Introduction

Polarization is one primary characteristics of electromagnetic wave. It can provide information different from intensity, spectral and coherence, which related inherent physical and chemical characteristics of targets, such as target material, surface texture, and structure1,2,3. Therefore, polarization detection, has been widely studied, particularly in remote sensing4,5, where it enhances the detection and analysis of environmental features; in medical diagnostics6,7, where it aids in the detection of tissue abnormalities; in underwater detection8,9, where it improves visibility and object identification; and in military applications10, where it enhances target recognition and tracking capabilities.

Modelling and studying the polarization characteristics of the reflection is crucial for polarization detection, target recognition, and polarization simulation. The Bidirectional Reflectance Distribution Function (BRDF) describes the distribution of the reflection. Since the 1980s, numerous scholars have conducted research on BRDF11,12. In 2000, Priest and Germer13 combined the Torrance-Sparrow model (T-S model) with the Fresnel reflection equation and Mueller matrix to study a macroscopic surface and proposed P-G pBRDF, which achieved the polarization of BRDF. In 2009, Hyde et al.14 assumed that the diffuse reflection process was completely depolarized in the P-G pBRDF, theoretically derived the mathematical expression of the diffuse reflection component, and introduced the geometric attenuation factor (GAF) into the pBRDF to subdivide the influence of shadowing and masking effects between microfacets on the specular reflection. The error of the model was effectively reduced. However, Hyde model considered the diffuse reflection to be an ideal Lambert diffuse reflection, which was only effective for highly reflective materials. Zhu et al.15. proposed a three-component pBRDF for metallic surfaces, which can describe the polarization reflection characteristics of metals in the upper hemispherical space, but the volume scattering component was considered to be an ideal Lambert reflection, ignoring the influence of the surface roughness and reflected angle. Fu et al.16 established pBRDF for iron plates, glass plates, sod plates, and green-painted iron plates, which associated Fresnel equation with the complex index of refraction but did not further explore the polarization spectral characteristics of the four targets. Based on the P-G model, Sun et al.17 introduced scattering and phase functions to optimize the model and carried out multi-angle polarization measurement experiments on dark green coated aluminium. Based on the existing P-G model and parameter inversion, the output results of the model are compared with the experimental data through simulation, and the results show that the relative error of the target's linear polarizability is reduced under the improved model. However, the models above only described the spatial distribution of the polarization characteristics of the target at a single wavelength and ignored the polarization characteristics of the target changing with the spectrum, which is defined as polarization spectral correlation in this paper.

To describe the polarization spectral correlation of the target, a pSBRDF was established based on the three-component model by combining the Fresnel reflectance in the specular reflection component with Cauchy's empirical dispersion equation and introducing a set of dispersion constants that do not vary with wavelength. The Northeast Normal University Laboratory Goniospectrometer System (NENULGS) was used to measure the polarization of two types of coated fabric samples. The parameters of the model were inverted using the nonlinear least-square algorithm with DoLP as the objective function. The results indicated that the pSBRDF can well describe both the spatial distribution and the spectral correlation of polarization characteristics.

Theory

The concept of BRDF was first proposed by Nicodemus18 in 1965, which was used to characterize the directional scattering and energy distribution of light in the upper hemispherical space on targets. The spatial representation of BRDF is shown in Fig. 1.

$$f_{{\text{r}}} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ) = \frac{{dL_{{\text{r}}} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda )}}{{dE_{{\text{i}}} (\theta_{{\text{i}}} ,\phi_{{\text{i}}} ,\lambda )}}$$
(1)

where \(\lambda\) is the wavelength of light (nm), \(\phi_{{\text{i}}}\) and \(\phi_{{\text{r}}}\) are the azimuth angle of incident light and reflected light respectively, \(\phi = \left| {\phi_{{\text{r}}} - \phi_{{\text{i}}} } \right|\) is the relative azimuth angle;\(\theta_{{\text{i}}}\) and \(\theta_{{\text{r}}}\) are the zenith angle of incident light and reflected light, respectively; Z is the normal direction of macroscopic surface; the reflected radiance \({\text{d}}L_{{\text{r}}} (\theta_{{\text{r}}} ,\theta_{{\text{r}}} ,\phi ;\lambda )\) is the radiation flux emitted per unit area and per unit solid angle in the direction of reflection \({\text{(W/(m}}^{{2}} \cdot {\text{sr))}}\); the incident irradiance \(dE_{{\text{i}}} (\theta_{{\text{i}}} ,\phi_{{\text{i}}} ;\lambda )\) is the radiation flux arriving per unit area in the incident direction \({\text{(W/m}}^{{2}} {)}\).

