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假设入射平面波Ei(r)=exp(ik·r),k=k·$\hat{\boldsymbol{k}}$为波矢,$\hat{\boldsymbol{k}}$表示单位波矢,k=2π/λ为自由空间波数,λ为波长,散射情况如图 1所示。其中,S′是粗糙目标的平均面,SΣ是凸体目标的起伏表面,下标Σ表示起伏;rs为观察点P与目标坐标系原点之间的矢量距离,rΣ′为目标原点到面SΣ的矢径;r′为目标原点到面S′的矢径;n′为平均面S′的单位外法向矢量,N′为起伏表面SΣ的外法向矢量,ξ(r′)是沿光滑面S′外法线方向的随机高度起伏,且为高斯零均值分布;θi为r′位置的局部入射角,θi′为rΣ′位置的入射角;ki是入射波矢,ks是散射矢量。为了计算过程方便,省略入射波的时间因子exp(-iωt)。
Figure 1. Schematic diagram of scattering of rough objects[6]
根据标量Helmholtz方程[6],散射场表达式为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\int_{S_{\varSigma}}\left[E\left(\boldsymbol{r}_{\varSigma}^{\prime}\right) \frac{\partial G\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}-\right. \\ \left.G\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}_{\varSigma}^{\prime}\right) \frac{\partial E\left(\boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}\right] \mathrm{d} S_{\varSigma} \end{gathered} $
(1) 式中,$E\left(\boldsymbol{r}_{\varSigma}^{\prime}\right) \text { 和 } \frac{\partial E\left(\boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}$为粗糙面SΣ上的总电场和它的法向导数; G(rs, rΣ′)为自由空间标量格林函数,其表达式为:
$ G\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}_{\varSigma}^{\prime}\right)=\frac{\exp \left[\mathrm{i} k\left(\left|\boldsymbol{r}_s-\boldsymbol{r}_{\varSigma}^{\prime}\right|\right)\right]}{4 \pi\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}_{\varSigma}^{\prime}\right|} $
(2) 观察点P处的散射场表达式为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\int_{S_{\varSigma}}\left[E^{\prime}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right) \frac{\partial G\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}-\right. \\ \left.G\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}_{\varSigma}^{\prime}\right) \frac{\partial E^{\prime}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}\right] \mathrm{d} S_{\varSigma} \end{gathered} $
(3) 当表面任一点的主曲率半径远大于入射波长,可采用切平面近似。粗糙面上任一点rΣ′上的散射场和法向导数可分别为:
$ E^{\prime}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right)=\left(1+R_{\mathrm{i}}\right) E_{\mathrm{i}}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right) $
(4) $ \frac{\partial E^{\prime}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right)}{\partial \boldsymbol{N}^{\prime}}=\mathrm{i}\left(1-R_{\mathrm{i}}\right) \boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{N}^{\prime} E_{\mathrm{i}}\left(\boldsymbol{r}_{\varSigma}^{\prime}\right) $
(5) 式中,Ri为菲涅耳反射系数。采用远场近似,(1)式可简化为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\frac{\mathrm{i}}{4 \pi} \int_{S_{\varSigma}}\left\{\left(\boldsymbol{R}_{\mathrm{i}} \boldsymbol{V}-\boldsymbol{M}\right) \cdot \boldsymbol{N}^{\prime} \exp \left(\mathrm{i} \boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right. \\ \left.\frac{\exp \left[\mathrm{i} k\left(\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}_{\varSigma}^{\prime}\right|\right)\right]}{\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}_{\varSigma}^{\prime}\right|}\right\} \mathrm{d} S_{\varSigma} \end{gathered} $
