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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 10 — May. 19, 2003
  • pp: 1156–1165
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Intrinsic eigenstate spectrum of planar multilayer stacks of two-dimensional photonic crystals

K. H. Dridi  »View Author Affiliations


Optics Express, Vol. 11, Issue 10, pp. 1156-1165 (2003)
http://dx.doi.org/10.1364/OE.11.001156


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Abstract

A specific class of planar photonic crystals is investigated that provides a condensed matter combining the properties of planar multilayer stacks and two-dimensional photonic crystals in order to achieve large partial bandgaps in the eigenstate spectrum. These gaps are larger than the directionally dependent and polarization-dependent partial gaps of photonic crystal slabs. Full in-plane gaps are demonstrated numerically. Strong dispersion, waveguide confinement, high Q-cavities, and alternative photonic signal processing are feasible with these structures.

© 2003 Optical Society of America

1. Introduction

Photonic crystals (PCs) made of dielectric materials are structures with a spatially periodic variation of dielectric constant. In 1987 it was suggested that photons in subwavelength periodic dielectric structures might be controlled in pretty much the same way that electrons are in a periodic electrical potential of atoms and molecules in a crystal lattice of semiconductor material [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

, 2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

]. In the past ten years scientific research has clarified some of the remarkable properties of one-, two-, and three-dimensional PCs through extensive theoretical and numerical studies [3

3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

] and through experimental validation [4

4. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

, 5

5. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

, 6

6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ=1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001). [CrossRef]

, 7

7. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Direct measurement of the quality factor in a two-dimensional photonic-crystal microcavity,” Opt. Lett. 26, 1903–1905 (2001). [CrossRef]

]. Photonic bandgaps in the spectrum of allowed photonic energy states in condensed matter arise from multiple destructive interference of electromagnetic waves resulting from multiple scattering and diffraction effects in periodic dielectric structures with geometric features in the subwavelength regime of operation. The energy flow in such structures shows that photons can experience a photonic potential barrier in multidimensional periodic gratings or PCs by diffraction [8

8. T. Sϕndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B. 61, 15688–15696 (2000). [CrossRef]

] just as electrons experience an electrical potential barrier in a crystal lattice of atoms.

Two-dimensional (2D) PCs allow the characterization of modes into TE (electric field vector parallel to the 2D plane) and TM (magnetic field vector parallel to the 2D plane) polarization states and can exhibit complete, polarization-dependent 2D bandgaps, or partial bandgaps in the spatial frequency-frequency domain {(k,ω)} in which we study the spectrum of spatiotemporal signals of the space-time domain {(r, t)}. However, these crystals cannot confine electromagnetic radiation in the third dimension, because of their 2D nature. A series of three-dimensional (3D) PCs that exhibit complete 3D bandgaps for all polarized states have been proposed and studied theoretically in the past 12 years [9

9. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

, 10

10. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991). [CrossRef] [PubMed]

, 11

11. S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000). [CrossRef]

, 12

12. S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, “Design of three-dimensional photonic crystals at submicron lengthscales,” Appl. Phys. Lett. 65, 1466–1468 (1994). [CrossRef]

, 13

13. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000). [CrossRef] [PubMed]

, 14

14. O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001). [CrossRef] [PubMed]

, 15

15. E. Kuramochi, M. Notomi, T. Tamamura, T. Kawashima, S. Kawakami, J. Takahashi, and C. Takahashi, “Drilled alternating-layer structure for three-dimensional photonic crystals with a full band gap,” J. Vac. Sci. Technol. B 18, 3510–3513 (2000). [CrossRef]

, 16

16. S. Kawakami, E. Kuramochi, M. Notomi, T. Kawashima, J. Takahashi, C. Takahashi, and T. Tamamura, “A new fabrication technique for photonic crystals: nanolithography combined with alternating-layer deposition,” Opt. Quantum Electron. 34, 53–61 (2002). [CrossRef]

]. All these structures have a true 3D nature with complex unit cells and are made of materials with high dielectric contrast with a high filling fraction of low index material. They require very precise three-dimensional alignment and numerous intermediate steps in the material processing whose physical chemistry implies several processing temperatures, gaseous byproducts, material and geometric grain impurities inducing mechanical stress that can disturb the optical quality of the final product. These facts make the microfabrication of these structures rather difficult and the use of electron beam lithography instead of photolithography does not make them easily amenable for mass production. Furthermore, the creation of localized cavities and line defects in these structures is not an obvious task. Instead, some effort has lately been directed at using quasi-3D structures where the planar confinement of light is achieved through a 2D PC while the vertical confinement results from total internal reflection [17

17. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B. 60, 5751–5758 (1999). [CrossRef]

, 18

18. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B. 62, 8212–8222 (2000). [CrossRef]

, 5

5. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

, 6

6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ=1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001). [CrossRef]

]. However, the bandgaps in PC slabs that have a symmetry plane parallel to the slab surface are guided mode gaps and have been found to be polarization dependent and only cover a reduced area in (k,ω) space, i.e., they are partial gaps under the light cone, meaning that radiation in the top and lower cladding is allowed even for frequencies in the partial bandgap. This is problematic because guided modes localized in line defects in such structures are operating in a pseudogap that covers an even smaller area of (k,ω) space [18

18. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B. 62, 8212–8222 (2000). [CrossRef]

] and it has been shown experimentally that light in such defects is not strongly confined [6

6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ=1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001). [CrossRef]

] as the efficiency of waveguide bends is only acceptable for a very narrow frequency range within the frequency band of the pseudogap.

Recently, a very important observation was made by noticing that a complete three-dimensional bandgap is not a necessary requirement for obtaining high-omnidirectional reflectivity [19

19. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

, 20

20. J. N Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998). [CrossRef]

, 21

21. M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, “Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001). [CrossRef]

, 22

22. B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, “Low-loss infrared dielectric material system for broadband dual-range omnidirectional reflectivity,” Opt. Lett. 26, 1370–1372 (2001). [CrossRef]

]. External high omnidirectional reflectivity has been demonstrated for waves travelling in air and incident upon the one-dimensional multilayer stack of alternating layers with high index contrasts. The symmetries of this structure have been exploited in the design of a cylindrical coaxial omnidirectional waveguide with very promising properties [23

23. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000). [CrossRef] [PubMed]

, 24

24. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotycally single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

], but the manufacture of such a fiber remains a challenge as good accuracy is needed in order to maintain the periodicity in the rotationally symmetric structure. The bandgaps in this fiber are only possible because the angular wave-vector component, k ϕ, goes to zero as the observation point moves away from the fiber-axis (p. 752 in [24

24. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotycally single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

]).Without this fact there would be no bandgap. One-dimensional planar multilayer stacks exhibiting external omnidirectional reflection relative to a surrounding filled with air are efficient reflectors, but their eigenstates do not result in band diagrams with large in-plane bandgaps as shown in [25

25. C. Hooijer, D. Lenstra, and A. Lagendijk, “Mode density inside an omnidirectional mirror is heavily directional but not small,” Opt. Lett. 25, 1666–1668 (2000). [CrossRef]

]. Hooijer et al. have shown that omnidirectional reflection is not a sufficient signature of a photonic andgap [25

25. C. Hooijer, D. Lenstra, and A. Lagendijk, “Mode density inside an omnidirectional mirror is heavily directional but not small,” Opt. Lett. 25, 1666–1668 (2000). [CrossRef]

]. Although dramatic angular redistribution takes place, the mode density of the electromagnetic field is hardly altered within the planar omnidirectional reflection range. The latter is due to the huge dielectric contrast, and the mode characteristics resemble that of a waveguide. This is seen in band diagrams. Therefore, air or low index channel defects positioned within a bulk planar 1D omnidirectional stack cannot localize energy via bandgaps. A planar technology that would allow similar electromagnetic localization as that of the cylindrical omniguide fibers in a massive background of high omnidirectional reflection and avoid the complexity of truely 3D PCs might enable advanced all-optical signal processing in reduced volumes of condensed matter. For many photonic applications, energy transport aside, the cylindrical form is not necessarily an advantage. In this article we present a highly generic class of planar PCs that exhibit large partial bandgaps in the eigenstate spectrum. These gaps are larger than the directionally dependent and polarization-dependent partial gaps of photonic crystal slabs. These wider gaps are obtained at the expense of a little extra processing complexity. Full in-plane gaps are demonstrated numerically. It is not claimed nor demonstrated that total external and internal reflectivity can be achieved in defects created inside the presented structures, but the large intrinsic bandgaps suggest that strong dispersion, waveguide confinement, high Q-cavities, and alternative photonic signal processing can be expected.