Fig. 1
figure 1

The spatial representation of BRDF.

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Because the BRDF is a scalar form, it can’t adequately describe the distribution of polarized scattered light in space. The pBRDF \({\mathbf{F}}^{{{\text{pBRDF}}}}\) was defined as a 4 × 4 transformation of the Stokes vectors of incident \({\mathbf{S}}^{{{\text{in}}}}\) and reflected light \({\mathbf{S}}^{{{\text{out}}}}\).

$${\mathbf{S}}^{{{\text{out}}}} = {\mathbf{F}}^{{{\text{pBRDF}}}} \cdot {\mathbf{S}}^{{{\text{in}}}}$$
(2)

In Fig. 2, the microfacet theory considers that the surface of an object is composed of the adjacent microfacets, each of which follows Fresnel’s equation. Based on the three-component pBRDF, this paper considers that the reflection process is divided into specular reflection \({\mathbf{F}}_{{\text{s}}}\), multiple reflection \({\mathbf{F}}_{{\text{m}}}\), and volume scattering \({\mathbf{F}}_{{\text{v}}}\). The specular component obeys Fresnel equation. The multiple reflection is formed by the light rebounding many times on the uneven surface of the sample. The volume scattering is formed because when the incident light reaches the surface, part of it participates in the specular reflection and multiple reflection, and the rest is transmitted to the inner material, while part of the transmitted light is selectively absorbed by the particles of the inner material, and the other part is scattered in the inner material and returns to the original transmitted material (air) through the interface. \({\mathbf{I}}\) is the incident light vector, and \({\mathbf{R}}\) is the reflected light vector. \({\mathbf{F}}^{{{\text{pBRDF}}}}\) can be expressed as the sum of specular reflection, multiple reflection, and volume scattering.

$${\mathbf{F}}^{{{\text{pBRDF}}}} = k_{{\text{s}}} {\mathbf{F}}_{{\text{s}}} + k_{{\text{m}}} {\mathbf{F}}_{{\text{m}}} + k_{{\text{v}}} {\mathbf{F}}_{{\text{v}}}$$
(3)

where, \(k_{{\text{s}}}\) is the specular reflection component coefficient, \(k_{{\text{m}}}\) is the multiple reflection component coefficient, and \(k_{{\text{v}}}\) is the volume scattering component coefficient.

Fig. 2
figure 2

The three-component reflection based on microfacet theory.

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In Fig. 2, \(\theta\) is the incident angle of the relative the microfacet element normal N, \(\alpha\) is the angle between the normal direction of macroscopic surface Z and the macroscopic surface N, which is also the inclination angle of the microfacet, and satisfies:

$$\theta = \frac{{\theta_{{\text{i}}} + \theta_{{\text{r}}} }}{2}$$
(4)
$$\alpha = \frac{{\left| {\theta_{{\text{i}}} - \theta_{{\text{r}}} } \right|}}{2}$$
(5)
$$\cos \alpha = \frac{{\cos \theta_{{\text{i}}} + \cos \theta_{{\text{r}}} }}{2\cos \beta }$$
(6)
$$\cos (2\beta ) = \cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} + \sin \theta_{{\text{i}}} \sin \theta_{{\text{r}}} \cos \left( {\phi_{{\text{r}}} - \phi_{{\text{i}}} } \right)$$
(7)

Specular reflection component (introduced Cauchy’s empirical dispersion)

The expression of the specular reflection component after introducing Cauchy’s empirical dispersion equation is as follows:

$$F_{ij}^{{\text{s}}} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\Delta \phi ;\lambda ,D) = f_{{\text{s}}} \cdot M_{ij}^{{\text{s}}} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D) = \frac{{P(\alpha ) \cdot G(\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma )}}{{4\cos \theta \cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} }}{\mathbf{M}}_{ij} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D)$$
(8)

where, the variation of specular reflection component with wavelength is embodied in the elements of pBRDF Mueller matrix \({\mathbf{M}}_{ij} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D)\), and the dispersion constants \(A_{0}\), \(A_{1}\), \(A_{2}\), \(B_{0}\), \(B_{1}\) and \(B_{2}\) are collectively referred to as D. \(G(\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma )\) is the geometrical attenuation factor (GAF), which refers to the effect of shadows and occlusions on the object’s surface when light reflects off the object19,20, used to describe the geometric attenuation of reflection caused by the shadowing and masking effect of the incident light and reflected light. In this paper, the integral geometric attenuation factor (IGAF) is adopted21.

