FDTD Method for Electromagnetic Simulations in Media Described by Time-Fractional Constitutive Relations

Abstract

In this paper, the finite-difference time-domain (FDTD) method is derived for electromagnetic simulations in media described by the time-fractional (TF) constitutive relations. TF Maxwell’s equations are derived based on these constitutive relations and the Grünwald–Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced. Finally, one-dimensional signal propagation in such electromagnetic media is considered. The proposed FDTD method is derived based on a discrete approximation of the Grünwald–Letnikov definition of the fractional derivative and evaluated in a code. The stability condition is derived for the proposed FDTD method based on a numerical-dispersion relation. The obtained numerical results are compared with the outcomes of reference frequency-domain simulations, proving the accuracy of the proposed approach. However, high spatial resolution is required in order to obtain accurate results. The developed FDTD method is, unfortunately, computation and memory demanding when compared to the ordinary FDTD algorithm.

1. Introduction

Fractional calculus has been used in modeling of electromagnetic phenomena for many years. In particular, time-fractional (TF) derivatives allow for modeling materials with memory effects and energy dissipation [1]. Nonlocality is a crucial property of fractional-order (FO) operators as it allows for including the effects of the electromagnetic-field history on the current response of the media [2]. Notably, the FO models of dielectric media, such as the Cole–Cole or Davidson models, can be used to describe some biological tissues more accurately and over a broader frequency range than the integer-order (IO) ones, e.g., the Debye model [3]. Fractional derivatives can also be applied in modeling of capacitors, which has been experimentally confirmed to be more accurate than the IO classical approach [4,5]. Spatial fractional derivatives, in the form of a fractional curl operator [6], are applied to the description of chiral media [7]. Another possible application of the FO calculus is the formulation of fractional multipoles, which has proved to be useful in solving some electrostatic problems, especially those including perfectly conducting wedges and cones [8].

The finite-difference time-domain (FDTD) method [9] remains one of the most popular techniques of electromagnetic simulation, which is successfully applied to FO models of media. For the Debye, Drude and Lorentz media, whose relative complex permittivity is described by a rational function involving integer powers of $j\omega$, the time-domain expression of the constitutive relation can easily be formulated and implemented inside the FDTD algorithm [10]. On the other hand, the dielectric response involving non-integer powers of $j\omega$ requires implementation of fractional derivatives in the time domain, which introduces difficulties in terms of domain discretization and computational overhead. In the early approach [11,12], dielectrics with fractional relaxation of the Cole–Cole, Cole–Davidson and Havriliak–Negami models are implemented based on approximation of the dielectric response by the Debye functions. Unfortunately, this technique does not correctly represent the singularity of time-domain susceptibility, which arises from dielectric models with fractional relaxation [13]. Hence, it is not possible, with the use of this technique, to accurately compute the impulse or step response of spatially complex dispersive scatterers. Other approaches employ the Z-transformation for implementation of the Cole–Cole, Cole–Davidson and Havriliak–Negami models of dispersive media. The method based on the Z-transformation and a second-order Taylor approximation of the Cole–Cole formula is proposed in [14]. This approach is then applied to the problem of the penetration of a short electromagnetic pulse into biological matter. The algorithm of the Cole–Cole model implementation in FDTD [15] is developed based on the Z-transformation and the bilinear transformation. The FO differentiators in this model are approximated by polynomials, whose coefficients are found using a least-squares fitting method. In the approach [16], fractional derivatives in the time-domain representation of the dielectric response are circumvented by the use of fast inverse Laplace transformation and Prony’s method. The Laplace transformation is used to transform the frequency-domain response of a medium into the time domain, and Prony’s method is employed to extract the parameters and transform further the time-domain response into the Z-domain in order to incorporate it directly into the FDTD method. The obtained numerical results are found to be in good agreement with those obtained by an analytical method over a broad frequency range, demonstrating the validity of this FDTD scheme. In other approaches [17,18], the FDTD simulations of the Cole–Cole media employing the auxiliary differential equation (ADE) are proposed. Whilst the former approach [17] approximates the FO differentiation with the use of a rational function derived based on the Padé approximation, the latter approach [18] approximates the FO derivatives linearly between time-steps. Both techniques, therefore, provide efficient ADE schemes with straightforward implementation within the FDTD algorithm. In the approach [19], FDTD is successfully applied to the more general Raicu medium. The proposed method solves Maxwell’s equations directly in the time domain with the use of a Riemann–Liouville FO derivative. The obtained results prove the accuracy of this technique in the ultra-wideband frequency range. In [20], a general FDTD scheme, able to incorporate the model of the medium based on the general fractional polynomial series approximation, is presented. Its applicability is further considered in several examples demonstrating its usefulness for understanding a variety of electromagnetic phenomena involving media with power-law frequency dispersion.