Fig. 1. Multilayer stacks of 2D photonic crystals that exhibit large bandgaps in their mode spectrum (from left to right MS2DPC1 and MS2DPC2). MS2DPC1 is made of a 1D dielectric multilayer stack of alternating planar layers A and B filled with materials A and B, respectively, in which a 2D crystal lattice of cylindrical holes is etched and where the holes might be filled with material C. MS2DPC2 is made of a 2D crystal lattice of cylindrical pillars grown in a background material C. These pillars are made of a 1D dielectric multilayer stack of alternating planar layers with materials A and B.

2. Planar multilayer stacks of photonic crystal slabs

Figure 1 shows two types of multilayer stacks of 2D photonic crystals that exhibit large bandgaps in their mode spectrum (from left to right MS2DPC1 and MS2DPC2). MS2DPC1 is made of a 1D dielectric multilayer stack of alternating planar layers of materials A and B, respectively, in which a 2D crystal lattice of cylindrical holes is etched, where these might be filled with materialC.MS2DPC2 is made of a 2D crystal lattice of cylindrical pillars grown in a background material C. These pillars are made of a 1D dielectric multilayer stack of alternating planar layers with materials A and B.

With reference to the Cartesian coordinate system of Fig. 1, a MS2DPC operates in a way that the 1D structure creates a local bandgap for electromagnetic modes of radiation propagating along the z-axis, while the 2D structure creates a local bandgap for electromagnetic modes of radiation propagating along directions parallel to the (x,y) plane, i.e., parallel to the layer surfaces in the stack. When these two bandgaps overlap in {(k,ω)} space, the eigenstates are strongly redistributed in a way that large bandgaps are possible for many directions of propagation. This redistribution does not automatically mean that the total density of states is low, but at least modes are locally suppressed. The 1D periodicity along the z-axis is characterized by Λz=tAz +tBz, where tAz and tBz are the respective layer thicknesses. The 2D crystal lattice is characterized by the nature of the lattice (quadratic, triangular, honeycomb, etc …), the lattice constant Λ, and the material distribution in the spatial unit cell. In the structures of Fig. 1 we define r 0 to be the radius of the filled cylinders, and denote by nA, nB, and nC the respective refractive indices, and by NzΛz the total height of these structures. The following theoretical and numerical investigations relate to structures where Nz is infinite in order to understand the general electrodynamics. In reality Nz is finite and something between four and twenty units will suffice depending on the application and the spatiotemporal radiation rate in a specific configuration. The theoretical investigations in this article rely on numerical results from the solution of a magnetic field formulation of Maxwell’s equations for determining electromagnetic eigenstates of general polarization in structures with spatial inversion symmetry where the fields are expanded in a plane wave basis, and where an iterative method for minimizing the energy functional is applied. There is good agreement between our own results and those obtained with a widely used package for photonic band calculation [26

26. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

].

Fig. 2. Projected band diagrams for all polarization states and dependency of the bandgap on the normalized kz component of the propagation vector for a specific MS2DPC1 design (relevant kz values belong to the interval [0;0.5Λ/Λz]). This structure has a triangular lattice of air holes, nA=4.6, nB=4.1, nC=1, r 0/Λ=0.475 with Λz/Λ=0.5 and tAzz=1- tBzz=0.8. The light blue region represents modes for which electromagnetic radiation is allowed in the bulk MS2DPC1 structure, while the dark cyan region represents forbidden modes where electromagnetic radiation is not allowed in the bulk MS2DPC1 structure. The region hatched with dark lines is denoted R 1, and will be the subject of a discussion in a later paragraph.

The MS2DPC partly emulates purely 2D PCs thanks to the stack of alternating layers that inhibit certain out-of-plane propagating modes, and enable large intrinsic bandgaps as shown in Fig. 2. The latter shows the projected band diagrams for all polarization states and the dependency of the bandgap on the normalized kz component of the propagation vector for a specific MS2DPC1 design (relevant kz values belong to the interval [0;0.5Λ/Λz]). This structure has a triangular lattice of air holes, nA=4.6, nB=4.1, nC=1, r 0/Λ=0.475 with Λz/Λ=0.5 and tAzz=1-tBzz=0.8. The light blue region represents modes for which electromagnetic radiation is allowed in the bulk MS2DPC1 structure, while the dark cyan region represents forbidden modes where electromagnetic radiation in the bulk MS2DPC1 structure is not allowed.