In the specular reflection component, the model assumes that consists of a set of random set of microfacets whose angle of inclination \(\alpha\) satisfy the Gaussian distribution:

$$P(\alpha ) = \frac{C}{{2{\uppi }\sigma^{2} \cos^{3} \alpha }}\exp ( - \frac{{\tan^{2} \alpha }}{{2\sigma^{2} }})$$
(9)

where, \(\sigma\) is the surface roughness; C is the normalization factor that integrates \(P(\alpha )\) into 1 over the entire space.

In order to obtain the Mueller matrix \({\mathbf{M}}_{ij} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D)\),the Jones matrix elements are calculated by Fresnel's equation first:

$$\begin{aligned} \left( {\begin{array}{*{20}c} {E_{{\text{r}}}^{{\text{s}}} } \\ {E_{{\text{r}}}^{{\text{p}}} } \\ \end{array} } \right) = & \left( {\begin{array}{*{20}l} {T_{{{\text{ss}}}} } \hfill & {T_{{{\text{ps}}}} } \hfill \\ {T_{{{\text{sp}}}} } \hfill & {T_{{{\text{pp}}}} } \hfill \\ \end{array} } \right)\left( {\begin{array}{*{20}l} {E_{{\text{i}}}^{{\text{s}}} } \hfill \\ {E_{{\text{i}}}^{{\text{p}}} } \hfill \\ \end{array} } \right) \\ = & \left( {\begin{array}{*{20}c} {\cos \eta_{{\text{r}}} } & {\sin \eta_{{\text{r}}} } \\ { - \sin \eta_{{\text{r}}} } & {\cos \eta_{{\text{r}}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r_{{{\text{ss}}}} } & 0 \\ 0 & {r_{{{\text{pp}}}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\cos \eta_{{\text{i}}} } & { - \sin \eta_{{\text{i}}} } \\ {\sin \eta_{{\text{i}}} } & {\cos \eta_{{\text{i}}} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {E_{{\text{i}}}^{{\text{s}}} } \\ {E_{{\text{i}}}^{{\text{p}}} } \\ \end{array} } \right) \\ \end{aligned}$$
(10)

where, \(E_{{\text{i}}}^{{\text{s}}}\) and \(E_{{\text{r}}}^{{\text{s}}}\) are the complex electric field components of the vertical polarization of the incident light and reflected light respectively; \(E_{{\text{i}}}^{{\text{p}}}\) and \(E_{{\text{r}}}^{{\text{p}}}\) are the complex electric field components of the parallel polarization of the incident light and reflected light respectively. Since \(E_{{\text{i}}}^{{\text{s}}}\), \(E_{{\text{r}}}^{{\text{s}}}\), \(E_{{\text{i}}}^{{\text{p}}}\) and \(E_{{\text{r}}}^{{\text{p}}}\) are defined relative to the macroscopic coordinate system, the coordinate transformation is involved:

$$\cos \eta_{{\text{i}}} = \frac{{\left( {\cos \alpha - \cos \theta_{{\text{i}}} \cos \beta } \right)}}{{\left( {\sin \theta_{{\text{i}}} \sin \beta } \right)}}$$
(11)
$$\cos \eta_{{\text{r}}} = \frac{{\left( {\cos \alpha - \cos \theta_{{\text{r}}} \cos \beta } \right)}}{{\left( {\sin \theta_{{\text{r}}} \sin \beta } \right)}}$$
(12)

The reflectance of vertical and parallel polarized electric fields \(r_{{{\text{ss}}}}\) and \(r_{{{\text{pp}}}}\) in Fresnel equation is a function of the complex refractive index and the incident angle.

