Effect of Geometry on the Electromagnetic Wave Transport Properties of Photonic Crystals: Comparison of Top-Flat and Top-Curved Refractive Index Proles

In the current paper, we try to engineer the refractive index profile in a one-dimensional photonic crystal as a powerful tool to manage the electromagnetic wave transmission properties. For this purpose, we have compared four sinusoidal, rectangular, triangular, and saw-tooth refractive index profile types. In this way, we have used a transfer matrix method accompanied by the discretization of the spatial domain. This method can readily be applied to any arbitrary continuous refractive index profile. Then, we have tried to address the effects of different geometrical and physical parameters, including the photonic crystal length L, dielectric permittivity 𝜀 𝑑 , number of layers and plasma density n p , etc. on the light propagation through the mentioned photonic crystals. In the proposed two-layer plasma/dielectric photonic crystals we could observe acceptable ranges of Omni-directional photonic band gaps that their position width and their number can be regulated. We determine the most and least tunable systems.


Introduction
Photonic crystals are defined as artificial structures with a periodic refractive index that can change the electromagnetic wave propagation similar to the periodic potentials in the atomic crystal. They can alter the electron motion by creating an allowed and forbidden electronic spectrum. If the frequency of a propagating wave meets a photonic bandgap, electromagnetic wave propagation is prohibited. In other words, the photonic crystal periodicity forbids the propagation of some frequencies throughout the structure. The photonic band gaps essentially depend on the contrast of the dielectric functions of materials in a photonic crystal. They can be interpreted using the interference of the forward and backward traveling waves in the system. Photonic structures have so far been realized in one [1], two [2], and three dimensions [2]. One-dimensional photonic crystals can be fabricated using different techniques, including electrochemical etching, spin coating, and various physical vapor deposition [3,4]. The photonic crystals importance arise from their intense applications in optical devices such as optical filters [5], optical switches [6], optical logic devices [7], optical buffers [8], solar cells, 3D fabrication matrices, sensors, color displays fluorescent enhancement devices [9], infrared reflectors [10], etc.
To consider the transmission properties of a physical system, different methods can be employed. Among these approaches, the transfer matrix has a certain situation. So far this approach has been used in the solution of non-homogeneous anisotropic wave equations [43], localization of elastic waves [44], wave propagation in layered anisotropic media [45], prediction of transmission loss of multilayer acoustic materials [46], diffuse-field transmission through multilayered cylinders [47], etc. This method can also be used along with tight-binding models [48], Ising model [49], Monte Carlo schema [50], etc. In addition, the gradient index photonic crystals are considered in some available references such as parabolic refractive index profile [51], hyperbolic-secant refractive index profile [52], saw-tooth refractive index [53], the refractive index of graded layers with linear, exponential, and parabolic profiles [54], and sinusoidal densities [55].
In the current work, we have compared the rectangular, sinusoidal, saw-tooth, and triangular refractive index profiles. A difference between our work and previously studied such as refractive index of graded layers with linear, exponential, and parabolic profiles in [54], is that in their work each layer has a thickness of L, and the refractive index inside this layer changes gradually. However, in our work, we have continuous refractive index profiles thorough the total system spatial domain. Besides in the existing continuous refractive index profiles such as saw-tooth refractive index in ref. [53] or sinusoidal densities of ref. [55], the transmission coefficient has been found using complicated analytical evaluation of the transfer matrix elements. However, in this study, we have used a numerical schema that employs the discretization of the spatial domain. This helps us to study an arbitrary refractive index profile without out much overwhelming effort.
In this research, we consider the electromagnetic wave propagation in plasma photonic crystals containing alternative plasma and dielectric layers. using the transfer matrix method, we have tried to investigate the effects of the plasma layer refractive index, total system length, dielectric constants in the dielectric layers, etc. on the omnidirectional bandgap intervals and the transmission properties of the proposed structures.

