Professor

Physics

TDPG College, VBS Purvanchal University

 Jaunpur, U.P., India 

Associate Professor

Physics

TDPG College, VBS Purvanchal University

Jaunpur, U.P., India

Since the concept of photonic crystals (PCs) was theoretically proposed by Yablonovitch [1] and John [2] in 1987, it has gathered a considerable attention of the researchers and over the last couple of decades, the advancement in photonic crystals research has established the fact that photonics technologies have been playing a major role in the rapid development of information and communications technology [3, 4]. Moreover, due to their fascinating unique characteristics, PCs have gone far beyond the realization of low-threshold lasers, low-loss resonators, optical switches, waveguides, and optical fibers, gas sensing to optical filters, photonic papers, inkless printing, and reflective flat displays, with innovative sensing designs for environmental monitoring, medical diagnosis, defense, and food quality control [5, 6].

Photonic crystals are microstructural optical materials with a spatial periodicity of their dielectric constant on a wavelength scale, that interact with light in a manner analogous to that in which crystal lattices interact with electrons. The essential property of these structure is the existence of allowed and forbidden frequency bands for light, in analogy to the energy bands and band-gaps of semiconductors. This suggests that the propagation of light within the frequency range of the photonic band gap (PBG), will be forbidden [1]. Nevertheless, the periodicity of this dielectric structure will be broken, if some defects are introduced in PCs, which makes it possible for PCs to present strong electromagnetic field confinement, small mode volume, and low extinction loss [2]. On the other hand, by adjusting the structural parameters of PC or infiltrating suitable materials in the air holes of PC, the propagation of light can be modified and engineered as desired. Therefore, many PC based devices have been widely used in the applications of light flow control, such as filters [7, 8], electro-optical modulators [9, 10], switches [11, 12], and delay devices [13]. Specially, PC based sensors seem to be much more popular due to their promising characteristics like ultra-compact size, high measurement sensitivity, flexibility in structural design, and more suitable for monolithic integration [14-16]. Besides, the PC based sensors can also inherit the favorable characteristics of optical sensors, such as safety in flammable explosive environment, immunity to electromagnetic interference, long-distance monitoring, and rapid response speed.   

Photonic Crystals

A photonic crystal (PC) is a periodic structured material that exhibits strong interaction with light. It is composed of a multilayer stack of alternating high and low index of refraction dielectric materials. Strong interaction with the light occurs in such a material due to interference of the reflected and refracted light at all interfaces inside the material. The complex pattern of superimposing beams will reinforce or cancel out one another according to the wavelength of light, direction of incidence, refractive index, size and arrangement of the building stacks. Due to this, one can find a frequency band for which the propagation of light is forbidden in a certain direction (Fig.1). The forbidden frequency region is called photonic band gap [7].

                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 (Fig. 2).  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.

Origin of the Photonic Band gap

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 the so-called lattice constant ‘a’, and the basic step vector is called primitive lattice vector ‘a’. Because of this symmetryfor 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:

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 Bloch state. The wave vectors kr that differ by integral multiples m of 2π/a are not different from a physical point of view. In fact, all the modes with wave vector of the

that 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. In this way, we array at the description of the modes of the photonic bands w_nk 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.

 

Fig. 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).

The above figure 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 which 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 scalar waves, while photons are vectorial ones. This implies that polarization must be taken into account while dealing with photons.

Important Crystal Parameters

The Parameters on which the optical features of a photonic crystal will depend are indicated below-

1. The type of symmetry of the structure- The position of the building blocks of the PCs will set the symmetry of the lattice. Its examples are the sc (simple cubic), bcc, fcc,  hcp and diamond structures.

2. 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.

3. 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.

4. Filling fraction- It is the ratio between the volumes occupied by each dielectric with respect to the total volume of the composite.

5. Refractive index contrast (δ) - It is defined as the ratio between the refractive index of the high dielectric constant material and the low dielectric constant material.

6. The shape of the scattering centers.

7. Scalability.

8. 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 (Fig.4).