$$r_{{{\text{ss}}}} = \frac{{E_{{\text{r}}}^{{\text{s}}} }}{{E_{{\text{i}}}^{{\text{s}}} }} = \frac{{\left( {\tilde{n}^{2} - \sin^{2} \theta } \right)^{\frac{1}{2}} - \cos \theta }}{{\left( {\tilde{n}^{2} - \sin^{2} \theta } \right)^{\frac{1}{2}} + \cos \theta }} \,$$
(13)
$$r_{{{\text{pp}}}} = \frac{{E_{{\text{p}}}^{{\text{r}}} }}{{E_{{\text{i}}}^{{\text{r}}} }} = \frac{{\tilde{n}^{2} \cos \theta - \left( {\tilde{n}^{2} - \sin^{2} \theta } \right)^{\frac{1}{2}} }}{{\tilde{n}^{2} \cos \theta + \left( {\tilde{n}^{2} - \sin^{2} \theta } \right)^{\frac{1}{2}} }}$$
(14)

where, \(\tilde{n} = n + {\text{i}}k\) is the complex refractive index of the target material, n and k are the refractive index and extinction coefficient, respectively.

The classical Cauchy dispersion equation only describes the refractive index n of the target, without considering the extinction ability of the target. Therefore, based on the classical Cauchy dispersion equation, this paper adds the equation of the change of the extinction coefficient k with wavelength, whose expression is as follows:

$$n = A_{0} + \frac{{A_{1} }}{{\lambda^{2} }} + \frac{{A_{2} }}{{\lambda^{4} }}$$
(15)
$$k = B_{0} + \frac{{B_{1} }}{{\lambda^{2} }} + \frac{{B_{2} }}{{\lambda^{4} }}$$
(16)

The Jones matrix elements \(T_{{{\text{ss}}}}\), \(T_{{{\text{ps}}}}\), \(T_{{{\text{sp}}}}\), and \(T_{{{\text{pp}}}}\) are descriptions of the polarization state of the light, and the relation with wavelength can be obtained by combining Eqs. (9–16). Here, The complex refractive index \(\hat{n}\) can be regarded as function of wavelength \(\lambda\) and dispersion constant D, so the specific expression of matrix elements is as follows:

$$\begin{aligned} T_{{{\text{ss}}}} = & \frac{{(\tilde{n}(\lambda ,D)^{2} + \sin^{2} \theta )^{\frac{1}{2}} }}{{(\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \cos \eta_{{\text{r}}} \cos \eta_{{\text{i}}} \\ \quad & + \frac{{\tilde{n}(\lambda ,D)^{2} \cos \theta - (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }}{{\tilde{n}(\lambda ,D)^{2} \cos \theta + (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \sin \eta_{{\text{r}}} \sin \eta_{{\text{i}}} \\ \end{aligned}$$
(17)
$$\begin{aligned} T_{{{\text{ps}}}} = & - \frac{{(\tilde{n}(\lambda ,D)^{2} + \sin^{2} \theta )^{\frac{1}{2}} }}{{(\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \cos \eta_{{\text{r}}} \sin \eta_{{\text{i}}} \\ \quad & + \frac{{\tilde{n}(\lambda ,D)^{2} \cos \theta - (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }}{{\tilde{n}(\lambda ,D)^{2} \cos \theta + (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \sin \eta_{{\text{r}}} \cos \eta_{{\text{i}}} \\ \end{aligned}$$
(18)
$$\begin{aligned} T_{{{\text{sp}}}} = & - \frac{{(\tilde{n}(\lambda ,D)^{2} + \sin^{2} \theta )^{\frac{1}{2}} }}{{(\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \sin \eta_{{\text{r}}} \cos \eta_{{\text{i}}} \\ & \quad + \frac{{\tilde{n}(\lambda ,D)^{2} \cos \theta - (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }}{{\tilde{n}(\lambda ,D)^{2} \cos \theta + (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \cos \eta_{{\text{r}}} \sin \eta_{{\text{i}}} \\ \end{aligned}$$
(19)
$$\begin{aligned} T_{{{\text{pp}}}} = & - \frac{{(\tilde{n}(\lambda ,D)^{2} + \sin^{2} \theta )^{\frac{1}{2}} }}{{(\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \sin \eta_{{\text{r}}} \sin \eta_{{\text{i}}} \\ \quad & + \frac{{\tilde{n}(\lambda ,D)^{2} \cos \theta - (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }}{{\tilde{n}(\lambda ,D)^{2} \cos \theta + (\tilde{n}(\lambda ,D)^{2} - \sin^{2} \theta )^{\frac{1}{2}} }} \cdot \cos \eta_{{\text{r}}} \cos \eta_{{\text{i}}} \\ \end{aligned}$$
(20)

according to Eqs. (17–20), the Mueller matrix elements \({\mathbf{M}}_{ij} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D)\) containing wavelength variable and the dispersion constants can be deduced22.

