## Abstract

Finite-difference time-domain (FDTD) algorithm with a pulse wave excitation is used to investigate the wide-band composite scattering from a two-dimensional(2-D) infinitely long target with arbitrary cross section located above a one-dimensional(1-D) randomly rough surface. The FDTD calculation is performed with a pulse wave incidence, and the 2-D representative time-domain scattered field in the far zone is obtained directly by extrapolating the currently calculated data on the output boundary. Then the 2-D wide-band scattering result is acquired by transforming the representative time-domain field to the frequency domain with a Fourier transform. Taking the composite scattering of an infinitely long cylinder above rough surface as an example, the wide-band response in the far zone by FDTD with the pulsed excitation is computed and it shows a good agreement with the numerical result by FDTD with the sinusoidal illumination. Finally, the normalized radar cross section (NRCS) from a 2-D target above 1-D rough surface versus the incident frequency, and the representative scattered fields in the far zone versus the time are analyzed in detail.

© 2011 OSA

## 1. Introduction

Composite electromagnetic scattering between the target and randomly rough surface has been of interest for extensive applications of radar surveillance, target tracking, oceanic remote sensing, landmine detection and so on. Usually the radar cross section (RCS) of the composite model is calculated and analyzed for the different parameters of the target and the rough surface, as well as the different incident conditions. Due to the complicated interactions between the target and the rough surface, the conventional analytical approaches for the rough surface scattering based on some approximations [1–3] are unable to deal with the problem. Fortunately, with the rapid development of computation technology, the electromagnetic scattering from the composite model has been extensively researched by the numerical method. The multilevel fast multipole algorithm [4] and the fast inhomogeneous plane wave algorithm [5] have been analyzed the scattering from a target above the rough surface. The method of moments [6–7] have been employed to deal with the scattering of a target partially embedded in a dielectric rough surface interface. The mode-expansion method [8], the multiple sweep method of moments [9], and the rigorous fast method [10] have been utilized to analyze the scattering from a target located above an ocean-like rough surface.

In all the references mentioned above, however, the electromagnetic scattering from the composite model is investigated with only one frequency of interest, and the scattering results of wide-band frequency response are not presented. Moreover, the complete RCS signature of the composite model also includes the computation of its frequency response. Some methods [11–13] have been dealt with the wide-band scattering from the buried object. In the paper, the finite-difference time-domain (FDTD) method is utilized to analyze the wide-band composite scattering from a 2-D target above 1-D rough surface. This can be solved using the FDTD method in one of two ways.

The continuous wave (CW) FDTD solution can be adopted in the steady state for the sinusoidal wave excitation. In our previous work, this method has been utilized to solve the scattering from a randomly rough surface [14,15], and the composite scattering from a target above a rough surface [16–18].For the wide-band frequency response of the composite model, the method requires an individual FDTD run for every frequency of interest. Another solution is to use FDTD with a pulsed plane wave illumination (pulsed FDTD), which can provide the complete frequency response from a single FDTD run.

In this study, the FDTD computation with a pulsed wave excitation is performed, and the 2-D representative far zone scattered field in the time domain is obtained by directly extrapolating the currently calculated data on the output boundary. Then transform the representative time-domain scattered field to the frequency domain and multiply the result by a factor. This yields the actual far zone scattered field in the frequency domain. It is worth noting that the relative dielectric constant of the composite model is assumed to be irrelevant to the change of frequency in this study (i.e. non-dispersive medium). The paper is organized as follows: the theoretical formulae of calculating wide-band scattering fields by pulsed FDTD are developed in Section 2. The frequency response of the composite model and the representative far zone scattered field versus time are presented and discussed in Section 3 for different parameters, such as the incident angle, the electrical permittivity, the rms height and the correlation length of the rough surface, as well as the size of the target. Section 4 ends with the conclusions of the paper and proposed the further investigation in this topic.

## 2. Wide-band scattering of composite model

Figure 1
shows the geometry of the composite model from a 2-D infinitely long target above a Gaussian rough surface. A pulse wave propagates in the direction of ** k_{i}**, which makes angle ${\theta}_{i}$ relative to the

*y*-axis. And the scattered direction

**is rotated clockwise by the angle ${\theta}_{s}$ (i.e. scattered angle) from**

*k*_{s}*y*-axis. And one-dimensional randomly rough surface profile with Gaussian spectrum is simulated by Monte Carlo method [19].