Let neff,z=kz/ω be the z-component of the effective refractive index with ω=Λ/λ, λ being the free-space wavelength. In a homogeneous isotropic bulk dielectric material where nd=nd(λ) is the refractive index, neff,z=ndcosθ, where θ is the angle of incidence with respect to the z - axis. Two lines corresponding to kz/ω=nd=1 (air: solid line), and kz/ω=nd=1.46 (SiO 2: dashed line) are shown. The region defined by kz/ω < nd above the pure-dielectric line for which kz/ω=nd would host all the propagating modes allowed in a corresponding homogeneous dielectric medium. The region hatched with dark lines is denoted R 1, and will be the subject of a discussion in a later paragraph. There is a region of forbidden modes (RFM) for 0.3780 < Λ/λ < 0.4288 approximately (region with white line hatching) for defects made of air created in this MS2DPC1 design, while a RFM for defects made of pure silica is non-existent. However, complete three-dimensional omnidirectional reflection is not a necessary requirement for photonic-bandgap-guiding in channel defects created in a MS2DPC. That is because the (k,ω) mode with localized energy guided by the photonic bandgap need not couple energy into radiation states with higher kz values. This is observed in photonic crystal slabs for in-plane propagating modes [17

17. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B. 60, 5751–5758 (1999). [CrossRef]

, 18

18. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B. 62, 8212–8222 (2000). [CrossRef]

, 5

5. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

, 6

6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ=1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001). [CrossRef]

] although the radiation in line defect bends in the latter is substantial because the in-plane gap for kz=0, which is polarization dependent, is only partial and coupling to radiation modes at line defect bends is allowed. This radiation loss might be minimized in a MS2DPC background because of a full in-plane bandgap for kz=0, which is not the case for photonic crystal slabs. Whether the full in-plane bandgap for kz=0 enables strong in-plane localization and guiding or not in the xy-plane in channels filled with dielectric materials with low or high refractive index shall be investigated in future work. It should be possible to exploit the large RFM for strong localization and dispersion as well as effective redistribution of electromagnetic signals in waveguides created inside or adjacent to a MS2DPC.

The rotationally symmetric omniguides of [23

23. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000). [CrossRef] [PubMed]

, 24

24. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotycally single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

] are based on the 1D omnidirectional reflector of [19

19. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

]. In these structures the bandgap of the bulk material is large and only depends on the 1D multilayer stack design, but rotational symmetry is needed in order to guide electromagnetic modes and avoid significant leakage of energy. In the MS2DPC the bandgap mostly depends on the 2D PC in layers A and B, and are therefore a little narrower, but rotational symmetry in a cylindrical fiber is no longer a requirement for energy-guiding defects with low leakage, at least for strongly localized bound modes. The proposed planar structures offer other photonic storage/manipulation and signal processing opportunities because miniaturized high Q cavities and (k,ω) dispersion managing components can be processed locally on a chip through deposition and etching techniques. This is advantageous for e.g. Wavelength Division Multiplexing (WDM) in telecommunications and sensor technologies.

In this article we primarily focus on MS2DPC1 with a triangular lattice of holes filled with air. Let ωl(kz) and ωu(kz) denote the lower and upper normalized frequencies on the edges of the bandgap, and ωe(kz) represent the eventual end of the bandgap region where ωl(kz)=ωu(kz). The two bandgap edges ωl(kz) and ωu(kz) are ascending functions of kz because larger kz values imply modes that experience more propagation in the regions with lower index of refraction (the modes experience a lower effective index). The curves of ωl(kz) and ωu(kz) are not parallel and the narrowing and eventual closing of the bandgap region at ωe(kz) happens because two fields of two different frequencies propagating in a direction out-of plane experience different effective periodicities along that direction (two such fields incident from an air interface would experience different bending and diffraction angles). One of the more attractive features of a MS2DPC is that creating internal defects filled with a dielectric material might enable energy to be efficiently trapped/localized for a long time in resonators, cavities, or waveguides where strong dispersion enables alternative signal processing.