Multiple reflection and volume scattering component

For the multiple reflection component, the light undergone multiple collisions in the process, and its polarization state changed many times, so the polarization direction has been complex. It is generally believed that the process is completely depolarized, that is, randomly polarized light is generated, which only contributes to the total energy of reflection. Moreover, the multiple reflection component is not only related to the incident and reflected angle but also the surface roughness of the target. Then the pBRDF multiple reflection component matrix is:

$${\mathbf{F}}_{ij}^{{\text{m}}} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma ,c) = f_{{\text{m}}} \cdot M_{ij}^{{\text{m}}} = \frac{1}{{\uppi }}(\cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} )^{c} (\exp^{\sigma } - 1)\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]$$
(21)

where c is the undetermined constant, whose value range is \(( - 1,0)\).

For the volume scattering component, because the light is transmitted twice on the surface of the sample and has an extremely complex scattering process with the inner material, the scattering light propagates randomly when it reaches the interface. In general, the refractive index of the sample is always greater than air. According to Fresnel equation, the volume scattering light occurs when the angle between the direction of the scattered light returned to the upper layer and the horizontal position is smaller than the inclination of the adjacent surface; that is, the greater the surface roughness, the greater the probability of volume scattering. The specific expression is as follows:

$${\mathbf{F}}_{ij}^{{\text{v}}} (\theta_{{\text{r}}} ,\sigma ) = f_{{\text{v}}} \cdot {\mathbf{M}}_{ij}^{{\text{v}}} = \cos^{\sigma } \theta_{{\text{r}}} \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]$$
(22)

In summary, combining the specular reflection component \({\mathbf{F}}_{ij}^{{\text{s}}}\) and the multiple reflection component \({\mathbf{F}}_{ij}^{{\text{d}}}\), and the volume scattering component \({\mathbf{F}}_{ij}^{{\text{v}}}\) gives a complete pSBRDF expression \({\mathbf{F}}_{ij}\), where the first element \({\mathbf{F}}_{00}\) represents the total intensity of the reflected light (the sum of the specular and diffuse components). Therefore, the complete pSBRDF expression can be obtained by combining the above three components, which is as follows:

$$\begin{aligned} {\mathbf{F}}_{ij}^{{{\text{pSBRDF}}}} = & k_{{\text{s}}} \cdot {\mathbf{F}}_{ij}^{{\text{s}}} + k_{{\text{m}}} \cdot {\mathbf{F}}_{ij}^{{\text{m}}} + k_{{\text{v}}} {\mathbf{F}}_{ij}^{{\text{i}}} \\ = & k_{{\text{s}}} \cdot \frac{{P(\alpha ) \cdot G(\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma )}}{{4\cos \theta \cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} }}{\mathbf{M}}_{ij} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D) \\ \quad & + k_{{\text{m}}} \cdot \frac{1}{{\uppi }}(\cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} )^{c} (e^{\sigma } - 1)\left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right) + k_{{\text{v}}} \cdot \cos^{\sigma } \theta_{{\text{r}}} \left( {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right) \, \\ \end{aligned}$$
(23)

Experiment

The NENULGS23 was used to measure the DoLP of samples under laboratory conditions, as shown in Fig. 3.

Fig. 3
figure 3

The Northeast Normal University Laboratory Goniospectrometer System.

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The system is composed of a light source system, goniometer, ASD spectrometer, and polarization lens, among other components. The light source system consists of a 100W halogen lamp and two lenses to produce non-polarized parallel light. The detector is equipped with a field angle constraint tube that changes the field angle during testing by moving up and down. The setting range is 8°–25°. In this paper, the field angle is set to 12°, and the vertical distance from the polarizer to the sample is 28 cm. The goniometer adopts stepper motor control to detect the change of zenith angle and azimuth angle, where the detection zenith angle can move within 90° with an accuracy of 0.25°, and the detection azimuth angle can rotate in the range of 0°–360° with an accuracy of 0.25°. The ASD spectrometer, FieldSpec®3, has an effective detection range of 350 nm–2500 nm, a distance of 0.2 m from the fiber optic probe to the sample surface, and a field of view angle size of 8°. When the detection zenith angle varies between 0° and 60°, the fiber optic probe records data from a circular footprint with a diameter of 2.8 cm to an elliptical footprint with a long axis of 5.7 cm24. The polarizer uses a Glan–Thompson polarizer with an extinction ratio of 10,000:1 and an effective band of 350 nm–2200 nm, which is mounted on a rotating device in front of the detector and can rotate the polarizer at 0°–360° with an accuracy of 1°. Because the polarization characteristic of the sample in the visible wavelength is the focus of the study. Combined with the effective band of ASD spectrometer and polarizer, the detection range is selected from 400 to 760 nm.

