Optical Properties of GaAs 2 D Archimedean Photonic Lattice Tiling With the p 4 g Symmetry Đ

In this paper we present our investigation of 2D Archimedean lattice photonic crystals with p4g space group symmetry. The structures are made of GaAs both as air holes and dielectric rods in air. In order to analyze the photonic crystal optical properties we performed calculations of the band structures, equi-frequency contours and electromagnetic propagation through the basic structures and waveguides. In addition, we investigated negative refraction and left-handedness in the p4g photonic crystal.


Introduction
Photonic crystals (PhC) are periodically structured electromagnetic media, generally possessing photonic band gaps i.e. frequency ranges where light cannot propagate through the structure.This periodicity, whose lengthscale is proportional to the wavelength of light in the band gap, is the electromagnetic analogue of a crystalline atomic lattice, where the latter acts on the electrons to produce the familiar band gaps as in semiconductors.Photonic crystals are also known as the semiconductor counterparts for light, since they exhibit the ability, when engineered appropriately, to mold and control the propagation of electromagnetic (EM) waves [1,2].Until now, based on properties mentioned above, PhCs have found many applications in optics and electronics.Additionally PhCs, similarly to metamaterials [3], exhibit interesting properties regarding negative refraction, left-handedness and light focusing [4][5][6][7][8][9][10][11].Some of these properties are the subject of this paper too.
Already mentioned, in PhCs, light propagation is forbidden in a certain wavelength range, due to bandgap presence.Unlike in classic optical waveguides where light is guided because of total internal reflection in a two-dimensional crystal composed of parallel highrefractive index rods in a low-index background a line defect can be formed by removing a row of these rods, which can act as a waveguide for frequencies in the bandgap of the crystal (photonic crystal optical waveguides) [12].
Among the main research topics in PhC physics are certainly the effects of negative refraction and left-handedness that can be realized both in metamaterials and photonic crystals.Left-handed materials are first proposed in the seminal work by Veselago [3] who proposed materials with simultaneously negative permittivity and permeability.Such a material is often called a left-handed material (LHM), since the electric field, the magnetic field and the wave vector of an electromagnetic wave propagating in it form a left-hand system.In contrast to Veselago's metamaterials, photonic crystals are usually lossless dielectric materials with locally positive permittivity and permeability where diffraction effects make them to be LHM.If the most of the propagating energy goes into the zero-order diffraction beam [10], one can apply Snell's law for an effective index of refraction in those materials.The necessary condition to be fulfilled for LHM in PhCs is v ph ⋅ v gr < 0 (v ph and v gr are phase and group velocities) or equivalently k⋅S<0 (S is Pointing vector), and additional condition is λ o ≡ c/f ≥ 2⋅a s (a s is the surface lattice constant) in order to avoid higher order Bragg diffractions out of the crystal.
In this paper we present research of optical properties (negative and left-handed refraction) of the GaAs (ε = 12.96) 2D photonic (3 2 , 4, 3, 4) structure.This structure falls in the class of eleven Archimedean lattice tiling lattices that were first thoroughly described by Kepler [13] in Harmonices Mundi II (1619) and presented in Fig. 1.Archimedean lattices are regular patterns of polygonal tessellation in plane using regular polygons.Among Archimedean lattices square, triangular, honeycomb and Kagome structures are well known.The last three are called regular lattices since they consist of equal polygons.Others are made of different polygons and are called irregular Archimedean lattices.

Results
PhC lattices were made as air hole structures (r/a=0.35,where ''r'' is the rod radius and ''a'' is the lattice constant equal to 1 μm) for research of negative left-handed materials (LH -), whereas for a waveguide application the structure consists of dielectric rod arrangement (r/a=0.3) in air.As a tool for the analysis we used RSOFT [15] software to calculate band structures and equi-frequency contours (EFC) as well as for EM propagation through PhCs.The theoretical models are the well-known plane wave expansion method (PWE) for the first two calculations and a finite-difference time-domain (FDTD) method for the latter.
In  To confirm that assumption we analyzed EFC for the TE4 band and the results are presented in Fig. 4. In Fig. 4 the full normal and parallel line are the normal and parallel component of k wave vector.The parallel component of k is conserved in refraction (Snell law).The symmetry points Г, X and M of the first Brillouin zone are also marked.In addition, the values of EFC`s are presented with numbers of descending order in the Г-M direction.
Blue circles and arrows stand for the air EFC with f (=ωa/2πc) = 0.205, (λ/a = 4.88) and the air wave vector with an angle of incidence of 30 0 .The green arrow corresponds to the kvector in PhC and the yellow one denotes the group velocity, which is determined as an inward gradient (negative group velocity) of corresponding EFC with the same frequency of the incident wave.All EFC values are in f units.
The EFC`s are convex in the vicinity of Г point having the inward-pointing group velocity.Since the inner product of k ph and v gr is negative, we can conclude that n eff < 0 leads to a LH -type of refraction.Also, directions of the group and phase velocities are almost antiparallel like in Veselago metamaterials.
In order to confirm LH -negative refraction in our structure we made FDTD simulations.The results of performed simulations of wave propagation are presented in Fig. 5 for the same parameters of the EFC analysis shown in Fig. 4. In this figure the red arrow belongs to the incident and refracted wave in air, blue is for group velocity and yellow is the phase one.There is very good agreement between EFC calculations and FDTD simulations comparing Figs 4 and 5. Also, we proved that the refraction in this structure is of the LH - type.
Fig. 5 FDTD simulations of (3 2 , 4, 3, 4) PhC (air-holes in GaAs) for λ/a = 4.88( f =0.205) and in α = 30 0 As an example of possible applications we briefly analyzed the (3 2 , 4, 3, 4) structure as a candidate for the fabrication of waveguides.Here we used GaAs rods in air with r/a = 0.3 and apply the same methods for analysis as before.Band structures, for TM polarization, are presented in Fig. 6 where we found that waveguiding in this structure is possible for TM modes inside the bandgap range of f = 0.21-0.3.

Conclusion
In this paper we present our research of optical properties of GaAs 2D Archimedean Photonic lattice tiling with the p4g symmetry.We investigated possibilities of negative and left-handed refraction and possible applications of (3 2 , 4, 3, 4) PhC lattice.As numerical tools PWE and FDTD are used.Negative left-handed refraction was found in the PhC made of GaAs with air holes.A useful waveguide based on the same lattice (GaAs rods in air) was suggested.

Fig 3 a
band structure of the air hole PhC structure, of the first 5 bands and for the TE polarization is presented.From this figure, from the frequency dependences with k of the TE4 band near the Г symmetry point and from our previous work[6][7][8], we supposed that negative refraction will occur, a little bit below the first gap.

Fig. 6
Fig.6 Band structure of GaAs ladybug dielectric material rods in air FDTD simulations of the (3 2 , 4, 3, 4) PhC lattice waveguide are presented Fig. 7.The waveguide is made as a line defects formed by removing some rows and columns of GaAs