A research on the electromagnetic properties of Plasma Photonic Crystal based on the Symplectic Finite-Difference Time-Domain method

doi:10.1016/j.ijleo.2015.11.089

Optik

Volume 127, Issue 4, February 2016, Pages 1838-1841

https://doi.org/10.1016/j.ijleo.2015.11.089 Get rights and content

Abstract

The Symplectic Finite-Difference Time-Domain (SFDTD) method was applied to simulate the one-dimensional unmagnetized Plasma Photonic Crystal (PPC) with a sandwich structure. Compared with the traditional Finite-difference Time-Domain (FDTD) method, the results from the SFDTD method are reliable and accurate. Also the SFDTD method has better numerical dispersion characteristics than the traditional FDTD method and the Multi-Resolution Time-Domain (MRTD) method too. Electromagnetic propagation process by differential Gaussian pulse is given in the PPC band gap with the SFDTD method used. The influence of the plasma frequency and the thickness ratio of plasma and medium to PPC on the band gap structure is analyzed. The plasma frequency most affects the PPC in low-frequency range. Also the propagation of an electromagnetic wave can be controlled by the thickness ratio of plasma and medium. The results above prove that the SFDTD method can give the right and better results on the research of PPC.

Introduction

Plasma Photonic Crystal (PPC) [1] refers to the plasma and dielectric which have artificially periodic structure. The density of plasma structure changes periodically in the space, and the PPC alternately contains plasma and other media. Due to its unique characteristics of Photonic Band Gap (PBG) [2], [3], the PPC functions as photonic crystal. However, the PPC has the property of localized states of photons as a result of dispersion and dissipation of plasma, which the conventional photonic crystal does not have.

Compared with the traditional FDTD method used for simulating the PPC, the high-order FDTD method can reduce the numerical dispersion effectively [4], [5]. The high-order FDTD scheme (Fang [6] and Turkel and Yefet [7]) retains the simplicity of the original algorithm described by Yee [8] and can save computational resources with coarse grids compared with the traditional FDTD method. Cole [9] applies nonstandard finite difference scheme to construct the high-order. Forgy [10] uses an overlapped lattice to improve the numerical dispersion. Krumpholz [11] proposes a new time-domain method which is similar to the Method of Moment (MOM) based on multiresolution analysis. Namiki [12] improves the Alternating-Direction Implicit (ADI) FDTD method by eliminating Courant–Friedrichs–Levy (CFL) limit with poorer numerical dispersion. Meanwhile, other methods are also developed, i.e. the Pseudospectral Time-Domain (PSTD) algorithm [13] requires only two cells per wavelength with the Fast Fourier Transform (FFT). The Discrete Singular Convolution (DSC) [14] method uses delta cores, such as Shannon core, Poisson core or Lagrange core. And the Multi-Resolution Time-Domain (MRTD) method [15] is based on the orthonormal Harr wavelet expansions. Because these high-order FDTD approaches have destroyed the symplectic structure of the Maxwell's equations, the computed result is not satisfactory. As most traditional algorithm is non-symplectic evolved with time, and the total energy of the Hamiltonian system changes linearly, the result of the traditional algorithm is distorted. Thus, it is necessary to add the symplectic integrator into the high-order FDTD algorithm. Since Maxwell's equations can be written as an infinite dimensional Hamiltonian system, one stable and accurate solution is the Symplectic Finite-Difference Time-Domain (SFDTD) method. The SFDTD method is based on the dynamic system expressed in Hamilton and it can maintain the energy of the Hamiltonian system constant. The right mathematical formulation for the evolution of the system state is the symplectic transformation.

Though the SFDTD method has been used to solve the guided-wave [16], electromagnetic radiation and scattering problems [17], it has not been used to improve the accuracy and decreasing dispersion in the PPC. In this research, we apply the SFDTD method for numerical simulation of the one-dimensional unmagnetized PPC. And the influence of plasma frequency, dielectric constant, and thickness ratio of plasma and medium to PPC on the band gap structure is presented.