In this paper, two kinds of coated fabric samples in desert and soil colors were used, as shown in Fig. 4.

Fig. 4
figure 4

The experimental samples.

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To reduce the geometric impact, the sample were laid flat on a board. The surface roughness of the sample was measured using the Ra option of the surface roughness tester TR10025, where Ra represents the arithmetic mean of the absolute distance between each point on the measured profile and the midline of the profile within a sampling length of 0.25mm. The specific measurement process is to obtain the result by averaging 10 measurements of the two samples at different positions. The surface roughness of two samples are \(\sigma_{{{\text{desert}}}} = 2.27{\mu m}\) and \(\sigma_{{{\text{soil}}}} = 2.84{\mu m}\), respectively.

The NENULGS was used to measure the spatial light intensity distribution in the upper hemispherical space of the target under different incident and reflected conditions. The measurement principle is shown in Fig. 5.

Fig. 5
figure 5

The measurement schematic diagram.

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As shown in Fig. 5, the azimuth angle of the light source ___location is set to 0°. When the incident zenith angle is \(\theta_{{\text{i}}} =\) 30°, 40°, and 50°, there are a total of 8 detection azimuth angles (the relative azimuth angle) \(\phi_{{}}\) for the sample, which are 180°, 165°, 150°, 120°, 90°, 60°, 30°, 15°, and 0° respectively. Because the position of the detector overlaps with the position of the light source, the method of obtaining the linear difference was chosen at the position of 0°. The reflected zenith angle \(\theta_{{\text{r}}}\) is measured at the intervals of 10°in range of 0°–60°, and the polarization lens is rotated at 0°, 45°, 90°, and 135°, respectively. The measurement results are \(I_{{0^{ \circ } }}\), \(I_{{45^{ \circ } }}\), \(I_{{90^{ \circ } }}\), and \(I_{{135^{ \circ } }}\), respectively.

The DoLP can be calculated by experimental measurement results, and the specific expression is as follows:

$$DoLP_{{{\text{Exp}}{.}}} = \frac{{\sqrt {(S_{1} )^{2} + (S_{2} )^{2} } }}{{S_{0} }} = \frac{{\sqrt {(I_{{0^{ \circ } }} - I_{{90^{ \circ } }} )^{2} + (I_{{45^{ \circ } }} - I_{{135^{ \circ } }} )^{2} } }}{{I_{{0^{ \circ } }} + I_{{90^{ \circ } }} }}$$
(24)

Model verification and analysis

Since the light source system in the multi-angle polarization spectrum measurement device simulates non-polarized parallel light, that is, the incident light is unpolarized, which can be represented as \({\mathbf{S}}^{{{\text{in}}}} = [\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ \end{array} ]^{{\text{T}}}\) by Stokes vector, then the Stokes vector of scattered light \({\mathbf{S}}^{{{\text{out}}}}\) can be obtained from Eq. (2):

$$\begin{aligned} {\mathbf{S}}^{{{\text{out}}}} = &{\varvec{F}}_{ij}^{{{\text{pBRDF}}}} \cdot {\mathbf{S}}^{{{\text{in}}}} \\ = & \left[ {\begin{array}{*{20}c} {F_{00}^{{}} } \\ {F_{10}^{{}} } \\ {F_{20}^{{}} } \\ {F_{30}^{{}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {k_{{\text{s}}} \cdot F_{00}^{{\text{s}}} + k_{{\text{d}}} \cdot F_{00}^{{\text{m}}} + k_{{\text{i}}} \cdot F_{00}^{{\text{v}}} } \\ {k_{{\text{s}}} \cdot F_{10}^{{\text{s}}} } \\ {k_{{\text{s}}} \cdot F_{20}^{{\text{s}}} } \\ {k_{{\text{s}}} \cdot F_{30}^{{\text{s}}} } \\ \end{array} } \right] \\ \end{aligned}$$
(25)

