Scattering of targets over layered half space using a semi-analytic method in conjunction with FDTD algorithm

Le Cao; Bing Wei

doi:10.1364/OE.22.020691

1. Introduction

There are many interests in the scattering and radiation problem of targets over a half space or layered half space. Its application can be found in many domains such as OTHR (Over-the-Horizon-Radar), radar stealth technology, target recognition, remote sensing and so on [1–8].Many modeling methods for solving such problems are explored deeply in spatial domain. The key of the integral equation method is the evaluation of half space or layered half space Green function. However, the spatial domain half space Green function is expressed in the form of Sommerfeld Integral (SI) whose expression is very complex and numerical integration is extremely time-consuming.

The scattering and radiation by buried objects are studied in [9], in which SI is evaluated by method of steepest descent. In the given Green function, there is a singularity when the source and field point are in transverse coincidence. The half space Green function in a four layer medium is given in [10], which is a complicated mathematic formula involving SI. To the author’s knowledge, there are no reports about more than four layers because the Green function is hard to acquire. It is much harder to deal with the case of anisotropic medium [11,12].

As we know, it takes a lot of memory and time to evaluate Green function when analyzing the scattering characteristics of electrically large size targets over a layered half space. In this paper, a new semi-analytic method is proposed to calculate the RCS characteristics of targets over a layered half space by the use of FDTD algorithm. This new method avoids using the complex Green function, thus, the calculation complexity, memory and time requirements will be greatly decreased in comparison with traditional FDTD method. Furthermore, it also can be used in anisotropic medium. The computing process includes mainly three parts. First, the introduction of bidirectional incident electromagnetic plane wave is completed. Then, the surface electric and magnetic current is acquired by FDTD algorithm. Finally, the far field RCS is calculated by a new half space near-far field output method.

2. The introduction of bidirectional incident wave

2.1 The traditional approach

FDTD method can be applied in the scattering analysis of the targets flying near sea surface and ground (e.g. Tomahawk cruise missile, aircraft flying near ground) and the targets over sea or earth surface (e.g. armored vehicle, water surface craft and aircraft carrier). The model of half space is shown in Fig. 1.According to the four-path model of half space problem [13], the incident wave consists of two parts: the direct incident wave and singly reflected wave. Similarly, the scattering field of observation point P is the sum field of direct wave from the target and reflected wave from the half space interface. It must be noted that the polarization of singly reflected wave generally changes which must be considered.

Fig. 1 The scattering problem of targets over a half space.

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The traditional approach of introducing an incident plane wave in half space case can be classified into two categories. First, the three-wave FDTD approach was proposed by Wong [14] which only can be used in a lossless medium. Second, the one-dimensional modified Maxwell’s equation was given by Winton [15] which describes a direct way to introduce a time domain plane wave into the TF/SF (Total Field/Scattered Field) boundary. This method can be used in a lossy medium. The computational region is shown in Fig. 2. It contains target (Region 1), part of half space interface (Region 3) and the region between them (Region 2).It can be seen that the requirement of time and memory will be greatly raised with the increase of the height of the target from the half space interface.

Fig. 2 The traditional FDTD computational domain.

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2.2 A semi-analytic method of introducing a bidirectional incident wave

To decrease the memory requirement, a new computational region pattern which is shown in Fig. 3 is proposed in this paper. It only contains the target itself (Region 1 in Fig. 2). Hence, the memory size of the process is much smaller. According to Fig. 3, the total incident wave consists of two parts: the direct incident wave and the singly reflected wave. The direct incident wave introduction can be realized by TF/SF method. With the consideration of polarization problem, the singly reflected wave introduction can be realized in the same way.

Fig. 3 The new FDTD computational region.

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In order to make a comparison between the traditional FDTD method and proposed method, take the Tomahawk cruise missile for example. Suppose the wavelength of incident wave is 0.4m. The dimension of the missile is given in Fig. 4.

Fig. 4 Modeling diagram of Tomahawk cruise missile.

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The discrete grid in FDTD calculation is $δ = 0.02 m$ according to the Courant stability condition [16]. Suppose the field component is single precision variable which takes 4 Bytes memory, the memory usage of three dimensions FDTD calculation can be estimated by

Memory = 30 N Bytes

where N is the number of discrete grids. Usually, in order to have a stable and reliable result, the electromagnetic wave has to travel back and forth along the diagonal line of the computational region for 5~6 times. There are

\sqrt{3} N

discrete grids on the diagonal line. According to the Courant stability condition, the total time steps can be estimated by

Timestep=2 \sqrt{3} \times [2 \times (3 \sim 5)] N^{1 / 3}

Consider the Tomahawk cruise missile is located 30m over a half space, the dimension of the computational region, the total number of discrete grids (including absorbing boundary, connective boundary and output boundary), the memory usage and required time steps are given in Table1.

