Degree of Polarization Calculation for Laser Backscattering from Typical Geometric Rough Surfaces at Long Distance

Abstract

Measurement of the degree of polarization of backscattering light from rough surfaces plays an important role in targets-detection applications. The polarization bidirectional reflectance function is the key to establish the relation between the polarization states of incidence and backscattering light. For the purpose of obtaining a polarized bidirectional reflectance distribution function (pBRDF) of a realistic, complicated target, it is decomposed as typical geometric surfaces and analytically calculated as the degree of polarization of the backscattering light, using a microfacet model, under conditions in which the scale of the target is far less than the target distance. In an experiment testing several typical geometric models, the results coincided with the theoretical calculation. The degree of polarization varied substantially as the rotation angle of the target changed, but showed little dependence on the size of target. The results have potential in applications discriminating between targets at different spatial orientations.

1. Introduction

Rough surfaces scattering light can carry varied information about the target characteristics, such as the surface material, the spatial form, or the movement status. Measuring the parameters, such as the echo waveform, the intensity, the phase, or the polarization states of scattering light to extract these target characteristics facilitates varied application in the realm of target discrimination [1,2,3,4,5], vegetation detection [6,7], the earth surface investigation [8,9,10], ocean water monitoring [11,12,13], atmospheric remote sensing [14,15], light detection and ranging (LiDAR) [16,17,18,19,20], hyperspectral data analysis [21] and topography analysis [22].

Polarization is the mode of spatial orientations in the electric field. An arbitrary polarization state can be described by four Stokes parameters. More explicitly, the Stokes parameter $S_0$ represents the total light intensity, while $S_1$, $S_2$ and $S_3$ represent the intensity difference between horizontal and vertical polarized light, ${45}^{\circ }$ and $-{45}^{\circ }$ polarized light, and right and left circular polarized light, respectively. Polarization detection can obtain these Stokes parameters as well as the degree of polarization (DOP) [13,23], or an angle of polarization (AOP) [24,25]. It has the following major advantages against intensity detection. Firstly, it can discriminate between different surface materials since the surfaces with different roughness parameters or refractive indices have significant differences in DOP [3]. Secondly, atmospheric turbulence affects polarization much less than the light intensity or the phase in question. Therefore, polarization detection has higher visibility and thus a longer imaging distance in foggy or hazed atmospheres. The polarization detection can thus be used in dehazing imaging technology [25,26].

The theoretical model for rough surface scattering is generally based on the bidirectional reflectance distribution function (BRDF) [27,28,29], which describes the distribution characteristics of scattered light in the upper hemisphere space of a rough surface. Such a function includes many complex physical processes such as specular reflection, diffuse reflection, bulk scattering, and others involving many parameters that affect the characteristic of scattering light, making the exact model for rough surfaces scattering impractical in the analysis or calculations. Various approximate models are proposed to simplify the numerical calculation in practical applications according to the scale of rough surfaces parameters such as the standard deviation of roughness ${\sigma }_h$ and correlation length l. When wavelength $\lambda \ll {\sigma }_h,l$ or $\lambda \gg {\sigma }_h,l$, the Kirchhoff–Beckmann’s approach [30,31] or perturbation method [32], respectively, are suitable approximation theories. The most practical one for objective scattering measurement is the Torrance–Sparrow model, also known as microfacet theory [28,33], treating macro rough surfaces as innumerable small perfectly smooth facets, the size of which is sufficiently large compared with the wavelength, such that reflection on them obeys Fresnel’s law. The normal directions of microfacets are random, but are usually assumed to obey normal distribution.

On the other hand, the polarization characteristic of scattering light is described by the polarization BRDF (pBRDF), connecting the Stokes vectors of incident and scattering light, this idea was firstly proposed by D. Flynn [34]. G. Priest [35] extended the unpolarized BRDF to the polarization version using the Mueller matrix, giving an explicit method to obtain the Stokes vector of backscattered light from rough surfaces. Priest’s model considered only the specular component of scattered light, which failed to describe the depolarization effect. Various modified models are proposed to take the contribution of diffuse scattering into account [23,36,37]. M. Hyde introduced the directional hemispherical reflectance function and treated the rough surfaces as ideal Lambertian [36], which approximately coincides the experiment result. I. Renhorn presented a four-parameter pBRDF model, accurately fitting the s and p components of polarization [38,39]. H. Liu proposed another empirical model, considering not only the reflectance, but also refraction on rough surfaces, to describe the subsurface scattering according to Fresnel’s law [40].

Measuring the polarization states and DOP from them is necessary for the remote polarization detection of realistic targets. However, in application, the current literature is mainly focused on polarization imaging [41] or discrimination between different kinds of surface materials [42,43] based on the degree of polarization. In the theoretical research area, pBRDF models are concerned with dependence on the incidence or reflection angles [36,44,45,46]. This is meaningful and essential in improving the accuracy of pBRDF. However, pBRDF models rely on so many variables, making it difficult to perform calculations in the practical applications. For example, for a given satellite model, the general method for obtaining the pBRDF is measuring the polarization states of backscattering light at different incidence and reflection angles. This is not feasible when the target is a long-distance target.

Therefore, it is necessary to give a numerical or even analytical calculation for the pBRDF of targets. The numerical calculation of the Stokes vector or Mueller matrix could be complicated due to the complication of the geometry of the target surfaces. However, the irregular objectives can be simplified to be the combination of several typical regular geometric surfaces, which easily calculate the backscattering Stokes vector and thus the DOP. For example, Figure 1 shows a model of LIDAR imaging satellite, which consists of typical geometric shapes: two rectangle solar panels (blue parts), two cuboid or prism main bodies (yellow parts), and an antenna (gray part). Each color corresponds to different surface-coating materials. For detecting the polarization of backscattering light from the remote satellite, it is useful to calculate from the individual parts. If the DOP and polarization states of these regular geometric surfaces are already known, then the discrimination for complicated targets can be simplified, since fewer pulses are required for transmitting between laser and target.

