On Time-Domain NRBC for Maxwell’s Equations and Its Application in Accurate Simulation of Electromagnetic Invisibility Cloaks

Abstract

In this paper, we present analytic formulas of the temporal convolution kernel functions involved in the time-domain non-reflecting boundary condition (NRBC) for the electromagnetic scattering problems. Such exact formulas themselves lead to accurate and efficient algorithms for computing the NRBC for domain reduction of the time-domain Maxwell’s system in ${\mathbb {R}}^3$. A second purpose of this paper is to derive a new time-domain model for the electromagnetic invisibility cloak. Different from the existing models, it contains only one unknown field and the seemingly complicated convolutions can be computed as efficiently as the temporal convolutions in the NRBC. The governing equation in the cloaking layer is valid for general geometry, e.g., a spherical or polygonal layer. Here, we aim at simulating the spherical invisibility cloak. We take the advantage of radially stratified dispersive media and special geometry, and develop an efficient vector spherical harmonic-spectral-element method for its accurate simulation. Compared with limited results on FDTD simulation, the proposed method is optimal in both accuracy and computational cost. Indeed, the saving in computational time is significant.

Appendices

Appendix A. Vector Spherical Harmonics

We adopt the notation and setting as in Nédélec [28]. The spherical coordinates $(r,\theta ,\varphi )$ are related to the Cartesian coordinates ${\varvec{r}}=(x,y,z)$ via

$$\begin{aligned} x=r\sin \theta \cos \varphi ,\quad y=r\sin \theta \sin \varphi ,\quad z=r\cos \theta , \end{aligned}$$

(A.1)

where $r\ge 0,\theta \in [0,\pi ]$ and $\phi \in [0,2\pi ).$ The corresponding moving (right-handed) orthonormal coordinate basis $\{{\varvec{e}}_r, {\varvec{e}}_\theta , {\varvec{e}}_\varphi \}$ is given by

$$\begin{aligned} \begin{aligned}&\varvec{e}_r={\varvec{r}}/{r},\;\; {\varvec{e}}_\theta =(\cos \theta \cos \varphi , \ \cos \theta \sin \varphi , \ -\sin \theta ), \;\; {\varvec{e}}_\varphi = (-\sin \varphi , \ \cos \varphi , \ 0). \end{aligned} \end{aligned}$$

(A.2)

Let $\{Y_{l}^m\}$ be the spherical harmonics as normalized in [28], and let S be the unit sphere. Recall that

$$\begin{aligned} \nabla _S Y_l^m =\frac{\partial Y_l^m}{\partial \theta }{\varvec{e}}_\theta +\frac{1}{\sin \theta } \frac{\partial Y_l^m}{\partial \varphi }{\varvec{e}}_\varphi . \end{aligned}$$

(A.3)

The VSH family $\big \{ \varvec{Y}_l^m , \varvec{\Psi }_{l}^m, \varvec{\Phi }_l^m\big \}:=\big \{ Y_l^m {\varvec{e}}_r, \nabla _SY_{l}^m, \nabla _S Y_l^m \wedge {\varvec{e}}_r\big \},$ which has been used in the Spherepack [37] (also see [27]) forms a complete orthogonal basis of ${\varvec{L}}^2(S):=(L^2(S))^3$ under the inner product:

$$\begin{aligned} \langle \varvec{u}, \varvec{v}\rangle _S=\int _S \varvec{u}\cdot \bar{\varvec{v}}\, dS=\int _0^{2\pi }\quad \int _0^{\pi } \varvec{u}\cdot \bar{\varvec{v}}\, \sin \theta \, d\theta d\varphi . \end{aligned}$$

(A.4)

Define the subspace of ${\varvec{L}}^2(S),$ consisting of the tangent components of the vector fields on S:

$$\begin{aligned} {\varvec{L}}_T^2(S)= \big \{\varvec{u}\in {\varvec{L}}^2(S) : \varvec{u}\cdot \hat{\varvec{x}}=0\big \}. \end{aligned}$$

(A.5)

The VSH $\{\varvec{\Psi }_{l}^m, {\varvec{\Phi }}_l^m\}$ forms a complete orthogonal basis of ${\varvec{L}}_T^2(S).$ Consequently, the vector field expanded in terms of VSH has a distinct separation of tangential and normal components. For any vector fields $\varvec{u}\in {\varvec{L}}^2(S)$, we write

$$\begin{aligned} {\varvec{u}}=u_{00}\varvec{Y}_0^0+\sum _{l=1}^\infty \sum _{|m|=0}^l\big \{u_{lm}^{r}\, \varvec{Y}_l^m\, +u_{lm}^{(1)}\varvec{\Psi }_{l}^m+u_{lm}^{(2)}\, {\varvec{\Phi }}_l^m\big \}, \end{aligned}$$