The division model of FDTD computation region is shown in reference [17].The incident wave is generated on the connective boundary, and the convolutional perfectly matched layer (CPML) absorbing medium is the outer boundary of FDTD region. In addition, the output boundary must be set to do a near-to-far transformation to obtain the far fields.

#### 2.1 Near fields

For the 2-D Maxwell’s equations, only ${E}_{z}-$, ${H}_{x}-$, ${H}_{y}-$ field components (p polarization)/ ${H}_{z}-$, ${E}_{x}-$,${E}_{y}-$field components(s polarization) are nonzero. Using Yee’s central difference scheme in both time and space for the p polarization, we obtain [20]

*m*is denoted as $\left(i,j\right)$, $\left(i,j+1/2\right)$, $\left(i+1/2,j\right)$. The coefficients $CA\left(m\right)$, $C\text{B}\left(m\right)$, $CP\left(m\right)$ and $CQ\left(m\right)$ are related to the electrical permittivity

*ε*, magnetic permeability

*μ*, electric conductivity

*σ*, magnetic conductivity ${\sigma}_{m}$, respectively. $\mathrm{\Delta}x$, $\mathrm{\Delta}y$ are the spatial increments in the

*x*-and

*y*-directions, and $\mathrm{\Delta}t$ is the time increment. To ensure the stability and accuracy of FDTD algorithm [21], the spatial increment and temporal increment could be set as $\mathrm{\Delta}x=\mathrm{\Delta}y=\mathrm{\Delta}={\lambda}_{\mathrm{min}}/20$, $\mathrm{\Delta}t=0.5\times \mathrm{\Delta}/c$, where ${\lambda}_{\mathrm{min}}$ is the minimum incident wavelength and

*c*is the light speed in vacuum, respectively. Similarly, the finite-difference equations for the s polarization can be obtained by dual relations [20].

In the numerical simulations of the composite scattering, the finite-length rough surface must be used to model the scattering from the infinite surface. When a plane wave impinges on a finite-length rough surface, the boundary reflection occurs. One way of minimizing the reflection is to construct an incident wave that tapers to very small values at the surface edges. Reflection still occurs, but it makes negligible contributions to the scattered field. To solve this problem, Fung .*et al*. [22] put forward the Gauss window function to guard against the truncation effect. And the Gauss window function is expressed as

*T*is a constant which determines the tapering width of the window function so chosen that the tapering drops from unity to ${10}^{-3}$ at the edge. And in this case $\mathrm{cos}{\theta}_{i}/T=2.6/{\rho}_{m}$, ${\rho}_{m}$ is the minimum distance from the center point $\left({x}_{0,}{y}_{0}\right)$ to the edge of the connective boundary.

In our study, the CPML medium [23] is used to terminate the FDTD lattices. The application of the CPML is completely independent of the host medium. Thus, no modifications are necessary when applying it to inhomogeneous, lossy, anisotropic, dispersive, or nonlinear media. Moreover, the CPML is efficient for highly evanescent waves or for late-time, low-frequency interactions. Taken the p-polarized wave for example, the difference formulation of the electric field ${E}_{z}$on the basis of Eq. (1) is updated as follows [23]

*x*- and

*y*-directions. It is assumed that the thickness of the CPML is

*d*, with the front planar interface located in the $x=0$ plane, and the PEC outer boundary at $x=d$. Thus, ${\sigma}_{x}$, ${k}_{x}$ and ${a}_{x}$ are defined as

*M*, ${M}_{a}$ are all the constants, referring to [23].

The difference formulation of magnetic fields ${H}_{x}$ and ${H}_{y}$ in the CPML medium is obtained by the similar derivation.

#### 2.2 Far fields

To obtain the far zone scattered characteristic of 1-D rough surface, it is necessary to implement the near zone to far zone transformation on the output boundary. The solution of pulsed FDTD method is to perform a 2-D FDTD calculation and use the currently calculated data on output boundary to extrapolate to the far field at each time step. For the 3-D FDTD extrapolation, the frequency domain transformation equations in the far zone can be transformed to the time domain and used to derive an approach to transform near zone FDTD fields to the far zone directly in the time domain [24]. But the approach above cannot be conveniently applied to the 2-D case due to the factor of $1/\sqrt{\text{j}k}$ ($\text{j=}\sqrt{\text{-1}}$and *k* is the incident wavenumber) in the 2-D frequency domain far zone transformation equations [25]. However, it is easily determined that in the frequency domain, the relationship between far zone electric fields obtained from a 3-D transformation with no *z* variation and the 2-D far zone fields is given by