where, \(F_{00}^{{}}\), \(F_{10}^{{}}\), \(F_{20}^{{}}\), and \(F_{30}^{{}}\) are elements of the pSBRDF Mueller matrix of the target, we assume that the light source is unpolarized. In addition, the circular polarization of light reflected from most naturally illuminated surfaces is negligible, which simplifies the Mueller matrix \(F_{ij}^{{{\text{pSBRDF}}}}\) from \(4 \times 4\) to \(3 \times 3\). Based on these assumptions, the DoLP can be expressed as:

$${\mathbf{S}}^{{{\text{out}}}} = \left[ {\begin{array}{*{20}c} {F_{00}^{{}} } \\ {F_{10}^{{}} } \\ {F_{20}^{{}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {k_{{\text{s}}} \cdot F_{00}^{{\text{s}}} + k_{{\text{m}}} \cdot F_{00}^{{\text{m}}} + k_{{\text{v}}} \cdot F_{00}^{{\text{v}}} } \\ {k_{{\text{s}}} \cdot F_{10}^{{\text{s}}} } \\ {k_{{\text{s}}} \cdot F_{20}^{{\text{s}}} } \\ \end{array} } \right]$$
(26)

According to Eq. (21), the DoLP expression of the model can be obtained as follows:

$$\begin{aligned} DoLP_{{\text{M}}} = & \frac{{k_{{\text{s}}} \sqrt {(F_{10}^{{\text{s}}} )^{2} + (F_{20}^{{\text{s}}} )^{2} } }}{{k_{{\text{s}}} \cdot F_{00}^{{\text{s}}} + k_{{\text{m}}} \cdot F_{00}^{{\text{m}}} + k_{{\text{v}}} \cdot F_{00}^{{\text{v}}} }} \\ = & \frac{{k_{{\text{s}}} \cdot \frac{{P(\alpha ) \cdot G(\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma )}}{{4\cos \alpha \cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} }}\sqrt {(M_{10} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D))^{2} + (M_{20} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D))^{2} } }}{{k_{{\text{s}}} \cdot \frac{{P(\alpha ) \cdot G(\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\sigma )}}{{4\cos \alpha \cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} }} \cdot M_{00} (\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda ,D) + k_{{\text{m}}} \cdot \frac{1}{{\uppi }}(\cos \theta_{{\text{i}}} \cos \theta_{{\text{r}}} )^{c} (e^{\sigma } - 1) + k_{{\text{v}}} \cdot \cos^{\sigma } \theta_{{\text{r}}} }} \\ \end{aligned}$$
(27)

Spectral correlation verification

The unknown model parameters in the inversion process can be divided into the model component coefficient and the set of dispersion constants D. When constructing the inversion program, the nonlinear least-square algorithm was used for finding the best inversion value of the model and the corresponding objective function can be expressed as:

$$\min E = \mathop \Sigma \limits_{\lambda } \left[ {\frac{{\left( {DoLP_{{{\text{Exp}}{.}}} \left( {\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda } \right) - DoLP_{{\text{M}}} \left( {\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ;\lambda ,k_{{\text{s}}} ,k_{{\text{m}}} ,k_{{\text{v}}} ,\sigma ,c,D} \right)} \right)^{2} }}{{DoLP_{{{\text{Exp}}{.}}} \left( {\theta_{{\text{i}}} ,\theta_{{\text{r}}} ,\phi ,\lambda } \right)^{2} }}} \right]$$
(28)

In the inversion process, the DoLP data of the samples was used in the wavelength range of 400 nm–760 nm under the conditions that the incident angles are \(\theta_{{\text{i}}} =\) 30°, 40°, and 50°, the relative azimuth angle \(\phi { = 180}^{^\circ }\), and the reflected zenith angle \(\theta_{{\text{r}}}\) at an interval of 10° from 0° to 60°. The modeled results and the residual sum of squares (RSS) are shown below.

Figure 6 shows the DoLP experimental and modeled results of the samples at three incident angles. It is evident that at all angles, the lower wavelength bands exhibit a certain degree of ripple. This is due to the lower transmittance of the polarizer at lower wavelengths compared to higher wavelengths, resulting in a lower signal-to-noise ratio in the experimental data for the lower wavelength bands. It can be seen from the experimental results that the overall DoLP of the samples increases with the increase of the incident angle, and the overall DoLP of the desert sample is smaller than that of the soil sample. Under the condition of the same incident angle, the DoLP values of the two samples increase with the increase of detection zenith angle. Under the condition of the same incident angle and detection zenith angle, the DoLP values of the two samples decrease with the increase of wavelength, which is the same as the modeled results. Tables 1 and 2 respectively show the modeled errors RSS of two samples at the different incident and reflected angles. It can be seen that the maximum RSS of the desert sample and soil are 0.00743 and 0.0757, respectively. affirming the pSBRDF has good accuracy in describing the spectral behavior of the samples.