In this paper, based on the specular and diffuse components pBRDF model presented in Ref. [36], a novel method to access the Stokes parameter and DOP of backscattering light from rough surfaces is presented. By giving analytical mathematical formulas of pBRDF for typical geometric surfaces, the Stokes parameters and DOP can thus be calculated as functions of geometric and spatial parameters. Firstly, a practical fitting guidance is given for the diffuse component of BRDF as a function of the incident angle. The fitting coefficients depend only on the surface material, which can be easily measured in application scenarios. Then, the analytical calculations follow to show the dependence on backscattering light DOP, of geometric parameters, such as target size and spatial orientation. Finally, except for the case that the rotation angle of a planar surface is greater than around 50${}^{\circ }$, the experiment results of DOP of backscattering light from squares and discs agreed with the theoretical calculation within the experiment precision. The work is expected to pave a way for discrimination between the remote target based on the geometric size or spatial orientation, instead of materials.

2. Method

2.1. General Theories of pBRDF

This subsection explains the general theory for the Stokes parameters and microfacets model for pBRDF calculations. The polarization states are described by Stokes vectors,

$$\mathit{S}={\left[\begin{array}{cccc}{S}_0& S_1& S_2& S_3\end{array}\right]}^T$$

(1)

where the four parameters can be defined by the electric field components $E_x=a_1{e}^{i{\delta }_1}$, $E_y=a_2{e}^{i{\delta }_2}$, as:

$$\begin{array}{cc}\hfill S_0& =a_1^2+a_2^2\hfill \\ \hfill S_1& =a_1^2-a_2^2\hfill \\ \hfill S_2& =2a_1{a}_{2}cos({\delta }_1-{\delta }_2)\hfill \\ \hfill S_2& =2a_1{a}_{2}sin({\delta }_1-{\delta }_2)\hfill \end{array}$$

(2)

According to this, the degree of polarization is defined as:

$$DOP=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}$$

(3)

The input and output Stokes vectors can then be connected by the pBRDF matrix of the rough surfaces, whose elements are $F_{jk}$, as follows:

$$F_{jk}({\theta }_i,{\theta }_r,{\varphi }_r-{\varphi }_i)=\frac{d{S}_{out,j}}{S_{in,k}cos{\theta }_{i}d{\Omega }_i}$$

(4)

which is expressed by the differential Stokes parameter $d{S}_{out,j}$ and differential solid angle $d{\Omega }_i$, or equivalently, the integral form:

$${\mathit{S}}_{out}={\int \int }_{{\Omega }_i}\mathit{F}({\theta }_i,{\theta }_r,{\varphi }_r-{\varphi }_i)·{\mathit{S}}_{in}cos{\theta }_{i}d{\Omega }_i$$

(5)

where $S_{out,j}$ and $S_{in,k}$ are the jth and kth components of output and input Stokes parameters, respectively, ${\theta }_i$, ${\theta }_r$ are incidence and reflectance angles, respectively, ${\varphi }_i$, ${\varphi }_r$ are azimuth angles of incidence and reflectance directions, respectively, and the integral is over the solid angle ${\Omega }_i$ subtended by the target surface at the observation point. Furthermore, the reversibility of light requires the symmetry for $F_{jk}$ such that

$$F_{jk}({\theta }_i,{\theta }_r,\varphi )=F_{jk}({\theta }_r,{\theta }_i,\varphi )=F_{jk}({\theta }_i,{\theta }_r,-\varphi )$$

(6)

where $\varphi$ denotes ${\varphi }_r-{\varphi }_i$ for simplicity.

To express the explicit form of pBRDF matrix $\mathit{F}$, it is necessary to describe the scattering behavior of light on the rough surface. Each single reflection on a microfacet obeys Fresnel’s law, thus, for a fixed-incident angle ${\theta }_i$, the reflection angle ${\theta }_r$ can vary from 0 to $\pi$, since the random distribution of the normal of the microfacet, and vice versa. According to the reversibility of light, it is expected that the non-polarized BRDF is unity if the integrate area ${\Omega }_r$ is spread across the space, if there is no energy loss due to absorption of the surface material [35],

$${\int }_0^{2\pi }d\varphi {\int }_0^{\pi }f({\theta }_i,{\theta }_r,\varphi )cos{\theta }_{r}sin{\theta }_{r}d{\theta }_r=1$$

(7)

However, for the sake of backscattering light, the reflection angle is confined from 0 to $\pi /2$ only, then the physical meaning of that $\pi /2\le {\theta }_r\le \pi$ is that the light experiences multiple reflections on the microfacets. Therefore, the pBRDF matrix can be decomposed as two terms: the single reflection term and multiple reflection term [36], the former is the product of BRDF $f^s({\theta }_i,{\theta }_r,\varphi )$ and Mueller matrix $\mathit{M}$,

$$\mathit{F}={\mathit{F}}^s+{\mathit{F}}^d=f^s({\theta }_i,{\theta }_r,\varphi )\mathit{M}({\theta }_i,{\theta }_r,\varphi )+{\mathit{F}}^d$$

(8)

It is of course quite difficult to evaluate the multiple reflection BRDF; however, as a good approximation, for an ideal Lambertian, it can be treated as isotropic diffusion in every direction, which implies $f^d$ is a constant for a fixed incident angle. As a result, the diffusion term of BRDF can be expressed as [36]

$$f^d\left({\theta }_i\right)=\frac{1}{\pi }\left[1-DHR\left({\theta }_i\right)\right]$$

(9)

where the directional hemispherical reflectance (DHR) is defined as

$$DHR\left({\theta }_i\right)={\int }_0^{2\pi }d\varphi {\int }_0^{\pi /2}{f}^s({\theta }_i,{\theta }_r,\varphi )cos{\theta }_{r}sin{\theta }_{r}d{\theta }_r$$

(10)

which depends only on the incident angle ${\theta }_i$ and the roughness parameters. Then, the corresponding diffuse component of pBRDF is

$$F_{00}^d=f^d\left({\theta }_i\right)M_{00}({\theta }_i,{\theta }_r,\varphi )$$

(11)

while other matrix elements vanish.