3. Polarization characteristics of electromagnetic eigenstates

In the 1D omnidirectional reflector the region of omnidirectional reflection is limited by the TM-like polarization [19

19. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

]. By filtering away certain polarization states, or by efficiently coupling into specific polarization states in structures that inhibit the propagation of certain polarized states in well determined directions or planes, we may take advantage of a larger effective bandgap. In order to analyse the effective polarization of eigenstates we estimate the average magnetic/electric energy stored in the Cartesian components of the magnetic/electric field in the unit cell of the crystal such that EFd=d̂·Fd̂|d̂·Fd̂F|F, where d is one of the Cartesian coordinates x, y, or z, and F is the magnetic or the electric field. Figure 3 shows the band diagram for kz=0 for all polarization states of the specific MS2DPC of Fig. 2 near the gap. The eigenstates are clearly polarized in this window of the spectrum, and are categorized into the P 1 and the P 2 polarization types. In Fig. 3 the P 1 bands are even modes, while the P 2 bands are odd modes. Numerically we define the P 1 states as those having EEz<0.1 , and the P 2 states as those having EEz>0.1 . The P 1 polarization roughly has the non-zero components (Hz,Ex,Ey) for kz=0 (TE-like or HE), and the P 2 polarization roughly has the non-zero components (Ez,Hx,Hy) (TM-like or EH) for kz=0 (in-plane propagation). If we define the incident plane as being the plane which is span by the k and ẑ vectors, the classical s polarization (Ez=0 and H parallel to the incident plane) is a subgroup of the P 1 eigenstates. In structures involving photonic crystal slabs however the classification into s and p polarizations is not that practical because of the many dielectric interfaces and surface normals [17

17. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B. 60, 5751–5758 (1999). [CrossRef]

, 18

18. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B. 62, 8212–8222 (2000). [CrossRef]

]. The total polarization-independent bandgap for kz =0 is determined by the TM-like P 2 polarization whereas P 1 states experience a larger bandgap covering 0.2870 < Λ/λ <0.4996 approximately because most of the stored electric energy oscillates in the plane where the dielectric contrast is largest and where the photonic potential barrier is greatest due to many more air-dielectric interfaces which provide strong subwavelength scattering and destructive interference.

Fig. 3. Band diagram for kz=0 for all polarization states of the specific MS2DPC of Fig. 2 near the gap. The eigenstates are clearly polarized in this window of the spectrum, and are categorized into the P 1 and the P 2 polarization types.

The bandgap and polarization characteristics of a MS2DPC for kz=0 are sufficient for a multitude of interesting scientific and technological applications, but what happens to the polarization of eigenstates for kz > 0 ? Unfortunately, the fields are not completely polarized, and lack a clear parity. However, we may consider a scenario where we only couple energy into states where most of the stored electric energy is in the plane with EEz<0.1 (which is the case of P 1). Figure 4 shows the RFM for states with EEz<0.1 resulting from filtering the band diagrams with energy constraints for the MS2DPC1 structure with D=2r 0=0.95Λ and D=2r 0=0.8Λ. Other parameter values are the same as those for the structure of Fig. 2. The light grey region is the RFM for D=0.95Λ with respect to SiO 2 defects, while the dark grey region is the RFM for D=0.8Λ with respect to SiO 2 defects. The upper region hatched with lines with positive slope is the RFM for D=0.95Λ with respect to air defects, while the lower region hatched with lines with negative slope is the RFM for D=0.8Λ with respect to air defects. If we create waveguide defects in a MS2DPC supporting modes with EEz<0.1 and only couple energy into such states we can take advantage of a large region of polarization-dependent energy gap for a wide range of materials in the defect region. Line defects that allow localized guided modes with EEz<0.1 and whose propagation strongly relies on localization by the photonic potential barrier render the MS2DPC even more interesting. Figure 4 shows that if we solely operate with the right polarization the region of forbidden radiation relative to many dielectric materials is enlarged. Furthermore, the r 0/Λ parameter enables the movement of the bandgap region and flexible tailoring of photonic bands in the design procedure. The lower r 0/Λ value implies a higher effective refractive index which is why the energy bands move toward lower ω values. Air defects yield a region of forbidden modes for P 1 states in 0.3147 < ω < 0.4996 for D=0.95Λ and in 0.1990 < ω < 0.3437 for D=0.8Λ, while SiO 2 defects yield a region of forbidden modes for P 1 states in 0.3502 <ω < 0.4996 for D=0.95Λ and in 0.2105 <ω < 0.3437 for D=0.8Λ.

Fig. 4. Regions of forbidden modes for states with EEz<0.1 resulting from filtering the band diagrams with energy constraints for the MS2DPC1 structure with D=2r 0=0.95Δ and D=2r 0=0.8Λ. Other parameter values are the same as those for the structure of Fig. 2. The light grey region is the RFM for D=0.95Λ with respect to SiO 2 defects, while the dark grey region is the RFM for D=0.8Λ with respect to SiO 2 defects. The upper region hatched with lines with positive slope is the RFM for D =0.95Λ with respect to air defects, while the lower region hatched with lines with negative slope is the RFM for D=0.8Λ with respect to air defects.