(6) 式中,令$\boldsymbol{V}=\boldsymbol{k}_{\mathrm{i}}-\boldsymbol{k}_{\mathrm{s}}, \boldsymbol{M}=\boldsymbol{k}_{\mathrm{i}}+\boldsymbol{k}_{\mathrm{s}}$, 则: $\boldsymbol{k}_{\mathrm{i}}=k\left(\sin \theta_{\mathrm{i}} \cos \varphi_{\mathrm{i}}, \right.\left.\sin \theta_{\mathrm{i}} \sin \varphi_{\mathrm{i}}, -\cos \theta_{\mathrm{i}}\right)$, $\boldsymbol{k}_{\mathrm{s}}=k\left(\sin \theta_{\mathrm{s}} \cos \varphi_{\mathrm{s}}, \sin \theta_{\mathrm{s}} \sin \varphi_{\mathrm{s}}, \right.\left.\cos \theta_{\mathrm{s}}\right)$。
假设目标为导体,粗糙目标的远场散射场可简化为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\frac{-\mathrm{i}}{2 \pi} \int_{S^{\prime}}\left\{\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}^{\prime} \exp \left[\mathrm{i} \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right] \times\right. \\ \left.\frac{\exp \left[\mathrm{i}\left(k\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right|+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right]}{\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right|}\right\} \mathrm{d} S^{\prime} \end{gathered} $
(7) 利用远场近似,(7)式中距离为:
$ \left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right| \approx \boldsymbol{R}-\frac{\boldsymbol{r}^{\prime} \cdot \boldsymbol{r}_{\mathrm{s}}}{k} $
(8) (7) 式中指数项的分母|rs-r′|可以近似为R,R是观察点P与光滑面S′原点之间的距离,则粗糙目标的远区散射场可表示为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\frac{-\operatorname{iexp}(\mathrm{i} k \boldsymbol{R})}{2 \pi \boldsymbol{R}} \int_{S^{\prime}} \boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}^{\prime} \exp \left[\mathrm{i} \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right] \times \\ \exp \left(\mathrm{i} \boldsymbol{V} \cdot \boldsymbol{r}^{\prime}\right) \mathrm{d} S^{\prime} \end{gathered} $
(9) -
根据(9)式可知,平均散射场可表示为:
$ \begin{gathered} \left\langle E_{\mathrm{s}}\right\rangle=\frac{-\operatorname{iexp}(\mathrm{i} k \boldsymbol{R})}{2 \pi \boldsymbol{R}} \int_{S^{\prime}} \boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}^{\prime}\left\langle\exp \left[\mathrm{i} \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right]\right\rangle \times \\ \exp \left(\mathrm{i} \boldsymbol{V} \cdot \boldsymbol{r}^{\prime}\right) \mathrm{d} S^{\prime} \end{gathered} $
(10) 散射场场量的二阶统计特征即互相关函数表达式[15]为:
$ \begin{gathered} I_{\mathrm{c}}=\left\langle E_{\mathrm{s}}\right\rangle\left\langle E_{\mathrm{s}}{ }^*\right\rangle=\frac{\exp \left[\mathrm{i}\left(k_1-k_2\right) \boldsymbol{R}\right]}{(2 \pi \boldsymbol{R})^2} \times \\ \int_{S^{\prime}} \mathrm{d} \boldsymbol{r}_1{ }^{\prime} \int_{S^{\prime \prime}} \mathrm{d} \boldsymbol{r}_2{ }^{\prime}\left(\boldsymbol{k}_{\mathrm{i}, 1} \cdot \boldsymbol{n}_1{ }^{\prime}\right)\left(\boldsymbol{k}_{\mathrm{i}, 2} \cdot \boldsymbol{n}_2{ }^{\prime}\right) \times \\ \quad \exp \left[\mathrm{i} \boldsymbol{V}\left(\boldsymbol{r}_1{ }^{\prime}-\boldsymbol{r}_2{ }^{\prime}\right)\right] \times \\ \left\langle\exp \left[\mathrm{i} \boldsymbol{V} \boldsymbol{n}_1{ }^{\prime} \xi\left(\boldsymbol{r}_1{ }^{\prime}\right)\right]\right\rangle\left\langle\exp \left[-\mathrm{i} \boldsymbol{V} \boldsymbol{n}_2{ }^{\prime} \xi\left(\boldsymbol{r}_2{ }^{\prime}\right)\right]\right\rangle \end{gathered} $
