Scattering center models of backscattering waves by dielectric spheroid objects

Kun-Yi Guo; Xiao-Zhe Han; Xin-Qing Sheng

doi:10.1364/OE.26.005060

1. Introduction

Scattered waves by objects at high frequency can be approximated as scattered waves from a series of scattering centers, which rely upon the local nature of the scattering phenomenon [1]. Scattering center models have been proved to be useful in some applications, such as radar echo simulation, synthetic aperture radar (SAR) imagery interpretation and radar target recognition [2–5].

For higher accuracy of radar signal simulation or parameter estimation, the scattering center model need to be built based on the scattering characteristics of the extended target rather than the signatures shown in radar or optical images. The early point model simply assumes the targets are constituted by a set of ideal scattering centers without considering the effects of the geometry and material components. Whereas the attributes of scattering centers, such as their localizations, scattering amplitudes and dependencies on aspect angle, frequency and polarization are strongly dependent on the geometry and material of the target. Several parametric models for scattering centers have been developed, including Prony model [6–8], GTD (geometrical theory of diffraction)-based model [9], attributed model [10,11], sliding scattering center model [12,13] and creeping wave model [14,15]. Besides, the scattering center extracting methods have also received considerable attention [16–18]. However, all above models and extracting methods are proposed considering the conducting objects only. There is no research on scattering center models considering scattering characteristics of dielectric objects or a complex object with dielectric parts.

Dielectric materials are used more and more in radar targets, such as thermal-protective coated aircraft, stealth coated low detectable target and antenna radome. The study on scattering center models of dielectric objects is of great value in applications. A dielectric spheroid object has rich scattering mechanisms, and is a good example demonstrating the difference between conducting scattering centers and dielectric ones. Therefore, the mechanisms of the scattering from dielectric spheroid objects are investigated in detail. Based on the analysis on scattering mechanisms, scattering center models for dielectric spheroid are proposed in this paper. The proposed models can characterize the scattered waves by scattering centers with sparse and physical parameters.

To validate these models, the rigorous solution of Mie series for a dielectric sphere, and the computation results by the full-wave numerical method for a dielectric spheroid are used. The full-wave numerical method used in this paper is the multilevel fast multipole algorithm (MLFMA) [19, 20]. The high resolution range profiles (HRRP) [21] obtained by scattering center models, Mie series and MLFMA are compared, the results demonstrate that the signatures of each scattering center have achieved good agreements. It is found that for the lossless and low-loss media, the major contributors to scattering field are surface waves and multiple internal reflections instead of the direct reflection. It is also found that the number of scattering centers of dielectric objects is far larger than that of conducting objects because of the more complicated mechanisms of dielectric objects and that most of the scattering centers are distributed beyond the object because multiple reflections inside the object extend the propagation path. The method of scattering center modeling in this paper are of referential values for the further insight into dielectric objects with other geometry structures.

The rest of this paper is organized as follows. Section 2 presents the parametric scattering center models of a dielectric sphere. They are expanded to a dielectric spheroid in section 3. Then in order to validate these models, numerical validations are presented in section 4 by the HRRPs of a sphere with the index of refraction n = 5 − i0.005 and a spheroid with n = 3. At last, section 5 concludes this paper.

2. Scattering center models of dielectric sphere

2.1. Analysis on scattering mechanisms

Scattering mechanisms of a dielectric sphere are far more complex than those of a conducting sphere. The backscattered waves of a dielectric sphere include four dominant scattering contributors [22–26], front axial reflection, rear axial reflection, glory refraction and internal surface wave, as shown in Fig. 1(a). In comparison, a conducting sphere only includes two scattering contributors, front axial reflection and creeping wave, as shown in Fig. 1(b).

Fig. 1 Scattering mechanisms of a dielectric and a conducting sphere. (a) Dielectric sphere. (b) Conducting sphere.

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Dielectric objects have distinct differences from conducting objects that multiple reflections inside the medium are generally encountered, which results in a series of scattering centers. For example, the scattering contributions of rear axial reflections include the transmitted waves of one and multiple reflected rays at the rear axial points, as shown in Fig. 2. A series of scattering centers with one diameter interval are formed. The series of scattering centers are taken as a group of scattering centers in this paper. Dielectric objects have various groups of scattering centers rather than multiple individual scattering centers as conducting objects. A dielectric sphere has four types of scattering centers which are described in details as follows.