It is therefore important to distinguish between polarization-dependent bandgaps and polarization-independent bandgaps, and between partial bandgaps and extended bandgaps for kz=0 and kz > 0. In these terms a complete bandgap is a polarization-independent extended bandgap independent of material and k. Recalling results from purely 2D PCs, the TE polarization experiences bandgaps in structures with high index backgrounds with relatively high filling fractions of low index material (thin connecting veins of high index), while the TM polarization experiences bandgaps in structures with low index backgrounds with relatively low filling fractions of high index material (isolated spots of high index). Complete bandgaps in 2D PCs can be found in structures based on the triangular lattice and the honeycomb lattice [3

3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

].

Larger bandgaps are achievable if we disregard the second main polarization type which we from now on call P 2, and concentrate on the P 1 states which we define as states with EEz<0.1 . This allows us to work with lower index materials and lower r 0/Λ values. The dependency of the electrodynamics on the 2D PC parameters is well understood thanks to the theoretical and numerical investigations of the past decade. We might consider using combinations of air, SiO 2, SiN, TiO 2, Si, Ge, and Te depending on the application’s intended operational electromagnetic spectrum and functionality. For the optical regime around the communication wavelength 1.55µm these approximately have a refractive index of 1, 1.46, 2.1, 2.6, 3.44, 4.1, and 4.6, respectively, of course depending on the molecular doping, impurity level, physical/chemical processing, and the thin-film quality (interdiffusion and homogeneity) both optically and mechanically. Te is not the most obvious optical material as it is a britle uniaxial solid with an ordinary and an extraordinary refractive index [27

27. Z. Y. Li and Y. Xia, “Omnidirectional absolute band gaps in two-dimensional photonic crystals,” Phys. Rev. B. 64, 153108–153112 (2001). [CrossRef]

], and metals are lossy at optical frequencies, although in a thin film photonic reflector without waveguide defects lossy metals might not have an impact so catastrophic on the energy flow (this remains to be demonstrated). However, it has recently been demonstrated that low attenuation can be achieved through structural design rather than high-transparency material selection in a photonic bandgap fiber [28

28. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002). [CrossRef] [PubMed]

]. The latter did guide optical power in air surrounded by a cylindrical Bragg-stack containing a highly lossy material at optical frequencies.

Figure 5 shows bandgap regions for states with EEz<0.1 (P 1) together with the air line and the SiO 2 line for several material systems (MA,MB) (material A and material B) with (Ge,Si) (filled square symbols), (Si,SiN) (open square symbols), (Si,SiO 2) (filled circle symbols), (SiN,SiN) (Si-rich SiN with index 2.35, and SiN with index 2.1) (open circle symbols), and (SiN,SiO 2) (filled triangle symbols), in a MS2DPC where nC=1, r 0/Λ=0.41, Λz/Λ=0.4, and tAzz=1 -tBzz=0.5. In parentheses we write the index contrasts of the respective material systems. We define n¯eff=(tAznA +tBznB)/Λz and Δ=nA -nB. We have ( eff, Δ)=(3.704,0.66) for (Ge,Si), ( eff, Δ)=(2.636,1.34) for (Si,SiN), ()=(2.252,1.98) for (Si,SiO 2), ( eff, Δ)=(2.2,0.25) for (SiN,SiN), and ( eff, Δ)=(1.716,0.64) for (SiN,SiO 2). Higher eff values cause an increase of the effective refractive index of eigenstates, move the band of forbidden energies toward lower frequencies, and widens the 2D bandgaps. Furthermore, the RFM relative to many dielectric materials is enlarged and reaches higher kz values. The (SiN,SiN) and (Si,SiO 2) material systems yield similar eff, but the (Si,SiO 2) structure has a higher Δ value and results in a larger RFM. Similar electrodynamic behaviour is found for MS2DPC2, where a TM-like polarization state inherits the properties of P 1 in MS2DPC1. The honeycomb lattice of high index pillars in air is relevant for a polarization-independent RFM in MS2DPC2.