The aim of this paper is to present the FDTD method for simulation of wave propagation in the media described by the TF constitutive relations [21]. The considered medium is analysed based on TF Maxwell’s equations. In [22], an analysis of Poynting’s vector and propagation velocity in such a medium is performed. The causality for a plane wave propagating in the considered medium is confirmed by verification of the Kramers–Krönig relations. In [23], the uniqueness of solutions and the reciprocity are proved. In both papers, the analysis is supported by the results of numerical simulations. These are performed using the frequency-domain simulation method, employing the Fourier and Hilbert transformations, or using the time-domain method, which is based on Green’s function computations. In this paper, we investigate an alternative simulation method in an attempt to replicate the existing results and evaluate the applicability of FDTD for the considered media with the TF constitutive relations. The novelty of the paper relies on formulation of FDTD for TF Maxwell’s equations, and its evaluation in a code. Furthermore, the stability condition is derived for the proposed FDTD method based on a numerical-dispersion relation.

The paper begins with a short introduction to the fractional-derivative definition and its approximation used throughout the paper. In Section 3, TF Maxwell’s equations are derived based on the TF constitutive relations. In Section 4, the FDTD method is formulated for the media described by the TF constitutive relations, and its stability analysis is performed. Long derivations of dispersion relations and stability conditions are collected in Appendix A, Appendix B and Appendix C. In Section 5, the obtained numerical results are presented in reference to the existing frequency-domain method presented in the literature. In Section 6, an analysis of the computational complexity of the developed method is performed. Finally, the conclusions are drawn in Section 7.

2. Fractional Derivative

Let us consider a fractional derivative of the time-variable function $f\left(t\right)$, where $f:\mathbb{R}\to \mathbb{R}$. There are many definitions of a fractional derivative, which are not equivalent in the general case [24]. The most prominent ones include the Riemann–Liouville, Caputo, Marchaud and Grünwald–Letnikov definitions. For modeling electromagnetic systems, it is desirable to choose the definition which reflects certain properties, like the semigroup property

$$D_t^{\alpha }{D}_t^{\beta }f\left(t\right)=D_t^{\beta }{D}_t^{\alpha }f\left(t\right)=D_t^{\alpha +\beta }f\left(t\right)$$

(1)

and the trigonometric function invariance (derivative of the complex exponent)

$$D_t^{\alpha }{e}^{\mathsf{\Omega }t}={\mathsf{\Omega }}^{\alpha }{e}^{\mathsf{\Omega }t}.$$

(2)

In (1) and (2), $D_t^{\alpha }$ denotes the $\alpha$-order fractional derivative, $\alpha ,\beta \in \mathbb{R}$ and $\mathsf{\Omega }\in \mathbb{C}$ [25]. The Riemann–Liouville and Caputo definitions do not meet the conditions (1) and (2). The Grünwald–Letnikov and Marchaud definitions, however, satisfy both properties, making them more appropriate for applications in the electromagnetic theory [25]. In this paper, we employ the Grünwald–Letnikov definition given by [2]

$$D_t^{\alpha }f\left(t\right)=\underset{h\to 0^{+}}{lim}{\left(\frac{1}{h}\right)}^{\alpha }\sum _{l=0}^{\infty }{\left(-1\right)}^l\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)f(t-lh).$$

(3)

In the above formula, the symbol $\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)$ for non-integer $\alpha$ is given by

$$\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)=\frac{\alpha (\alpha -1)(\alpha -2)\dots (\alpha -l+1)}{l!}.$$

(4)

Apart from satisfying the desired conditions, the Grünwald–Letnikov definition can also be naturally approximated using finite differences. It is also consistent with the Marchaud definition used by us in previous investigations [22].

