INVESTIGATION OF REFLECTION BANDS OF 1D ANNULAR PHOTONIC CRYSTAL CONTAINING DOUBLE NEGATIVE INDEX MATERIAL AND NON-MAGNETIZED PLASMA

Lithuanian Journal of Physics, Vol. 62, No. 3, pp. 138–147 (2022)
© Lietuvos mokslų akademija, 2022

In this paper, optical reflectance properties of an annular photonic crystal (APC) composed of alternate layers of double negative (DNG) material and a non-magnetized plasma (NMP) layer, immersed in free space, have been theoretically investigated and studied. The reflectance spectra of the annular PC have been obtained by employing the transfer matrix method (TMM) for the cylindrical waves in the case of TE-polarized wave only. It has been found that the spectral position and width of the reflection bands of APC are greatly influenced by the variation in the thickness of DNG metamaterial and NMP layer, respectively. Interestingly, it is observed that the presence of NMP layer causes the increase in photonic band gap (PBG) whereas the DNG layer reduces the PBG. Further, the effect of azimuthal mode number (m) on the reflectance spectra shows that for m > 0, splitting in the reflection bands occurs at the frequency corresponding to the zero permeability value of DNG metamaterial layer. Moreover, with the increase in azimuthal mode number one PBG is red-shifted and the second one is blue-shifted. Finally, the effect of change in the starting radius parameter of curved surface and plasma electron density on the reflectance spectra of APC has also been studied and very interesting results have been observed.

Keywords: annular photonic crystal, circular photonic crystal, cylindrical Bragg waves, reflectance spectra, double negative index material, non-magnetized plasma, transfer matrix method

1. Introduction

The periodic profile of alternate dielectric constant of the material on the optical wavelength scale has the properties of controlling and manipulating the flow of light wave and is called photonic crystal (PC) [1–6]. When the optical wave propagates through such novel class of optical materials, then a certain range of frequencies (or wavelengths) are prohibited. The forbidden frequency (or wavelength) range is called stop band or photonic band gap (PBG) and arises due to interference of Bragg scattering. Since in the PC propagation of light wave can be controlled and manipulated very effectively, therefore, it has various applications in the field of photonics and optoelectronics [7–14].

The introduction of double negative (DNG) materials in designing and fabrication of photonic crystals has focused much interest in the scientific community during the last two decades [15–19]. DNG materials, also called negative index materials (NIMs) or left-handed materials (LHMs), are artificial composites with a simultaneous negative permittivity and permeability. DNG material was theoretically investigated by Veselago [15] and experimentally demonstrated by Smith et al. [16]. When the electromagnetic wave propagates through the DNG material, the direction of Poynting vector S = E × H becomes opposite to that of the wave vector k, so that k, E and H form a left-handed set of vectors. Thus, when such types of metamaterial are used in a PC structure, a very distinct feature is observed [20–27].

PCs containing the non-magnetized and magnetized plasma have also been the subject of attraction to researchers and scientists in recent years. By changing the parameters of plasma materials, the position and width of transmission and reflection bands can be tuned at a particular frequency and a different range of frequencies [28–31].

In recent times, the propagation of electromagnetic wave in a cylindrical multilayer structure (CMS) has also received much attention and generated lots of interest among the scientific community [32–34]. A photonic crystal having a periodic cylindrical multilayer structure is called a circular photonic crystal (CPC) or an annular photonic crystal (APC). Due to the technological advancement of modern fabrication technique it is possible to design and fabricate a photonic crystal in different geometrical shapes [35–38]. An annular Bragg laser (ABR) or resonators consisting of ABRs surrounding a radial defect have been developed, which shows a specific feature of vertical emission. Such types of lasers have a better sensing ability in the biological and chemical function than planar PC resonators, in addition to the application in the optical communication system [39–42].

Kaliteevski et al. [43] have developed the version of transfer matrix method (TMM), like in a planar PC that is applied to analyze the propagation of electromagnetic wave in APC. The optical characteristics of APC (or CPC) composed of different types of materials have been studied by many researchers [44–50].

In this work, we theoretically investigate and study the reflectance properties of an annular photonic crystal (APC) composed of alternate layers of DNG material and non-magnetized plasma immersed in free space. In order to obtain reflectance spectra we use the transfer matrix method (TMM) for cylindrical waves that was developed by Kaliteevski et al. [42]. We will study the effects of change in the thickness of DNG metamaterial and NMP layer, the azimuthal mode number (m), the electron density of plasma material and the starting radius (ρ₀) on the reflectance spectra of the proposed APC structure for the TE-wave only.