In the long-distance backscattering-detection scenario, where the “long-distance” condition is defined as: the ratio of target size a to light transmission distance L is much smaller than unity, for example, $a/L\le {10}^{-2}$, and it is proper to assume that the laser source and detector are coaxial. In this case, the incident and reflection angles are treated exactly the same, ${\theta }_i={\theta }_r$ and $\varphi =0$, resulting in the independence of the Mueller matrix element of an incident angle,

$$\mathit{M}=\left[\begin{array}{cccc}{M}_{00}& 0& 0& 0\\ 0& M_{00}& 0& 0\\ 0& 0& M_{22}& M_{23}\\ 0& 0& -M_{23}& M_{22}\end{array}\right]$$

(12)

where the elements $M_{00}$, $M_{22}$ and $M_{23}$ can be determined only by the surface roughness parameters, more explicitly, as the standard deviation of roughness ${\sigma }_h$, correlation length l and the complex refractive index, respectively.

To sum up, the pBRDF matrix in case of coaxial system has the following form:

$$\mathit{F}=\left[\begin{array}{cccc}(f^s+f^d)M_{00}& 0& 0& 0\\ 0& f^s{M}_{00}& 0& 0\\ 0& 0& f^s{M}_{22}& f^s{M}_{23}\\ 0& 0& -f^s{M}_{23}& f^s{M}_{22}\end{array}\right]$$

(13)

If the input light is horizontally linear polarized, then

$${\mathit{S}}_{in}=I_z{\left[\begin{array}{cccc}1& 1& 0& 0\end{array}\right]}^T$$

(14)

where $I_z$ is light intensity at distance L along the propagating direction, by assuming that the laser beam profile is Gaussian, then

$$I_z=\frac{I{w}_0}{kL}{e}^{-\frac{x^2+y^2}{k^2{L}^2}}$$

(15)

where $w_0$ is the beam waist and k is the full divergence angle of the laser, and x and y are the coordinates perpendicular to the light propagating direction. Thus, $kL$ is the beam radius at the distance of L.

$$\begin{array}{cc}\hfill S_{out,0}& ={\int \int }_{{\Omega }_i}{F}_{00}({\theta }_i,{\theta }_r,\varphi )I_{z}cos{\theta }_{i}d{\Omega }_i\hfill \\ \hfill S_{out,1}& ={\int \int }_{{\Omega }_i}{F}_{11}({\theta }_i,{\theta }_r,\varphi )I_{z}cos{\theta }_{i}d{\Omega }_i\hfill \\ \hfill S_{out,2}& =S_{out,3}=0\hfill \end{array}$$

(16)

The degree of polarization can thus be calculated according Equation (3):

$$DOP={\left[1+\frac{{\int \int }_{{\Omega }_i}{f}^d\left({\theta }_i\right)I_{z}cos{\theta }_{i}d{\Omega }_i}{{\int \int }_{{\Omega }_i}{f}^s({\theta }_i,{\theta }_i,0)I_{z}cos{\theta }_{i}d{\Omega }_i}\right]}^{-1}$$

(17)

which is independent of the Muller matrix elements. Denote these two surface integrals in Equation (17) as $I^d$ and $I^s$. Then, the task of this paper is to calculate the degree of polarization of different typical geometric surfaces at a given incident direction to the target objects.

2.1.1. Specular Component

The incident (reflection) angle ${\theta }_i$ (${\theta }_r$) is the polar angle between the macro surface’s normal and irradiation (radiation) light. Meanwhile, the angle between the normal of microfacet and macro surface is $\theta$, which is randomly oriented, and is supposed to have a Gaussian distribution probability density function [28,35],

$$p\left(\theta \right)=\frac{1}{2\pi {\sigma }^2{cos}^3\theta }{e}^{-\frac{{tan}^2\theta }{2{\sigma }^2}}$$

(18)

where $\sigma =\sqrt{2}{\sigma }_h/l$. According to the microfacets model, the specular component of BRDF is [36]

$$f^s({\theta }_i,{\theta }_r,\varphi )=\frac{G({\theta }_i,{\theta }_r,\varphi )e^{-\frac{{tan}^2\theta }{2{\sigma }^2}}}{8\pi {\sigma }^2{cos}^4\theta cos{\theta }_{i}cos{\theta }_r}$$

(19)

where

$$G({\theta }_i,{\theta }_r,\varphi )=min\left(1,\frac{2cos{\theta }_{i}cos\theta }{cos\beta },\frac{2cos{\theta }_{r}cos\theta }{cos\beta }\right)$$

(20)

is called the shadowing/masking function, which could keep the BRDF finite while the incident or reflection angle is close to $\pi /2$. Each single reflection on a microfacet obeys Fresnel’s law, thus these geometric relations connecting all angles are defined:

$$\begin{array}{cc}\hfill cos2\beta & =cos{\theta }_{i}cos{\theta }_r+sin{\theta }_{i}sin{\theta }_{r}cos\varphi \hfill \\ \hfill cos\theta & =\frac{cos{\theta }_i+cos{\theta }_r}{2cos\beta }\hfill \end{array}$$

(21)

In the practical case of the coaxial system, ${\varphi }_i={\varphi }_r$, ${\theta }_i={\theta }_r=\theta$, and $\beta =0$, thus Equation (19) can be simplified for the integration of $I^s$, as follows:

$$I^s=\frac{1}{8\pi {\sigma }^2}{\int \int }_{\Omega }G(\theta ,\theta ,0)\frac{e^{-\frac{{tan}^2\theta }{2{\sigma }^2}}{I}_z}{{cos}^5\theta }d\Omega$$

(22)

In our previous work [47], the explicit specular component of pBRDF was calculated analytically or numerically on varied surfaces, such as planar discs, squares, cones and cylinders.

2.1.2. Diffuse Component

In the current work, for calculating the degree of polarization, the remain part is to give the expression for diffuse component $I^d$, and firstly the result of diffuse BRDF $f^d$ in Equation (9), based on Equations (10) and (19). It is impossible to obtain an analytic form for $f^d$ because of the complexity of the relations of the angles. However, there is an approximate approach to give a relation between $f^d$ and $cos{\theta }_i$. Since $f^d$ is finite as $cos{\theta }_i$ varies in the interval $\left[0,1\right]$, one can expand $f^d$ as the Taylor series of $cos{\theta }_i$, as follows:

$$f^d\left({\theta }_i\right)cos{\theta }_i=\sum _{m=1}^{\infty }{a}_m{cos}^m{\theta }_i$$

(23)

where the coefficients $a_m$ are determined only by the roughness parameter $\sigma$. The numerical simulation of $f^d\left({\theta }_i\right)cos{\theta }_i$ is plotted as Figure 2.