(A.6)

where we denote $\beta _l=l(l+1),$ and have

$$\begin{aligned} u_{00}=\langle \varvec{u}, \varvec{Y}_0^0 \rangle _S,\;\;\;u_{lm}^r= \langle \varvec{u}, \varvec{Y}_l^m \rangle _S, \;\;\; u_{lm}^{(1)}= {\beta _l^{-1}}\langle \varvec{u}, \varvec{\Psi }_{l}^m \rangle _S, \;\;\; u_{lm}^{(2)}= {\beta _l^{-1}}\langle \varvec{u}, {\varvec{\Phi }}_l^m\rangle _S. \end{aligned}$$

(A.7)

It is noteworthy that given $\varvec{u}$, we can be computed $\{u_{lm}^r, u_{lm}^{(1)}, u_{lm}^{(2)}\}$ via the discrete VSH-transform using the Spherepack [37], and vice versa by the inverse transform. Moreover, the normal component solely involves the first term while the tangential component $\varvec{E}_T$ of $\varvec{E}$ involves the last two terms in (A.6).

Now, we collect some frequently used vector calculus formulas. Define the differential operators:

$$\begin{aligned} d_l^{\pm }=\frac{d}{dr}\pm \frac{l}{r}, \quad {\hat{\partial }}_r=\frac{d}{dr}+\frac{1}{r}, \quad {{\mathcal {L}}}_l={\hat{\partial }}_r^2-\frac{\beta _l}{r^2}=\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr} -\frac{\beta _l}{r^2}, \end{aligned}$$

(A.8)

where $\beta _l:=l(l+1)$. For any given f(r), the following properties can be derived from [15]:

• For divergence operator

  $$\begin{aligned} {\mathrm{div}}\big (f \varvec{Y}_{l}^m\big )=\Big (\frac{d}{dr}+\frac{2}{r}\Big ) f\,Y_l^m,\quad {\mathrm{div}}\big (f \varvec{\Psi }_{l}^m\big )=-\beta _l\frac{f}{r} \,\varvec{Y}_l^m,\quad {\mathrm{div}}\big (f \varvec{\Phi }_{l}^m\big )=0; \end{aligned}$$

  (A.9)

• For curl operator

  $$\begin{aligned} \nabla \times \big (f \varvec{Y}_l^m \big )=\frac{f}{r} \,\varvec{\Phi }_l^m,\quad \nabla \times \big (f \varvec{\Psi }_{l}^m\big )=-{\hat{\partial }}_r f \,\varvec{\Phi }_l^m, \quad \nabla \times \big (f \varvec{\Phi }_l^m\big )={\hat{\partial }}_r f \,\varvec{\Psi }_{l}^m+\beta _l\frac{f}{r} \varvec{Y}_l^m; \end{aligned}$$

  (A.10)

• For Laplace operator

  $$\begin{aligned} \Delta \big (f{\varvec{\Phi }}_l^m\big )={{\mathcal {L}}}_{l}(f){\varvec{\Phi }}_l^m. \end{aligned}$$

  (A.11)

Appendix B. Proof of Proposition 3.2

Proof

Recall that if ${\mathrm{div}} {\varvec{u}}=0,$ then $\nabla \times \nabla \times {\varvec{u}}=-\Delta {\varvec{u}}.$ Thus, from (A.8)–(A.11), we derive

$$\begin{aligned}&\nabla \times \nabla \times \big ( u{\varvec{\Phi }}_l^m\big )=-\Delta \big ( u {\varvec{\Phi }}_l^m\big )=-{\mathcal {L}}_l(u){\varvec{\Phi }}_l^m,\\&\nabla \times \nabla \times \nabla \times \big ( v {\varvec{\Phi }}_l^m\big )=-\nabla \times \big ( \Delta \big ( v {\varvec{\Phi }}_l^m\big ) \big )=-\nabla \times \big ( {\mathcal {L}}_l\big (v\big ){\varvec{\Phi }}_l^m\big ). \end{aligned}$$

Therefore, (3.28a) can be reduced to:

$$\begin{aligned} \frac{\partial ^2u^i_{lm}}{\partial t^2}-c^2{\mathcal {L}}_l( u^i_{lm} )=f_{1,l}^{i,m}\quad \frac{\partial ^2v^i_{lm}}{\partial t^2}-c^2{\mathcal {L}}_l( v^i_{lm} )=f_{2,l}^{i,m}, \quad r\in I_i,\quad i=0, 2, 3, \end{aligned}$$

(B.1)

for $|m|\le l$, $l=1, 2, \cdots$, by using the expansions (3.32).