FASC - Front axial scattering center
Front axial reflection is an axial return from the front specular portion of the sphere. The location of FASC is the local specular portion on the spherical surface. In order to acquire the relative locations of the other scattering centers, the phase delays of the other scattered waves are counted relative to this reflection point.
RASC - Rear axial scattering center
The propagation process of rear axial reflection is shown in Fig. 2. It is an axial return from the rear surface after one or multiple internal reflections. According to propagation paths, the phase delay of the scattering waves of each scattering center in RASC group can be expressed as below.
$exp (j ϕ_{R}) = exp (- j 2 k a p n)$ where p is positive even and (p − 1) corresponds to the number of the internal reflection. $n = \sqrt{μ_{r} ε_{r}}$ is the index of refraction, a is the radius of the sphere and k is the wave number in free space. In this paper, we suppose that the material is non-magnetic, μ_r = 1. Thus the location of RASCs is apn with respect to FASC, i.e., the interval of two adjacent RASCs is the product of diameter and n.
GSC - Glory scattering center
The glory ray represents the ray that hits the sphere at an incidence angle of α, enters the sphere with the refraction angle β according to Snell’s law, encircles the center by an angle p (π − 2β), where p implies that the ray experiences (p − 1) internal reflections and then emerges in the backward direction, as shown in Fig. 3. Only the transmitted waves that are anti-parallel to the incident direction can be received by radar under mono-static mode. Therefore the incident angle and the refraction angle should satisfy the following relation.
$2 α + p (π - 2 β) = 2 N π$ where α and β obey the law of refraction, sin α = n sin β.
For given values of p and N (it is required that p ≥ 2N), α and β can be found only for certain n, which means that glory rays exist only for this range of n. Based on the paths of glory rays, the phase delays of GSCs can be expressed as below.
$exp (j ϕ_{G}) = exp {- j k [2 a p n cos β + 2 a (1 - cos α)]}$ where 2apn cos β is the propagation distance inside the sphere and 2a (1 − cos α) is the propagation distance outside the sphere.
ISSC - Internal surface scattering center
The aforementioned scattering centers are all induced by scattering contributors in the light area. However, the scattered waves from the shadow area are also dominant scattering contributors to the total scattered waves of dielectric objects [22]. In most cases, they are even stronger than the contributions of specular reflections, which is one distinct difference between dielectric objects and conducting objects.

Fig. 2 Ray paths of the front axial reflection and multiple rear axial reflections.

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Fig. 3 Ray paths of glory rays.

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The surface waves are excited by rays of a grazing incidence and propagate along the surface. The surface waves may enter the sphere at a critical angle β (point A), take a shortcut or a number of shortcuts through the sphere and reemerge as surface waves again (point B), as shown in Fig. 4. Surface waves will be generated at B due to the total reflection and parts of surface waves may be refracted inside the sphere again. The surface waves shed scattering rays while creeping on the surface. Only the scattered rays along the backscattering direction contribute to the backscattered waves. To differentiate it from general surface wave, the surface ray taking short cuts inside the sphere are called as internal surface wave. As the rays travel around the sphere, they may take multiple short cuts. Fig. 4 shows the internal surface waves of the cases (p = 1, N = 1) and (p = 4, N = 2).

Fig. 4 Ray paths of internal surface waves.

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According to Snell’s law, refraction angle of a tangent ray is $β = arcsin \frac{1}{n}$ . The difference between the path taking a shortcut through the sphere and the path traveling on the surface is d = a [(π − 2β) − 2n cos β]. Then for p shortcuts, the difference is h = pd. Therefore, the total propagation path with respect to that of FASC is L = 2a + a(2N − 1) π − h. According to n and N, it can be deduced that $p \leq ⌊ \frac{(2 N - 1) π}{π - 2 β} ⌋$ . The phase delay of ISSC can be derived as below.

exp (j ϕ_{I}) = exp (- j k L) = exp {- j k [2 a + a (2 N - 1) π - h]}

Based on phase delays of all aforementioned scattering centers, the distances of all scattering centers farther than FASC can be derived, then the equivalent locations along the incident direction can be obtained, which is $\frac{ϕ}{- 2 k}$ . The equivalent locations of all scattering centers with n = 5 are illustrated in Fig. 5. The locations are listed in Table 1. They are normalized by the equivalent diameter ED, ED = 2an.