Polynomial fitting of these curves shows that it is accurate enough to maintain the first 4 terms of series Equation (23), as shown in Figure 3.

Assume that the equation of the macro surface is $F(x,y,z)=0$, and that the light source is at point $Q=(0,0,1)$, then for an arbitrary point P on the surface, the incident angle ${\theta }_i$ can be expressed by the coordinates $(x,y,z)$:

$$cos{\theta }_i=\frac{\mathit{n}·\overrightarrow{PQ}}{|\overrightarrow{PQ}|}$$

(24)

where $\mathit{n}=\left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right)$ is the unit normal vector of the surface, assuming that it is normalized. The element solid angle $d{\Omega }_i$ is

$$d{\Omega }_i=\frac{\mathit{n}·\overrightarrow{PQ}}{|\overrightarrow{PQ}{|}^3}dS$$

(25)

Substituting Equations (23)–(25) into Equation (17), the approximate expression for $I^d$ is:

$$I^d\approx \frac{I{w}_0}{k}\sum _{m=1}^4{a}_m\int \int \frac{{(\mathit{n}·\overrightarrow{PQ})}^{m+1}{e}^{-\frac{x^2+y^2}{k^2{(1-z)}^2}}}{(1-z)|\overrightarrow{PQ}{|}^{m+3}}dS$$

(26)

To summarize, if the surface equation, laser parameters and roughness $\sigma$ are given, one can firstly numerically calculate the diffuse component of BRDF, $f^{d}cos{\theta }_i$, and perform polynomial fitting to obtain the series coefficients. Then, the diffuse component of pBRDF can be calculated through Equation (26).

3. Theoretical Calculations

In this section, the diffuse component of pBRDF is calculated, $I^d$, and thus the degree of polarization for specific typical geometric rough surfaces. The calculation of the specular component can be found in our previous work, Ref. [47].

3.1. General Planar Surfaces

The most representative regular planar surfaces are of course disc and rectangle. Without loss of generality, suppose the surface equation is $Ax+By+Cz+D=0$, and the unit normal vector is $\mathit{n}=\left(A,B,C\right)$. Therefore, according to the geometry,

$$\begin{array}{cc}\hfill & \mathit{n}·\overrightarrow{PQ}=C+D\hfill \\ \hfill & |\overrightarrow{PQ}|={\left[x^2+y^2+{(1-z)}^2\right]}^{1/2}\hfill \end{array}$$

(27)

Substitute Equation (27) into Equation (26), one can obtain

$$I^d=\frac{I{w}_0}{k}\sum _{m=1}^4{a}_m{(C+D)}^{m+1}\int \int \frac{e^{-\frac{x^2+y^2}{k^2{(1-z)}^2}}}{(1-z){\left[x^2+y^2+{(1-z)}^2\right]}^{\frac{m+3}{2}}}dS$$

(28)

In the practical applications, the distance between target and the laser source will be much larger than the target size, which implies that $x,y,z\ll 1$. Thus, the quadratic or higher terms in the expansion of denominator in Equation (28) can be omitted, which becomes:

$$\begin{array}{cc}\hfill & {(1-z)}^{-1}{\left[x^2+y^2+{(1-z)}^2\right]}^{-\frac{m+3}{2}}\approx 1+(m+4)z\hfill \\ \hfill & =1-\frac{(m+4)(D+Ax+By)}{C}\hfill \end{array}$$

(29)

and

$$e^{-\frac{x^2+y^2}{k^2{(1-z)}^2}}\approx e^{-\frac{x^2+y^2}{k^2}}$$

(30)

3.2. Disc

As the simplest example of planar surface, the disc is common and widely used in practical remote detection. Suppose that the disc center locates in the origin, and the unit normal vector and radius of the disc are $\mathit{n}=\left(0,-sin\alpha ,cos\alpha \right)$ and R, respectively. The planar equation is $-ysin\alpha +zcos\alpha =0$. To integrate Equation (28) analytically, a coordinates transformation was performed,

$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\alpha & -sin\alpha \\ 0& sin\alpha & cos\alpha \end{array}\right]\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]$$

(31)

where $(x^{\prime },y^{\prime },z^{\prime })$ are the local coordinates on the disc. In this coordinates system, the surface equation is $x^{\prime 2}+y^{\prime 2}\le R^2$ and $z^{\prime }=0$, or equivalently, the cylindrical coordinates version ${\rho }^{\prime }\le R$ and $z^{\prime }=0$. Then, Equation (28) becomes

$$I^d\approx \frac{I{w}_0}{k}\sum _{m=1}^4{a}_m{cos}^{m+2}\alpha \int \int e^{-\frac{x^{\prime 2}+y^{\prime 2}{cos}^2\alpha }{k^2}}\left[1+(m+4)y^{\prime }sin\alpha \right]d{x}^{\prime }d{y}^{\prime }$$

(32)

The integral area is the disc, $x^{\prime 2}+y^{\prime 2}\le R^2$. Note that the second term of the integral kernel is odd about $y^{\prime }$, and thus vanishes through the integration. Thus, Equation (32) becomes

$$I^d=\frac{I{w}_0}{k}\sum _{m=1}^4{a}_m{cos}^{m+2}\alpha {\int }_0^R{e}^{-\frac{{\rho }^{\prime 2}}{k^2}}{\rho }^{\prime }d{\rho }^{\prime }{\int }_0^{2\pi }{e}^{\frac{{\rho }^{\prime 2}{sin}^2\alpha {sin}^2{\varphi }^{\prime }}{k^2}}d{\varphi }^{\prime }$$

(33)

Using the well-known Poisson’s integral expression for the Bessel function of order $\nu$,

$$J_{\nu }\left(z\right)=\frac{{(z/2)}^{\nu }}{\sqrt{\pi }\Gamma (\nu +1/2)}{\int }_0^{\pi }{e}^{izcos\varphi }{sin}^{2\nu }\varphi d\varphi$$