The approximation scheme for the derivative $D_t^{\alpha }f\left(t\right)$ is natural for a sufficiently small time-step size $T>0$, and is given by the infinite series

$$D_t^{\alpha }f\left(t\right)\approx {\left(\frac{1}{T}\right)}^{\alpha }\sum _{l=0}^{\infty }{\left(-1\right)}^l\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)f(t-lT).$$

(5)

In the case of causal functions (i.e., vanishing for times $t<0$), it naturally simplifies to the finite sum

$$D_t^{\alpha }f\left(nT\right)\approx {\left(\frac{1}{T}\right)}^{\alpha }\sum _{l=0}^n{\left(-1\right)}^l\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)f\left((n-l)T\right)$$

(6)

when calculated for $t=nT$.

3. FO Model of Dielectric Media

In a homogeneous and isotropic medium, the electromagnetic field satisfies Maxwell’s equations

$$\mathbf{\nabla }·\mathbf{D}=\rho$$

(7)

$$\mathbf{\nabla }\times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$

(8)

$$\mathbf{\nabla }·\mathbf{B}=0$$

(9)

$$\mathbf{\nabla }\times \mathbf{H}=\mathbf{J}+\frac{\partial \mathbf{D}}{\partial t}$$

(10)

where $\mathbf{D}$ and $\mathbf{B}$ are, respectively, the electric- and magnetic-flux densities, $\mathbf{E}$ and $\mathbf{H}$ are, respectively, the electric- and magnetic-field intensities, $\mathbf{J}$ is the current density and $\rho$ is the volume-charge density.

Let us consider the electromagnetic medium described by the FO constitutive relations

$${\epsilon }_{\beta }\mathbf{E}=D_t^{1-\beta }\mathbf{D}$$

(11)

$${\mu }_{\gamma }\mathbf{H}=D_t^{1-\gamma }\mathbf{B}$$

(12)

$$\mathbf{J}={\sigma }_{\eta }{D}_t^{1-\eta }\mathbf{E}$$

(13)

where ${\epsilon }_{\beta }$, ${\mu }_{\gamma }$, ${\sigma }_{\eta }$ ($\beta ,\gamma ,\eta \in (0,1]$) are parameters corresponding to the permittivity, permeability and conductivity, with units $\frac{\mathrm{F}}{{\mathrm{s}}^{1-\beta }\mathrm{m}}$, $\frac{\mathrm{H}}{{\mathrm{s}}^{1-\gamma }\mathrm{m}}$ and $\frac{\mathrm{S}}{{\mathrm{s}}^{\eta -1}\mathrm{m}}$ (F—Farad, H—Henr, S—Siemens), respectively. Assuming there is no energy dissipation due to Joule heating (${\sigma }_{\eta }=0$), we can write the set of constitutive relations in the form

$${\epsilon }_{\beta }{D}_t^{\beta }\mathbf{E}=\frac{\partial \mathbf{D}}{\partial t}$$

(14)

$${\mu }_{\gamma }{D}_t^{\gamma }\mathbf{H}=\frac{\partial \mathbf{B}}{\partial t}.$$

(15)

Using (14) and (15), one obtains from (7)–(10) Maxwell’s equations with the TF derivatives as follows:

$$\mathbf{\nabla }·\mathbf{E}=0$$

(16)

$$\mathbf{\nabla }\times \mathbf{E}=-{\mu }_{\gamma }{D}_t^{\gamma }\mathbf{H}$$

(17)

$$\mathbf{\nabla }·\mathbf{H}=0$$

(18)

$$\mathbf{\nabla }\times \mathbf{H}={\epsilon }_{\beta }{D}_t^{\beta }\mathbf{E}.$$

(19)

Let us calculate the curl of (17). Then, one obtains

$$\mathbf{\nabla }\times \mathbf{\nabla }\times \mathbf{E}=-{\mu }_{\gamma }{\epsilon }_{\beta }{D}_t^{\beta +\gamma }\mathbf{E}.$$

(20)

Using the vector calculus identity $\mathbf{\nabla }\times \mathbf{\nabla }\times \mathbf{E}=\mathbf{\nabla }(\mathbf{\nabla }·\mathbf{E})-{\nabla }^2\mathbf{E}$ and (16), one obtains from (20) the fractional diffusion-wave equation for the electric field

$${\nabla }^2\mathbf{E}-{\mu }_{\gamma }{\epsilon }_{\beta }{D}_t^{\beta +\gamma }\mathbf{E}=\mathbf{0}.$$

(21)

The fractional diffusion-wave equation interpolates between the diffusion and the wave processes, which are completely different with regard to their response to a localized disturbance [22]. That is, the diffusion equation describes a process in which a disturbance spreads infinitely fast, whereas the wave equation describes a process in which a disturbance propagates with a constant finite velocity. For the fractional diffusion-wave equation, a disturbance spreads with an infinite velocity, but its fundamental solution possesses a maximum which propagates with a finite velocity [26].

