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, 2]. 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] and through experimental validation [4, 5, 6, 7]. 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] 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, 10, 11, 12, 13, 14, 15, 16]. 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, 18, 5, 6]. 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] and it has been shown experimentally that light in such defects is not strongly confined [6] 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, 20, 21, 22]. 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, 24], 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]).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]. Hooijer et al. have shown that omnidirectional reflection is not a sufficient signature of a photonic andgap [25]. 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.

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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 Λ_z=t_Az +t_Bz, where t_Az and t_Bz 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 n_A, n_B, and n_C the respective refractive indices, and by N_zΛ_z the total height of these structures. The following theoretical and numerical investigations relate to structures where N_z is infinite in order to understand the general electrodynamics. In reality N_z 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].

 

Fig. 2. Projected band diagrams for all polarization states and dependency of the bandgap on the normalized k_z component of the propagation vector for a specific MS2DPC1 design (relevant k_z values belong to the interval [0;0.5Λ/Λ_z]). This structure has a triangular lattice of air holes, n_A=4.6, n_B=4.1, n_C=1, r ₀/Λ=0.475 with Λ_z/Λ=0.5 and t_Az/Λ_z=1- t_Bz/Λ_z=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.

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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 k_z component of the propagation vector for a specific MS2DPC1 design (relevant k_z values belong to the interval [0;0.5Λ/Λ_z]). This structure has a triangular lattice of air holes, n_A=4.6, n_B=4.1, n_C=1, r ₀/Λ=0.475 with Λ_z/Λ=0.5 and t_Az/Λ_z=1-t_Bz/Λ_z=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 n_eff,z=k_z/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 n_d=n_d(λ) is the refractive index, n_eff,z=n_dcosθ, where θ is the angle of incidence with respect to the z - axis. Two lines corresponding to k_z/ω=n_d=1 (air: solid line), and k_z/ω=n_d=1.46 (SiO ₂: dashed line) are shown. The region defined by k_z/ω < n_d above the pure-dielectric line for which k_z/ω=n_d 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 k_z values. This is observed in photonic crystal slabs for in-plane propagating modes [17, 18, 5, 6] although the radiation in line defect bends in the latter is substantial because the in-plane gap for k_z=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 k_z=0, which is not the case for photonic crystal slabs. Whether the full in-plane bandgap for k_z=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, 24] are based on the 1D omnidirectional reflector of [19]. 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 ω_l(k_z) and ω_u(k_z) denote the lower and upper normalized frequencies on the edges of the bandgap, and ω_e(k_z) represent the eventual end of the bandgap region where ω_l(k_z)=ω_u(k_z). The two bandgap edges ω_l(k_z) and ω_u(k_z) are ascending functions of k_z because larger k_z 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 ω_l(k_z) and ω_u(k_z) are not parallel and the narrowing and eventual closing of the bandgap region at ω_e(k_z) 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]. 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 $E_{F_d}=\frac{〈\hat{\mathbf{d}}·\mathbf{F}\hat{\mathbf{d}}|\hat{\mathbf{d}}·\mathbf{F}\hat{\mathbf{d}}〉}{〈\mathbf{F}|\mathbf{F}〉},$ 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 k_z=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 $E_{E_z}<0.1$ , and the P ₂ states as those having $E_{E_z}>0.1$ . The P ₁ polarization roughly has the non-zero components (H_z,E_x,E_y) for k_z=0 (TE-like or HE), and the P ₂ polarization roughly has the non-zero components (E_z,H_x,H_y) (TM-like or EH) for k_z=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 (E_z=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, 18]. The total polarization-independent bandgap for k_z =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 k_z=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.

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The bandgap and polarization characteristics of a MS2DPC for k_z=0 are sufficient for a multitude of interesting scientific and technological applications, but what happens to the polarization of eigenstates for k_z > 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 $E_{E_z}<0.1$ (which is the case of P ₁). Figure 4 shows the RFM for states with $E_{E_z}<0.1$ 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 $E_{E_z}<0.1$ 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 $E_{E_z}<0.1$ 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 $E_{E_z}<0.1$ 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.

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It is therefore important to distinguish between polarization-dependent bandgaps and polarization-independent bandgaps, and between partial bandgaps and extended bandgaps for k_z=0 and k_z > 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].

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 $E_{E_z}<0.1$ . 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], 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]. 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 $E_{E_z}<0.1$ (P ₁) together with the air line and the SiO ₂ line for several material systems (M_A,M_B) (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 n_C=1, r ₀/Λ=0.41, Λ_z/Λ=0.4, and t_Az/Λ_z=1 -t_Bz/Λ_z=0.5. In parentheses we write the index contrasts of the respective material systems. We define n¯_eff=(t_Azn_A +t_Bzn_B)/Λ_z and Δn¯=n_A -n_B. We have (n¯ _eff, Δn¯)=(3.704,0.66) for (Ge,Si), (n¯ _eff, Δn¯)=(2.636,1.34) for (Si,SiN), (n¯,Δn¯)=(2.252,1.98) for (Si,SiO ₂), (n̄ _eff, Δn¯)=(2.2,0.25) for (SiN,SiN), and (n¯ _eff, Δn¯)=(1.716,0.64) for (SiN,SiO ₂). Higher n¯ _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 k_z values. The (SiN,SiN) and (Si,SiO ₂) material systems yield similar n¯ _eff, but the (Si,SiO ₂) structure has a higher Δn¯ 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 $E_{E_z}<0.1$ (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 n_C=1, r ₀/Λ=0.41, Λ_z/Λ=0.4, and t_Az/Λ_z=1-t_Bz/Λ_z=0.5. In parentheses we write the index contrasts of the respective material systems.

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