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Abstract
A channeled spectropolarimeter is a powerful tool for the simultaneous measurement of the intensity, spectral, and polarization information of a target. However, the fore-optics introduce additional polarization information, which leads to inaccurate reconstruction of the Stokes parameters. In this study, we propose a simple method for polarimetric calibration and Stokes parameters reconstruction for a fieldable channeled spectropolarimeter. The polarization effects of the fore-optics and phase factors of the high-order retarders at varying view angles are considered and calibrated independently using a single reference beam. Moreover, the misalignment of the retarders is also considered. Simulation results demonstrate that the polarization effects of fore-optics can be precisely determined, enhancing the measurement accuracy of the Stokes parameters by approximately an order of magnitude. The effectiveness of the proposed method is also verified experimentally.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A spectropolarimeter is an instrument that simultaneously measures the intensity, spectral, and polarization information of a target. It has various applications, including remote sensing [1–3], biomedicine [4–6], and material characterization [7]. Channeled spectropolarimetry (CSP) is a powerful snapshot technique among the various spectropolarimeter instruments developed previously [8,9]. It utilizes two high-order retarders and a polarizer to achieve polarimetric spectral intensity modulation (PSIM) in the wavenumber domain, then the four components of the Stokes parameters are encoded into a single modulated spectrum, achieving the snapshot nature.
Typically, the angle of incidence of light onto the PSIM module should be below 5° to avoid more complicated propagation behavior [1]. Therefore, a front lens group is essential to collect and collimate the light to reduce the angle of incidence onto the retarders. However, the polarization effects of the fore-optics, including retardance and diattenuation, will alter the polarization state of the incident light [10]. Moreover, lenses are usually coated to improve their transmission, and complex coating systems can also introduce additional phase retardation. These polarimetric errors can lead to inaccurate reconstruction of the Stokes parameters.
To counteract polarization effects, a common method involves measuring the fore-optics Mueller matrix and using its inverse during Stokes parameters measurement to deduce the input before the fore-optics. This method is simple; however, it requires large amounts of accurate experimental data, which can be time-consuming and limits its application. Another approach is to calibrate the entire fieldable CSP system. Bin Yang et al. [11] used an additional high-order retarder and three reference beams to determine the errors of the PSIM module and the polarization effects of the fore-optics, respectively, and compensated for them when reconstructing the Stokes parameters. This method allows for simultaneous calibration of the entire system after instrument assembly; however, using additional elements complicates the calibration process, and requiring three reference beams for each field of view (FOV) is still slightly cumbersome. Moreover, this method presumes that the second retarder's retardation is uniform across the FOV, an assumption invalidated if the retarder is not aligned exactly at 45° or its material exhibits significant birefringence, which will be demonstrated in the simulation section.
To address this problem, we propose a simple method to calibrate the polarization effects of fore-optics. A single reference beam is used to calibrate both the polarization effects of fore-optics and the phase retardation at different view angles. Moreover, we consider the alignment errors of the PSIM module. By employing our proposed methods, the calibration process remains simple while improving the accuracy of Stokes parameters reconstruction at each view angle for fieldable CSP.
The remainder of this paper is organized as follows: Section 2 provides an overview of CSP principles. Section 3 describes the polarimetric calibration and reconstruction methods. Simulation and experimental verification are presented in Sections 4 and 5, respectively. Finally, Section 6 concludes the study.
2. Principle of the channeled spectropolarimeter
The optical schematic of the fieldable CSP is depicted in Fig. 1. The light from the target object is collected and collimated by the fore-optics, passed through the PSIM module, and then emitted into the spectrometer via the imaging optics and a slit. The PSIM module consists of two retarders, R1 and R2, and an analyzer A. The transmission axis of A and the fast axis of R2 intersect at an angle of 45°, while the fast axis of R1 is aligned with the transmission axis of A. In this study, the thickness ratio of R1 and R2 is set to 3:1, as it can prevent signal leaking from the baseband into the first heterodyne band and also creates two additional channels that contain useful calibration data if the system’s retarders are misaligned [12].