2. Theoretical modelling and formulation

The schematic diagram of one-dimensional annular photonic crystal is illustrated in Fig. 1. The structure consists of the inner core region with the refractive index n₀ (starting medium) and the starting radius ρ₀. Green and orange colours represent the DNG metamaterial and NMP layer, respectively. The first layer A has the refractive index n₁(permittivity ε₁ and permeability µ₁) with the thickness d_A and radius ρ₁ whereas the second layer B has the refractive index n₂, thickness d_B and radius ρ₂, respectively. The outer region (final medium) of the structure has the refractive index n_f with the radius ρ_f.

Fig. 1. Schematic diagram of the APC structure consisting of the inner core region with the refractive index n₀ and starting radius ρ₀. Green and orange colours represent the DNG metamaterial and the NMP layer, respectively, having refractive indices, thicknesses and radius n₁, n₂, d_A, d_B and radii ρ₁ and ρ₂, respectively. The outer region (final medium) of the structure has refractive index n_fwith radius ρ_f.

Here we consider the physically realizable DNG metamaterial from Ref. [22], with effective permittivity ε₁ and permeability µ₁ given by

ε_{1} (f) = 1 + \frac{5^{2}}{{(0.9)}^{2} - f^{2} - i f γ_{e}} + \frac{10^{2}}{{(10.5)}^{2} - f^{2} - i f γ_{e}} (1)

μ_{1} (f) = 1 + \frac{3^{2}}{{(0.902)}^{2} - f^{2} - i f γ_{m}}, (2)

where f is the angular frequency, γ_e and γ_m are electric and magnetic damping frequencies (also called electric and magnetic loss factors) measured in GHz.

The refractive index of non-magnetized plasma is given by the equation [51]

n_{2} (f) = \sqrt{ε_{2} (f)} = \sqrt{1 - \frac{{(f_{p e})}^{2}}{f^{2}},} (3)

where $f_{p e} = {(\frac{n_{e} e^{2}}{4 π^{2} m_{e} ε_{0}})}^{\frac{1}{2}}$ is the plasma frequency, n_e is the electron plasma density, and m_e and e are the electron mass and charge, respectively.

It is assumed that the cylindrical wave would diverge from the axis of symmetry ρ = 0 and then interfere normally on the first interface at ρ = ρ₀. The reflectance at the first circular boundary ρ = ρ₀is obtained by employing the transfer matrix method (TMM) in the cylindrical Bragg wave [42, 45] which is the most useful technique to analyze the transmission and reflection properties of PC structure.

Assuming that the time dependent part of all the electromagnetic fields is exp(jωt), the source-free two curl Maxwell’s equations are given as

\nabla \times E = - i ω μ H, (4)

By using the method of separation of variables, we can get the solution for E_z which is expressed as

\nabla \times H = i ω ε E . (5)

There are two possible modes (or polarizations) including TE and TM modes for the cylindrical Bragg wave. E_z, H_φ and H_ρ are the non-zero fields for the TE-wave and satisfy the following three equations in each single layer

\frac{1}{ρ} \frac{\partial E_{z}}{\partial φ} = - i ω μ H_{ρ}, (6 a)

\frac{\partial E_{z}}{\partial ρ} = - i ω μ H_{φ}, (6 b)

\frac{\partial (ρ H_{φ})}{\partial ρ} - \frac{\partial H_{ρ}}{\partial φ} i ω ε ρ E_{z} . (6 c)

The governing equation for E_z can be obtained from Eqs. (6a–c) and is given by

\begin{matrix} ρ \frac{\partial}{\partial ρ} (ρ \frac{\partial E_{z}}{\partial ρ}) - ρ^{2} \frac{1}{μ} \frac{\partial μ}{\partial ρ} \frac{\partial E_{z}}{\partial ρ} \\ + \frac{\partial}{\partial φ} (\frac{\partial E_{z}}{\partial φ}) + ω^{2} μ ε ρ^{2} E_{z} 0 \end{matrix} (7)

By using the method of separation of variables, we can get the solution for E_z which is expressed as

\begin{matrix} E_{z} (ρ, φ) = V (ρ) ϕ (φ) \\ = [A J_{m} (k p) + B Y_{m} (k p) e x p (i m φ), \end{matrix} (8)

where A and B are considered as constants. J_m and Y_m are the Bessel function and Neumann function, respectively, m is the azimuthal number, $k = ω \sqrt{μ ε}$ is the wave number of the medium and ω is the angular frequency.