(34)

the second integral of Equation (33) can be explicitly expressed by Bessel function:

$${\int }_0^{2\pi }{e}^{\frac{{\rho }^{\prime 2}{sin}^2\alpha {sin}^2{\varphi }^{\prime }}{k^2}}d{\varphi }^{\prime }=2\pi I_0\left(\frac{{\rho }^{\prime 2}{sin}^2\alpha }{2k^2}\right)e^{\frac{{\rho }^{\prime 2}{sin}^2\alpha }{2k^2}}$$

(35)

where $\Gamma \left(x\right)$ is the gamma function, $I_0\left(x\right)$ is the modified Bessel function of order 0 of the first kind. Let $\xi ={\rho }^{\prime 2}/k^2$, finally, the diffuse component of backscattering light is

$$\begin{array}{cc}\hfill I^d& =\frac{2\pi I{w}_0}{k}\sum _{m=1}^4{a}_m{cos}^{m+2}\alpha {\int }_0^R{I}_0\left(\frac{{\rho }^{\prime 2}{sin}^2\alpha }{2k^2}\right)e^{-\frac{{\rho }^{\prime 2}}{k^2}\left(1-\frac{{sin}^2\alpha }{2}\right)}{\rho }^{\prime }d{\rho }^{\prime }\hfill \\ \hfill & =\pi Ik{w}_0{f}^d\left(\alpha \right){cos}^3\alpha {\int }_0^{R^2/k^2}{I}_0\left(\frac{{sin}^2\alpha }{2}\xi \right)e^{-\left(1-\frac{{sin}^2\alpha }{2}\right)\xi }d\xi \hfill \end{array}$$

(36)

which in general cannot be simplified further.

From Equation (36), it is easy to see the dependence of $I^d$ on all parameters. Firstly, the coefficients $a_m$ are dependent on surface roughness $\sigma$. Secondly, the laser divergence angle k contributes the upper bound of the integral and overall value. Finally, the geometric parameter, disc radius R, also determines the integral upper bound, while rotation angle $\alpha$ contributes the integral kernel.

Figure 4 shows the curves of diffuse component $I^d$ as a function of the disc rotation angle $\alpha$ and the radius R, at four different surface roughness parameters $\sigma$, according to Equation (36). As expected, at a fixed geometric configuration ($\alpha ,R$ fixed), as $\sigma$ increases, the diffuse backscattering light intensity will become stronger, this is in contrast to the specular component, see Ref. [47]. From the physical explanation, $\sigma$ is the standard variance of the tangent of microfacets normal vector direction, $\theta$. When $\sigma$ increases, the probability of a large angle of $\theta$ is higher, thus the multi-reflection becomes substantial. In Figure 4b, the diffuse component backscattering light intensity tends to be asymptotic as the disc radius increases close to the beam diameter, which coincides with that of the specular component [47].

The degree of polarization of planar discs is plotted in Figure 5. The DOP is high when the rotate angle $\alpha$ is small, and decreases monotonically and rapidly as $\alpha$ increases when the surface is “smooth”, i.e., $\sigma$ is small, as shown with the blue line in Figure 5a. This is because, in this condition, the specular component of pBRDF dominates. However, when the surface becomes “rougher”, the diffuse component of pBRDF dominates. However, when the surface becomes “rougher”, the diffuse component dominates, and thus DOP becomes low when $\alpha$ is small. The other three lines in Figure 5a imply that DOP first increases before a maximum value then drops rapidly to 0 as $\alpha$ increases. The physical explanation of this phenomenon is that the specular component is more sensitive to the rotation angle than the diffuse component. Therefore, the former contributes to DOP at a relative narrow range of $\alpha$ than the latter. When $\alpha$ is larger than about ${60}^{\circ }$, $I^s$ rapidly vanishes, so that DOP tends to 0.

On the other hand, note that the DOP is calculated in the approximation condition of long distance, $R\ll 1$, for example, $R\sim {10}^{-2}$. The relative error of DOP is less than ${10}^{-2}$ orders of magnitude. Within this accuracy, the DOP is essentially a constant at a certain roughness $\sigma$, as Figure 5b suggested. Figure 6 plots the DOP increases with disc radius. The relative change rate of DOP is only about $0.239066/0.239050-1\approx 6.7\times {10}^{-5}$, as the radius varies from 0 to 0.1, which is 3 orders of magnitudes lower than that of $R^2\sim 0.01$. Thus, in this accuracy condition, the contribution of higher order, $O\left(x^{\prime 2}\right)$ and $O\left(y^{\prime 2}\right)$, in Equation (32) must be taken into account.

3.3. Rectangle

Another typical planar surface is a rectangle. Suppose the side lengths are a and b, respectively, then the surface equation is the same as the disc discussed in the previous subsection. According to Equation (32), the diffuse component of backscattering light intensity is

$$\begin{array}{cc}\hfill I^d& =\frac{I{w}_0}{k}\sum _{m=1}^4{a}_m{cos}^{m+2}\alpha {\int }_{-\frac{a}{2}}^{\frac{a}{2}}d{x}^{\prime }{\int }_{-\frac{bcos\alpha }{2}}^{\frac{bcos\alpha }{2}}{e}^{-\frac{x^{\prime 2}+y^{\prime 2}{cos}^2\alpha }{k^2}}\left[1+(m+4)y^{\prime }sin\alpha \right]d{y}^{\prime }\hfill \\ \hfill & =\pi I{w}_{0}k{f}^d\left(\alpha \right){cos}^2\alpha \mathsf{\Phi }\left(\frac{a}{2k}\right)\mathsf{\Phi }\left(\frac{bcos\alpha }{2k}\right)\hfill \end{array}$$

(37)

where

$$\mathsf{\Phi }\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$$

(38)

is the error function.

It is still worthy considering a spacial case that $a=b$, a square. Figure 7 shows the diffuse component of backscattering light intensity, $I^d$, as a function of the planar rotation angle $\alpha$ and square length a, respectively. The square length in Figure 7a is the same as the disc diameter in Figure 4a. The behavior of these curves are consistent with that of discs, based on the same physical principles, except the absolute intensity differs, which results from the irradiance areas of the two surfaces being difference.