Fig. 5 The equivalent locations of scattering centers along the incident direction.

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Table 1. Range distances of scattering centers (times of ED)

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2.2. Scattering center models

Based on the scattering mechanism analysis, the equivalent locations can be derived. However, to build complete parametric models for scattering centers, the scattering amplitudes of each contribution should be described by appropriate mathematic models, and the models are required to agree with the physical features of scattering centers to the best.

FASC
The scattering amplitude of FASC can be expressed as below referring to [23].
$E_{F} = - \frac{n - 1}{2 (n + 1)} [1 - j k a (1 - \frac{j}{3 k a})] \frac{cos ϕ}{j k}$ where ϕ is the azimuth angle of line of sight (LOS). At sufficiently high frequency, Eq. (5) can be approximated as $E_{F} \approx \frac{a}{2} R_{12}$ where $R_{12} = - \frac{n - 1}{n + 1}$ is the reflection coefficient on the surface of the sphere. FASC can be expressed as $E_{F}^{s} \approx \frac{a}{2} R_{12} exp (j ϕ_{F})$ where ϕ_F is the phase delay according to the origin of local coordinate system, which is set at the front axial boundary in this paper. Then the location of FASC is r′_F = 0, therefore ϕ_F = 2kr′_F · r̂_los = 0, r̂_los is the direction of line of sight (LOS) of radar.
RASC
The scattering fields of RASC can be expressed as below referring to [23].
$\begin{array}{l} E_{R}^{s} = & \frac{cos ϕ}{j k} \frac{2 n (n - 1)}{{(n + 1)}^{3}} exp [- j 2 k (2 n - 1) a] \cdot \\ [\frac{1}{1 - {(\frac{n - 1}{n + 1})}^{2} exp (- j 4 n k a)} + \frac{j k a (1 - \frac{j}{3 k a})}{1 + {(\frac{n - 1}{n + 1})}^{2} exp (- j 4 n k a)}] \end{array}$ At sufficiently high frequency, the scattering amplitude of RASC can be approximated as $E_{R} \approx \frac{a}{2} j^{p} T_{12} R_{21}^{p - 1} T_{21}, p = 2, 4, 6, \dots$ $T_{12} = \frac{2}{n + 1}, R_{21} = \frac{n - 1}{n + 1}, T_{21} = \frac{2 n}{n + 1}$ where T₁₂ and T₂₁ are respectively the refraction coefficients when the wave is incident on the outside and the inside surface. R₂₁ is the reflection coefficient on the internal surface of the sphere. The derivation is given in Appendix A.
By Eq. (9), it can be seen that the scattering amplitude of RASC are related with the dielectric parameters and the size of the sphere. Thomas [27] proposed a modifying factor $\frac{n}{2 - n}$ which can improve the results when p = 2. Repeating application of the modified method leads to a factor $\frac{n}{p - n}$ for multiple-bounce rear axial rays [28]. The modification factor has a singular value when n ≈ p, therefore extra modification should be used in this case. The scattering center model of RASC then can be derived as below.
$E_{R}^{s} = \sum_{p = 2, 4, 6, \dots}^{\infty} \frac{n}{p - n} \frac{a}{2} j^{p} T_{12} R_{21}^{p - 1} T_{21} exp (j ϕ_{R})$
GSC
The scattering amplitude of GSC can be expressed as below referring to [23].
