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In this paper we theoretically study refraction effects in the 2D square-like Archimedean photonic crystals (32, 4, 3, 4) and (4, 82) made of dielectric rods in air. In addition, we investigated a band isotropy and band gap structure in these lattices. We compared the square and square-like structures as well, their refraction characteristics, zone structures and the level of band and band gap isotropy (bandwidth and band gap dependence on the wave vector). We found that square-like structures can have some advantages over the square ones regarding the completeness of the gap, its isotropy and the gap width. Also, due to the same square primitive unit cell and the first Brillouin zone, the square and square-like lattices have similar optical response in lower bands.
©2008 Optical Society of America
1. Introduction
Photonic crystals (PhC) have intensively been studied these years taking into account different possibilities for control of light. The main concept of PhCs concerns an analogy with electrons in conventional crystals. Namely, for both electrons in a material with periodic arrangement of atomic potentials (e.g. semiconductors) and photons in PhCs with a periodically modulated dielectric medium band-gaps exist. There are many interesting phenomena in PhCs which have been both theoretically and experimentally studied including a lot of already tested applications (antennas, waveguides, resonators, filters, lasers, light-emitting diodes or photonic integrated circuits) that have been emerged so far [1–10]. For example, PhCs are the source of several attractive effects like suppression of spontaneous light emission, localization of light, negative refraction, left-handedness [11–18], the light focusing below the diffraction limit [20–23] and band isotropy [24–27] which are partially studied in this communication.
The Archimedean lattices or tilings shown in Fig. 1 make an infinite tessellation consisting of regular convex polygons (Bravais lattices). The polygons are placed edge-to-edge to each other and are not necessarily identical. Although the Archimedean 2D lattices have been known since ancient times, Johannes Kepler was the first who gave a complete description of all 11 possible tilings in his Harmonices Mundi in 1619 [28]. Last three lattices in Fig. 1 are regular ones (formed by just one type of polygons). These are well known, square, triangular and honeycomb lattices which have numerously been exploited so far in research of photonic crystals. The others, although more complicated and far less used in PhC research comparing to regular Archimedean lattices [18, 19, 24, 25, 29], could also be very useful structures at least as an alternative or even as structures with improved PhC characteristics. Until now Archimedean lattices have mostly been studied in mathematics and statistical mechanics [30]. Therefore, the mathematical Grünbaum-Shephard notation [31] is used regarding shapes and number of polygons around each vertex, (m1 n1, m2 n2,…). Starting from the smallest polygon and going clockwise around a vertex, the numbers mi denote the number of sides of each polygon, and the superscript ni refers to the number of equal adjacent polygons. For example the (3, 3, 4, 3, 4) or (32, 4, 3, 4) and (3, 2, 4, 3, 4) lattices are of the same species but of different type. In this work we study the square-like Archimedean lattices (32, 4, 3, 4) and (4, 82) and compare them with a standard square lattice concerning, zone structure, band and band gap isotropy, right-handed negative and left-handed refraction.
Materials with isotropic optical properties are desired for some special applications like LED’s. The isotropy of bandgaps in some Archimedean and Archimedean-like lattices (quasicrystals) was found in structures with higher order of local symmetries than in the Bravais lattices [24–26]. It was found that isotropy increases with the order of symmetry meaning that those crystals are much less optical sensitive to light propagation directions [24].
In this communication we are also focused on negative refraction and left-handednes which were first theoreticaly predicted in the seminal paper by Veselago [3] who analysed materials which had both the electrical permittivity and the magnetic permeability µ simultaneously negative. In those materials index of the refraction is negative while the Pointing vector S and the wave vector k (phase velocity) are anti-parallel producing backward waves. Since E, H and k form a left-handed set, Veselago called the new materials ‘left-handed’ (LH). Nowadays these materials are commonly called metamaterials. First artificial magnetic material with negative µ was devised for microwaves by Pendry [11], and followed by a fabrication of a metamaterial in the Veselago sense (both ε and µ were negative) which resulted in experimental verification of negative refraction [12]. In contrast to Veselago’s meta-materials, photonic crystals consist usually of periodically modulated dielectric lossless material where µ>0. Also, in PhCs diffraction effects (λ~ the lattice constant) are pronounced and can produce effective negative refraction or even negative index and LH effects as in metamaterials. The necessary condition to be fulfilled for left-handedness in PhCs is ν ph·ν gr<0 or equivalently k·S<0, since v gr is parallel to S in large enough PhCs, and for right-handedness (RH) is ν ph·ν gr>0 or equivalently k·S>0. Also, necessary conditions for negative refraction require that the equifrequency contours (EFCs) in PhCs are both convex (with an inward gradient) and larger than the corresponding EFCs in air. Additionally, λo≡c/f≥2·as (as is the surface lattice constant) in order to avoid higher order Bragg diffractions out of the crystal.