Table 1. The Memory Usage and Time Step for Missile

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It is obvious that the proposed method has an absolute advantage in terms of economizing memory and computational time.

2.2.1 The introduction of incident wave

In the FDTD scattering computation, the incident wave is introduced by TF/SF method [16]. The TF/SF formulation is based on the linearity of Maxwell’s equations. The total electric and magnetic field can be decomposed in the following manner:

E = E_{i} + E_{s} H = H_{i} + H_{s}

As shown in Fig. 5, the computation region can be zoned into two distinct regions: total fields region and scattered fields region. These two regions are separated by a virtual surface that serves to connect the fields in the two regions, and thereby generates the incident wave.

Fig. 5 The total and scattered fields region.

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For the incident wave to be only in total fields region, equivalent surface electric and magnetic current are set on the connecting surface A shown in Fig. 6.

Fig. 6 Introduction of incident wave .

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The equivalent electric and magnetic current on the virtual surface can be computed as

J = - e_{n} \times Η_{i} J_{m} = e_{n} \times E_{i}

where

e_{n}

is the outer normal vector of surface A,

H_{i}

and

E_{i}

are the field values of incident wave.

2.2.2 The introduction of reflected wave

It is known that the frequency spectrum of the singly reflected wave is the product of the direct incident frequency spectrum and the reflection coefficient of the corresponding frequency. The incident wave field can be decomposed into TE wave and TM wave. Generally, the reflection coefficients of TE and TM are different from each other. Hence, the polarization of reflected wave is different from incident wave. The study shows that both the amplitude and phase of reflected wave change if reflection coefficient is a complex number. Normally, for a linear polarized incident wave, the reflected wave is elliptically polarized.

Suppose $θ$ is the angle between wave vector $k$ and the normal vector of the interface. The incident plane is defined as the plane organized by $k$ and z axis. $\hat{h}$ and $\hat{v}$ are defined as directions perpendicular to the wave vector $k$ and the incident plane, respectively. And the equation $\hat{k} = \hat{h} \times \hat{v}$ must be satisfied. $α_{0}$ is the angle between $\hat{h}$ and the polarization direction of incident wave. Hence, the electric field of incident wave can be decomposed into a horizontal (to incident plane) component and a vertical component.

\begin{array}{l} E_{h} = e_{h} \cos (ω t - π / 2 + ψ_{h}) \\ E_{v} = e_{v} \cos (ω t - π / 2 + ψ_{v}) \end{array}

where

e_{h} = E_{0} \cos α_{0}, e_{v} = E_{0} \sin α_{0}

ψ_{h}

,

ψ_{v}

is the phase of incident wave. The phase difference between

E_{h}

and

E_{v}

is

ψ_{i} = ψ_{v} - ψ_{h}

.

The horizontal and vertical electric field component of reflected wave can be written as a complex form

\begin{matrix} E_{r h} = e_{r h} \exp [- i (ω t - π / 2 + ψ_{r h})] \\ E_{r v} = e_{r v} \exp [- i (ω t - π / 2 + ψ_{r v})] \end{matrix}

ψ_{r h}

,

ψ_{r v}

is the phase of reflected wave.

\begin{array}{l} ψ_{r} = ψ_{r v} - ψ_{r h} \\ e_{r h} = | R^{T M} | E_{0} \cos α \\ e_{r v} = | R^{T E} | E_{0} \sin α \end{array}

For a lossy medium, the polarizing angle of the reflected wave is [17]

α = \frac{1}{2} \tan^{- 1} (\frac{2 e_{h} e_{v} \cos ψ_{r}}{e_{h}^{2} - e_{v}^{2}})

From Eq. (3) to Eq. (7), the field value of reflected wave is obtained.

After obtaining both the direct incident wave and singly reflected wave, we can introduce them into the FDTD calculation region. From 2.3.1, we know that the introduction into FDTD lattice of any direction incident wave excitations can be accomplished by TF/SF method. For a multi-direction incident wave, we can use the same approach. As shown in Fig. 7, a bidirectional incident wave can be introduced into FDTD calculation region by setting two one-dimensional FDTD iterative expressions [16].

Fig. 7 Introduction of multi-direction incident wave.

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In this approach, the far radiation field can be calculated by near-far field output after FDTD iterative computation of near field is finished.