In spherical coordinates (cf. [1]):

$$\begin{aligned} \nabla \times {\varvec{v}}= \frac{1}{r\sin \theta }\Big ( \frac{\partial \big (\sin \theta v_\varphi \big )}{\partial \theta } -\frac{\partial v_\theta }{\partial \varphi }\Big ){\varvec{e}}_r+ \frac{1}{r} \Big (\frac{1}{\sin \theta } \frac{\partial v_r}{\partial \varphi }- \frac{\partial \big (r v_\varphi \big )}{\partial r} \Big ){\varvec{e}}_\theta +\frac{1}{r} \Big (\frac{\partial \big (r v_\theta \big )}{\partial r}-\frac{\partial v_r}{\partial \theta } \Big ){\varvec{e}}_\varphi , \end{aligned}$$

(B.2)

for any vector field $\varvec{v}=v_r{\varvec{e}}_r+v_{\theta }{\varvec{e}}_\theta +v_{\varphi }{\varvec{e}}_\varphi$. Apparently, we have $\nabla \times (u_{00}^i(r, t)\varvec{Y}_0^0)=\varvec{0},$ as $\varvec{Y}_0^0={\varvec{e}}_r/\sqrt{4\pi }$. For the coefficient $u_{00}^i$, we then have

$$\begin{aligned} \frac{\partial ^2u_{00}^i}{\partial t^2}=f_{00}^i,\quad r\in I_i,\quad i=0, 2, 3. \end{aligned}$$

(B.3)

We now turn to the governing equation (3.28b) in the cloaking layer $I_1=(R_1,R_2)$. According to (A.10), the vector spherical harmonic expansion of $\varvec{D}^1$ can be rewritten as

$$\begin{aligned} {\varvec{D}}^1=u_{00}^1\varvec{Y}_0^0+\sum _{l=1}^\infty \sum _{|m|=0}^l \Big \{ u_{lm}^1\varvec{\Phi }_l^m+ {\hat{\partial }}_r v_{lm}^1 \varvec{\Psi }_{l}^m+\frac{\beta _l}{r}v_{lm}^1 \varvec{Y}_{l}^m \Big \}. \end{aligned}$$

(B.4)

Using (B.4) and the fact that ${\mathscr {D}}_1$ defined in (3.9) is uniaxial, we have

$$\begin{aligned} {\mathscr {D}}_1[{\varvec{D}}^1]=\big (u_{00}^1+\theta _1*u_{00}^1\big )\varvec{Y}_{0}^0+\sum _{l=1}^\infty \sum _{|m|=0}^l\Big \{\epsilon ^{-1}u_{lm}^1\varvec{\Phi }_l^m+\epsilon ^{-1}{\hat{\partial }}_r v_{lm}^1 \varvec{\Psi }_{l}^m +\frac{\beta _l}{r}\big (v_{lm}^1+\theta _1*v_{lm}^1\big )\varvec{Y}_{l}^m\Big \}. \end{aligned}$$

(B.5)

Using formula (B.2), we have

$$\begin{aligned} \nabla \times \Big (\big (u_{00}^1+\theta _1*u_{00}^1\big )\varvec{Y}_{0}^0\Big )=\varvec{0}. \end{aligned}$$

(B.6)

Then, we calculate from (B.5) that

$$\begin{aligned} \begin{aligned} \nabla \times \big ({\mathscr {D}}_1[{\varvec{D}}^1 ] \big )=&\sum _{l=1}^\infty \sum _{|m|=0}^l\Big (\frac{\beta _l}{r^2}\big (v_{lm}^1+\theta _1*v_{lm}^1\big )-\epsilon ^{-1} {\hat{\partial }}_r^2 v_{lm}^1\Big )\varvec{\Phi }_l^m\\&+\sum _{l=1}^\infty \sum _{|m|=0}^l\Big (\epsilon ^{-1} {\hat{\partial }}_ru_{lm}^1 \varvec{\Psi }_{l}^m+\epsilon ^{-1}\frac{\beta _l}{r}u_{lm}^1\varvec{Y}_{l}^m\Big ) \end{aligned} \end{aligned}$$