$E_{G} = \frac{cos π}{j k} {(\frac{π}{b})}^{\frac{1}{2}} sin α cos α {(k a)}^{2} \sum_{N} {(- 1)}^{N - 1} Q (α)$ where $b = \frac{k a}{2} f^{″} (α)$ , f(α) = 2 cos α − 2np cos β + sin α [2α + p (π − 2β) − 2Nπ]. Q (α) can be rewritten according to the laws of optics as below. $Q (α) = \frac{1}{2} \sum_{p} [T_{12}^{⊥} T_{21}^{⊥} {(R_{21}^{⊥})}^{p - 1} - T_{12}^{‖} T_{21}^{‖} {(R_{21}^{‖})}^{p - 1}]$ $\begin{array}{l} T_{12}^{⊥} = \frac{2 cos α}{n cos β + cos α}, T_{21}^{⊥} = \frac{2 n cos β}{n cos β + cos α} \\ R_{21}^{⊥} = \frac{n cos β - cos α}{n cos β + cos α}, T_{12}^{‖} = \frac{2 cos α}{cos β + n cos α} \\ T_{21}^{‖} = \frac{2 n cos β}{cos β + n cos α}, R_{21}^{‖} = \frac{cos β - n cos α}{cos β + n cos α} \end{array}$ where $T_{12}^{⊥}$ , $T_{21}^{⊥}$ , $R_{21}^{⊥}$ , $T_{12}^{‖}$ , $T_{21}^{‖}$ , $R_{21}^{‖}$ are refraction and reflection coefficients. The superscripts ⊥ and ‖ indicate vertical and parallel polarization respectively. Compared with the form in [23], Eq. (13) shows clear physical mechanism, e.g., it shows that the ray experiences two refractions and (p − 1) times of reflections.
After simplification, the scattering center model of GSC can be expressed as below.
$E_{G}^{s} = e^{j \frac{π}{2}} \sqrt{\frac{π k}{2}} \sum_{p} [A (α) exp (j ϕ_{G}) \sum_{N} {(- 1)}^{N - 1} Q (α)]$ where $A (α) = a^{\frac{3}{2}} sin 2 α {(f^{″} (α))}^{- \frac{1}{2}}$ . In scattering modeling, in order to simplify the expression, A(α) is set as an unknown real to be estimated. The estimation approach used in this work is optimal matching of this model with the scattered waves computed by MLFMA using genetic algorithm [29]. Numerical results show that the estimated A(α) has better results than the analytical expression.
ISSC
The scattering amplitude of ISSC in [23] are too complex to be included in the scattering center model. The scattering model proposed here is based on an approximate model developed from physical considerations [30]. As shown in Fig. 4, when N is given, the smaller p is, the larger creeping distance (denoted by l) on the spherical surface will be, consequently, the less scattering amplitude. Assuming that the scattering amplitude attenuates with l as the function of $D_{0} exp (- γ \frac{l}{λ})$ , where D₀ and γ are unknown positive reals to be estimated. According to the ray path of the internal surface wave, it can be derived that l = a (2N − 1) π − pa (π − 2β). Besides, only parts of total reflected wave are refracted into the sphere at point B, therefore the loss of scattering amplitude should be considered, it is denoted by δ(< 1). δ is also an unknown real to be estimated. Then the scattering amplitude of ISSC is expressed as below. By combining it with Eq. (4), the complete scattering center model can be derived.
$E_{I} = D_{0} δ^{p - 1} exp (- γ \frac{l}{λ})$ $E_{I}^{s} = \sum_{p} E_{I} exp (j ϕ_{I})$

3. Scattering center models of dielectric spheroid

The scattering by a spheroid is more complex than a sphere, as shown in Fig. 6. The generating conditions of GSC and ISSC are stricter than the case of a sphere. In most situations, they have none contribution to backscattered waves [31]. In the case when incident direction is parallel to the axes of the spheroid, GSC and ISSC may be formed under certain conditions, which are discussed in the following analysis. The backscattered waves of a spheroid on axis also include four types of dominant scattering centers, FASC, RASC, GSC and ISSC. FASC and RASC are similar to those of a sphere and will not be discussed in this section. Besides, the hybrid rays of glory ray and surface wave may also contribute to the backscattered waves. The corresponding scattering center is named as hybrid scattering center (HSC).