3. The spatial domain half space near-far field output formula

The radiation field of the target can be seen as sum radiation fields of all current elements flowing along the output surface. Therefore, after obtaining the surface electric and magnetic current, the scattered field can be calculated.

Then the radiation problems of electric and magnetic current over half space can be converted to the radiation problems of electric and magnetic dipole over half space. The study of Wait [18] indicated that the main radiation far field of a current element over a layered half space is the total field of direct wave and reflected wave from the interface when the observation point is far away from the interface. Based on the reciprocity theorem, the far field formulation of an arbitrarily oriented dipole over a layered half space including anisotropic medium is given in [19] with the consideration of direct wave and reflected wave, which can be applied to space-based radar and OTHR. In this paper, this formula can be used in the fast calculation of the radiation of the electric and magnetic current flowing along the output surface. Figure 8 is the comparison of results from [19] and analytic method. It is obvious that they are in good agreement.

Fig. 8 Far radiation of a vertical electric dipole over half space.

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According to [19], for an electric dipole $I l$ which is located at $(θ_{0}, φ_{0})$ in spherical coordinates system, its far radiation at observation point $P (θ, φ)$ can be written as

E_{θ} (P) = {\begin{array}{l} [1 - \exp (i k Δ) R^{TM}] \sin θ_{0} \cos θ \cos (φ_{0} - φ) \\ - [1 + \exp (i k Δ) R^{TM}] \cos θ_{0} \sin θ \end{array}} i ω μ I l \frac{exp (i k r)}{4 π r}

E_{φ} (P) = \sin θ_{0} \sin (φ_{0} - φ) [1 + \exp (i k Δ) R^{TE}] i ω μ I l \frac{exp (i k r)}{4 π r}

The spherical wave factor is

\frac{exp (i k r)}{4 π r} \approx \frac{exp [i k (r_{0} - r^{'} \cdot {\hat{r}}_{0})]}{4 π r_{0}}

where

r^{'}

is the position vector of the dipole, r is the magnitude of position vector from location of the dipole to the observation point, and

r_{0}

is the position vector from coordinate origin O to the observation point. The wave path-difference of direct wave and reflected wave is

Δ = h (1 + \cos 2 θ) / \cos θ

R^{TE}

and

R^{TM}

are the reflection coefficients of TE wave and TM wave. h is the height from the interface.

According to the duality principle, the formula of magnetic dipole is

E_{φ} (P) = - i k I l {\begin{array}{l} [1 - \exp (i k Δ) R^{TM}] \sin θ_{0} \cos θ \cos (φ_{0} - φ) \\ - [1 + \exp (i k Δ) R^{TM}] \cos θ_{0} \sin θ \end{array}} \frac{exp (i k r)}{4 π r}

E_{θ} (P) = i k I l \sin θ_{0} \sin (φ_{0} - φ) [1 + \exp (i k Δ) R^{TE}] \frac{exp (i k r)}{4 π r}

It must be noted that the length of dipole l in Eq. (8)-(11) is equal to the discrete grid

δ

in FDTD calculation.

4. Numerical results

4.1 The introduction of incident wave

To observe the introduction of incident wave, we consider a conducting sphere with a radius of 0.12m located over a lossy half space with the subsurface characterized by $ε_{r} = 10, μ_{r} = 1$ , $σ = 1.0. S / m$ . The wavelength and polarizing angle of incident wave is 0.003m and $α = 45 °$ . The incident angle is ( $θ_{i} = 135 °, φ_{i} = 90 °$ ). The discrete grid $δ$ in FDTD is 0.12m. It is worth noting that $θ_{i}$ is the angle between incident wave vector and the positive z axis.

In the case of free space, the horizontal and vertical component of field value at any field point is equal to each other since the polarizing angle is $45 °$ . However, in the case of half space, generally, they are different because the polarizing angle of singly reflected wave changes. Figure 9 gives the variation of reflected wave polarizing angle with incident angle $θ_{i}$ . It can be seen that the reflected wave polarizing angle is not $45 °$ in most cases. Hence, the horizontal and vertical component of field value at field point is different from each other.

Fig. 9 Variation of reflected wave polarizing angle with incident angle.

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The two dimensional near field distribution patterns when time step is 1000 are shown in Fig. 10 and Fig. 11, where (a) and (b) are the horizontal and vertical components. The center circle is the conducting sphere. It can be seen that the two components are same in the case of free space while different in the case of half space. The red arrow represents the direction of incident wave.

Fig. 10 Near field distribution pattern of free space.

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Fig. 11 Near field distribution pattern of half space.