2. Results
In this paper we, for the first time, present a comprehensive analysis of optical properties in square-like Archimedean, ‘ladybug’ lattice (32, 4, 3, 4) and ‘bathroom’ tile (4, 82) PhCs (see Fig. 1). We compare their PhC properties with the characteristics of a regular Archimedean square lattice. In Fig. 2 the square-like structures are showed together with their primitive unite cells and the first Brillouin zones. They have a primitive unit cell and the first Brillouin zone of the same shape like a normal square one. The symmetry points also stay the same. The squre-like structures belong to the family of the square plane symmetry groups according to International tables for Crystalography [31]. Short Hermann-Mauguin symbols for the (32, 4, 3, 4) is p4gm and p4mm for the (4, 82) structure. The square-like structures in Fig. 2, in contrast to the square one, have 4 ‘atoms’ - rods per unit cell (lattice with a basis) and the same square Bravais lattice. These four rods make a square which is rotated by 15° (32, 4, 3, 4) and 45° (4, 82) with respect to the unit cell mesh.
Fig. 2. Structure, primitive unit cell and the first Brillouin zone for a) (32, 4, 3, 4) (ladybug) and b) (4, 82) (bathroom tile).
We analysed optical properties (band structures and refraction) in 2D dielectric PhCs with a dielectric constant ε=12.96 (GaAs), or n=3.6. The lattice are made of dielectric rods in air with the ratio, r/a, spanning the range (0.1–0.5) (where ‘r’ is rod radius and ‘a’ is the smallest distance between the rods which is the same for all three lattices). Also, it is assumed that a dielectric material is linear, homogeneous and lossless. As a tool for analysis we used RSOFT [33] package (BandSolve for the plane wave method (PWE) and FullWave for the finite-difference time-domain (FDTD) simulations). The package has its own CAD system which greatly facilitates the seting-up of the new structures. Both methods are well described in standard textbooks e.g. [2] for PWE and [34, 35] for FDTD.
The band gap maps for the first 15 bands, TM/TE polarization and all three lattices are presented in Fig. 3 as a function of rod radius. From this figure one can see that complete band gaps emerge in the (4, 82) and the square PhC. Broad TM gaps appear in the (32, 4, 3, 4) lattice. The (4, 82) one has a very large complete band gap for f̄≈0.45 (all frequencies are normalized as f̄=ωa/2πc=a/λ) for which Δf̄ max=0.03, and a small one for f̄≈0.33, with Δf̄ max=0.003. A complete band gap (a gap that exists for both the TM and TE polarizations) for square lattice is found for f̄≈0.5, with Δf̄ max=0.005.
Fig. 3. Gap maps for a) (32, 4, 3, 4); b) (4, 82) and c) square PhC lattices for the TE/TM polarization. The insets in b) and c) represent the enlarged areas where complete gaps take place in these structures.
Because the TM gaps dominate, like in the other rods in air structures [1], we perform additional analyses. The gap-to-midgap ratios and the gap widths of all three analyzed structures are presented in Fig 4. This figure shows that the largest gap-to-midgap ratio (Δf̄/f̄ 0)max is 0.475 for the (32, 4, 3, 4) lattice, wheras these ratios are 0.236 and 0.411 for the (4, 82) PhC and the square lattice, respectively. Furthermore, the maximal value of the gap-to-midgap ratio for all three structures are around r/a≈0.17. Figure 4 shows that the maximal gap (Δf̄ max) is around r/a≈0.16 for all three structures and that the largest maximal gap has been observed in the (32, 4, 3, 4) lattice with value of Δf̄ max=0.18. The other gap values are 0.095 and 0.15 for the (4 82) and square ones, respectively. In addition, based on the PWE calculations, we found the threshold values of dielectric constants which enable gaps to appear in both polarizations. The values of the εmin are 2.11, 2.88 and 2.8 for (32, 4, 3, 4), (4, 82) and square lattices, respectively and all of them appear for the TM polarization. Regarding the size of the gaps and their completeness, from Figs. 3 and 4 we conclude that the (32, 4, 3, 4) and (4, 82) structures have an advantage over the square one for applications.
Fig. 4. The TM mode gap-to-midgap ratio (Δf̄/f̄ 0) and gap width for a) the (32, 4, 3, 4); b) (4, 82) and c) square PhC lattices. Different gaps between the bands are denoted by appropriate band numbers.