4. Conclusion

Both polarization-dependent and polarization-independent forbidden frequency gaps are possible in multilayer stacks of two-dimensional (2D) photonic crystals. In such structures the eigenstate spectrum exhibits larger bandgaps than photonic crystal slabs for a large set of propagation directions. The presented structures are amenable to large-scale industrial production, are less complex than truly three-dimensional (3D) crystals, and can host local intrinsic defects created within them. Such defects and the large intrinsic bandgaps suggest that strong dispersion, waveguide confinement, high Q-cavities, and alternative photonic signal processing can be exploited. Future investigations should clarify such matters together with the internal and external reflectivity of the presented structures. Energy distributions due to different excitations and mode coupling must be investigated in future work.

Fig. 5. Bandgap regions for states with EEz<0.1 (P 1) together with the air line and the SiO 2 line for severalmaterial systems (A,B) with (Ge,Si) (filled square symbols), (Si,SiN) (open square symbols), (Si,SiO 2) (filled circle symbols), (SiN,SiN) (Si-rich SiN with index 2.35, and SiN with index 2.1) (open circle symbols), and (SiN,SiO 2) (filled triangle symbols), in a MS2DDPC where nC=1, r 0/Λ=0.41, Λz/Λ=0.4, and tAzz=1-tBzz=0.5. In parentheses we write the index contrasts of the respective material systems.

Acknowledgments

I thank NKT Research & Innovation for supporting the photonics program, and particularly K. E. Mattsson, B. H. Larsen, and L. P. Nielsen for valuable discussions in connection with the present work.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

3.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).

4.

S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998). [CrossRef] [PubMed]

5.

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-µm wavelength,” Opt. Lett. 25, 1297–1299 (2000). [CrossRef]

6.

E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ=1.55µm wavelengths,” Opt. Lett. 26, 286–288 (2001). [CrossRef]

7.

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Direct measurement of the quality factor in a two-dimensional photonic-crystal microcavity,” Opt. Lett. 26, 1903–1905 (2001). [CrossRef]

8.

T. Sϕndergaard and K. H. Dridi, “Energy flow in photonic crystal waveguides,” Phys. Rev. B. 61, 15688–15696 (2000). [CrossRef]

9.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990). [CrossRef] [PubMed]

10.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991). [CrossRef] [PubMed]

11.

S. G. Johnson and J. D. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000). [CrossRef]

12.

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, “Design of three-dimensional photonic crystals at submicron lengthscales,” Appl. Phys. Lett. 65, 1466–1468 (1994). [CrossRef]

13.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional photonic bandgap crystals at near-infrared wavelengths,” Science 289, 604–606 (2000). [CrossRef] [PubMed]

14.

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133–1135 (2001). [CrossRef] [PubMed]

15.

E. Kuramochi, M. Notomi, T. Tamamura, T. Kawashima, S. Kawakami, J. Takahashi, and C. Takahashi, “Drilled alternating-layer structure for three-dimensional photonic crystals with a full band gap,” J. Vac. Sci. Technol. B 18, 3510–3513 (2000). [CrossRef]

16.

S. Kawakami, E. Kuramochi, M. Notomi, T. Kawashima, J. Takahashi, C. Takahashi, and T. Tamamura, “A new fabrication technique for photonic crystals: nanolithography combined with alternating-layer deposition,” Opt. Quantum Electron. 34, 53–61 (2002). [CrossRef]

17.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B. 60, 5751–5758 (1999). [CrossRef]

18.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B. 62, 8212–8222 (2000). [CrossRef]

19.

Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

20.

J. N Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998). [CrossRef]

21.

M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, “Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001). [CrossRef]

22.

B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, “Low-loss infrared dielectric material system for broadband dual-range omnidirectional reflectivity,” Opt. Lett. 26, 1370–1372 (2001). [CrossRef]

23.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000). [CrossRef] [PubMed]

24.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotycally single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

25.

C. Hooijer, D. Lenstra, and A. Lagendijk, “Mode density inside an omnidirectional mirror is heavily directional but not small,” Opt. Lett. 25, 1666–1668 (2000). [CrossRef]

26.

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

27.