(11) 式中, r1′为目标原点到光滑面S′的矢径,r2′为目标原点到光滑面S″的矢径,S″是粗糙目标散射场共轭后的光滑面,k1为散射场Es的自由空间波数,k2为散射场Es共轭后的自由空间波数,ki, 1表示散射场Es的入射波数矢量,n1′为光滑面S′的单位外法向矢量,ki, 2表示散射场Es共轭后的入射波数矢量,n2′为光滑面S″的单位外法向矢量,ξ(r1′)是沿光滑面S′外法线方向的随机高度起伏,ξ(r2′)是沿光滑面S″外法线方向的随机高度起伏。
散射场量强度的非相干分量可表示为:
$ I_{\mathrm{f}}=\left\langle E_{\mathrm{s}} E_{\mathrm{s}}^*\right\rangle-\left|\left\langle E_{\mathrm{s}}\right\rangle\right|^2=K \int_{\mathrm{S}^{\prime}} \mathrm{d} \boldsymbol{r}_1{ }^{\prime} \int_{S^{\prime \prime}} \mathrm{d} \boldsymbol{r}_2{ }^{\prime}\left(\boldsymbol{k}_{i, 1} \cdot \boldsymbol{n}_1{ }^{\prime}\right) \times \\ \begin{gathered} \left(\boldsymbol{k}_{\mathrm{i}, 2} \cdot \boldsymbol{n}_2{ }^{\prime}\right) \exp \left[\mathrm{i} \boldsymbol{V}\left(\boldsymbol{r}_1{ }^{\prime}-\boldsymbol{r}_2{ }^{\prime}\right)\right] \times \\ \left(\chi_{\mathrm{f}}-\chi_1 \chi_2\right) \end{gathered} $
(12) 式中, $K=\exp \left[\mathrm{i}\left(k_1-k_2\right) \boldsymbol{R}\right] /(2 \pi \boldsymbol{R})^2$,粗糙面起伏函数ξ(r′)服从高斯分布,粗糙面的粗糙度为σ,相关长度为lc,令Vz=V·n′,则高斯分布的1阶、2阶特征函数[14]分别为χ1=exp(-k12σ2Vz2/2), χ2=exp(-k22σ2Vz2/2), χf=exp[-k2σ2Vz2(1-〈ξ1ξ2〉)]。
定义比值系数γ12为:
$ \gamma_{12}=\frac{I_{\mathrm{f}}}{I_{\mathrm{c}}} $
(13) 由(11)式、(12)式可定义γ12的数学含义,其为高斯粗糙球体非相干分量与相干分量的比值。如图 2所示,入射平面波沿着-z轴方向入射在粗糙球体上,球体中心位于坐标系的原点。假设平面波入射波长λ=1.06μm,球体半径为a,σ为粗糙面的粗糙度,lc为相关长度,入射角θi=0°,方位角φi=0°,其轴线与ks的夹角为θs,即ks=(sinθs, 0, cosθs)。
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假设平面波入射在圆锥体侧面,取圆锥底面中心为坐标原点,如图 3所示,圆锥体的底面半径为b,高为h,半锥角为α,方位角为φ,则锥面方程[7]为:
$ \left\{\begin{array}{l} x=(h-z) \tan \alpha \cos \varphi \\ y=(h-z) \tan \alpha \sin \varphi \\ z=z \end{array}\right. $
(14) 则圆锥曲面上的微元$\mathrm{d} s=(h-z) \tan \alpha \cdot \sec \alpha \mathrm{d} \varphi \mathrm{d} z$。假设入射波在平面xOz内,即方位角为0°。圆锥的散射波矢量为$\boldsymbol{k}_{\mathrm{s}}=\sin \theta_{\mathrm{s}}\left(\cos \varphi_{\mathrm{s}} \cdot \boldsymbol{x}+\sin \varphi_{\mathrm{s}} \cdot \boldsymbol{y}\right)+\cos \theta_{\mathrm{s}} \cdot \boldsymbol{z}$,法向矢量为$\boldsymbol{n}^{\prime}=\cos \alpha \cos \varphi \cdot \boldsymbol{x}+\cos \alpha \sin \varphi \cdot \boldsymbol{y}+\sin \alpha \cdot \boldsymbol{z}$,入射波矢量为$\boldsymbol{k}_{\mathrm{i}}=-\sin \theta_{\mathrm{i}} \cdot \boldsymbol{x}-\cos \theta_{\mathrm{i}} \cdot \boldsymbol{z}, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}$分别为单位正交基向量。
