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The dispersion characteristics of binary 1D-PPCs having inhomogeneous plasma in the unit cell are studied. Using the transfer matrix method the required dispersion relations are obtained. Here the linear and exponential plasma density profiles are considered and compared with the homogeneous plasma having uniform density profile. It is observed that the inhomogeneity in plasma layer highly affect dispersion curves. By comparing the dispersion curves obtained in all considered cases, it is found that the widths of band gaps and phase velocities are always larger for exponential density profile than the linear uniform density profiles in the considered frequency range.

Photonic crystals (PCs) are periodically structured electromagnetic media in which certain range of electromagnetic (EM) waves are not allowed to propagate through the structure. This range of frequencies is called photonic band gaps (PBGs). The periodicity of the structure and the periodic variation of dielectric constant of different materials are the essential parameters for the formation of these PBGs [

Recently, plasma photonic crystals (PPCs) have attracted the attention of researchers because the properties of these PPCs are externally controllable and it possesses the characteristics of conventional PCs and plasma. The unit cell of PPCs consists of periodic arrangement of plasma and dielectric/air. The dispersion relation of binary one dimensional plasma photonic crystals 1D-PPCs by solving Maxwell equations using a method analogous to the Kronig-Penny’s problem in quantum mechanics is derived by Hojo et al. [

Similarly, the dispersion characteristics and optical properties of PPCs can also be controlled by considering inhomogeneous plasma in the unit cell. This consideration is more practical also because homogeneous plasma having uniform density is rarely realized in the laboratory plasma [

A plane EM waves with angular frequency ω is assumed to obliquely incident on the 1D-PPCs structure. The unit cells of considered 1D-PPCs have periodic arrangement of inhomogeneous plasma and homogeneous dielectric material. The unit cell is shown in

Similarly, the exponential plasma density profile is given as:

where b is the width of plasma layer, p is gradation parameter for controlling variation of density in the plasma layer and n_{cr} is the critical density [

For linear plasma density profile, the permittivity is written as:

with condition that. Here ε_{1} is the dielectric constant of dielectric layer, , a and b are widths of dielectric and plasma layer respectively. The permittivity profile is linearly varying with space along x-direction. The electric field in the case of TE mode is in y-z plane. Along the z-direction there is no change in permittivity, so z-component of wave vector is conserved. The one dimensional wave equation for the spatial part of:

Typical field solution can be expressed as and using this in Equation (3) we can write the above equation both regions: inhomogeneous plasma layer and dielectric medium in the n^{th} unit cell as:

where β is the z-component of wave-vector and is given by. Using Snell’s law, angle θ_{1 }can be calculated. If θ is the angle of incidence then

with. Here, it is assumed that EM waves incident from vacuum at certain angle θ.

The solutions of above equations for electric fields in the n^{th} unit cell are given by

where

and

Ai and Bi are Airy functions. Now, using the continuity condition of electric field E(x) and its derivatives at boundaries and and TMM, we obtain following matrix relation:

where m_{11}, m_{12}, m_{21 }and m_{22} are elements of unit translation matrix [

The constant K is known as the Bloch wave number. Here the subscript of k_{1x} is dropped for simplicity in notations and k_{1x} is K. Now the problem at hand is to determine K and as functions of w and b. In terms of column vector representation and using Equation (6), the periodic condition (8) for Bloch wave is simply given:

It follows from Equations (7) and (8) that the column vector of Bloch wave satisfies the following eigen-value problem

The phase factor e^{iK}^{Λ} is the eigen-value of the unit translation matrix (m_{11}, m_{12}, m_{21} and m_{22}) and is given by:

So using Equation (11), the dispersion relation for proposed structure can be written as:

Similarly, we can also find the unit translation elements: m_{11}, m_{12}, m_{21} and m_{22} for exponential density profiles. The elements of unit translation elements for exponential density profile and linear density are given in the Appendix 1 and Appendix 2, respectively. In case of uniform (constant) density, the unit translation element can be easily obtained [

The Equation (12) is known as dispersion relation which will give all the information regarding PBGs of the considered (linear plasma density and exponential plasma density) profiles. The unit cell of proposed PPCs structure contains inhomogeneous plasma of width b and homogeneous dielectric of width a. Here the thickness of inhomogeneous plasma b is related with homogeneous dielectric layer width a by the relation. Here d is the ratio of thickness of two layers. The permittivity profile in inhomogeneous plasma layer is and homogeneous dielectric layer is glass with dielectric constant e_{1} = 2.25. In the case of homogeneous plasma layer, the plasma frequency ω_{p} is constant and its value is. There are five selection parameters; incident angles θ, gradation parameter p, a, d and e_{1} involved in the numerical calculations.

and second band gap are nearly same for exponential density (solid line) and linear density (dash line) profiles at incident angle. Also, the width of band gap is large for exponential plasma density profile than the linear plasma density profile. By comparing these results with the case of homogeneous plasma having uniform density profile (dotted line), it is observed that the width of band gap of uniform density profile and linear density profile is approximately equal for first band gap but differ in higher order band gaps.

increases in all considered profiles. It is also clear from

The interesting feature of this graph is that the lower edge frequency, of first band gap, of linear density profile is now intact with uniform density profile (dotted line). As the thickness of plasma layer increases from d = 0.5 to 1.0, as shown in Figures 3(a) and (b), the lower edge frequency of first band gap for linear density profile and uniform density profile remains intact and are shifted towards lower frequency. At d = 2.0 (shown in

_{1} = 2.25. In this case, the considered volume average permittivity is 0.8394. By comparing

From this study, it is found that position and width of

band gaps strongly depend on the density profile and by choosing suitable profile; one can tune the band gaps in desired frequency range.

The dispersion characteristics of a plasma photonic crystal having inhomogeneous plasma density profile have been studied at the first time in our knowledge. The exponential and linear types of inhomogeneity in the plasma density profile are considered because homogeneous plasma having uniform density is rarely realized in the laboratory plasma. A comparison is also made with 1D-PPCs having homogeneous plasma with uniform density profile. It is observed that, for a constant volume average permittivity, the exponential density profile will always give larger band gaps widths than linear density profile and uniform density profile. With increase in angle of incidence, the widths of band gaps increase and at the same time band gaps are shifted towards higher frequency for all considered profiles. The number of band gaps also increases with increase of the thickness of inhomogeneous/homogeneous plasma layer. This analysis shows that with increase in the volume average permittivity, the widths of band gaps are increased and the lower edge frequency of first band gap remains intact but shifted towards lower frequency for all profiles. The important finding of this study is that the lower edge frequency of first band gap for exponential density profile and linear density profile remains intact and shifted towards higher frequency with increase in incident angle.

Finally, it is concluded that the width and position of band gaps strongly depend on the type of density profile of the plasma layer. Therefore, for correct prediction about the width and position of band gaps, the type of density profile should be ensured in PPCs.

The authors are grateful to Dr. B. Prasad and Dr. R. D. S. Yadava for their continuous encouragement and supports in many ways.

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