Of course, a detailed formulation of the problem in electromagnetics requires not only the Equation (21), but also boundary and initial conditions. While the boundary conditions result from the assumed behavior of the electromagnetic field on the boundary of the spatial domain, the initial conditions are assumed to be a zero electromagnetic field for $t<0$. For the electromagnetic system described by (16)–(19), employing the Grünwald–Letnikov derivative with an infinite memory of the past, it is important how the current state of the system was obtained. Therefore, the only option is to assume the zero initial conditions, which allow for including the entire history of earlier events in calculations of the current state.

Let us now focus our analysis on the one-dimensional case of a wave propagating along the z-axis, $E_x$ being the electric field, and $H_y$ being the magnetic field. Equations (17) and (19) can then be simplified to

$$\frac{\partial E_x}{\partial z}=-{\mu }_{\gamma }{D}_t^{\gamma }{H}_y$$

(22)

$$-\frac{\partial H_y}{\partial z}={\epsilon }_{\beta }{D}_t^{\beta }{E}_x$$

(23)

whereas (21) simplifies to

$$\frac{{\partial }^2{E}_x}{\partial z^2}-{\mu }_{\gamma }{\epsilon }_{\beta }{D}_t^{\beta +\gamma }{E}_x=0.$$

(24)

Let us further assume that $\beta =\gamma =\alpha \in (0.5,1)$, which means that the cases intermediate between diffusion and wave propagation are considered. This assumption and consideration of the one-dimensional propagation allow us to easily refer to the results generated by the method already presented in the literature.

4. FDTD Algorithm

In this section, the FDTD method for the media described by the TF constitutive relations is presented.

4.1. Update Equations

The FDTD method is based on a leapfrog time-stepping scheme with update equations for the electric and magnetic fields derived from discretized Maxwell’s equations [9]. Below, for any discretized function $F:{\mathbb{R}}^4\to \mathbb{R}$ representing one of the components $E_x$, $E_y$, $E_z$, $H_x$, $H_y$, $H_z$, we denote

$$F^N(I,J,K)=F(I·\mathsf{\Delta }x, J·\mathsf{\Delta }y, K·\mathsf{\Delta }z, N·\mathsf{\Delta }t)$$

where $\mathsf{\Delta }t$ is the time-step size and $\mathsf{\Delta }x$, $\mathsf{\Delta }y$, $\mathsf{\Delta }z$ are the spatial-step sizes along the x, y and z directions, respectively. After approximating in (17) and (19) the spatial derivatives by central differences [9] and the TF derivatives by the finite sum (6) (under the assumption $\beta =\gamma =\alpha$), one obtains the following relations, which are used to compute new values of the electric and magnetic fields