由$\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right| \approx\left|\boldsymbol{r}_{\mathrm{s}}\right|-\left|\boldsymbol{r}^{\prime}\right| \cdot \cos \left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime}\right)$可知,在稳相点处有$\tan \varphi=\sin \theta_{\mathrm{s}} \sin \varphi_{\mathrm{s}} /\left(\sin \theta_{\mathrm{s}} \cos \varphi_{\mathrm{s}}+\sin \theta_{\mathrm{i}}\right)$,在高频极限条件下,$\boldsymbol{k}_{\mathrm{s}} \cdot \boldsymbol{n}^{\prime}=-\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}^{\prime}=\cos \theta_0$,即$\cos ^2 \theta_0=\frac{\left(1-\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{k}_{\mathrm{s}}\right)}{2}, \theta_0$为镜向点处的入射角,则粗糙圆锥体的散射场为:
$ \begin{gathered} E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)=\frac{-\mathrm{i}}{2 \pi} \int_{S^{\prime}} R_{\mathrm{i}} \cos \theta_0 \exp \left[\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right] \times \\ \frac{\exp \left[\mathrm{i}\left(k\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right|+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right]}{\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right|} \mathrm{d} S^{\prime} \end{gathered} $
(15) 令$\boldsymbol{R}\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime}\right)=\left|\boldsymbol{r}_{\mathrm{s}}-\boldsymbol{r}^{\prime}\right|, \boldsymbol{R}$是观察点P与光滑面S′原点之间的距离,则平均散射场可表示为:
$ \begin{aligned} & \left\langle E_{\mathrm{s}}\left(\boldsymbol{r}_{\mathrm{s}}\right)\right\rangle=\frac{-\mathrm{i} R_{\mathrm{i}} \cos \theta_0 \tan \alpha \sec \alpha}{2 \pi}\langle\exp [\mathrm{i} k \boldsymbol{V} \cdot \\ & \left.\left.\boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right]\right\rangle \int_0^h(h-z) \frac{\exp \left[\mathrm{i}\left(k \boldsymbol{R}+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right]}{\boldsymbol{R}} \mathrm{d} z \end{aligned} $
(16) 利用高频近似,则(16)式的积分可简化为:
$ \begin{gathered} \int_0^h(h-z) \frac{\exp \left[\mathrm{i}\left(k \boldsymbol{R}+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right]}{\boldsymbol{R}} \mathrm{d} z=-\frac{\exp \left\{\mathrm{i} k \boldsymbol{R}-\mathrm{i} k a\left[\sin \theta_{\mathrm{s}} \cos \left(\varphi-\varphi_{\mathrm{s}}\right)+\sin \theta_{\mathrm{i}} \cos \varphi\right]\right\}}{\mathrm{i} \boldsymbol{R} k\left\{\tan \alpha\left[\sin \theta_{\mathrm{s}} \cos \left(\varphi-\varphi_{\mathrm{s}}\right)+\sin \theta_{\mathrm{i}} \cos \varphi\right]-\cos \theta_{\mathrm{s}}-\cos \theta_{\mathrm{i}}\right\}} \times \\ \left\{h \cdot \exp \left[\mathrm{i} k h\left[\tan \alpha\left[\sin \theta_{\mathrm{s}} \cos \left(\varphi-\varphi_{\mathrm{s}}\right)+\sin \theta_{\mathrm{i}} \cos \varphi\right]-\cos \theta_{\mathrm{s}}-\cos \theta_{\mathrm{i}}\right]\right]\right\} \end{gathered} $
(17) 故锥体目标的相干散射分量为:
$ I_{\mathrm{c}}=\left|\left\langle E_{\mathrm{s}}\right\rangle^2\right| $
(18) 根据粗糙面散射理论及(15)式、(16)式,粗糙圆锥体对平面波的非相干散射分量为:
$ \begin{aligned} & I_{\mathrm{f}}=\left\langle E_{\mathrm{s}} E_{\mathrm{s}}^*\right\rangle-\left|\left\langle E_{\mathrm{s}}\right\rangle\right|^2= \\ & \frac{1}{4 \pi} \int_{S^{\prime}} \int_{S^{\prime \prime}} \frac{\left|R_{\mathrm{i}}\right|^2 \cos ^2 \theta_0 \exp \left[\mathrm{i}\left(k \boldsymbol{R}\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime}\right)+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime}\right)\right]}{\boldsymbol{R}\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime}\right) \boldsymbol{R}\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime \prime}\right)} \times \\ & \exp \left[\mathrm{i}\left(k \boldsymbol{R}\left(\boldsymbol{r}_{\mathrm{s}}, \boldsymbol{r}^{\prime \prime}\right)+\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{r}^{\prime \prime}\right)\right] \times \\ & \left\{\left\langle\exp \left[\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)-\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime \prime} \xi\left(\boldsymbol{r}^{\prime \prime}\right)\right]\right\rangle-\right. \\ & \left\langle\exp \left[\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right]\right\rangle \times \\ & \left\langle\exp \left[\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime \prime} \xi\left(\boldsymbol{r}^{\prime \prime}\right)\right]\right\} \mathrm{d} S^{\prime} \mathrm{d} S^{\prime \prime} \end{aligned} $
(19) 由图 4可知,引入新的积分变量,假设S′面上任一点处的曲率半径近似等于圆锥体底面半径b,且曲率半径远大于波长λ和粗糙面的相关长度lc,并满足条件Vlc2/a≪1,则V·n′≈V·n″,对dR的积分可以远近似用在r′处的切平面S⊥′(r′)内的积分代替,即积分变量dR≈dR⊥。
Figure 4. Tangential plane approximate calculation of the incoherent scattering intensity of the object[7]
当观察点位于散射场远场时,令ρ(rs, r′)≈R(rs, r′)+ki·r′,将(19)式进一步简化为:
$ \begin{aligned} I_{\mathrm{f}} & =\frac{\left|\boldsymbol{R}_{\mathrm{i}}\right|^2 \cos ^2 \theta_0}{4 \pi} \int_{S^{\prime}} \mathrm{d} S^{\prime} \int_{S^{\prime \prime}} \frac{\exp (\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{R})}{\rho^2\left(\boldsymbol{r}_{\mathrm{s}}\right)} \cdot \\ & {\left[\chi_2\left(\boldsymbol{V} \cdot \boldsymbol{n}^{\prime} ; \boldsymbol{R}_{\perp}\right)-\chi^2\left(\boldsymbol{V} \cdot \boldsymbol{n}^{\prime}\right)\right] \mathrm{d} \boldsymbol{R}_{\perp} } \end{aligned} $
(20) 式中,$\chi_2\left(\boldsymbol{V} \cdot \boldsymbol{n}^{\prime} ; \boldsymbol{R}_{\perp}\right)=\left\langle\operatorname { e x p } \left\{\mathrm { i } k \boldsymbol { V } \cdot \boldsymbol { n } ^ { \prime } \left[\xi\left(\boldsymbol{r}^{\prime}\right)-\xi\left(\boldsymbol{r}^{\prime}+\right.\right.\right.\right.\left.\left.\left.\left.\boldsymbol{R}_{\perp}\right)\right]\right\}\right\rangle, \chi\left(\boldsymbol{V} \cdot \boldsymbol{n}^{\prime}\right)=\left\langle\exp \left[\mathrm{i} k \boldsymbol{V} \cdot \boldsymbol{n}^{\prime} \xi\left(\boldsymbol{r}^{\prime}\right)\right]\right\rangle$为1维、2维特征函数。
根据ISHIMARU[4]的描述,平面波对粗糙物体的非相干散射强度可以看作是平面波入射场的单位面积粗糙平面非相干射强度的叠加,则:
$ I_{\mathrm{f}}=A \cdot I_{\mathrm{f}, 0} $
(21) 式中, A=(2L)2为被照射面积,If, 0为单位面积粗糙面的非相干散射强度。
对于圆锥的非相干散射有:
$ I_{\mathrm{f}}=\int_{S^{\prime}} I_{\mathrm{f}, 0} S\left(\theta_{\mathrm{i}}, \theta_{\mathrm{s}}\right) \mathrm{d} S^{\prime} $
(22) 式中, S(θi, θs)为遮蔽函数,且:
$ \begin{gathered} S\left(\theta_{\mathrm{i}}, \theta_{\mathrm{s}}\right)=S\left(\theta_{\mathrm{i}}\right) S\left(\theta_{\mathrm{s}}\right)= \\ \left\{\begin{array}{l} 1, \left(-\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}_{\mathrm{s}}>0 \text { 且 } \boldsymbol{k}_{\mathrm{s}} \cdot \boldsymbol{n}_{\mathrm{s}}>0\right) \\ 0, \left(-\boldsymbol{k}_{\mathrm{i}} \cdot \boldsymbol{n}_{\mathrm{s}}<0 \text { 或 } \boldsymbol{k}_{\mathrm{s}} \cdot \boldsymbol{n}_{\mathrm{s}}<0\right) \end{array}\right. \end{gathered} $
(23) (23) 式表示曲面积分只在照射区域进行。If, 0为单位面积粗糙平面的非相干散射强度,ns为粗糙物体平均表面外法向单位矢量。将(17)式代入(20)式得:
$ I_{\mathrm{f}}=\frac{h^2}{2} \tan \alpha \sec \alpha \int I_{\mathrm{f}, 0} S\left(\theta_{\mathrm{i}}, \theta_{\mathrm{s}}\right) \mathrm{d} \varphi $
(24) 由(18)式、(20)式和(24)式可计算出粗糙圆锥体目标的非相干散射分量比。
粗糙球体和锥体目标激光散射非相干分量比
Incoherent component ratio of laser scattering from rough sphere and cone targets
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摘要: 为了研究激光散斑对目标探测的影响,采用物理光学近似方法,进行了平面波激光照射在粗糙球体和圆锥体目标时对散射场统计特性的理论分析,推导了粗糙体目标散射场量的二阶统计矩, 数值计算了粗糙球体和锥体的非相干散射分量比随粗糙度、散射角、半径及目标材料的变化情况。结果表明,散射角的变化对粗糙球体散射非相干分量比有影响,粗糙度变大,目标的非相干分量占总散射分量的比重越大;随着粗糙球体半径变小,球体表面越粗糙;圆锥体目标散射非相干分量比的峰值位置随粗糙度变化而不同,但其峰值均位于镜反射方向上;金属类材料比非金属抛光铝材料的非相干分量比小,且半径变化与非相干分量比成正比。该研究结果可为更复杂目标激光散射特性和激光散斑探测、识别的研究提供一定的参考价值。Abstract: In order to study the effect of laser speckle on target detection, the theoretical analysis of the statistical characteristics of the scattering field when the plane wave laser irradiates the rough sphere and cone target was carried out by using the physical optics approximation method, and the second order statistical moment of the scattering field quantity of rough targets was derived. The variation of incoherent scattering component ratio of rough sphere and cone with roughness, scattering angle, radius, and target material is calculated numerically. The results show that the change of scattering angle has an effect on the incoherent component ratio of rough sphere scattering. The larger the roughness, the larger the proportion of incoherent component of target to the total scattering component. As the radius of the rough sphere becomes smaller, the surface of the sphere becomes rougher. The peak position of incoherent component ratio of cone target scattering varies with roughness, but its peaks are all located in the direction of specular reflection. The incoherent component ratio of metallic materials is smaller than that of non-metallic polished aluminum materials, and the radius change is proportional to the incoherent component ratio. The research results provide some reference value for the study of laser scattering characteristics of more complex targets and laser speckle detection and identification.
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Figure 1. Schematic diagram of scattering of rough objects[6]
Figure 4. Tangential plane approximate calculation of the incoherent scattering intensity of the object[7]
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