*ω*is the incident angular frequency. Thus, the 2-D time domain transformation of near zone fields to the far zone can be obtained by the modifications to the 3-D transient transformation. The 3-D time-domain electric vector potential ${w}_{3D}(t)$ and magnetic vector potential ${u}_{3D}(t)$ on the output boundary are [20]

*j*and ${j}_{m}$ are the time-domain electric and magnetic surface currents on the output boundary ${S}^{\prime}$. ${r}^{\prime}$ is the vector from the origin to the source point of integration, and

*r*is the distance from the origin in the reference coordinate to the observed point in the far zone. The corresponding equations for the 3-D time-domain far zone scattered fields are where $\eta =\sqrt{{\mu}_{0}/{\epsilon}_{0}}$ is the wave impedance in free space. Let ${w}_{3D}(t)$ and ${u}_{3D}(t)$ integrate the

*z*variable over a unit distance, and the representative 2-D time-domain far zone scattered fields ${e}_{z,2D}^{\prime}(t)$ and ${h}_{z,2D}^{\prime}(t)$ are obtained by

**Simultaneously, the incident pulse wave**${e}_{i}(t)$ is also transformed to ${E}^{i}$ by the Fourier transform. And the 2-D normalized radar cross section (NRCS) in the far zone [26] is written as

*L*is the length of rough surface.

## 3. Numerical results and discussions

In this Section, for the composite model of an infinitely long cylinder located above a Gaussian rough surface as an example, the wide-band scattering characteristic for different parameters are investigated. And the Gaussian pulse wave is adopted and expressed as

Where*τ*determines the width of Gaussian pulse wave. In performing the calculations, some parameters are given as following: $\mathrm{\Delta}=0.005$m, $\tau =60\mathrm{\Delta}t$, ${t}_{0}=0.8\tau $, the thickness of the CPML medium is $10\mathrm{\Delta}$, the length of the randomly rough surface is $L=2048\mathrm{\Delta}$, and the perfect electrical conductor (PEC) cylinder is considered. The numerical results are averaged by 20 Monte Carlo realizations.

In order to ensure the validity of the pulsed FDTD method presented in the paper, the NRCS of an infinitely long PEC cylinder above rough surface versus the frequency by the pulsed FDTD method is depicted in Fig. 2 by the solid line. And the numerical result by the CW FDTD method is also plotted by circle for the comparison. For the CW FDTD method, the NRCS for each frequency needs an individual FDTD run (In Fig. 2 every circle corresponds a frequency), but using the pulsed FDTD method can obtain results over a wide frequency band from a single FDTD run. In the Ref [17], the NRCS from a cylinder above the rough sea surface by the CW FDTD method have been validated by the results of the method of moments (MOM). It should be noted that the electrical properties of the target and the rough surface must be identical for the pulsed FDTD and the CW FDTD, where the rms height and the correlation length of Gaussian rough surface are $\delta =$0.02m and $l=0.15$m, respectively. The altitude and the radius of the cylinder are $h=0.25$m and ${r}_{0}=0.1$m. The electrical permittivity of the rough surface is $\epsilon =16.16{\epsilon}_{0}$ and the electric conductivity of the rough surface is a constant and equal to $\sigma =1.15{\epsilon}_{0}\times 2\pi \times 3.0\times {10}^{9}$. Figure 2 (a) represent the wide-band scattering of composite model in the backward direction, and Fig. 2(b) is the result in the specular direction. It is obvious that the NRCS by the pulsed FDTD is in good agreement with that obtained by CW FDTD for different incident angles and different polarizations, which demonstrates the effectiveness and accuracy of the presented method.

The effect of the incident angle on the NRCS in the backward direction is analyzed in Fig. 3 , where the p polarization is considered, and the previous data in Fig. 2 are used except for the incident angles. Figure 3 (a) and Fig. 3 (b) represent the backscattering results of the rough surface only and the composite model, respectively. As the scattering of rough surface only is concerned, it is easily observed that the NRCS decreases with the increase of the incident angle over the whole frequency range. However, this phenomenon is not obvious for the composite model, especially for the low-frequency backscattering.