(B.7)

by using formulas (A.10). Repeating the above calculation and using the definition of ${\mathscr {D}}_2$ and (A.10), we obtain

$$\begin{aligned} \begin{aligned}&\nabla \times \big ( {\mathscr {D}}_2\big [\nabla \times \big ({\mathscr {D}}_1[{\varvec{D}}^1] \big )\big ] \big )=\sum _{l=1}^\infty \sum _{|m|=0}^l\epsilon ^{-1}\Big ( \frac{\beta _l}{r^2}(u_{lm}^1+\theta _2*u_{lm}^1)-\epsilon ^{-1}{\hat{\partial }}_r^2 u_{lm}^1 \Big )\varvec{\Phi }_l^m\\&\quad +\sum _{l=1}^\infty \sum _{|m|=0}^l\epsilon ^{-1}\nabla \times \Big (\Big ( \frac{\beta _l}{r^2}(v_{lm}^1+\theta _1*v_{lm}^1)- \epsilon ^{-1}{\hat{\partial }}_r^2 v_{lm}^1 \Big )\varvec{\Phi }_l^m \Big ), \end{aligned} \end{aligned}$$

(B.8)

Inserting the above equation into (3.28b), one immediately shows that the expansion coefficients $\{u_{lm}^1$, $v_{lm}^1\}, |m|\le l, l=1, 2, \cdots$ satisfy the same governing equation (3.40b) with different convolution kernels $\theta _2$ and $\theta _1$, respectively. As in (B.3), $u_{00}^1$ satisfies the same differential equation.

According to (B.4) and (B.5) and the facts

$$\begin{aligned} \varvec{\Psi }_l^m\times \varvec{e}_r=\varvec{\Phi }_l^m,\quad \varvec{\Phi }_l^m\times \varvec{e}_r=-\varvec{\Psi }_l^m, \end{aligned}$$

(B.9)

we have

$$\begin{aligned} \begin{aligned}&{\varvec{D}}^i\times \varvec{e}_r= \sum _{l=1}^\infty \sum _{|m|=0}^l\big (-u_{lm}^i\varvec{\Psi }_l^m+ {\hat{\partial }}_r v_{lm}^i \varvec{\Phi }_{l}^m\big ),\quad i=0, 2,3,\\&\quad (\nabla \times {\varvec{D}}^i)\times \varvec{e}_r= \sum _{l=1}^\infty \sum _{|m|=0}^l\Big \{ \Big ({\hat{\partial }}_r^2 v_{lm}^i- \frac{\beta _l}{r^2}v_{lm}^i\Big )\varvec{\Psi }_l^m+\hat{\partial }_ru_{lm}^i \varvec{\Phi }_{l}^m\Big \},\quad i=0, 2,3, \end{aligned} \end{aligned}$$

(B.10)

and

$$\begin{aligned} \begin{aligned}&{\mathscr {D}}_1[{\varvec{D}}^1]\times \varvec{e}_r=\sum _{l=1}^\infty \sum _{|m|=0}^l\big (-\epsilon ^{-1}u_{lm}^1\varvec{\Psi }_l^m+\epsilon ^{-1}{\hat{\partial }}_r v_{lm}^1 \varvec{\Phi }_{l}^m\big ),\\&\quad \big (\nabla \times ({\mathscr {D}}_1[{\varvec{D}}^1])\big )\times \varvec{e}_r= \sum _{l=1}^\infty \sum _{|m|=0}^l\Big \{\Big (\epsilon ^{-1} {\hat{\partial }}_r^2 v_{lm}^1- \frac{\beta _l}{r^2}\big (v_{lm}^1+\theta _1*v_{lm}^1\big )\Big )\varvec{\Psi }_l^m +\epsilon ^{-1} {\hat{\partial }}_ru_{lm}^1 \varvec{\Phi }_{l}^m\Big \}. \end{aligned} \end{aligned}$$

