### I. Introduction

### II. FDTD Formulation for Time-Varying Dusty Plasma

*N*

_{e}is defined as follows:

*f*(t) represents the time-varying function, and

*N*

_{e,0}is the constant of the electron density. The time-varying characteristic of electron density affects two dusty parameters. Therefore, angular plasma frequency

*ω*

_{pe}_{,}

_{0}and

*η*

_{ed}_{,}

_{0}are the constants of the angular plasma frequency and the charging response factor, respectively. In the equations above,

*ɛ*

_{0}is the permittivity of free space,

*e*is the electric charge of an electron,

*m*

*is the mass of an electron,*

_{e}*r*

*is the radius of dust particles, and*

_{d}*N*

*is the density of dust particles. Note that other dusty parameters are time invariant.*

_{d}### 1. SO-FDTD

*ɛ*

_{r}_{,}

_{∞}is relative permittivity at an infinite frequency,

*c*

_{0}is the speed of light in free space,

*ν*

*is the dust charging frequency, and*

_{ch}*ν*

*is the effective collision frequency.*

_{eff}*t*represents the FDTD time step size. The update coefficients

*B*

*–*

_{a}*B*

*are the functions of time, and they can be rewritten by decomposing the time-varying and the time-invariant parts. For example,*

_{d}*B*

*is expressed as*

_{a}**E**in Eq. (10) can be written as

##### (11)

**W**

_{1}represents the grouped field variables at the time step

*n*in Eq. (10), and

**W**

_{2}and

**W**

_{3}represent the variables at

*n*–1 and

*n*–2, respectively. Note that five field components are required in the SO-FDTD method with the state-space technique, while seven field components are needed in the standard SO-FDTD method.

### 2. BT-FDTD

**E**can be obtained by applying the inverse Fourier transform to Eq. (12) and then the CDS to the resulting equation:

*ɛ*

_{∞}=

*ɛ*

_{0}

*ɛ*

_{r}_{,}

_{∞}. The FDTD update for

**J**is derived by utilizing the BT approach to Eq. (13):

*jω*is approximated as

**H**is the same as that for SO-FDTD. As the

**E**

^{n}^{+1}field in Eq. (14) cannot be updated explicitly, we plug Eq. (15) into Eq. (14).

**E**is expressed as follows:

##### (17)

**J**can be expressed as follows:

**E**field update equation by using the state-space approach. In Eq. (17), we can replace the variables at time step

*n*–1 in

**E**field update equation of BT-FDTD is drastically reduced, enhancing the computational efficiency of BT-FDTD.

### III. Numerical Examples

*N*

*, following [14]. The other dusty plasma parameters are*

_{e}*N*

*= 10*

_{d}^{12}m

^{3},

*ν*

*= 10 GHz, and*

_{eff}*ν*

*= 8.7 GHz. The simulation frequency range is 1–100 GHz. We define the FDTD space step size as Δ*

_{ch}*z*= 40 μ

*m*and the FDTD time step size as Δ

*t*=0.125 ps. The computational domain is terminated by 10-cell perfectly matched layers [19–21]. All FDTD simulations are performed using an Intel i7-10700 CPU.

*f*(

*t*) to validate our FDTD simulations. As shown in Figs. 1 and 2, both FDTD simulations agree well with [22]. We then simulate the FDTD formulations with the state-space approach for time-varying dusty plasma. In this work, two time-varying functions are considered:

*t*<

*T*

*[1] and*

_{r}*f*

_{2}(

*t*)=

*t*/

*T*

*for 0 <*

_{r}*t*<

*T*

*[14]. In both cases,*

_{r}*T*

*(=8000Δt) is the time limit when dusty plasma becomes stable and the time-varying function becomes one after*

_{r}*T*

*. As shown in Figs. 1 and 2, the BT-FDTD simulations are in good agreement with the SO-FDTD simulations, regardless of the time-varying characteristic and the state-space approach. The reflection coefficients decrease because of the time-varying functions in the entire frequency range (Fig. 1). Conversely, the transmission coefficients increase for time-varying dusty plasma compared to time-invariant dusty plasma (Fig. 2).*

_{r}### IV. Conclusion

**J**field update equation of BT-FDTD by grouping the variables at the same time step as the

**W**variables. We optimize the

**E**field update equation of BT-FDTD using the defined

**W**variables. As a result, the FDTD update formulation of BT-FDTD is much simpler than that of SO-FDTD and has better computational efficiency in time and memory, with the same accuracy. Numerical examples are used to validate the computational efficiency improvement in the proposed FDTD modeling of time-varying dusty plasma. The proposed FDTD modeling approach using the combination of BT and the state-space approach can be extended to other time-varying dispersive media in nanophotonics and metamaterials.