GSC
The path of a glory ray is related with n, two axes a and b. a is the semi-axis of the spheroid along the propagation direction and b is the semi-axis perpendicular to the propagation direction. The propagation path is computed by the principle of the ray tracing method.
The incident wave with direction r̂_i enters the spheroid at point P_i, then reflects at point P_r with direction r̂_r, and finally refracts back into the free space at point P_e along direction r̂_e. Only when r̂_e = −r̂_i, the glory ray will contribute to the backscattered waves. According to the laws of optics, the coordinates of P_i, P_r and P_e are derived as given in Appendix B. Then the propagation distance L_G relative to FASC can be expressed and the scattering model of GSC is presented referring to the case of the sphere.
$L_{G} = L_{1} + n L_{2}$ $E_{G}^{s} = e^{j \frac{π}{2}} \sqrt{\frac{π k}{2}} \sum_{p} [\bar{A} (θ_{i}) exp (- j k L_{G}) \sum_{N} {(- 1)}^{N - 1} Q (θ_{i})]$ where L₁ and L₂ indicate the propagation distances in the free space and the medium respectively. For the case shown in Fig. 7, $L_{1} = (\bar{{OP}_{i}} + \bar{{OP}_{e}}) \cdot \hat{x}$ , $L_{2} = \bar{P_{i} P_{r}} + \bar{P_{r} P_{e}}$ . More complex L₂ for multiple internal reflections are computed by ray tracing. Ā(θ_i) is an unknown real to be estimated.
ISSC
Two possible ray paths of internal surface waves for a dielectric spheroid are shown in Fig. 8. Similar to the case of Fig. 4, the surface wave takes short cuts inside the spheroid. The surface wave enters the medium, and reaches the boundary. When the incident angle of transmitted wave is not less than the critical angle, the surface wave will be formed; when the incident angle is less than critical angle, the transmitted wave will be reflected. The surface waves parallel to the axis then be received by the radar.
The propagation distances of ISSCs are too complex to be derived into a closed form, they are computed numerically through ray tracing in this work. According to the computed propagation distances, the scattering center model of ISSC is given as below.
$E_{I}^{s} = \sum_{p} D_{s} δ^{p - 1} exp (- γ_{s} \frac{l_{s}}{λ}) exp (- j k L_{s})$ where D_s, δ and γ_s are unknown reals to be estimated. L_s is the extra propagation distance relative to FASC, and l_s is the creeping distance on the surface, they are computed by tracing the rays.
HSC
There are possible hybrid scattering centers (HSC) that are generated by hybrid scattering process of glory rays and surface waves for a dielectric spheroid. For example, the incident wave refracts into the spheroid, and reaches the boundary. When the incident angle is not less than the critical angle, surface wave will be generated, as shown in Fig. 9.

Fig. 6 The scattering mechanisms of a dielectric spheroid.

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Fig. 7 The glory ray of the dielectric spheroid (N = 1, p = 2).

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Fig. 8 The possible ray paths of internal surface waves.

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Fig. 9 The hybrid scattering of glory rays and surface waves.

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Similar to GSC and ISSC, the propagation distance of HSC is computed numerically by ray tracing. The scattering center model of HSC is expressed as below.

E_{H}^{s} = \sum_{p} D_{h} δ^{p - 1} exp (- γ_{h} \frac{l_{h}}{λ}) exp (- j k L_{h})

where D_h, δ and γ_h are unknown reals to be estimated. L_h is the farther propagation distance with respect to FASC and l_h indicates the creeping distance on the surface.

4. Validations

4.1. Dielectric sphere

The scattering amplitudes and locations of scattering centers can be well represented by the high resolution range profile (HRRP). Therefore, the HRRPs of each scattering contributors and the sum of all scattering centers are obtained in this section. The parameters used in the simulation are: n = 5 − i0.005, a = 0.2 m, f_c = 2 GHz, B = 2 GHz. f_c is the center frequency of the wide band signal and B is the bandwidth. To present the relationships of the locations of scattering centers more clearly, the range of HRRP is normalized by the equivalent diameter, ED = 2an.