The azimuthal part of the magnetic field H_φ can be obtained from Eq. (6b) and is given by

\begin{matrix} H_{φ} (p, φ) = U (ρ) φ (φ) \\ = - ip [A J_{m} (k ρ) + B Y_{m} (k ρ)] e x p (i m φ) . \end{matrix} (9)

In Eq. (9), $p = \sqrt{μ / ε}$ represents the intrinsic admittance/impedance of the medium. The single layer matrix relationship corresponding to the electric field and magnetic field at its two interfaces can be obtained with the help of Eqs. (8) and (9).

The matrix for the first layer M₁ with the refractive index n₁ and interfaces at ρ = ρ₀ and ρ₁ can be expressed as [42]

[\begin{matrix} V (ρ_{1}) \\ U (ρ_{1}) \end{matrix}] = M_{1} [\begin{matrix} V (ρ_{0}) \\ U (ρ_{0}) \end{matrix}], (10)

where the single layer matrix M₁ is given by

M_{1} = [\begin{array}{l} m_{11} & m_{12} \\ m_{11} & m_{22} \end{array}] .

The matrix elements of M₁ are given by

\begin{array}{r} m_{11} = \frac{π}{2} k_{1} ρ_{0} [Y_{m}^{'} (k_{1} ρ_{0}) J_{m} (k_{1} ρ_{1}) \\ - J_{m}^{'} (k_{1} ρ_{0}) Y_{m} (k_{1} ρ_{1})] \\ m_{12} = i \frac{π}{2} \frac{k_{1}}{ρ_{1}} ρ_{0} [J_{m} (k_{1} ρ_{0}) Y_{m} (k_{1} ρ_{1}) \\ - Y_{m} (k_{1} ρ_{0}) J_{m} (k_{1} ρ_{1})], \\ m_{21} = i \frac{π}{2} k_{1} ρ_{0} p_{1} [Y_{m}^{'} (k_{1} ρ_{0}) J_{m}^{'} (k_{1} ρ_{1}) \\ - J_{m} (k_{1} ρ_{0}) Y_{m}^{'} (k_{1} ρ_{1})] \end{array}

\begin{array}{r} m_{22} = \frac{π}{2} k_{1} ρ_{0} [J_{m} (k_{1} ρ_{0}) Y_{m}^{'} (k_{1} ρ_{1}) \\ - Y_{m} (k_{1} ρ_{0}) J_{m}^{'} (k_{1} ρ_{1})] \end{array}

In the above equations, $ρ_{1} = \sqrt{μ_{1} / ε_{1}}$ It is obvious that the matrix elements are dependent on the radii of the two interfaces. For the ith layer the matrix M_i can be obtained by replacing $ρ_{0} \to ρ_{i - 1},$ $ρ_{1} \to ρ_{i},$ $k_{1} \to k_{i},$ $p_{1} \to p_{i},$ For the N-period bilayer periodic structure there are total 2N layers, therefore there should be 2N matrices to establish the total system matrix M. This makes a connection between the first and final interfaces as

\begin{array}{l} [\begin{array}{l} V (ρ_{f}) \\ U (ρ_{_{f}}) \end{array}] = M_{2 N} \dots \dots \dots M_{2} M_{1} [\begin{array}{l} V (ρ_{0}) \\ U (ρ_{0}) \end{array}] \\ = M [\begin{array}{l} V (ρ_{0}) \\ U (ρ_{0}) \end{array}] . \end{array} (11)

The reflection coefficient of the proposed APC (or CPC) structure would be determined by the following equation

\begin{matrix} (M_{21} + i p_{0} C_{m 0}^{(2)} M_{11}) \\ r_{d} = \frac{- i p_{f} C_{m f}^{(2)} (M_{22}^{'} + i p C_{m 0}^{(2)} M_{12}^{'})}{(- i p_{0} C_{m 0}^{(1)} M_{11}^{'} - M_{21}^{'})} . \\ - i p_{f} C_{m f}^{(2)} (- i p_{0} (- i p_{0} C_{m 0}^{(1)} M_{12}^{'} - M_{22}^{1}) \end{matrix} (12)