$$\begin{array}{cc}\hfill H_x^{n+\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2})=& -\sum _{l=1}^n{w}_l^{\alpha }{H}_x^{n-l+\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2})+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\mu }_{\alpha }}\left(\frac{E_y^n(i,j+\frac{1}{2},k+1)-E_y^n(i,j+\frac{1}{2},k)}{\mathsf{\Delta }z}-\right.\hfill \\ & \left.\frac{E_z^n(i,j+1,k+\frac{1}{2})-E_z^n(i,j,k+\frac{1}{2})}{\mathsf{\Delta }y}\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill H_y^{n+\frac{1}{2}}(i+\frac{1}{2},j,k+\frac{1}{2})=& -\sum _{l=1}^n{w}_l^{\alpha }{H}_y^{n-l+\frac{1}{2}}(i+\frac{1}{2},j,k+\frac{1}{2})+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\mu }_{\alpha }}\left(\frac{E_z^n(i+1,j,k+\frac{1}{2})-E_z^n(i,j,k+\frac{1}{2})}{\mathsf{\Delta }x}-\right.\hfill \\ & \left.\frac{E_x^n(i+\frac{1}{2},j,k+1)-E_x^n(i+\frac{1}{2},j,k)}{\mathsf{\Delta }z}\right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill H_z^{n+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k)=& -\sum _{l=1}^n{w}_l^{\alpha }{H}_z^{n-l+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k)+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\mu }_{\alpha }}\left(\frac{E_x^n(i+\frac{1}{2},j+1,k)-E_x^n(i+\frac{1}{2},j,k)}{\mathsf{\Delta }y}-\right.\hfill \\ & \left.\frac{E_y^n(i+1,j+\frac{1}{2},k)-E_y^n(i,j+\frac{1}{2},k)}{\mathsf{\Delta }x}\right)\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill E_x^{n+1}(i+\frac{1}{2},j,k)=& -\sum _{l=1}^n{w}_l^{\alpha }{E}_x^{n-l+1}(i+\frac{1}{2},j,k)+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\epsilon }_{\alpha }}\left(\frac{H_z^{n+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k)-H_z^{n+\frac{1}{2}}(i+\frac{1}{2},j-\frac{1}{2},k)}{\mathsf{\Delta }y}-\right.\hfill \\ & \left.\frac{H_y^{n+\frac{1}{2}}(i+\frac{1}{2},j,k+\frac{1}{2})-H_y^{n+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k-\frac{1}{2})}{\mathsf{\Delta }z}\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill E_y^{n+1}(i,j+\frac{1}{2},k)=& -\sum _{l=1}^n{w}_l^{\alpha }{E}_y^{n-l+1}(i,j+\frac{1}{2},k)+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\epsilon }_{\alpha }}\left(\frac{H_x^{n+\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2})-H_x^{n+\frac{1}{2}}(i,j+\frac{1}{2},k-\frac{1}{2})}{\mathsf{\Delta }z}-\right.\hfill \\ & \left.\frac{H_z^{n+\frac{1}{2}}(i+\frac{1}{2},j+\frac{1}{2},k)-H_z^{n+\frac{1}{2}}(i-\frac{1}{2},j+\frac{1}{2},k)}{\mathsf{\Delta }x}\right)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill E_z^{n+1}(i,j,k+\frac{1}{2})=& -\sum _{l=1}^n{w}_l^{\alpha }{E}_z^{n-l+1}(i,j,k+\frac{1}{2})+\hfill \\ \hfill & \frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\epsilon }_{\alpha }}\left(\frac{H_y^{n+\frac{1}{2}}(i+\frac{1}{2},j,k+\frac{1}{2})-H_y^{n+\frac{1}{2}}(i-\frac{1}{2},j,k+\frac{1}{2})}{\mathsf{\Delta }x}-\right.\hfill \\ & \left.\frac{H_x^{n+\frac{1}{2}}(i,j+\frac{1}{2},k+\frac{1}{2})-H_x^{n+\frac{1}{2}}(i,j-\frac{1}{2},k+\frac{1}{2})}{\mathsf{\Delta }y}\right).\hfill \end{array}$$

(30)

The coefficients of the Grünwald–Letnikov derivative $w_l^{\alpha }={\left(-1\right)}^l\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)$ can be computed in a recursive manner, which allows for reducing the computational overhead as follows:

$$w_l^{\alpha }=\left(1-\frac{1+\alpha }{l}\right)w_{l-1}^{\alpha },w_0^{\alpha }=1.$$

(31)

This relation holds true because

$$\begin{array}{c}\hfill w_l^{\alpha }={\left(-1\right)}^l\left(\genfrac{}{}{0pt}{}{\alpha }{l}\right)={\left(-1\right)}^l\frac{\alpha (\alpha -1)(\alpha -2)\dots (\alpha -l+1)}{l!}=\hfill \\ \hfill (-1)·{\left(-1\right)}^{l-1}\frac{\alpha (\alpha -1)(\alpha -2)\dots (\alpha -l+2)}{(l-1)!}\frac{\alpha -l+1}{l}=\left(1-\frac{\alpha +1}{l}\right)w_{l-1}^{\alpha }.\hfill \end{array}$$

(32)

In the one-dimensional case, the update equations simplify to

$$H_y^{n+\frac{1}{2}}\left(k+\frac{1}{2}\right)=-\sum _{l=1}^n{w}_l^{\alpha }{H}_y^{n-l+\frac{1}{2}}\left(k+\frac{1}{2}\right)-\frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\mu }_{\alpha }}\frac{E_x^n(k+1)-E_x^n\left(k\right)}{\mathsf{\Delta }z}$$