The change of the NRCS from the PEC cylinder ($h=$0.25m, ${r}_{0}=0.1$m) above the lossless Gaussian rough surface ($\delta =$0.02m, $l=0.15$m) with the electrical permittivity *ε* is shown in Fig. 4
. Where the incident angle is ${40}^{\circ}$, and the s polarization is considered. It is apparent that for the most frequency band, the NRCS from the composite model increases with increasing of *ε* for both the backward and specular direction. This phenomenon should be not surprising, and the primary reason for this is that the scattering from the rough surface becomes stronger with the increase of *ε*.

Figure 5
shows the NRCS from the cylinder ($h=$0.25m, ${r}_{0}=0.1$m) located above a randomly rough surface for different *δ* and *l* with the s-polarized wave illumination. The incident angle is ${40}^{\circ}$, and the electromagnetic parameters of the rough surface are $\epsilon =16.16{\epsilon}_{0}$ and $\sigma =1.15{\epsilon}_{0}\times 2\pi \times 3.0\times {10}^{9}$. In Fig. 5(a), it is readily found that the NRCS in the specular direction decreases obviously with larger *δ* except for the low frequency. We attribute this behavior to the fact that the roughness of rough surface increases with increasing of *δ*. Consequently, the incoherent scattering enhances with increasing of the surface roughness, which leads to the smaller specular scattering. In addition, it is also seen that the NRCS in the specular direction become gently larger with the increase of *l* in Fig. 5(b). This is due to the fact that by keeping the rms height constant and by increasing the correlation length, the electromagnetic roughness is constant, but the rms slope decreases, leading to a narrower distribution of the scattered energy, which implies an increase of the scattered energy in the specular direction.

To further explore the important scattering characteristics, the representative time-domain far zone scattering fields ${e}_{z,2D}^{\text{'}}(t)$multiplied by the factor ${r}^{1/2}$ versus time is illustrated for the rough surface only and the composite model in Fig. 6 (a)-(b) . The incident angle is ${50}^{\circ}$, and the other parameters are same to those given in Fig. 2. It is clear that the amplitude of ${r}^{1/2}{e}_{z,2D}^{\text{'}}(t)$ in Fig. 6 (a) is distinctly much less than the result in Fig. 6 (b).

The dependence of the representative time-domain scattering field ${h}_{z,2D}^{\text{'}}(t)$ multiplied by the factor $\eta {r}^{1/2}$ on the radius ${r}_{0}$ of the cylinder is depicted in Fig. 7 . Where the incident angle is ${40}^{\circ}$, the parameters of the rough surface are $\delta =0.01$m, $l=0.15$m, and the height of cylinder is $h=0.6$m. Figure 7(a) shows that in the specular direction the amplitude of $\eta {r}^{1/2}{h}_{z,2D}^{\text{'}}(t)$ decreases with the larger ${r}_{0}$ .But in the backward direction, the amplitude of $\eta {r}^{1/2}{h}_{z,2D}^{\text{'}}(t)$ increases with the increase of ${r}_{0}$ in Fig. 7(b). This reason for this is that the coupled scattering increases when the coupled area between cylinder and rough surface becomes greater with the increase of ${r}_{0}$, which leads to the total scattering increases. In this case, the incoherent scattering increases, and the coherent scattering decreases.

## 4. Conclusion

This paper presents a study of wide-band composite scattering from an infinitely long target with arbitrary cross section above a randomly rough surface by using pulsed FDTD algorithm. Firstly, the basic theory of pulsed FDTD method for calculating the wide frequency response of composite model is developed. Then taking the composite scattering from the PEC cylinder above a randomly rough surface for the example, a good agreement is achieved for the wide-band composite scattering by the CW FDTD and the pulsed FDTD, which illustrates the effectiveness and accuracy of present method. Finally, the effect of the incident angle, the rms height and the correlation length of the rough surface, and the size of target on the wide-band scattering characteristic is investigated and analyzed in detail. The major advantage of pulsed FDTD over CW FDTD is that pulsed FDTD can obtain results over a broad frequency band from a single FDTD run. Future investigation will include the wide-band scattering from the 2-D target and 1-D dispersive rough surface, and the scattering from the 3-D arbitrary target and the 2-D randomly rough surface by the pulsed FDTD method.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 60971067), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20100203110016), and the Fundamental Research Funds for the Central Universities. The authors would like to thank the reviewers for their helpful and constructive suggestions.

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