In the above expression M₁₁', M₁₂', M₂₁' and M₁₂' are the matrix elements of the inverse of matrix M of Eq. (11). $p_{0} = \sqrt{μ_{0} / ε_{0}}$ and $p_{f} = \sqrt{μ_{f} / ε_{f}}$ are the admittances of the initial and final medium, $K = ω \sqrt{μ_{0} / ε_{0}}$ is the free-space wave number and

C_{m l}^{(1, 2)} = \frac{H_{m}^{{(1, 2)}^{'}} (k_{l} ρ_{l})}{H_{m}^{(1, 2)} (k_{l} ρ_{l})}, l = 0, f (13)

where H_m⁽¹⁾ and H_m⁽²⁾ are the Hankel function of the first and second kind. The reflectance (R) of APC (or CPC) may be obtained by using the expression

R = {| r_{d} |}^{2} . (14)

The equation of reflectance/transmittance in the case of TM-wave can be obtained just by simply replacing ε ↔ μ and i ↔ –i in the equation of TE-wave.

3. Numerical results and discussions

In this section, we numerically calculate the optical reflectance response of the proposed APC structure having alternate cylindrical layers of DNG metamaterial and non-magnetized plasma (NMP), immersed in air, i.e. n₀= n_f= 1.0. For the DNG metamaterial layer we choose electric and damping frequencies γ_e = γ_m= 1 × 10^–3 GHz. The values of permittivity and permeability of the DNG metamaterial are calculated by using Eqs. (1) and (2), respectively, and the refractive index of NMP layer is calculated by using Eq. (3). Figure 2 shows the variation in the real part of ε₁ (f) and μ₁ (f) of DNG metamaterial with frequency measured in GHz. The total number of periods taken is N = 10 and the effect of various parameters on the reflectance has been investigated in the frequency range 1.5–7.0 GHz.

Fig. 2. Variation in the real part of permittivity and permeability of DNG metamaterial with frequency at electric and damping frequencies γ_e = γ_m = 1 × 10^–3 GHz.

First, we investigate the effect of change in the thickness of DNG metamaterial (d_A) and NMP layer (d_B), respectively, on the optical reflectance spectra of APC at the normal incidence for the TE-polarized wave keeping other parameters fixed. Figure 3 shows the optical reflectance spectra at a different thickness of the DNG material such as d_A = 2, 4, 6 and 8 mm, respectively, for the azimuthal mode number m = 2, starting radius ρ₀ = 30 mm, electron density of plasma layer n_e = 6 × 10¹⁷ m^–3 and d_B = 3 mm. At the right side of the figure, an indicator of colours is shown which represents the colour dependent reflectance of electromagnetic wave. If colours are close to blue (dark), they show that the amount of reflectance decreases (transmittance increases), and if they become yellow (light), the amount of reflectance increases (transmittance decreases). The position of the band gap (BG) centre lies at 3.45 GHz at m = 0 that is equivalent to the BG centre of 1D planar PC and is shown by a dashed line in all the figures. This is so because the geometric difference due to the curved interfaces in APC has no effect on the reflectance properties compared to the 1D planar PC at m = 0.

Fig. 3. (a) Optical reflectance spectra at d_A = 2, 4, 6 and 8 mm, respectively, for azimuthal mode number m = 2, starting radius ρ₀= 30 mm, electron density of plasma layer n_e = 6 × 10¹⁷ m^–3 and d_B = 3 mm. (b) Zoom portion of the reflectance panel of (a) in the frequency range 2.9 to 3.2 GHz.

From Fig. 3(a) it can be observed that there exist two reflection bands (photonic band gaps) in a given frequency range for the selected parameters of APC. When the thickness of NMP layer is kept fixed at 3 mm, then by increasing the thickness of DNG metamaterial, the width of both reflection bands decreases. The left edges of the first PBGs are shifted towards the low frequency side but the right edges remain almost at the same frequency, while in the second PBGs the left band edges lie almost at the same frequency but the right edges shift towards a higher frequency side. One can say that the first PBG is red-shifted whereas the second PBG is blue-shifted by increasing the thickness of DNG material. The zoom portion of the reflectance panel in the frequency range 2.9 to 3.2 GHz is shown in Fig. 3(b). As can be seen from the panel (3b), there few transmission peaks all have origin at frequency nearly equal to 3.13 GHz where the permeability of metamaterial layer becomes zero. The transmission peaks are very sharp for a small thickness of the DNG metamaterial layer but they increase slightly with an increase in the thickness.