Firstly, the unknown parameters of scattering models are estimated by matching the HRRPs simulated by the models with those computed by Mie series. The HRRPs of each scattering contributor simulated by scattering center models with estimated parameters are shown in Fig. 10. The comparison between the simulation results and the rigorous results is shown in Fig. 11. The two HRRPs agree well with each other, which verifies the validity of scattering center models. The statistical errors are given as follows. The cross correlation coefficient is 98.4%, the root-mean-square error (RMSE) of scattering amplitude is 0.32 dB, and the RMSE of locations of scattering centers is less than 0.001 ED.

Fig. 10 The HRRPs of scattering contributions of scattering centers (n = 5 − i0.005).

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Fig. 11 The comparison between the HRRP of total fields computed by Mie series and that simulated by the scattering center models (n = 5 − i0.005).

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The HRRPs of individual scattering contributions show interesting features of scattering centers of dielectric sphere. Distinct characteristics of a dielectric sphere from a conducting sphere can be seen clearly.

A dielectric sphere has far more scattering centers than a conducting sphere, and most of them are distributed beyond the sphere. A dielectric sphere has three types of scattering center groups, RASCs, GSCs and ISSCs. The intervals of RASCs are the same, as the equivalent diameter, whereas the intervals of GSCs and ISSCs are different.
For n with large real part and small imaginary part, the strongest scattering center is the first scattering center in the group of ISSCs (as shown in Fig. 10). The FASC is relatively weak compared to the first ISSC. For n with large imaginary part, due to the attenuation as wave propagating, generally the strongest scattering center is the first one along the incident direction, i.e., the FASC.

4.2. Dielectric spheroid

The parameters used in the simulation are: n = 3, a = 0.15 m, b = 0.1 m, f_c = 3 GHz, B = 4 GHz. Firstly, the unknown parameters of the models are estimated by matching the HRRPs simulated by these models with those computed by MLFMA. The HRRPs of each scattering contributor simulated by scattering center models with estimated parameters are shown in Fig. 12. The comparison between the simulation results and the rigorous results is shown in Fig. 13. The cross correlation coefficient is 97.7%, the RMSE of scattering amplitude is 0.53 dB, and the RMSE of locations of scattering centers is less than 0.001 ED.

Fig. 12 The HRRPs of scattering contributions of scattering centers (n = 3).

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Fig. 13 The comparison of the HRRP of total fields computed by MLFMA and that simulated by the scattering center models (n = 3).

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The HRRPs of individual scattering contributions clearly show the features of scattering centers of a dielectric spheroid. The number of scattering centers seems less than a dielectric sphere. The reason is that the more critical conditions results in the less GSCs and ISSCs. As shown in Fig. 12, there are five types of scattering centers, one ISSC (denoted by SC2), two GSCs (denoted by SC3 and SC4), one HSC (denoted by SC1), one FASC, and one RASC. The ray paths of four dominant scattering centers are shown in Fig. 14.

Fig. 14 The ray paths of SC1, SC2, SC3 and SC4.

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5. Conclusions

Dielectric spheroid objects have rich scattering mechanisms, which makes them a good example demonstrating the difference between conducting scattering centers and dielectric ones and establishing that scattering center model for dielectric targets is possible. A dielectric spheroid has distinct characteristics of scattering centers from a conducting spheroid. The number of scattering centers of a dielectric spheroid is far larger than that of a conducting one, and most of these scattering centers are beyond the geometry of the object. Five types of scattering centers are found, as FASC, RASC, GSC, ISSC and HSC. Except for FASC, the other four types include several scattering centers, called as scattering center groups. Among these scattering centers, for the lossless and low-loss media, the strongest scattering center is the first scattering center in ISSC group or the first scattering center in RASC group depending on n and a, rather than the FASC for the case of conducting objects. The accuracy of scattering center models is validated by comparing the HRRPs simulated by this model with those obtained by Mie series and MLFMA. The conclusion in this work is of the reference value for the study on scattering centers of other canonical objects, such as dielectric cuboids and cylinders, and more complex dielectric objects.