Substituting the above equations into jump condition (3.28c) and (3.28d), we obtain jump conditions

$$\begin{aligned} \begin{aligned} \epsilon u_{lm}^0=u_{lm}^1,\quad \partial _r v_{lm}^1=\epsilon \partial _r v_{lm}^0+(\epsilon -1)r^{-1}v_{lm}^0 \quad {\mathrm{at}}\;\;\; r=R_1,\\ \epsilon u_{lm}^2=u_{lm}^1,\quad \partial _r v_{lm}^1=\epsilon \partial _r v_{lm}^2+(\epsilon -1)r^{-1}v_{lm}^2 \quad {\mathrm{at}}\;\;\; r=R_2, \end{aligned} \end{aligned}$$

(B.11)

and

$$\begin{aligned} \partial _r u_{lm}^1=\epsilon ^2 \partial _r u_{lm}^0+\epsilon (\epsilon -1)r^{-1}u_{lm}^0 \quad {\mathrm{at}}\;\;\; r=R_1, \end{aligned}$$

(B.12)

$$\begin{aligned} \partial _r u_{lm}^1=\epsilon ^2 \partial _r u_{lm}^2+\epsilon (\epsilon -1)r^{-1}u_{lm}^2 \quad {\mathrm{at}}\;\;\; r=R_2, \end{aligned}$$

(B.13)

$$\begin{aligned} {\hat{\partial }}_r^2 v_{lm}^0- \frac{\beta _l}{r^2}v_{lm}^0=\epsilon ^{-2} {\hat{\partial }}_r^2 v_{lm}^1- \frac{\beta _l}{\epsilon r^2}\big (v_{lm}^1+\theta _1*v_{lm}^1\big ) \quad {\mathrm{at}}\;\;\; r=R_1, \end{aligned}$$

(B.14)

$$\begin{aligned} {\hat{\partial }}_r^2 v_{lm}^2- \frac{\beta _l}{r^2}v_{lm}^2=\epsilon ^{-2} {\hat{\partial }}_r^2 v_{lm}^1- \frac{\beta _l}{\epsilon r^2}\big (v_{lm}^1+\theta _1*v_{lm}^1\big ) \quad {\mathrm{at}}\;\;\; r=R_2. \end{aligned}$$

(B.15)

Noting that $\hat{\partial }_r^2u=\frac{1}{r^2}\frac{\partial }{\partial r}\big (r^2\frac{\partial u}{\partial r}\big )$, the governing equations (3.40a) and (3.40b) then gives

$$\begin{aligned} \begin{aligned} \epsilon ^{-2} {\hat{\partial }}_r^2 v_{lm}^1- \frac{\beta _l}{\epsilon r^2}\big (v_{lm}^1+\theta _1*v_{lm}^1\big )=\frac{1}{c^2}\frac{\partial ^2v^1_{lm}}{\partial t^2},\quad {\hat{\partial }}_r^2 v_{lm}^i- \frac{\beta _l}{r^2}v_{lm}^i=\frac{1}{c^2}\frac{\partial ^2v^i_{lm}}{\partial t^2}, \end{aligned} \end{aligned}$$

(B.16)

for $i=0, 2, 3.$ Substituting (B.16) into the jump conditions (B.14) and (B.15) and integrate w.r.t. t and using homogeneous initial conditions (3.40g), we derive

$$\begin{aligned} v^0_{lm}=v^1_{lm}\quad {\mathrm{at}}\;\;\; r=R_1; \quad v^2_{lm}=v^1_{lm}\quad {\mathrm{at}}\;\;\; r=R_2. \end{aligned}$$

(B.17)

Note that the jump conditions at artificial interface $r=R_3$ are trivial. Thus, we consider the boundary condition at $r=b$.

Applying expansion (3.32) in (3.28f) and using identities (B.9), (B.10) and formulation (3.37), we obtain

$$\begin{aligned} \begin{aligned}&\sum _{l=1}^\infty \sum _{|m|=0}^l \!\Big (\partial _t{\hat{\partial }}_r v_{lm}^3+c\Big ({\hat{\partial }}_r^2 v_{lm}^3- \frac{\beta _l}{b^2}v_{lm}^3\Big )-\frac{c}{b^2}\omega _l*v_{lm}^{3}\Big )\varvec{\Psi }_l^m\\&\quad +\sum _{l=1}^\infty \sum _{|m|=0}^l\!\Big (\partial _tu_{lm}^3+c\hat{\partial }_ru_{lm}^3 -\frac{c}{b}\sigma _l*u_{lm}^{3}\Big )\varvec{\Phi }_{l}^m=\varvec{0}, \end{aligned} \end{aligned}$$