Figure 4 demonstrates the optical reflectance as a function of frequency at different thicknesses of the NMP layer d_B = 2, 4, 6 and 8 mm while the thickness of DNG material is kept fixed at 4 mm. The other parameters are the same as taken above. From the study of reflectance spectra it is found that the width of PBGs increases with increasing the thickness of NMP layer which is opposite to the case when the thickness of DNG material is increased. Further, it is noticed that for smaller values of the thickness (d_B < 5 mm) there are two PBGs, one with a smaller and other with a larger band width. But when the thickness of NMP layer becomes greater than 5 mm, both PBGs merge with each other and a very broad reflection band is observed.

Fig. 4. Optical reflectance at d_B= 2, 4, 6 and 8 mm, respectively, for azimuthal mode number m = 2, starting radius ρ₀= 30 mm, electron density of plasma layer n_e = 6 × 10¹⁷m^–3 and d_A= 4 mm.

The important feature of the above discussion is that the thickness of both layers has an appreciable influence on the position and width of reflection bands. The trend of change in PBG with an increase in the thicknesses of the DNG metamaterial and NMP layer is of a different nature and in contrast to each other. Hence, it can be concluded that the presence of NMP layer causes the increase in PBG whereas the DNG layer reduces the PBG when the thickness of a respective layer is increased. These results suggest that the annular PC structure consisting of the DNG metamaterial and NMP layer can be used to design a narrow band and a broad band optical reflector in a low and high frequency region by the proper choice of structure parameters.

Now we will investigate the effect of change in the azimuthal mode number m on the reflectance spectra. The azimuthal mode number is an additional parameter for the APC structure, in comparison to the plane photonic crystal, which controls the position and width of reflection bands. Since the field solutions of cylindrical Bragg waves depend on the azimuthal mode number for both polarizations, therefore, the reflectance spectra of APC are greatly influenced by changes in the azimuthal mode number.

The optical reflectance for different azimuthal mode numbers m = 0, 1, 2, 3, 4 and 5, respectively, is illustrated in Fig. 5, for the parameters ρ₀ = 30 mm, n_e = 6 × 10¹⁷ m^–3, d_A = 4 mm and d_B= 3 mm, respectively. It is obvious from Fig. 5 that for m = 0, there exists a single reflection band but for m > 0 the reflection band gets splitted into two bands separated by a narrow line at the frequency nearly equal to 3.13 GHz. For the TE-reflectance, splitting of PBG occurs at the frequency equal to 3.13 GHz corresponding to the zero permeability value for m > 0. The occurrence of zero-µ gap is due to the radial component of the magnetic field Hρ of TE-polarized wave. The first PBG (lies at the lower frequency side) has a much smaller band width than the second PBG which lies at the higher frequency side. The first band lies at nearly 2.3 GHz and the second lies at nearly 4.6 GHz, for m = 0 which is equivalent to the 1D PC case. Further, by increasing the value of m the width of the second PBG increases and is shifted towards the higher frequency side, but in the first PBG band the width remains almost constant for the lower order mode. But for the higher azimuthal mode number, small transmission peaks in the low frequency range disappear and cause the broadening in the PBG. Thus, the width of the first PBG also increases for the higher azimuthal mode number but it is red-shifted in contrast to the second PBG which is blue-shifted.

Fig. 5. Optical reflectance spectra at different azimuthal mode number m = 0, 1, 2, 3, 4 and 5, for other parameters ρ₀= 30 mm, n_e = 6 × 10¹⁷m^–3, d_A = 4 mm and d_B = 3 mm, respectively.

One more parameter which controls and affects the optical reflectance properties of APC structure is the starting radius parameter ρ₀ of the curved surface. The optical reflectance spectra for the parameters m = 2, n_e = 6 × 10¹⁷m^–3, d_A = 4 mm and d_B = 3 mm, respectively, at different starting radius ρ₀ = 10, 20, 30, 40 and 50 mm, respectively, are illustrated in Fig. 6. On examining the reflectance spectra it can be seen that for a small value of ρ₀ = 10 mm, a wider reflection band is observed. But as the value of starting radius increases, the band gets splitted in two parts, similarly to the previous case, by a narrow boundary lies at the frequency nearly equal to 1.13 GHz. It is further noticed that the first PBG (lies on the low frequency side) remains almost unaffected by increasing the value of starting radius, but the width of the second PBG (lies on the high frequency side) decreases by increasing the starting radius. Thus, by changing the size of the inner core region, the width of reflection band and its spectral position can be controlled without changing the other parameters of the structure. Since the starting radius parameter shows a strong dependence in controlling PBG, therefore it can be considered as a significant parameter (in addition to the thickness of constituent layers) in designing and producing the optical devices.