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4.2 Comparison with Green function method

The proposed method can be degraded into free space case. To demonstrate the correctness of the proposed output method when degraded into free space, we calculated the scattered fields of conducting sphere and cube. The incident wavelength is 0.032m and incident angle is ( $θ_{i} = 135 °, φ_{i} = 90 °$ ).The discrete grid $δ$ in FDTD calculation is 0.076m. The comparison plot is shown in Fig. 12 where the red solid line and blue points are the results of Green function method and proposed method. It can be seen that they are in good agreement.

Fig. 12 Comparison of Green function method and proposed method.

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4.3 Comparison with the results from the literature

To demonstrate the accuracy of the proposed method, we consider a dielectric cylinder ( $ε_{r} = 3.0, μ_{r} = 1, σ = 0$ ) of 3m length and 1m diameter is located 1.5m above the air-soil with the subsurface characterized by $ε_{r} = (5, - j 0.2)$ . A 200MHz plane wave is incident at $θ_{i} = 45 °, φ_{i} = 0 °$ and the observed scattering angles are $φ_{s} = 0 ° \sim 60 °, θ_{s} = 45 °$ . The bistatic HH-polarized RCS is given in Fig. 13. It is obvious that the results from proposed method and the literature are in good agreement.

Fig. 13 RCS of the cylinder over a lossy half space.

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4.3 Scattering field of missile

The modeling diagram of Tomahawk cruise missile is shown in Fig. 4. The wavelength of incident wave is 0.4m. The polarizing angle is $α = 0 °$ and it is a linear polarization wave. The schematic diagram of a missile over a layered half space is shown in Fig. 14. The first layer is dry soil ( $ε_{r} = 4.0, μ_{r} = 1, σ = 10^{- 5}$ ) and its thickness is $d_{1} = 1.0 m$ . The second layer is wet soil ( $ε_{r} = 10, μ_{r} = 1, σ = 10^{- 3}$ ) and its thickness is $d_{2} = 1.0 m$ . The last layer is underground water ( $ε_{r} = 81, μ_{r} = 1, σ = 10^{- 3}$ ) and its thickness is $d_{3} = 1.0 m$ . Apparently, for such a lossly layered medium, the reflected wave is elliptically polarized. The polarizing angle can be computed by Eq. (7). The missile is located 30m above the layered half space. The bistatic RCS of the missile is calculated. Figure 15 and Fig. 16 are the cases for observation point being located in the xoz and yoz planes. To compare with the free space case, the RCS of free space case is also calculated. It is worth noting that the $0 °$ in the following graph represents the direction of negative x or y axis.

Fig. 14 Schematic diagram of a missile over a layered half space.

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Fig. 15 RCS of xoz plane.

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Fig. 16 RCS of yoz plane.

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The variation of reflection coefficient with incident angle is given in Fig. 17. It is shown that for most incident angles, the reflected wave is very strong. Therefore, RCS of layered half space is larger than free space case under the same condition for the total incident wave is stronger.

Fig. 17 Variation of reflection coefficient with incident angle.

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5. Conclusion

The theoretical derivation and numerical results show that this new method which has a combination of FDTD method and semi-analytic method of introduction of incident wave is correct and effective in dealing with the scattering problem of target over a layered half space. The compared numerical examples show that this method is in good agreement with classic algorithm. However, comparing with the traditional method of introduction incident wave, the calculation memory requirement is greatly decreased. Meanwhile, the near-far field output approach proposed in this paper avoids using the complex and time-consuming half space Green function. Furthermore, this method can be used in any layered half space and even an anisotropic medium.

Acknowledgments

Project is supported by the National High Technology Research and Development Program of China (2012AA01308), the National Basic Research Program, and the National Natural Scientific Foundation of China (61231003).

References and links

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Algorithm	Dimension ( $m^{3}$ )	N	Memory usage	Time step
Traditional FDTD	6.79 × 2.62 × 33.1	9.3 × 10⁷	2661 MB	15588
Proposed method	6.79 × 2.62 × 3.14	7.3 × 10⁶	210 MB	2320

Scattering of targets over layered half space using a semi-analytic method in conjunction with FDTD algorithm

Abstract

1. Introduction

2. The introduction of bidirectional incident wave

2.1 The traditional approach

2.2 A semi-analytic method of introducing a bidirectional incident wave

2.2.1 The introduction of incident wave

2.2.2 The introduction of reflected wave

3. The spatial domain half space near-far field output formula

4. Numerical results

4.1 The introduction of incident wave

4.2 Comparison with Green function method

4.3 Comparison with the results from the literature

4.3 Scattering field of missile

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (17)

Tables (1)

Equations (15)

Optics Express