In Fig. 5 we present the dispersion relations for r/a=0.3 and analyse band and band gap isotropy. This figure shows that isotropic bands emerge for the (32, 4, 3, 4) lattice, for the frequencies around 0.53 and 0.73 for the TE modes and around 0.39 and 0.74 for the TM modes. In the (4, 82) lattice the TE modes are isotropic around 0.39 and 0.73 i.e. near 0.38 and 0.55 for the TM bands. In the square lattice the TM modes are isotropic around 1.09 and around 0.93 for TE. For all three lattices and both polarizations relative variations of these bands with propagation direction are below 1%. In addition, analyzing band gap isotropy for the TM polarization, we found that relative band gap variations with the propagation direction are only 3% for the square-like structures and 9% for the square one. Also, from Fig. 5, we conclude that for the TM polarization, all three lattices have three dominant gaps centered at the same normalized frequencies, f̄ 0≈0.25, 0.45 and 0.65. These values are independent from cell size and the number of atoms per unit cell and they are related to the distance between the neighbouring rods which is the same as in Archimedean-like tiling case [24]. This effect can also be explained taking into account Mie scattering on a single rod as considered in quasi-periodic photonic crystal structures [36, 37].
Fig. 5. Band and band gap isotropy for a) the (32, 4, 3, 4); b) (4, 82) and c) square lattices for r/a=0.3.
Using the PWE and FDTD calculations we analyzed refraction (right-handed negative and left-handed) in our three lattices trying to approve that lattice structure similarities are the basis for the same optical response. For this purpose we analyze the band structure and EFC calculations for all three lattices in the case of the first four TM/TE bands (since in the higher bands the Bragg diffraction condition prevents a single beam propagation) and for the ratio r/a between 0.1 and 0.5. The band structure analysis for the TM modes shows that the TM1, TM2 and TM3 bands have a similar dependance of the k-vector, for all three lattices, and are almost the same near the symmetry points. We found the same situation for the TE1, TE2 and TE3 modes. For some symmetry points and specific r/a, we noticed a minor deviation from this rule which is less than 10 % of all cases. The EFC calculations of the same bands, polarizations and the r/a ratios showed that the square-like lattices have very similar EFCs comparing to the square one, especially for the first three bands (see Figs. 6 and 7 with EFCs of the first three TM and TE bands). For example, the TM1 mode in the vicinity of the M point for all ratios, have a similar convex, round and inward gradient EFC for both the square-like and the square one structures making that way a necessary condition for right-handed negative (RH-) refraction [17]. The similar situation is observed for EFCs, in the vicinity of Γ symmetry point, where the TM2 mode leads to left-handed negative (LH-) refraction. The EFCs of TM3 mode around the M point in the square-like lattices are also similar to the square one, with an outward gradient, concave and star-like shapes. For these frequencies around M point an interesting left-handed positive (LH+) refraction emerges [17]. The EFCs for the TM4 modes are different for all three structure and r/a ratios. It is worth mentioning that negative refraction (both RH- and LH-) for TM4 mode only occurs in the square-like lattices.
Fig. 6. The EFC plots for the first three TM bands and r/a=0.48. The columns correspond to the (32, 4, 3, 4), (4, 82) and square lattices, respectively. The rows stand for the TM1, TM2 and TM3 band, respectively.
Similar situation arises in the first four TE bands (see Fig. 7) with some exceptions. Namely, in the case of the square PhC, the TE1 band enables just a very narrow incident angle RH- refraction around the M point since the corresponding EFCs are starlike. The pronounced TE1 RH- refraction exists only in the square-like structures. On the other hand, similarly as for the TM2 band, there is the LH- refraction in the TE2 band for frequencies around Γ. In addition, in the case of the TE2 band and r/a=(0.2–0.48), the square PhC shows the RH- refraction around the M point. It is worth emphasizing that the TE3 band exhibit the LH+ refraction wheras for the TE4 band there is no refraction similarities taking into account their EFCs. Concerning the TE4 band, LH- refraction exists just in the ladybug and the square PhC lattices.
Fig. 7. The EFC plots for the first three TE bands and r/a=0.48. The columns correspond to the (32, 4, 3, 4), (4, 82) and square lattices, respectively. The rows stand for the TE1, TE2 and TE3 band, respectively.
The analysis of refraction properties based on the EFC calculations are graphically presented in Fig. 8 and confirmed by FDTD simulations of wave propagation in Fig. 9. As an example, in the Fig. 8, the TE2 mode EFC plots for the square-like and square PhC lattices are presented. As it was said earlier, the EFCs for TE2 for all structures are convex and round in the vicinity of the Γ point (especially in the (4, 82) and square PhC lattices) with an inward gradient. According to the directions of kph and νgr, from the Fig. 8, we can conclude that k ph·ν gr<0 which lead to the LH- type of refraction. Also from Figs. 8 and 9, similarities of the EFCs in all three structures lead to similar optical responses (regarding similar incident, refracted angles and the same type of refraction, even if the structures have different interfaces). It is interesting there is the LH- refraction with negligible reflection in the square lattice, like in the negative-index metamaterials with n=-1 [see Fig. 9(c)].