1-D PC consists of alternating 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 crystal can be used as omnidirectional totally reflecting mirrors, frequency filters, microwave antenna substrates and enclosure coatings of waveguide etc. Its applications depend on the chosen geometry and frequency regions [8-10].

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 [11]. 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 [12]. 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-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 [12, 13] in 2-D photonic crystals. Also, the 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 [14-15]. In the recent years, the uses of 2D PCs have been widely explored in order to improve the overall performance of optoelectronic devices [15-20].  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 frequency ranges over which the ordinary linear propagation is forbidden irrespective of the direction [21, 22]. The existence of these complete photonic band gaps allows the complete control over the radiative dynamics of active material 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 [23, 24]. A face centered cubic lattice consisting of low dielectric inclusions in a connected high dielectric network, called inverse structure, can exhibit small photonic band gap [25]. The “wood pile” structure also represents a 3D-PBG structure that is built by the layer-by-layer fabrication technique [26]. Recently, Kosaka et al. [27-29] 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 [30].

Fig. 4 Schematic depictions of photonic crystals periodic in one, two, and three dimensions, where the periodicity is in the material structure of the crystals.

Theoretical Formalism: Maxwell on a Lattice

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 [30-32],

In the above equations, are the electric field vector and magnetic field vector respectively. Electromagnetic field is described by these two field vectors.  (Electric displacement) and  (magnetic induction) are introduced to include the effect of the field on matter. The quantities p (electric charge density) and (current density) may be considered as the sources of the fields.  These equations completely determine the electromagnetic field.

Maxwell’s equations can not be solved uniquely unless the relationship between are known to obtain a unique determination of the field vectors because these consist of 8 scalar equations that relate a total of 12 variables, 3 for each of the 4 vectors. The material equations are,

where e and m are known as the dielectric tensor (or permittivity tensor) and the permeability tensor (of rank 2) respectively, are electric and magnetic polarization, respectively. The electric field can perturb the motion of electron and produce a dipole polarization  per unit volume. The magnetic field can also induce a magnetization   in materials having a permeability which is different from m0. The constant e0 (permittivity of a vacuum) has a value of 8.854 ´10^-12  F/m. The constant 0 (permeability of a vacuum) has the exact value of 4p ´ 10^-7 H/m. For the isotropic medium, both  e and m tensors reduce to scalars.  The quantities  are assumed to be independent of the field strengths. But for the sufficiently strong field, the dependence of these quantities on   and   must be considered.

Throughout this thesis, linear dielectric materials (= 0) are considered that are independent of and non-magnetic (= 0),   are also zero. We have,. However, for the most dielectric materials of interest, the magnetic permeability is very close to unity and we may set,.

The electric field E and the magnetic field H can be written as product of a function of the position and a function depending only on the time as,

The Helmholtz equation (15 or 16) is the governing equation for the  electromagnetic phenomenon in the photonic crystal structure with different material constituents. The reflection and transmission coefficients of the electromagnetic wave at the dielectric interfaces between the layers can be calculated by using the continuity relationship of some of the components of field vectors at the dielectric boundary. This continuity condition can be derived directly from Maxwell’s equations.

Since the electromagnetic phenomenon in a dielectric medium have no fundamental length scale, the coefficients of the Helmholtz equation do not have length as its dimension. Due to this, the electromagnetic problems that differ only by an expansion or a contraction can be considered to be the same problem. This indicates that the master equation with its coefficients can be normalized with respect to a reference (length) parameter.

Calculation of Band Structure of the Photonic Crystals

The calculations on photonic crystals (PCs) 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. For the photonic crystals, the eigen value problem is to solve the master equation   which comes from a combination of the Maxwell’s equations (3 and 4). The operator in the left part of equation (19) is Hermitian for lossless material (e is purely real), which gives a set of orthogonal eigenstates as the solution. If there is a periodicity in e such that

Numerical Methods for the Calculation of Band Structure of Photonic Crystals         

Photonic crystals are studied numerically by the six main methods: (a) The Plane Wave Method, (b) the Finite Difference Time Domain (FDTD) method, (c) the Finite Element method, (d) the Transfer Matrix Method (TMM), (e) a method based on a rigorous theory of scattering by a set of rods (for a two-dimensional crystal), or a set of spheres (for a three-dimensional crystal), and (f) the study of diffraction gratings [35- 41].