Appendix

A

At sufficiently high frequency, k ≫ 1, Eq. (8) can be approximated to

E_{R}^{s} \approx \frac{cos ϕ}{j k} \frac{2 n (n - 1)}{{(n + 1)}^{3}} exp [- j k 2 (2 n - 1) a] {\frac{j k a}{1 + {(\frac{n - 1}{n + 1})}^{2} exp (- j 4 n k a)}}

The fractional formula in braces can be approximated as

\frac{- j k a}{2 (\frac{n - 1}{n + 1}) exp [- j 2 n k a + \frac{j}{2} π]} \sum_{p = 1}^{\infty} exp [(- j 2 n k a + \frac{j}{2} π) (p - 1)] {{(- \frac{n - 1}{n + 1})}^{p - 1} - {(\frac{n - 1}{n + 1})}^{p - 1}}

Then Eq. (22) can be derived as

\begin{array}{l} E_{R}^{s} = & \frac{cos ϕ}{j k} \frac{- n k a}{{(n + 1)}^{2}} exp [- j k 2 (n - 1) a] \cdot \\ \sum_{p = 1}^{\infty} exp [(- j 2 n k a + \frac{j}{2} π) (p - 1)] {{(- \frac{n - 1}{n + 1})}^{p - 1} - {(\frac{n - 1}{n + 1})}^{p - 1}} \\ = \frac{cos ϕ}{j k} \frac{2 n k a}{{(n + 1)}^{2}} \sum_{p = 2, 4, 6, \dots}^{\infty} j^{p - 1} exp [- j 2 k a (p n - 1)] {(\frac{n - 1}{n + 1})}^{p - 1} \end{array}

The amplitude of

E_{R}^{s}

assuming ϕ = π is

E_{R} = a j^{p} \frac{2 n}{{(n + 1)}^{2}} {(\frac{n - 1}{n + 1})}^{p - 1}

Eventually, E_R can be expressed as Eq. (9) by applying the laws of optics given in Eq. (10).

B

In our case, r̂_i = x̂, according to the laws of reflection and refraction, we can get that

{\hat{n}}_{i} \cdot {\hat{r}}_{t} = - cos θ_{t}, {\hat{r}}_{i} \cdot {\hat{r}}_{t} = cos (θ_{i} - θ_{t})

The incident angle, the refraction angle and the direction of refracted wave can be derived as

θ_{i} = π - arccos (\frac{x_{i}}{a^{2} Ω_{i}})

θ_{t} = arcsin (\frac{sin θ_{i}}{n})

{\hat{r}}_{t} = x_{t} \hat{x} + y_{t} \hat{y} = cos (θ_{i} - θ_{t}) \hat{x} - [cos θ_{t} + cos (θ_{i} - θ_{t}) \frac{x_{i}}{a^{2} Ω_{i}}] \frac{b^{2} Ω_{i}}{y_{i}} \hat{y}

where

Ω_{i} = \sqrt{{(\frac{x_{i}}{a^{2}})}^{2} + {(\frac{y_{i}}{b^{2}})}^{2}}

.

According to the function of the spheroid and r̂_t, the coordinates of P_r can be derived by solving the following set of equations.

{\begin{cases} \frac{x_{r} - x_{i}}{x_{t}} = \frac{y_{r} - y_{i}}{y_{t}} \\ {(\frac{x_{r}}{a})}^{2} + {(\frac{y_{r}}{b})}^{2} = 1 \end{cases}

The coordinates of P_e can be derived by the similar procedure.

Funding

National Natural Science Foundation of China (NSFC) (61471041, 61671059).

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No.	FASC	RASC	GSC	ISSC
1	0	1.012	2.998	0.600
2	-	1.986	7.420	1.611
3	-	2.998	-	2.661
4	-	4.010	-	4.010
5	-	-	-	5.022
6	-	-	-	6.071
7	-	-	-	7.083
8	-	-	-	8.469

Scattering center models of backscattering waves by dielectric spheroid objects

Abstract

1. Introduction

2. Scattering center models of dielectric sphere

2.1. Analysis on scattering mechanisms

2.2. Scattering center models

3. Scattering center models of dielectric spheroid

4. Validations

4.1. Dielectric sphere

4.2. Dielectric spheroid

5. Conclusions

Appendix

Funding

References and links

Cited By

Figures (14)

Tables (1)

Equations (30)

Optics Express