Section snippets

Theoretical derivations of the SFDTD method

For the temporal direction, the symplectic algorithm is approximated with different orders as [18]: $\exp (Δ t (U + V)) = \prod_{l = 1}^{m} \exp (d_{l} Δ t V) \exp (c_{l} Δ t U) + O (Δ t^{p + 1})$ where exp(Δt(U + V)) is the time evolution matrix of Maxwell's equations; c_l and d_l are the symplectic operators; m is the stages of the symplectic scheme; p is the order of the symplectic scheme and m ≥ p.

For the spatial direction, the explicit fourth-order-accurate difference expression is: $\frac{\partial f^{n} (i, j, k)}{\partial x} = \frac{9}{8} \frac{f^{n} (i + 1 / 2, j, k) - f^{n} (i - 1 / 2, j, k)}{Δ x} - \frac{1}{24} f^{n} (i + 3 / 2, j, k) -$

The SFDTD method's advantage on the characteristics of numerical dispersion

As mentioned above, compared with the traditional FDTD method, the high-order FDTD method can reduce the numerical dispersion effectively. And both the MRTD method and the SFDTD method are the improved high-order FDTD method. Thus, they all have better numerical dispersion than the traditional FDTD method too. The advantages of the MRTD method compared with the traditional FDTD method have been proved in the literature [15], [19]. So the compared results, which are calculated by the SFDTD

Characteristics of PBG in PPC with sandwich structure

The transmission coefficients of electromagnetic wave through PPC are calculated based on the SFDTD method. An one-dimensional Gaussian-derivative pulsed wave is used for the incident wave in the numerical simulation as: $E_{i} (t) = \frac{t - t_{0}}{t_{p}} \exp [- \frac{{(t - t_{0})}^{2}}{2 {t_{p}}^{2}}]$ where t_p = 5.4 ps, t₀ = 22 ps and the corresponding peak frequency is 30 GHz. The computing space step is 1 cm. Computational domain is subdivided into 3500 cells and eight-cells perfectly matched layer (PML) is used at the terminations of the space to

Conclusion

The transmission coefficients and PBG effect in 1-d PPC are analyzed by the SFDTD method and the traditional FDTD method. Compared with the traditional FDTD method, the results are reliable and consistent based on the SFDTD method. And the SFDTD method has its advantages in the numerical dispersion characteristics, with a lower relative phase error and better numerical dispersion error. At different plasma frequency and the thickness ratio of plasma and medium, the band gap structure of the PPC

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View full text

A research on the electromagnetic properties of Plasma Photonic Crystal based on the Symplectic Finite-Difference Time-Domain method

Abstract

Introduction

Section snippets

Theoretical derivations of the SFDTD method

The SFDTD method's advantage on the characteristics of numerical dispersion

Characteristics of PBG in PPC with sandwich structure

Conclusion

Dispersion relation of electromagnetic waves in one-dimensional plasma photonic crystals

J. Plasma Fusion Res.

Strong localization of photons in certain disordered dielectric superlattices

Phys. Rev. Lett.

Inhibited spontaneous emission in solid-state physics and electronics

Phys. Rev. Lett.

Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration. Part 1: theory

IEEE Antenna Propag. Mag.

Higher-order finite-difference schemes for electromagnetic radiation, scattering, and penetration. Part 2: applications

IEEE Antenna Propag. Mag.

Time Domain Finite Difference Computation for Maxwell's Equations

Fourth order method for Maxwell's equations on a staggered mesh

IEEE Antenna Propag. Soc. Int. Symp.

Numerical solution of initial boundary value problems involving Mxawell's equations in isotropic media

IEEE Trans. Antenna Propag.

A high-accuracy realization of the Yee algorithm using non-standard finite differences

IEEE Trans. Microw. Theor. Technol.