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 problem tackled. Some of these methods like methods (a to d) can simulate any doped or nondoped crystals [35-38] as they are highly flexible. Method (e) is limited to certain types of PCs that are parallel cylinders (for 2d photonic crystals) and spheres (for 3d) [39, 40]. Some of these methods as (a, d and f) can deal only with infinite crystals [35, 38, 41] and method (e) only with finite-sized structures [39, 40]. Finally, methods (a, d and f) use a super-cell to study the defect structures. On the contrary, methods (b, c and e) 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 [42-44]. 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 [45-47]. 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 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 [36].

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 well described [38]. The TMM consists on 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 [48-50]. It has also proved very useful and accurate when comparisons with experimental structures are undertaken [49, 50]. 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 [39, 40]. 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 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 because a large number of cylinders have been implemented (about one hundred).

The use of diffraction gratings theory [41] 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 can not simulate new PBG materials that are sophisticated 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.

Transfer Matrix Method for One Dimensional Photonic Crystals

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 [30, 31, 32, 38].

Let us consider a periodic arrangement of a multilayer film, with refractive indices n₁ and n₂ and each having thicknesses d₁ and d₂ respectively. The solution for the master equation will be the superposition of plane waves

Fig. 5: Schematic diagram of bilayers unit cell of refractive indices n₁ and n₂ with thicknesses d₁ and d₂ respectively.

traveling to the right and to the left. Say, for 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. Hence for layer with index n₁ the solution of equation (15) is, 

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 w, and it is unimodular (it is a square matrix with determinant equal to unity). Hence, for each w, the matrix M_i,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 Helmholtz equation (19) can be written in terms of Bloch waves [30-33, 51].

The equation (31) is known as 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 (w), which lies in the first Brilluion zone [0, p/d], E (x, K) is a periodic and travelling wave function. In this case, it is said that w is outside the band gap.

2. For imaginary K (w), defined by, K(w)=π/d+ip(w) E (x, K) is a standing wave function, a product of two periodic functions with an exponential increasing and a decreasing function, depending on the sign of p(w). In this case, w is inside the band gap.

3. For K (w) = p/d, E (x, K) is a periodic function of period 2^nd with special properties that it is a d-shift skew symmetric, E (x + d, K) = -E (x, K).

1. E. Yablonovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58, 2059-2062 (1987).

2. S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58, 2486-2489 (1987).

3. I.S. Amiri, S.R.B. Azzuhri, M.A. Jalil, H.M.Hairi, J. Ali, M. Bunruangses and P. Yupapin, Introduction to Photonics: Principles and the Most Recent Applications of Microstructures. Micromachines (Basel). 2018 Sep 11;9(9):452 (2018).

4. Z.Chen and M. Segev, Highlighting photonics: looking into the next decade. eLight 1, 2 (2021).

5. V. Melissinaki, O. Tsilipakos , M. Kafesaki , M. Farsari, and S. Pissadakis, Micro-Ring Resonator Devices Prototyped on Optical Fiber Tapers by Multi-Photon Lithography, IEEE Journal of selected topics in Quantum electronics, 27, 5900107, (2021).   

6.  E. Menard, M.A. Meitl, Y. Sun, J.U. Park, D. Jay-Lee Shir, Y.S Nam, S. Jeon, and J.A. Rogers, Micro- and Nanopatterning Techniques for Organic Electronic and Optoelectronic Systems, Chem. Rev, 107, 1117−1160 (2007)

7. J. D. Joannopoulos, R.D. Meade and J.N. Winn, Photonic Crystal: Molding the Flow of Light, (Princeton University Press, New Jersey 1995).

8. V. Rumyantsev, S.A., Fedorov and K.V. Gumennyk, Photonic Crystals: Optical Properties, Fabrication and Applications, Chapter 8, 183-200 (2011).