(33)

$$E_x^{n+1}\left(k\right)=-\sum _{l=1}^n{w}_l^{\alpha }{E}_x^{n+1-l}\left(k\right)-\frac{{\left(\mathsf{\Delta }t\right)}^{\alpha }}{{\epsilon }_{\alpha }}\frac{H_y^{n+\frac{1}{2}}\left(k+\frac{1}{2}\right)-H_y^{n+\frac{1}{2}}\left(k-\frac{1}{2}\right)}{\mathsf{\Delta }z}.$$

(34)

4.2. Stability

In the IO case of the FDTD method ($\alpha =1$), the time-step size is limited by the Courant condition [9]

$$\mathsf{\Delta }t\le \frac{1}{c_0\sqrt{\frac{1}{{\left(\mathsf{\Delta }x\right)}^2}+\frac{1}{{\left(\mathsf{\Delta }y\right)}^2}+\frac{1}{{\left(\mathsf{\Delta }z\right)}^2}}}$$

(35)

where $c_0$ denotes the light speed in a vacuum. Meeting this condition is required to ensure the stability of simulation. However, the condition (35) is not sufficient in the case of FDTD in the media described by the TF constitutive relations.

Following the standard derivation of the IO FDTD stability condition [9], the time-step limit is derived for FDTD in the media described by the TF constitutive relations based on (25)–(30)

$$\mathsf{\Delta }t\le 2^{1-\frac{1}{\alpha }}{\left(\frac{\sqrt{{\mu }_{\alpha }{\epsilon }_{\alpha }}}{\sqrt{\frac{1}{{\left(\mathsf{\Delta }x\right)}^2}+\frac{1}{{\left(\mathsf{\Delta }y\right)}^2}+\frac{1}{{\left(\mathsf{\Delta }z\right)}^2}}}\right)}^{\frac{1}{\alpha }}.$$

(36)

The detailed derivation of (36) is presented in Appendix A, Appendix B and Appendix C. This new time-step limit is obtained through analysis of the numerical-dispersion relation. When $\alpha =1$ is assumed, the condition (36) simplifies to the IO case (35). In the considered one-dimensional simulation, the stability condition simplifies to

$$\mathsf{\Delta }t\le 2^{1-\frac{1}{\alpha }}{\left(\sqrt{{\epsilon }_{\alpha }{\mu }_{\alpha }}\mathsf{\Delta }z\right)}^{\frac{1}{\alpha }}.$$

(37)

Let us verify numerically the condition (37). For this purpose, let us consider the simulation with an initial condition $E_x^0\left(k_s\right)=1$, while the field values at other spatial and temporal localizations are equal to zero. For a varying parameter $\frac{1}{2}<\alpha <1$, values of $\mathsf{\Delta }t$ around the analytical stability limit are considered. For each value of the time-step size, a short simulation is performed. In the case of the considered FO model, a decay in field amplitude is expected, which is further discussed in Section 5. Therefore, if the maximum value of the electric field at the end of the simulation is larger than one, it is considered as unstable. The results of numerical verification are presented in Figure 1, where the minimal unstable value of $\mathsf{\Delta }t$ is plotted along with the analytically obtained stability limit.

For each considered value of $\alpha$, all the verified values of the time-step size below the limit (37) resulted in stable simulations, while all the values above this limit resulted in unstable ones. This confirms the correctness of the derived stability condition.

5. Simulation Results

In this section, simulation results are presented for the FDTD method in the media described by the TF constitutive relations. For all the cases, unless stated otherwise, $\mathsf{\Delta }z=0.01 \mathsf{\mu }\mathrm{m}$ and the Courant coefficient is equal to $0.999$, which means that $\mathsf{\Delta }t$ is equal to $0.999$ of the stability limit given by (37). We focus on one-dimensional simulations due to the large computational overhead of the method, which includes the whole history of the electromagnetic field in the computations, as well as due to easy comparison with the results generated by the method presented in the literature.

5.1. Simulations of Wave Propagation

One-dimensional simulations of wave propagation are considered in a homogeneous FO medium, with the values of $\alpha$ slightly below the unit value. The source signal, in the form of a Gaussian-modulated sinusoidal pulse (presented in Figure 2), is injected $0.1\mathsf{\mu }\mathrm{m}$ from the left boundary of the computational domain and propagates rightwards.

Results of the FDTD simulations in the form of the electric-field waveforms measured at the distance $10\mathsf{\mu }\mathrm{m}$ from the source are presented in Figure 3.

