Chapter 1 An Overview: Photonic Band Gap Materials

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

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 - 2 -

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 Schroedingers and Helmholtzs equations, Eli Yablonovitch [5] had the idea in 1987 to built artificial periodic structures - 3 -

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 - 4 -

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]. Metallodielectric 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 metallodielectric 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, - 5 -

high impedance surfaces, compact uniplanar slow-wave lines; broad band filters [2030]. 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. - 6 -

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), twodimensional (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 - 7 -

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- - 8 -

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 microlasers 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 - 9 -

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(ik r.r) u k (r) (1.1) where, u k (r) is periodic in the real space lattice. This result is commonly known as Blochs theorem and the form of above equation is known as - 10 -

Bloch state. The wave vectors k r 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 k r +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 k r i.e. (k r )= (k r +m(2 /a)). In fact, we only need to consider k r to exist in the range / a k r / a. This region of nonredundant values of k r is called the Brillouin zone. Substituting the Bloch state into the Master Equation, we can get a reduced form of Master equation ( ik ) 1 ( ik ( r) ) 2 ( k) uk ( r) uk ( r) (1.2) c 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). - 11 -

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 - 12 -

of a plane wave (e ik.r ) times a function (u k ) 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. Maxwells 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) B 0 (1.4) B E (1.5) t D H J (1.6) t 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 - 13 -

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. D 0 ( r ) E and B 0 ( r) H (1.7) where (r ) and (r ) are the electric permittivity and magnetic permeability respectively. But for dielectric materials, (r ) 1; hence B 0 H. Applying these conditions Maxwells equations can be written as ( r ) E( r, t) 0 (1.8) H ( r, t) 0 (1.9) H ( r, t) E( r, t) 0 ( r) (1.10) t E( r, t) H ( r, t) 0 ( r) (1.11) t 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 where E ) i t ( r, t) E( r e (1.12) H ) i t ( r, t) H ( r e (1.13) represents the angular frequency. Substituting equations (1.9) and (1.10) in the above equations, we get E ( r, t) i 0H ( r ) 0 (1.14) H ( r, t) i o ( r ) E( r ) 0 (1.15) Now taking curl of equation (1.11), on both sides, we get [ E ( r, t)] i 0 H ( r ) (1.16) - 14 -

we get Now using equation (1.12) for eliminating H (r ) from equation (1.13), 2 2 E ( r ) ( r ) E( r ) 0 (1.17) 2 c 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 - 15 -

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 Maxwells 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 - 16 -

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 softwares 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 Maxwells 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 - 17 -

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 n 1 and n 2 with thicknesses d 1 and d 2 respectively. - 18 -

Let us consider a periodic arrangement of multilayer film (Figure 1.4), with refractive indices n 1 and n 2 and each having thicknesses d 1 and d 2 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 n 1, the right going and left going plane waves have amplitudes A 1 and B 1 respectively and the right going and left going plane waves have amplitudes C 1 and D 1 for layer with index n 2 respectively in the unit cell considered. Hence for layer with index n 1 the solution of equation (1.14) is, ik1xx ik1 xx E x) A1 e B1e ( (1.18) ik2 x ( x d1 ) ik2 x ( x d1) E ( x) C1e D1e (1.19) for the layer with index n 2. The wave numbers k 1x and k 2x are defined as, k jx n j cos j, j=1, 2 (1.20) c where 1 and 2 are the ray angles in the two mediums respectively. At the interface between layers (x = d 1 ), the solution and its derivative should be continuous. This gives a relation between plane wave amplitudes: C D 1 1 M 12 A B 1 1 (1.21) with, 1 2 1x 2x 1 x 1 2x 2x M (1.22) 12 1 k1x ik d 1 k1x ik d 2 1 1 k k k e e ik d 1x 1 1 2 2 1 1 k k k 1x 2x e e ik d 1x 1 1x 1 and, also at x = d, the continuity of the plane waves at the interface between layers with indices n 2 and n 1 and its derivative gives - - 19 -

A B 2 2 M 21 C D 1 1 (1.23) where the matrix M 21 is the same as (1.21) but with interchanging the indices. From the two matrix equations (1.21) and (1.23), we have, A B 2 2 M 21 M 12 A B 1 1 (1.24) A B 2 2 M A 1 i, j (1.25) B1 where, M i,j = M 21 M 12. The matrix element of the matrix M i,j are given by M M 1,1 1,2 e e ik 1 1 x b i cos( k2xd2) sin( k2xd2) (1.26) 2 ik 1 1 x d i 1 sin( k2 xd 2) (1.27) 2 k k M 2,1 M1,2 and 2,2 M1, 1 1x 2x M (1.28) 2 k1x. n2 for TE mode and 2 k. n 2 x 1 for TM mode (1.29) The matrix M i,j is called as the transfer matrix of one unit of the periodic lattice. The matrix M i,j depends on the frequency, and it is unimodular (it is a square matrix with determinant equal to unity). Hence, for each, the matrix M i,j defines a unique mapping for amplitudes of the plane waves in layer n 1 into the amplitude of the next layer with index n 2. 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]. - 20 -