Z. Y. Li and Y. Xia, “Omnidirectional absolute band gaps in two-dimensional photonic crystals,” Phys. Rev. B. 64, 153108–153112 (2001). [CrossRef]

28.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002). [CrossRef] [PubMed]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(130.0130) Integrated optics : Integrated optics
(230.0230) Optical devices : Optical devices

ToC Category:
Research Papers

History
Original Manuscript: March 24, 2003
Revised Manuscript: May 7, 2003
Published: May 19, 2003

Citation
Kim Dridi, "Intrinsic eigenstate spectrum of planar multilayer stacks of two-dimensional photonic crystals," Opt. Express 11, 1156-1165 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-10-1156


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References

  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  2. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R .D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N. J., 1995).
  4. S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, �??Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,�?? Science 282, 274-276 (1998). [CrossRef] [PubMed]
  5. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, �??Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1:5¡ mm wavelength,�?? Opt. Lett. 25, 1297-1299 (2000). [CrossRef]
  6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, �??Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at l= 1:55 mm wavelengths,�?? Opt. Lett. 26, 286-288 (2001). [CrossRef]
  7. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, �??Direct measurement of the quality factor in a two-dimensional photonic-crystal microcavity,�?? Opt. Lett. 26, 1903-1905 (2001). [CrossRef]
  8. T. Søndergaard and K. H. Dridi, �??Energy flow in photonic crystal waveguides,�?? Phys. Rev. B. 61, 15688-15696 (2000). [CrossRef]
  9. K. M. Ho, C. T. Chan, and C. M. Soukoulis, �??Existence of a photonic gap in periodic dielectric structures,�?? Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
  10. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, �??Photonic band structure: the face-centered-cubic case employing nonspherical atoms,�?? Phys. Rev. Lett. 67, 2295-2298 (1991). [CrossRef] [PubMed]
  11. S. G. Johnson and J. D. Joannopoulos, �??Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,�?? Appl. Phys. Lett. 77, 3490-3492 (2000). [CrossRef]
  12. S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, �??Design of three-dimensional photonic crystals at submicron lengthscales,�?? Appl. Phys. Lett. 65, 1466-1468 (1994). [CrossRef]
  13. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, �??Full three-dimensional photonic bandgap crystals at nearinfrared wavelengths,�?? Science 289, 604-606 (2000). [CrossRef] [PubMed]
  14. O. Toader and S. John, �??Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,�?? Science 292, 1133-1135 (2001). [CrossRef] [PubMed]
  15. E. Kuramochi, M. Notomi, T. Tamamura, T. Kawashima, S. Kawakami, J. Takahashi, and C. Takahashi, �??Drilled alternating-layer structure for three-dimensional photonic crystals with a full band gap,�?? J. Vac. Sci. Technol. B 18, 3510-3513 (2000). [CrossRef]
  16. S. Kawakami, E. Kuramochi, M. Notomi, T. Kawashima, J. Takahashi, C. Takahashi, and T. Tamamura, �??A new fabrication technique for photonic crystals: nanolithography combined with alternating-layer deposition,�?? Opt. Quantum Electron. 34, 53-61 (2002). [CrossRef]
  17. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B. 60, 5751-5758 (1999). [CrossRef]
  18. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystal slabs,�?? Phys. Rev. B. 62, 8212-8222 (2000). [CrossRef]
  19. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, �??A dielectric omnidirectional reflector,�?? Science 282, 1679-1682 (1998). [CrossRef] [PubMed]
  20. J. N.Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, �??Omnidirectional reflection from a one-dimensional photonic crystal,�?? Opt. Lett. 23, 1573-1575 (1998). [CrossRef]
  21. M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, �??Dielectric omnidirectional visible reflector,�?? Opt. Lett. 26, 1197-1199 (2001). [CrossRef]
  22. B. Temelkuran, E. L. Thomas, J. D. Joannopoulos, and Y. Fink, �??Low-loss infrared dielectric material system for broadband dual-range omnidirectional reflectivity,�?? Opt. Lett. 26, 1370-1372 (2001). [CrossRef]
  23. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, �??An all-dielectric coaxial waveguide,�?? Science 289, 415-419 (2000). [CrossRef] [PubMed]
  24. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Solja¡ci´c, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, �??Low-loss asymptotycally single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a> [CrossRef] [PubMed]
  25. C. Hooijer, D. Lenstra, and A. Lagendijk, �??Mode density inside an omnidirectional mirror is heavily directional but not small,�?? Opt. Lett. 25, 1666-1668 (2000). [CrossRef]
  26. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�?? Opt. Express 8, 173 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
  27. Z. Y. Li and Y. Xia, �??Omnidirectional absolute band gaps in two-dimensional photonic crystals,�?? Phys. Rev. B. 64, 153108-153112 (2001). [CrossRef]
  28. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, �??Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,�?? Nature 420, 650-653 (2002). [CrossRef] [PubMed]

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