Fig. 6
figure 6

The experimental and modeled results of DoLP.

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Table 1 The modeled error RSS of desert.
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Table 2 The modeled error RSS of soil.
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Since the relative azimuth angle is 180° at the mirror position, this is a special position. In order to ensure the applicability of the model at the non-mirror position, the DoLP data of 400–760 nm at 7 detection zenith angles at the incidence angle of the light source at 40° and the relative azimuth angle \(\phi = 165^{ \circ }\) are used for parameter inversion. Below are the model modeled results and errors.

Figure 7 shows the model fitting at the relative azimuth angle of 165°, and the maximum fitting error in Table 3 is 0.00748 for the RSS of desert and 0.0361 for the RSS of soil. It can be concluded that the model still has good accuracy in non-specific angles.

Fig. 7
figure 7

The experimental and modeled results of DoLP at \(\phi = 165^{ \circ }\).

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Table 3 The modeled error RSS of desert and soil at \(\phi = 165^{ \circ }\).
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Verification of spatial distribution of polarization characteristics

To prove the ability of the pSBRDF to describe the spatial distribution of the polarization characteristics of the target, the model parameter values fitted by the spectral data of the desert and the soil samples at the incident angle \(\theta_{{\text{i}}} = 40^{ \circ }\) is used, as shown in Table 4 below. The DoLP data measured in the upper hemispherical space at wavelength \(\lambda = 500{\text{nm}}\) and \(\lambda = 650{\text{nm}}\) are selected to compare with the modeled values in these parameters.

Table 4 The model parameter values of desert and soil at 40° incident angle.
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Figure 8 shows the DoLP experimental results, modeled results, and absolute errors of the DoLP in the upper hemispherical space of the two samples at the incident angle \(\theta_{{\text{i}}} = 40^{ \circ }\) and the wavelength of 500nm and 650nm. The measured results show that the DoLP values of forward scattering are greater than those of backscattering, and the maximum DoLP values appear at the detection zenith angle \(\theta_{{\text{r}}} = 60^{ \circ }\) and the relative azimuth angle \(\phi = 180^{ \circ }\), which is the same as the modeled results. The model fitting errors were controlled within 0.0947, 0.0451, 0.0868, and 0.0779, respectively. Compared with our previous work25, the absolute error results are shown in Fig. 9.

Fig. 8
figure 8

The upper hemispherical experimental and modeled results and absolute errors under 40° the incident angle.

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Fig. 9
figure 9

The absolute error of the model proposed and the model in Ref23.

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The absolute errors of the model in Ref26 are 0.0924, 0.0481, 0.0912, and 0.0744 respectively, which further proves that the model still has a good ability to describe the spatial distribution of the polarization characteristics of the samples.

Conclusion

Given the limited discussion on the polarization spectral correlation in pBRDF studies, the pSBRDF based on Cauchy's empirical dispersion equation is proposed, which considers the reflection process to be composed of three components: specular reflection, multiple reflection and volume scattering. In the pSBRDF model, the Fresnel equation in the specular reflection component is combined with the Cauchy’s empirical dispersion equation to derive the Jones matrix with wavelength variable and dispersion constant that does not change with wavelength, and the Mueller matrix in the specular reflection component is further derived. The multiple reflection is the Minnaert model with roughness index term, and the volume scattering component is the cosine function. On basis above, the DoLP model expression is derived, and the DoLP data of the upper hemispherical space of the two coated fabric samples are measured under the different incident and detection conditions. By using the nonlinear least-square method, the model parameters of DoLP of two samples can be obtained. The maximum values of RSS in the model describing the polarization spectral correlation of the desert and soil are 0.00743, 0.0757, respectively; Then, the model inversion values of the two samples were used to fit the DoLP spatial distribution characteristics of the two samples, and the absolute errors were controlled within 0.0947, 0.0451, 0.0868, and 0.0779 respectively. Compared with our previous work, the results proves that the model not only has a good ability to describe the polarization spectrum characteristics, but also has a good ability distribute the spatial distribution of the polarization characteristics of the samples.