Additionally, comparing the DOP of square and disc with the same length and diameter at different rotation angle, it is clear that the relative difference is at an order of magnitude of 0.01, as shown in Figure 8. Considering that the calculation accuracy is of ${10}^{-2}$ orders of magnitude, one can conclude that within this precision, the DOP of the disc and square of the same sizes are equal. The reasonable physical meaning is that the DOP of a planar surface is determined essentially by the material (the roughness $\sigma$) and incident angle $\alpha$, rather than the geometric size or shape.

3.4. Typical Curved Surfaces

For the typical curved surfaces, the most common ones are cone, hemisphere and cylinder. For calculation of the former one, first consider a general situation that the arbitrary revolved surface described by equation $F(\rho ,z)=0$. Assuming that ${\left({\partial }_{\rho }F\right)}^2+{\left({\partial }_{z}F\right)}^2=1$, according to Equation (26), the diffuse component backscattering light intensity is

$$I^d=\frac{I{w}_0}{k}\sum _{m=1}^4{a}_m\int \int \frac{{\left[{\partial }_{z}F-\left(\rho {\partial }_{\rho }F+z{\partial }_{z}F\right)\right]}^{m+1}}{(1-z){\left[{\rho }^2+{(1-z)}^2\right]}^{(m+3)/2}}{e}^{-\frac{{\rho }^2}{k^2{(1-z)}^2}}dS$$

(39)

where the element area is

$$dS=\frac{1}{|{\partial }_{\rho }F|}\rho d\varphi dz$$

(40)

For a cone with a half vertex angle $\alpha \in (0,\pi )$ and bottom radius R, or height H, the surface equation is $\rho cos\alpha +zsin\alpha =0$. Under this definition, the vertex of the cone is upward when $\alpha \in (0,\pi /2)$ and downward otherwise. Then, according to geometry relation of this configuration, the element area $dS=|sec\alpha |\rho d\varphi dz=csc\alpha \rho d\rho d\varphi$. Thus,

$$I^d=\frac{2\pi I{w}_0}{k}\sum _{m=1}^4{a}_m{sin}^m\alpha {\int }_0^R\frac{e^{-\frac{{\rho }^2}{k^2{(1+\rho cot\alpha )}^2}}}{(1+\rho cot\alpha ){(1+{\rho }^2{csc}^2\alpha +2\rho cot\alpha )}^{\frac{m+3}{2}}}\rho d\rho$$

(41)

Very carefully expand the denominator of Equation (41) as series. It can be carried out only under the long-distance condition that both $R\ll 1$ and $H\ll 1$. When $R\ge H$, or equivalently, $|cot\alpha |\le 1$,

$$\begin{array}{cc}\hfill I^d& \approx \frac{2\pi I{w}_0}{k}\sum _{m=1}^4{a}_m{sin}^m\alpha {\int }_0^R{e}^{-\frac{{\rho }^2}{k^2}}\left[1-(m+4)\rho cot\alpha \right]\rho d\rho \hfill \\ \hfill & =I_1^d+I_2^d\hfill \end{array}$$

(42)

where

$$I_1^d=\pi I{w}_{0}k\sum _{m=1}^4{a}_m{sin}^m\alpha \left(1-e^{-R^2/k^2}\right)$$

(43)

$$I_2^d=\pi I{w}_{0}kcot\alpha \sum _{m=1}^4(m+4)a_m{sin}^m\alpha \left[R{e}^{-R^2/k^2}-\frac{k\sqrt{\pi }}{2}\mathsf{\Phi }\left(\frac{R}{k}\right)\right]$$

(44)

are the first and second orders of approximation for $I^d$, respectively. As Figure 9 shows, the former is the diffuse component of backscattering light from the projection on $Oxy$ plane of the cone. If $\alpha =\pi /2$, the cone is reduced to a disc of radius R, which can be easily verified by comparing the results of Equation (43) with $\alpha =\pi /2$ and Equation (36) with $\alpha =0$. In the second order, Equation (44) is odd around $\alpha =\pi /2$, which is the consequence to the non-planar property.

Similarly, when $R\le H$, or equivalently, $|tan\alpha |\le 1$, Equation (42) can be expanded as

$$\begin{array}{cc}\hfill I^d& \approx \frac{2\pi I{w}_0}{k}\sum _{m=1}^4{a}_m{sin}^m\alpha {\int }_0^H{e}^{-\frac{z^2{tan}^2\alpha }{k^2}}{tan}^2\alpha \left[1+(m+4)z\right]zdz\hfill \\ \hfill & =I_1^d+I_2^d\hfill \end{array}$$

(45)

where now

$$I_1^d=\pi I{w}_{0}k\sum _{m=1}^4{a}_m{sin}^m\alpha \left(1-e^{-H^2{tan}^2\alpha /k^2}\right)$$

(46)

$$I_2^d=\pi I{w}_{0}kcos\alpha \sum _{m=1}^4(m+4)a_m{sin}^{m-1}\alpha \left[\frac{k\sqrt{\pi }}{2}\mathsf{\Phi }\left(\frac{Htan\alpha }{k}\right)-Htan\alpha e^{-\frac{H^2{tan}^2\alpha }{k^2}}\right]$$

(47)

Figure 10 shows the degree of polarization of backscattering light from the cone at different roughness varies as functions of the half-vertex angle and bottom radius, respectively. The DOP versus $\alpha$ exhibits a similar behavior as that of a disc, Figure 5a, since the major contribution portion is $I_1^d$. However, the DOP is not symmetrical around $\alpha =\pi /2$ due to the presence of $I_2^d$. It must be emphasized that in the realistic target detection, the difference between $DOP(\pi /2+\alpha )$ and $DOP(\pi /2-\alpha )$ is expected to be larger than the result given in this article. The reason is when the cone half-vertex angle $\pi /2<\alpha <\pi$, the multi-reflection occurs not only between the microfacets, but also the macro-internal surface of the cone. This effect will be studied in the future work. Figure 10b shows the dependence of DOP on the bottom radius of the cone. For a specific roughness $\sigma$, it exhibits a monotonic increase as R increases, until R reaches the beam radius $k(1-z)\approx k$, resulting from the increasing irradiated area by the light source.