Formalism
The periodicity of the dielectric permittivity of the materials in a typical photonic crystal may lead to the creation of the photonic bandgaps. Here, we produce the dielectric permittivity function using the function ( )  . Furthermore, m is the electron mass, e is the electron charge, and ne is the electronic density of the plasma [56]. For transverse electric (TE) and transverse magnetic (TM) waves, the electric field and magnetic fields are perpendicular to the plane of layers. The transfer matrix method (TMM) can connect the electric and magnetic fields at two sides of the neighboring layers using [57], . Besides, c, j  , n0 and j j j n    are the speed of light in vacuum, the beam angle in layer j, the refractive index of the free space, and the refractive index of the k th layer, respectively [58]. Also, i and i are the permittivity and the permeability of the materials in k th layer, respectively. Now, the total transfer matrix for N-layers reads, 11 1 12 21 22 The transmission coefficient of the proposed structures can now be written as Where we have The behavior of TM waves can be described using the previous relations with . The related transmittance T for the multilayer is obtained using

Results and Discussions
The transmission coefficient versus the incoming light frequency ω for rectangular refractive index profile is shown in Fig     Therefore, in this system, the number of OBGs can be changed using the plasma electron density variation. presented L=50 mm. In this panel, there are two forbidden bands in the frequency ranges 47 < ω < 54 GHz and 94< ω < 96 GHz. These bands do not exist in the system with a constant total length L=10 mm. Therefore, the system length has a great role in the creation of the OBGs.
Comparing this figure with figure 8 reveals that changing the plasma density is more efficient in the creation of OBGs. This is because greater OBGs can be obtained in Fig. 8 and in Fig. 8 the system size can do not grow too large. The later reason (i.e., the fixed system size) may be able to produce optical devices with smaller sizes. The effect of dielectric permittivity on the OBG for the plasma photonic crystal with a triangular refractive index profile is presented in Fig 10. We have used total system length L=30 mm, well number=20, plasma density np =1e10 18  , and 10 also show that the parameter is more efficient in tuning the OBG width and also regulating the number of OBGs than the plasma electron density and the total system length. This is because wider OBGs and also more OBGs can be achieved by changing the . Fig.10 Photonic band structure in terms of incident wave frequency and the incident angle for plasma photonic crystal with triangular refractive index profile. The total system length L=30 mm, well number=20, plasma density np =1e10 18 m-3 are used. Furthermore, in panels (A) and (B), the dielectric permittivities =2.5 and 9 are respectively used.
Finally, the transmission coefficient versus the incoming light frequency and the incident angle for plasma photonic crystal with triangular refractive index profile for both TE and TM polarization modes is shown in Fig. 11. Here, the total system length L=30 mm, dielectric permittivity =2.5, plasma density np =1e10 18 m -3 are utilized. In panels (A) and (B) of this figure, we consider well numbers 10 and 20, respectively. In panel (A), for well number=10, the total OBGs are in the interval between the two solid lines is in the range 39 < ω < 45 GHz and 78< ω < 80. For well number=20 in panel (B), the number of OBG is decreased but the width of OBG is increased. The width of OBG, in this case, is 13 GHz. Comparing the Figs. 8, 9, 10, and 11, reveal that the number of OBGs increases by increasing the dielectric constant, system length and the plasma electron density while decreases by increasing the number of wells. Also, we can conclude that the most and the least critical tools to tune the OBGs are the dielectric constant and the total system length, respectively.

Conclusion
In this work, we compared the transmission properties of plasma photonic crystals with top-flat rectangular and top-curved sinusoidal, triangular, and saw-tooth refractive index profile types. We showed that, for the rectangular refractive index profile, as increases, the frequency gaps redshift. the number of bandgaps increased by increasing the system length.
The plasma electron density had little effect on the position and bandwidth of the photonic gaps in higher frequency ranges. Therefore, one could easily use different plasma electron densities in his/her optical devices in a higher frequency regime without being worried about changing the photonic gaps when plasma electron density changes. Also, by increasing the well number, the position of the photonic bandgaps is blueshifted. For the saw-tooth refractive index profile, the widths of the transmission bands and the band gaps decreased by increasing the system length L. The transmission bandwidths were greater than the gap widths, while in the systems with saw-tooth-shaped profiles the gap widths were greater than the transmission bandwidths.
We also concluded that the behavior of the top-curved refractive index profiles is similar.
However, they are different from that of top-flat refractive index profiles. Finally, the most and the least critical tools to tune the OBGs were the dielectric constant and the total system length, respectively.