9.B. A. Chavez-Castillo, J. S. Pérez-Huerta, J. Madrigal-Melchor, S. Amador-Alvarado, I. A. Sustaita-Torres, V. Agarwal and D. Ariza-Flores,  A wide band porous silicon omnidirectional mirror for the near infrared range. Journal of Applied Physics 127:20, 203106 (2020).

10.  D. N. Chigrin and C. M. S. Torres, Periodic thin-film interference filters as one-dimensional photonic crystals, Optics and spectroscopy 91, 484-489 (2001).

11. D V Karpov and I G Savenko, Polariton condensation in photonic crystals with high molecular orientation, New J. Phys. 20 013037(2018).

12. P. Villeneue and P. Michel, Photonic band gaps in two-dimensional square and hexagonal lattices, Phys. Rev. B. 46, 4969-4972 (1992).

13. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Existence of a photonic band gap in two dimensions, Appl. Phys. Lett. 61, 495-497 (1992).

14. O. Painter, A. Husain, A. Scherer, P. T. Lee, I. Kim, J. D. O'Brien and P. D. Dakpus, Lithographic tuning of a two-dimensional photonic crystal laser array, IEEE  Photon. Technol. Lett. 12, 1126-1128 (2000).

15. P. Pottier, C. Seassal, X. Letartre, J. L. Leclercq. P. Viktorovitch, D. Cassagne, and C. Jounin, Triangular and Hexagonal High Q-Factor 2-D Photonic Bandgap Cavities on III-V Suspended Membranes, IEEE J. Lightwave Technol. 17, 2058-2062 (1999).

16. J. D. Joannopoulos, S. Fan, A. Mekis, and S. G. Johnson, Novelties of light with photonic crystals, in Photonic Crystals and Light Localization in the 21^st century; edited by C. M Soukoulis (Kluwer Academic, Dordrecht, 1-24, 2001).

17. S. Fan and J. D. Joannopoulos, Photonic crystals: towards large-scale integration of optical and optoelectronic circuits, Appl. Opt., Photon. News 11, 28-33 (2000).

18. Z. Rashki, M. Seyyed, S.J. Chabok, A New Proposal for PCRR- Based Optical Channel Drop Filter, Majlesi Journal of Telecommunication Devices, 6(1), 19-26 (2017).

19. L. Raffaele, R. M. De La Rue, J. S. Roberts, and T. F. Krauss, Edge-emitting semiconductor microlasers with ultrashort-cavity and dry-etched high-reflectivity photonic microstructure mirrors, IEEE Photon. Technol. Lett. 13, 176-178 (2001).

20. M. Tokushima, T. H. Kosaka, A. Tomita, and H. Yamada, Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide, Appl. Phys. Lett. 76, 952-954 (2000).

21. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, Sajeev John, Stephen W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J. P. Mondia, G. A. Ozin, O. Toader and H. M. van Driel, Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres, Nature 405, 437-439 (2000).

22. Y. A. Vlasov, Xiang-Zheng Bo, J. C. Sturm and D. J. Norris, On-chip natural assembly of silicon photonic bandgap crystals, Nature 414, 289-292 (2001).

23. I. I. Tarhan and G. H. watson, Photonic Band Structure of fcc Colloidal Crystals, Phys. Rev. Lett. 76, 315-318 (1996).

24. B. Q. Dong, X. H. Liu, T. R. Zhan, L. P. Jiang, H. W. Yin, F. Liu, and J. Zi, Structural coloration and photonic pseudo gap in natural random close-packing photonic structures, Opt. Express 18, 14430-14438 (2010).

25. Ming Che and Zhi-Yuan Li, Solution of photonic band diagrams by a plane-wave-based transfer-matrix method in combination with an interpolation method, J. Opt. Soc. Am. B 25, 1270-1276 (2008)

26. Kenji Ishizaki, Kou Gondaira, Yuji Ota, Katsuyoshi Suzuki, and Susumu Noda, Nanocavities at the surface of three-dimensional photonic crystals, Opt. Express 21, 10590-10596 (2013).