E ik ( ) x ( x, K) U K ( x). e (1.30) where U K (x) is a complex valued periodic function with the period of the lattice (d=d 1 +d 2 ), U K (x) = U K (x+d). The parameter K( ) is called the Bloch wave number for a periodic lattice with indices n 1 and n 2. The expression for K( ) is as follows, 1 1 1 K ( ) cos Tr( M i, j ) (1.31) d 2 with M i,j given in (1.25). After simplifying (1.31), one can obtain as, 1 1 1 1 K( ) cos cos( k1x d1)cos( k 2xd 2 ) sin( k1xd1)sin( k2xd 2 ) (1.32) d 2 k jx n j cos j, with j=1,2. (1.33) c The equation (1.32) is known as the dispersion relation of the periodic lattice with refractive indices n 1 and n 2 and thicknesses d 1 and d 2 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 the band gap. is outside 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). - 21 -

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 antireflection coating at a desired wavelength in application such as nonabsorbing 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 n 1 =( 1 ) 1/2 and n 2 =( 2 ) 1/2 with thickness d 1 and d 2 respectively and taken N unit of these cells and stack than as depicted in figure 1.5. - 22 -

Figure 1.5: Schematic diagram of bilayers unit cell of dielectric constant 1 and 2 with thickness d 1 and d 2 respectively. Figure 1.6: The dispersion and transmittance of one dimensional photonic crystal with n 1 (= 1 ) = 1.25, n 2 (= 2 ) = 2.5 and n 1 d 1 n 2 d 2 /4. - 23 -

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 n n 1 n 2, 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 n 0 =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. t 1 m 0 r (1.34) - 24 -

and m m 11 12 m with 11 11 1 N 2 m21 m22 m M U N U, m 21 M 21U N 1, m 12 M 12U N 1 U N 2, 22 M 22U N 1 U N 2 m and U N sin[( N 1). K( ). d] sin[ K( ). d] where M i,j are same as section (1.7.1) and transmission and reflection coefficients are given by m. m 12 21 t m11 (1.35) m22 m m 21 r (1.36) 22 The associated transmittance (T) and reflectance (R) are obtained by taking the absolute square of t and r respectively 2 T t and 2 R r (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 n 1 =1.5, n 2 =3.7 and d 1 =0.7d, d 2 =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]. - 25 -

Figure 1.7: Total ODR range for n 1 =1.5, n 2 =3.7 and d 1 =0.7d, d 2 =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 v g dk( d ) 1, 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, - 26 -

and it will represented as 1 v dk ( ) c g d n( ) dn / d, for the propagation of an electromagnetic pulse in a linear dispersive but nonabsorbing 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 g c n eff ( p) and v c n eff ( g). For example, it can be seen that the band structure in the 1D v photonic crystal with parameters n 1 =1.5 and n 2 =2.5 and n 1 d 1 = n 2 d 2, 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 ( v p K( ) ) p and group velocity ( v p d dk ( ) 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, v p exhibits a slight increase with frequency, while v g shows abnormal behavior and significantly slows down. In practice, the phase velocity (v p ) is often expressed in terms of the refractive index of the material v p =c/n eff (p); here n eff (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, n eff (p) is only weakly frequency dependent. When approaching the band edge, a large change of n eff (p) with frequency (frequency dispersion) occurs [82]. - 27 -

Figure 1.8 The phase velocity v p and effective refractive index (n eff (p)) and the group velocity v g and effective refractive index (n eff (g)) derived from the dispersion curves obtained n 1 =1.5, n 2 =2.5 and n 1 d 1 =n 2 d 2. Inside the band gap, n eff (p) is actually complex and also exhibits abnormal dispersion. Besides n eff (p) for the refractive index of the phase velocity in a PC, the refractive index of the group velocity, n eff (g), was introduced by Sakoda to describe the ratio of c/v g [31]. For the 1D PC, n eff (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), n eff (p) and n eff (g) are the same. Near the band edges n eff (g) exhibits sharp increase and is much larger than n eff (p), reflecting the marked slowing down of v g. Also, Ojha et al. have also studied the group velocity and effective group index using this concept. They have found a remarkable - 28 -

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. - 29 -

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 10 8 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. - 30 -

1.8 Yablonovite Structure This structure was given by Yablonovitch [20, 74, 92] in which 85% of lattice structure is taken as index n 1 = 1 for air and 15% of the lattice structure is taken as index n 2 = 1.5 for glass (SiO 2 ) 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 n 1 =1.25 and n 2 =2.5 the corresponding thickness of each layer are d 1 = /4n 1 = /5 and d 2 = /4n 2 = /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 2 R r where r m m 21 22. 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. - 31 -

Figure 1.9: Reflectance versus normalized frequency of one-dimensional photonic crystal with n 1 =1.25, n 2 =2.5 and n 1 d 1 =n 2 d 2 /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 - 32 -

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. - 33 -

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 - 34 -