(B.18)

which implies two boundary conditions

$$\begin{aligned} \frac{1}{c}\partial _tu_{lm}^3+\frac{{\partial }u_{lm}^3}{\partial r}+\frac{1}{b}u_{lm}^3 -\frac{1}{b}\sigma _l*u_{lm}^{3}=0\quad {\mathrm{at}}\;\; r=b, \end{aligned}$$

(B.19)

$$\begin{aligned} \frac{\partial }{\partial r}\frac{\partial v_{lm}^3}{\partial t}+\frac{1}{b}\frac{\partial v_{lm}^3}{\partial t}+c\Big ({\hat{\partial }}_r^2 v_{lm}^3- \frac{\beta _l}{b^2}v_{lm}^3\Big )-\frac{c}{b^2}\omega _l*v_{lm}^{3}=0\quad {\mathrm{at}}\;\; r=b. \end{aligned}$$

(B.20)

Here, the definition of differential operator $\hat{\partial }$ in (A.8) is applied. Obviously, the boundary condition for $u_{lm}^3$ is exactly the one we adopted in the model problem (3.40). Next, we will show that the equation (B.20) can be reformulated to the same form as (B.19). Indeed, we can directly calculate

$$\begin{aligned} \frac{c}{b^2}\omega _l(t)*v_{lm}^3(b, t)=\frac{1}{b}\bigg (\int _0^t\sigma _l'(t-\tau )v_{lm}^3(b, \tau )\,d\tau +\sigma _l(0)v_{lm}^3(b, t)\bigg )=\frac{1}{b}\partial _t(\sigma _l*v_{lm}^3(b,t)), \end{aligned}$$

(B.21)

by using the expression of $\omega _l(t)$ (2.36). Using the above equation and (B.16) in (B.20) gives

$$\begin{aligned} \frac{\partial }{\partial t}\Big \{\frac{ \partial v_{lm}^3}{\partial r}+\frac{1}{b}v_{lm}^3+\frac{1}{c}\frac{\partial v_{lm}^3}{\partial t}-\frac{1}{b}\sigma _l*v_{lm}^{3}\Big \}=0\quad {\mathrm{at}}\;\; r=b. \end{aligned}$$

(B.22)

Consequently, we obtain boundary condition (3.40f) by the zero initial data assumption.

Note that the initial boundary value problems for coefficients $u_{lm}^i$ and $v_{lm}^i$ have almost the same form except the interface conditions (B.11)–(B.13) and (B.17). Apparently, by introducing the variable substitution (3.39), $\{{\widetilde{u}}_{lm}^i\}$ satisfy the same governing equation as $\{u_{lm}^i$ and the same interface and boundary conditions as $v_{lm}^i$. $\square$

Appendix C. Normalisation of Maxwell’s Equations

Let us discuss the normalization of the Maxwell’s equations (1.1). Given the characteristic length $L_c$ and time $t_c$, we define dimensionless variables $\underline{\varvec{r}}$ and ${\underline{t}}$ such that

$$\begin{aligned} \varvec{r}=(x, y, z)=L_c\underline{\varvec{r}}=(L_c{\underline{x}}, L_c{\underline{y}}, L_c{\underline{z}}), \quad t={\underline{t}}/t_c. \end{aligned}$$

Then we have

$$\begin{aligned} \partial _t=t_c\partial _{{\underline{t}}},\quad \nabla =\frac{1}{L_c}\underline{\nabla }:=\frac{1}{L_c}(\partial _{{\underline{x}}},\partial _{{\underline{y}}}, \partial _{{\underline{z}}} )^{\top } \end{aligned}$$

and (1.1) can be transformed to

where

(C.2)

For simplicity, we consider the electromagnetic wave in the vacumm, i.e., $\underline{\varvec{D}}=\varepsilon _0\underline{\varvec{E}}$ and $\underline{\varvec{B}}=\mu _0\underline{\varvec{H}}$. From (C.1), we have

Let $L_c=1m$ and $t_c=1/\sqrt{\varepsilon _0\mu _0}\approx 2.9979e{+}7$. With a little abuse of notation, we still use the notation $\underline{\varvec{H}}({\underline{r}}, {\underline{t}})$ for scaled magnetic field $\sqrt{\frac{\mu _0}{\varepsilon _0}}\underline{\varvec{H}}({\underline{r}}, {\underline{t}})$. Then, we obtain the dimensionless Maxwell’s equations:

from (C.3).