4. Results

Figure 11 shows the experimental setup for testing the polarization state of backscattering light from typical geometric targets. A 532 nm pulsed laser of 10 $\mathsf{\mu }$J output energy and nanosecond pulse width is modulated by a signal generator at 10 Hz repetition frequency. An optical attenuator consists by a half-wave plate and two polarized beam splitters (PBSs), after which the light is horizontally polarized. The extinction ratio of each PBS is about 30 dB, then the total extinction ratio is 60 dB. The convex lens diverges the beam to about a 30 cm radius, which completely covers the target surface, located 15 m away from the laser. The beam divergence k, appearing in Equation (15), thus is k = 30 cm/15 m = 0.02. The laser angle of pitch was adjusted to be vertical to the reference board. The pedestal can be rotated to control the incident angle. The backscattering beam is collected by a optical camera with a photon detector integrated inside. Finally the wave pulses are recorded by an oscilloscope.

The Stokes parameter $S_0$, $S_1$ and $S_2$ can be measured by rotating the polarizer before the camera. Since the incidence light is horizontally polarized and the target surface material is non-conductive, the refractive index is real. According to Equation (13), the nonvanishing components of the backscattering Stokes parameter can only be $S_0$ and $S_1$. When rotating the polarizer, the detector response has maximum and minimum values, $I_{max}$ and $I_{min}$, which can be expressed by Stokes parameters as

$$\begin{array}{cc}\hfill I_{max}& =S_0+\frac{1}{2}\left(S_0-S_1\right)\hfill \\ \hfill I_{min}& =\frac{1}{2}\left(S_0-S_1\right)\hfill \end{array}$$

(48)

Thus, the degree of polarization is

$$DOP=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$$

(49)

The targets models are made by hardened white card paper, with a roughness parameter $\sigma$ estimated to be at 0.1∼1 order of magnitude. The geometric parameters are listed in

Table 1. Geometric parameters of target models.

Squares	Discs	Cones
Length	Radius	Radius	Height	$\mathit{\alpha }$
8 cm	4 cm	4 cm	2 cm	63${}^{\circ }$
10 cm	5 cm	5 cm	2.5 cm	63${}^{\circ }$
12 cm	6 cm	6 cm	3 cm	63${}^{\circ }$
14 cm	7 cm	7 cm	3.5 cm	63${}^{\circ }$
16 cm	8 cm	8 cm	4 cm	63${}^{\circ }$

. The first two columns are planar targets, for verifying DOP as either the size or rotating angle, corresponding to the theoretical calculation result of Figure 5. The vertex angles of cones are fixed at $\alpha ={63.4}^{\circ }$ to verify the influence of the bottom radius on the degree of polarization.

5. Discussion

To verify the dependency of DOP on the incident angle, the DOP of squares and discs were tested. Since the energy of pulses fluctuated, the measured pulse signal introduced errors, which could reduce SNR and affect the integration to obtain total pulse energy. To suppress the measurement error caused by laser power fluctuation, the experiment repeated 10 times for each geometric surface, the square and disc. Within each repetition, the pedestal angle $\alpha$ varies from 0 to 60${}^{\circ }$ in every 5${}^{\circ }$, while the polarizer angle $\psi$ independently varies from 0 to 90${}^{\circ }$ in every 5${}^{\circ }$. In every angle set $(\alpha ,\psi )$, 20 averaged pulses were measured to suppress the detector electronic noise. Figure 12 shows the measurement results of DOP of these two kinds of planar surfaces, which exhibits basically the same behavior that increases monotonically as the rotation angle increases. Comparing the theoretical plot and measured data in Figure 12, the tendency of DOP curves coincides with the theoretical calculation. Specifically, when $\alpha$ is less than about 20${}^{\circ }$, the mean measured results coincide with the theoretical calculation, while when $\alpha$ is larger than about 25${}^{\circ }$, the mean measured results are a bit less than the theoretical results. Additionally, with the rotation angle $\alpha$ becoming larger, the measured data (the dots in Figure 12) spread and variate from the mean value. There are two major reasons. One is that when the rotation angle is large, the total intensity of the backscattering light is very weak, while the environment light from the reference board (see Figure 11) is inevitable. While $\alpha$ increases, the intensity from target falls, then the component of the environment light dominates, which depolarizes the backscattering light substantially. This can be qualitatively analyzed as follows. If the unpolarized environment light intensity is $I_{en}$, then the measurement results are

$$\begin{array}{cc}\hfill I_{max}^{\prime }& =S_0+\frac{1}{2}\left(S_0-S_1\right)+\frac{1}{2}{I}_{en}\hfill \\ \hfill I_{min}^{\prime }& =\frac{1}{2}\left(S_0-S_1\right)+\frac{1}{2}{I}_{en}\hfill \end{array}$$

(50)

respectively. Thus, the realistic DOP is

$$DO{P}_{measure}=\frac{I_{max}-I_{min}}{I_{max}+I_{min}+I_{en}}<DO{P}_{theore}$$

(51)

On the other hand, the total light intensity rapidly decreases with $\alpha$ increasing, and the signal-to-noise ratio falls, making it difficult to extract the signal pulse from the backscattering light waveform. Thus, the measurement error is relatively large compared with small $\alpha$, which explains the spread of the dots of the measured data in Figure 12.

To verify the dependence of DOP on geometric size, five squares and five discs with different lengths were first tested. The pedestal angle was fixed such that the light was of normal incidence and measured backscattered light pulses in every 5${}^{\circ }$ of the polarizer angle. Each measurement result also contains 20 averaged laser pulses backscattered from the target. Furthermore, the experiment also repeated 10 times for each target. The experiment results are shown in Figure 13a,b. The experiment results agreed well with the theoretical calculations. However, when the size of squares or discs is small, the measured data show more fluctuations than that of the large-sized squares or discs. The reason is that, when the target size is small, the total light intensity that comes from the target surface is weak, leading to a low SNR. Thus, the extracted waveform from the signal pulse has a relatively larger deviation, as the dots suggested at a = 8 cm or r = 4 cm in Figure 13a and Figure 13b, respectively.