In the simulation with $\alpha =1$, the pulse propagates without deformation. It is expected, because this case is equivalent to wave propagation in a vacuum. In other cases, wave attenuation and distortion are observed, which is consistent with the results in [22]. Furthermore, the pulse maximum is also observed earlier than in a vacuum. All the described effects (i.e., wave attenuation, dispersion, and increased group velocity) increase with the decrease in the parameter $\alpha$.

The obtained results are compared with the results of the frequency-domain method presented in [22]. The electric-field waveforms are shown for different measurement positions in Figure 4. There are visible differences in the field values between the results of both methods, which diminish when the spatial-step size $\mathsf{\Delta }z$ is decreased for FO FDTD.

The differences between the waveforms are visible when the spatial resolution is insufficient in FO FDTD. It is clear, however, that the results of the FO FDTD simulations converge to those obtained using the frequency-domain method when the spatial-step size $\mathsf{\Delta }z$ is decreased. This confirms that the FO FDTD simulation method is developed correctly.

5.2. Simulations of Reflections

The FDTD method allows one to combine different materials in a single computational domain. Therefore, we consider a simulation of the wave propagating through a vacuum and impinging on a material described by the TF constitutive relations, with the parameter $\alpha =0.99$. The results, in the form of electric-field distributions in the space for varying time, are presented in Figure 5.

The incident wave is almost completely transmitted into the FO material, with only a small reflection, as can be seen in Figure 5b,c. The ratio of the amplitude of the reflected wave to the amplitude of the incident wave is equal to $0.0091$. The transmitted pulse is further attenuated while propagating through the FO medium. The incident Gaussian-modulated sinusoidal pulse is, however, also slightly distorted before arriving upon the boundary, as can be seen in Figure 5a. This is undesired behavior, as the wave should not experience any significant deformation or attenuation while travelling through a vacuum. This purely numerical distortion is a result of the Courant coefficient for a vacuum being significantly lower than 1, i.e., $\mathsf{\Delta }t$ is different than the magic time-step size, which provides accurate simulations in one-dimensional IO FDTD [9]. Such a value of the Courant coefficient is, however, necessary in order to ensure the stability of simulation in the FO medium.

6. Computational Complexity

Computational complexity is an important factor in numerical simulations. The IO FDTD algorithm has $O\left(n\right)$ complexity with regard to the number of iterations (i.e., time-steps). In the case of FO FDTD, complexity increases to $O\left(n^2\right)$ because, in every iteration, all the previous electromagnetic-field values are included in the update procedure. The quadratic dependence on the number of time-steps is confirmed numerically, with the results presented in Figure 6. All the presented execution times are obtained with the use of an Intel(R) Core(TM) i7-8750H processor. The OpenMP-based parallelization is used in our code in order to distribute computations among the CPU cores.

The execution time, depending on the size of the computational domain, is presented in Figure 7. The complexity is linear, i.e., the same as in the case of IO FDTD.

Apart from a larger time complexity, FO FDTD also suffers from a significantly larger memory demand. As all the previous field values need to be stored, the occupied memory grows linearly with time. In the IO case, on the other hand, it remains constant.

Higher computational and memory demands are further increased by the necessity of taking lower values of the time-step size in order to satisfy the stability condition (37). As can be seen in Figure 1, the stable $\mathsf{\Delta }t$ values are significantly lower than in the IO FDTD case, even for $\alpha$ near 1.

7. Conclusions

In this paper, the FDTD simulation method for wave propagation in the media described by the TF constitutive relations is presented. TF Maxwell’s equations are derived based on these constitutive relations and the Grünwald–Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced. The stability limit of the time-step size is derived for the proposed FDTD method which, to the best of our knowledge, has not been presented in the literature before. The obtained FDTD results are compared with those obtained with the use of the reference frequency-domain method already presented in the literature, and the compatibility between both methods is hereby demonstrated. Therefore, FDTD is proved to be an appropriate method for simulation of the FO media as well. High spatial resolution is required, however, to obtain accurate results. FO FDTD is, unfortunately, computation and memory demanding compared to the ordinary (i.e., IO) FDTD algorithm and, therefore, requires larger computational resources than in the IO case.

Our future research focus will include investigation of implicit FDTD schemes applied to the FO media, as well as consideration of their acceleration techniques.