han, X. Liu and J. Zi, J. Phys.: Condens. Matter 20,

255241 (2008).

[137] M. de Dios-Leyva and O. E. González-Vasquez, Phys. Rev. B 77,

125102 (2008).

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