Fig. 8. The EFC plots for the TE2 band and r/a=0.48. a), b) and c) are for (32, 4, 3, 4), (4, 82) and square lattices respectively. The black circles and arows indicates the incident air wave EFC for f̄=0.185 (λ/a=5.4), αair=27° (ΓX interface); f̄=0.145 (λ/a=6.9), αair=30° (ΓM) and f̄=0.27 (λ/a=3.7), αair=30° (ΓM) for (32, 4, 3, 4), (4, 82) and square lattices, respectively. Blue and green arrows correspond to the phase and group velocities. The parallel component of k is conserved in refraction (Snell’s law).
Fig. 9. Propagation of the wave patern for the TE2 mode made by FDTD simulations corresponding to the EFC analysis in Figs. 8(a), 8(b) and 8(c) are for the (32, 4, 3, 4), (4, 82) and square lattices, respectively. The red, blue and yellow vectors stand for air, group and phase wave velocity, respectively.
Our results are summarized in Tables 1 and 2 for both the TM and TE modes in the frequency ranges where the RH- and LH (LH- and LH+) types of refractions take place in the square-like Archimedean and square PhC lattices. The results are presented for the r/a=0.48, the first four bands, both polarizations and interfaces (ΓX and ΓM).
Table 1. Frequency ranges for RH- and LH in (32, 4, 3, 4), (4, 82) and the square lattice for the first four TM modes and the ΓM and ΓX incident interfaces.
Table 2. Frequency ranges for negative and left-handed refraction in (32, 4, 3, 4), (4, 82) and square lattices for the first four TM modes and incident interfaces ΓM and ΓX.
Values are presented in normalized units. From Table 1 one can see that in the square-like lattices, negative refraction (RH- and LH-) occurs in all TM bands except for TM3 where there is the LH+ refraction. In the square PhC, negative refraction appears just for the first two bands. Moreover, negative refraction appears in a lower frequency range in the square-like structures and in a higher one for the square lattices, mainly for the ΓM interface. For the ΓX interface there is no overlapping between the LH- and LH+ refraction in all three structures.
Taking into account the Brag diffraction condition (a single beam propagation through PhCs) the TE modes in the square-like lattices with RH- and LH refractions have a narrower frequency ranges than for the TM modes, especialy for the (4, 82) lattice. These refraction effects emerge for lower frequencies in the case of the square-like structures and for higher frequencies in the square lattice PhCs.
3. Summary
In this paper we analyse optical properties of dielectric 2D square-like Archimedean lattices having a square unit cell. As a model dielectric material we use GaAs (ε=12.96) rods (r/a=0.1–0.5) in air with same distance between neighboring cylinders. Since the square-like lattices, and the square one, have the same square primitive unit cell (the square first Brillouin zone as well) we predict that these structural similarities will influence the similar optical properties.
Analyzing gap maps for the different r/a ratios we conclude that square-like lattices, as a PhC materials, have certain advantages for applications over the square one (regarding the size of the gaps and their completeness). Concerning the band structure we found that all three structures possess, for both polarizations, isotropic bands in which relative frequency variations with propagation direction are below 1%. In the case of the TM polarization we found isotropic band gaps especial in the square-like lattices where relative band gap variation with propagation direction can only be 3%. As expected, the square lattice PhC has a higher gap variation of 9% due to lower local symmetry. The dominant lower gaps in all structures are centered around the same normalized frequencies.
The analysis of the refraction effects was performed calculating the band structure (for first four bands, r/a=0.1–0.5), EFCs, and the FDTD light propagation simulations through the lattices. We found that first three bands have a similar dependance of the k-vector, for all three lattices and are almost the same near the symmetry points. In addition, the square-like lattices have the EFCs very similar with square lattice especial near the same symetry points. As a result, the same type of refraction (RH- and LH) appear in the square-like lattices as well as in the square one for both polarizations and in the first three bands. The TE1 mode makes a single exception from that rule. Moreover, for both polarizations, all three lattices and the same incident angle we get similar angle of the refraction (in a particular band). Also, the FDTD simulations shows negative refraction without reflection in the square lattice (TE2) as in NIMs. In addition, the RH- and LH refractions appear in lower frequency ranges for square-like lattices and in higher frequency range for the square lattice and both polarizations. We confirmed that, the structural similarities among the structures cause similar optical response for lower bands. Based on above described properties the square-like Archimedean lattices can represent a good alternative or a supplement in applications to the well known square structure. The present work could help in the understanding and the design of the new optical devices.
Acknowledgments
This work was supported by the Serbian Ministry of Science Project 141047. K. H. is grateful for partial support from European Community Project N2T2 (IST-NMP2-2004-017481) and to the Linz Center for supercomputing (Johann Messner), and to Heinz Syringer from Photeon Technologies for financial support.
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