On the other hand, the degree of polarization of five cones with different bottom radii and fixed vertex angles were measured, as shown in Figure 13c. The DOP behavior is similar to that of the former two geometric shapes and basically coincides with the theoretical calculation. However, for the cones, there is another source of error. The cones are cylindrically symmetrical when the light is of normal incidence. The backscattered light from an arbitrary generatrix at the azimuth angle of $\varphi$ is equal to that from a plane whose normal direction is a function of $\varphi$ and $\alpha$. Thus, the DOP contribution is expected to be axial symmetric around the polarization direction of incident light. However, the beam profile deviates from Gaussian, destroying such axial symmetry. Therefore, the DOP of backscattered light depends on the beam profile, resulting in extra measurement errors for different sizes of cones.

The goal of the theoretical calculation of backscattering Stokes parameters and DOP is to discriminate remote targets by using the LIDAR. There are two major ways to give the pBRDF and thus the Stokes parameters in the current literature. One way is the measuring method [17,48,49,50,51]. For a specific target, for example, a plane with a coating material whose surface parameters like roughness or refractive index are unknown, measure the polarization states of backscattering light at different incidence and reflectance angles $({\theta }_i,{\theta }_r,\varphi )$. Therefore, according to Equation (4), the pBRDF as a function of these angles is given. However, the measurement method suffers from a disadvantage of un-universality. If the structures or surface parameters change, the measured pBRDF is invalid.

The other way to obtain the polarization state from the target is numerical simulation calculation by dividing the rough surface into smaller but finite facets [14,37,52,53]. There are two simulation methods, one is the Method of Moments (MoM) [36], which is to numerically solve integral equations about electromagnetic fields. It is an accurate solution to the electric field vector, however, it is confined to give the polarization state of backscattering light from a small area surface, as Ref. [36], millimeter level, inadequate for the purpose of realistic target detection. The other one is directly summing the backscattering light from all small planar facets. However, the parameters like normal directions or roughness are different in each facets. However, the numerical simulation method costs much computational time and resources due to the complexity of solving integral equations or calculating the 16 Mueller matrix elements for each small facet, as the number of facets increase, especially when the realistic targets are complex surfaces, while the simulation accuracy increases in contrast.

Fortunately, typical geometric surfaces have analytic pBRDF formulas. The previous work [47], as well as this current article, derived the mathematical formulas of pBRDF for some typical geometric surfaces that provide another possible way to access the pBRDF and Stokes parameter of the backscattering light with less computational time required. What is more essential, one can discriminate the target geometric size and spatial orientation. More explicitly, according to both theoretical and experimental results in this article, the DOP is dependent substantially on the rotation angle of the target, but insensitive to the size of the target. On the other hand, according to the results of our previous work [47], the Stokes parameter $S_0$ and $S_1$ stand in contrast. Based on this fact, in the further work of remote sensing applications, assuming that the targets are rigid bodies, and the geometric parameters, denoted by $(q_1,q_2,\dots )$, such as radius, heights, lengths, and vertex angles, are fixed for a given target, but the gesture parameters, denoted by $(p_1,p_2,\dots )$, such as spatial positions x, y, the rotation angles, can vary, then the Stokes parameters of backscattering light are the function of these parameters:

$$\mathbf{S}=\mathbf{S}(q_1,q_2,\dots ,p_1,p_2,\dots )$$

(52)

where the explicit form of the function is analytically calculated by the formulas shown in both current and previous works [47] for typical geometric surfaces. The DOP can be calculated by Equation (3) if the four Stokes parameters are measured. As the target moves, the gesture parameters vary, thus the measured Stokes parameters consequently change,

$$\begin{array}{cc}\hfill \mathbf{S}& =\mathbf{S}(q_1,q_2,\dots ,p_1,p_2,\dots )\hfill \\ \hfill {\mathbf{S}}^{\prime }& =\mathbf{S}(q_1,q_2,\dots ,p_1^{\prime },p_2^{\prime },\dots )\hfill \\ \hfill {\mathbf{S}}^{″}& =\mathbf{S}(q_1,q_2,\dots ,p_1^{\prime \prime },p_2^{″},\dots )\hfill \\ \hfill \dots & \end{array}$$

(53)

Then, according to the specific form of the function, one can recover the gesture parameters, expressed by the measured Stokes and geometric parameters, leading to the discrimination of target sizes and spatial status.

However, the mathematical formula method has its limitations as well. Firstly, the Stokes parameters of realistic targets with complicated geometric structures can usually be calculated by summing up the Stokes parameters from all typical geometric surface parts, as shown in Figure 1. The multi-reflection between different parts could further depolarize the backscattering light while the shadowing effect could also change the polarization, which is worth considering in the future work. Secondly, the accuracy of the pBRDF model, especially the shadowing and masking function, $G({\theta }_i,{\theta }_r,\varphi )$ model, plays a critical role in the theoretical calculation of the polarization states of the backscattering light, including DOP. Therefore, the development of a higher-precision pBRDF model is urgently needed to aid in further calculations, assessing more complicated targets, and for applications in remote sensing.

6. Conclusions

To summarize, this work provides an alternative method to access the pBRDF of a rough surface, and thus the Stokes parameters of backscattering light, by giving analytical formulas of typical geometric surfaces. Compared to the current ways, the direct measurement or numerical simulation, the method presented here has the advantage of universality and simplicity. For theoretical calculations, the model of diffuse components of BRDF was firstly simplified, giving a polynomial fitting, which is practical for the theoretical analysis of backscattering Stokes parameters. Then, by using this fitting model of diffuse component BRDF $f^d(cos{\theta }_i)$, the analytical calculation of DOP of backscattering light from several kinds of typical geometric rough surfaces is established. The theoretical calculations for different sizes of planar and cone targets agreed well with experiment results within our measurement accuracy, the latter represents a current method to obtain pBRDF. The results for planar targets at different rotation angles basically agreed with the theoretical calculations when the rotation angle is less than about 50${}^{\circ }$, but deviate when it increased. The imperfection of results is caused by the non-Gaussian distribution of the beam profile; the accuracy of the optical element, such as the polarizer; and the unpolarized environmental reflection light from background objects, such as the reference board, the pedestal or the connecting stick, which decreases the DOP. The agreement of the mathematical formulas and measurement results provides the reliability for the potential application in remote targets discrimination. However, the mathematical formula method of backscattering Stokes parameters still needs to be improved in the future practical application scenarios, considering the combination of typical geometric surfaces to the realistic complex target.