Fig. 6. Optical reflectance spectra at different starting radius ρ₀= 10, 20, 30, 40 and 50 mm, respectively, and other parameters m = 2, n_e = 6 × 10¹⁷m^–3, d_A = 4 mm and d_B = 3 mm, respectively.

Finally, we focus our discussion on the case of variation in plasma electron density for n_e = 2 × 10¹⁷, 4 × 10¹⁷, 6 × 10¹⁷ and 8 × 10¹⁷ m^–3, respectively, on the reflectance spectra of APC structure, for the parameters m = 2, ρ₀ = 30 mm, d_A= 4 mm and d_B = 3 mm, respectively, as shown in Fig.7. It can be observed from the reflectance spectra that the width of the first PBG changes very slightly but the second PBG is greatly influenced by the increase in the plasma electron density. When the electron density of NMP layer increases, the width of the second PBG also increases and it is blue-shifted. The left edge of the second PBG lies at the frequency nearly equal to 3.13 GHz for any value electron density, but the right edge of the PBG shifts towards the high frequency side. The enhancement in the reflection band occurs due to the resultant dielectric constant difference between the DNG metamaterial and NMP layer as the plasma electron density increases. The results predict that the use of NMP layer adds one more parameter for controlling the width and spectral position of reflection bands of the proposed APC structure.

Fig. 7. Optical reflectance spectra at different plasma density n_e = 2 × 10¹⁷, 4 × 10¹⁷, 6 × 10¹⁷ and 8 × 10¹⁷ m^–3, respectively, for parameters m = 2, ρ₀= 30 mm, d_A = 4 mm and d_B= 3 mm, respectively.

4. Conclusions

In summary, we have examined theoretically and studied the optical reflectance properties of an annular photonic crystal (APC) composed of the alternate layers of double negative (DNG) material and non-magnetized plasma (NMP) layer, immersed in free space. The numerical investigations are demonstrated using the transfer matrix method for cylindrical waves in the case of TE-polarized wave only. On examining the effect of variation in the thickness of DNG metamaterial and NMP layer alternately, by keeping other parameters fixed, on the optical reflectance of APC it is found that the spectral position and width of the reflection bands are greatly influenced. It is interesting to note that the trends of change in PBG with increase in the thicknesses of the DNG metamaterial and NMP layer are of different nature and opposite to each other. The investigation of increase in the azimuthal mode number m on the reflectance spectra shows the appearance of splitting in the PBG when m > 0 and bands occur at the frequency corresponding to the zero permeability value of DNG metamaterial layer. Moreover, with an increase in the azimuthal mode number one PBG is red-shifted and the second one is blue-shifted. In addition to the above investigation, we also analyzed the effect of increasing the size of the starting radius parameter on the reflectance spectra. The result predicts that the width of the reflection band and its spectral position can also be controlled by varying the size of the inner core region without changing the other parameters of the structure. Finally, the effect of plasma electron density on the reflectance spectra has been studied. The results show that the reflection band lies in the low frequency region and is affected very little while there is a significant enhancement in the second PBG with an increase in plasma electron density.

From the above discussions it may be inferred that the annular PC structure consisting of DNG metamaterial and NMP layer can be used to design a narrow band and a broad band optical reflector in a low and high frequency region by the proper choice of structure parameters. Moreover, the starting radius can be considered as a significant parameter in designing and producing the optical devices.

Acknowledgements

One of the authors, Dr. Sanjeev K Srivastava, is thankful to the Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India, for providing the necessary facility for this work.

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INVESTIGATION OF REFLECTION BANDS OF 1D ANNULAR PHOTONIC CRYSTAL CONTAINING DOUBLE NEGATIVE INDEX MATERIAL AND NON-MAGNETIZED PLASMA

1. Introduction

2. Theoretical modelling and formulation

3. Numerical results and discussions

4. Conclusions

Acknowledgements

References

VIENMACIO ŽIEDINIO FOTONINIO KRISTALO IŠ NEIGIAMU SKVARBU MEDŽIAGOS IR NEIMAGNETINTOS PLAZMOS ATSPINDŽIO JUOSTU TYRIMAS