26. H. Kosaka, T. Kawahima, A. Tomita, M. Notomi, T. Tamamura, T. Sato and S. Kawakami, Superprism phenomena in photonic crystals: toward microscale lightwave circuits, J. Lightwave Technol. 17, 2032-2038 (1999).

28. A. Gandhi and D. Alexei, Optical Modes of a Dispersive Periodic Nanostructure, Progress In Electromagnetics Research B, Vol. 52, 1-18, (2013)

29. H. Kosaka, T. Kawahima, A. Tomita, M. Notomi, T. Tamamura, T. Sato and S. Kawakami, Superprism phenomena in photonic crystals, Phys. Rev. B. 58, R10096-R10099 (1998).

30. T. Baba and M. Nakamura, Photonic crystal light deflection devices using the superprism effect, IEEE  J. Quantum Electron. 38, 909-914 (2002).

31. P. Yeh, Optics in Layered Media, (John Willey and Sons, New York, 1988).

32. 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).

33.  M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1964, 2^nd Edition).

34. S. G. Johnson, Photonic Crystal: From Theory to Practice, (Chapter 1, 2001).

35. K. Yasumoto, Electromagnetic theory and applications for photonic crystals, (Tailor & Francis Press, 2005).

36. J. B. Pendry, Calculating photonic band structure, J. Phys. Cond. Mat. 8, 1085–1108 (1996).

37. A. Taflove, Computational electrodynamics, The Finite-Difference Time-Domain Method, (Artech House, Boston, London, 1995).

38. G. Pelosi, R. Coccioli, and S. Selleri (Eds.), Quick Finite Elements for Electromagnetic Waves, (Artech House).

39. J. B. Pendry and A. MacKinnon, Calculation of photon dispersion relations, Phys. Rev. Lett. 69, 2772–2775 (1992).

40. D. Felbacq, G. Tayeb, and D. Maystre, Scattering by a random set of parallel cylinders, J. Opt. Soc. Am. A 11, 2526–2538 (1994).

41. F. de Daran, V. Vign´eras-Lefebvre, and J. P. Parneix, Modelling of electromagnetic waves scattered by a system of spherical particles, IEEE Trans. Magnetics 31, 1598–1601 (1995).

42. D. Maystre, Electromagnetic study of photonic bandgaps, Pure Appl. Opt. 3, 975–993 (1994).

43. Special issues on photonic crystals, J. Mod. Opt., 41 (1994); J. Opt. Soc. Am. B, 10 (1993).

44. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Photonic bound states in periodic dielectric materials, Phys. Rev. B. 44, 13772–13774 (1991).

45. Z. Y. Li, L.-L. Lin, and Z.-Q. Zhang, Spontaneous Emission from Photonic Crystals: Full Vectorial Calculations, Phys. Rev. Lett. 84, 4341–4344 (2000).

46. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, High Extraction Efficiency of Spontaneous Emission from Slabs of Photonic Crystals, Phys. Rev. Lett. 78, 3294–3297 (1997).

47. C. T. Chan, Q. L. Yu, and K. M. Ho, Order-N spectral method for electromagnetic waves, Phys. Rev. B. 51, 16635-16642 (1995).

48. Taflove (Ed.), Advances in Computational Electrodynamics, (Artech House).

49. J. B. Pendry, and L. Martin-Moreno, Energy loss by charged particles in complex media, Phys. Rev. B. 50, 5062–5073 (1994).

50. Jiro Kitagawa, Mitsuhiro Kodama, Shingo Koya, Yusaku Nishifuji, Damien Armand, and Yutaka Kadoya, THz wave propagation in two-dimensional metallic photonic crystal with mechanically tunable photonic-bands, Opt. Express 20, 17271-17280 (2012)

51. F. Gadot, A. Chelnokov, A. De Lustrac, P. Crozat, and J.-M. Lourtioz, D. Cassagne and C. Jouanin, Experimental demonstration of complete photonic band gap in graphite structure, Appl. Phys. Lett. 71, 1780–1782 (1997).