When the polarization effects of the optical systems are not considered, the spectrum obtained by the perfectly aligned CSP can be expressed as follows:
A fieldable CSP is generally composed of the fore-optics, PSIM module, imaging optics, and a spectrometer. The Stokes parameters for each FOV after the target light passes through the system can be expressed as follows:
3. Polarimetric calibration method
3.1 Derivation of the theoretical model
Since the detector is only capable of responding to the radiation intensity, the expression for the modulation spectrum of perfectly aligned fieldable CSP at view angle $\theta $ is:
The relationships described in Eq. (7) are for mathematical convenience only and do not have specific units. Furthermore, it is unnecessary to obtain the transmittance of the fore-optics separately, since the spectrum of Eq. (6) contains only the overall transmittance. By applying the inverse Fourier transform to Eq. (6), we can obtain the following seven channels:
To determine the polarization effects of the optical system, specifically the values of $A(\sigma ,\theta )$, ${D_{fore}}(\sigma ,\theta )$ and ${\delta _{fore}}(\sigma ,\theta )$, according to Eqs. (7) and (10), a reference beam with a known state of polarization (SOP) is required. The only condition is that S2 and S3 should not be simultaneously zero. For formulaic simplicity in this study, we utilize a 45° linearly polarized beam with known intensity, where ${X_{0,45^\circ }} = {X_{2,45^\circ }} = {S_{0,45^\circ }}$, ${X_{1,45^\circ }} = {D_{fore}}{S_{0,45^\circ }}$, and ${X_{3,45^\circ }} ={-} {\delta _{fore}}{S_{0,45^\circ }}$. By using the information from these channels and frequency filtering techniques, the polarization effects of optical systems can be calculated as follows:
Following the determination of the polarization effects, the Stokes parameters can be compensated and reconstructed as follows:
3.2 In the presence of alignment errors
Accurate alignment can be challenging even in the laboratory, even more in practice [14]. The angle between the fast axes of the two retarders and the transmission axis of the polarizer may not align perfectly at 0° and 45°, which can undermine the effectiveness of the proposed polarimetric calibration method. To address this issue, a generalized PSIM model has been established [15], in which the fast axes of R1 and R2 are aligned at angles α and β, respectively, with the transmission direction of A. Therefore, the Mueller matrix of the PSIM module becomes ${M_{PSIM}} = {A_{PSIM}} \cdot {M_A}(0^\circ ) \cdot {M_{{R_2}}}(\beta ,{\phi _2}) \cdot {M_{{R_1}}}(\alpha ,{\phi _1})$, the specific expressions for the PSIM module can be found in [15], here its transmittance is also considered. At this point, two additional channels, ${C_3}(L - {L_1})$ and ${C_{ - 3}}(L + {L_1})$, appear. The 45° reference beam calibration method remains effective, and the polarization effects of the fore-optics can be calculated as follows:
The variables a∼f represent trigonometric functions of retarder orientation:
In Eqs. (19)–(21), both the retardation and orientation of the retarders are required. The PSIM module is still capable of self-calibration and is not affected by the polarization effects of optical systems. This is because when misalignment occurs, the alterations in each channel are consistent in form, which does not impact the computation of the argument. Furthermore, any additional coefficients are eliminated when the channels are divided. Therefore, the calculations for retarder orientations and phase factors are consistent with those presented in [15]. This enables us to determine phase retardations for different view angles without relying on values from the central FOV as a substitute. The calculation method for the Stokes parameters remains unchanged, as described in Eq. (14), while the calculation of ${X_0}(\sigma ,\theta )\sim {X_3}(\sigma ,\theta )$ has become more complex:
By taking into account the polarization effects of the optical system and the errors in the PSIM module, the presented reconstruction method can produce more accurate reconstructed Stokes parameters.
4. Simulation analysis
In this section, we first analyze and explain why the phase retardations of each FOV of R2 cannot be replaced by the value of the central FOV. The expression for the phase factor of the high-order retarder is given by [16]
Figure 2 illustrates the variation of the phase retardation of R2 at different incident angles. According to [1], all incident angles on the high-order retarders of the PSIM module should be limited to less than approximately 5°. Our results demonstrate that when R2 is oriented at 45°, there is no significant change in phase retardation when the incident angle changes from 0° to 5°. However, when the orientation of R2 deviates from 45°, the difference in phase retardation at incident angles of 0° and 5° increases. Even a deviation of only 1° in the orientation of R2 can cause a retardation error of 0.011 rad, resulting in a non-negligible error in measurement accuracy (the error of S3/S0 increases from 4.4 × 10−4 to 9.6 × 10−3). Furthermore, in the model with arbitrary retarder orientation, φ2 cannot be considered constant across the entire FOV.
Fig. 2. Variation of the phase retardation of R2 (a) for different orientations when the incident angle changes from 0° to 5°, and (b) with orientation changes when the incident angle is fixed. The selected wavelength is 589.3 nm, and the thickness of R2 is 2 mm. The values of no(σ) and ne(σ) are 1.544 and 1.553 at the selected wavenumber [21].
Apart from the orientation of R2, this approximation is also invalid if the retarder material has a significant birefringence. Figure 3 illustrates the variation of the phase retardation of R2 for different materials. We have chosen yttrium orthovanadate (YVO4) [17,18] and calcite [13,19] as examples, both of which are commonly used waveplate materials [20]. The Sellmeier equation was used to calculate the refractive index of these materials in simulations. Our results indicate that when using materials with larger birefringence, there is a significant change in the retardation of R2 when the incident angle changes from 0° to 5°. In such cases, the largest root mean square errors (RMSEs) for S1/S0, S2/S0, and S3/S0 are 2.2 × 10−2, 3.8 × 10−2, and 2.5 × 10−1 respectively. The highest RMSE of the degree of polarization (DOP) is 4.4 × 10−2. Therefore, it is not feasible to substitute the retardation of R2 in different FOVs with that of the central FOV. The previous research [11] used such an approximation because it satisfied both conditions: the R2 orientation was 45° and the birefringence of the material was small, resulting in $[\frac{{n_e^2(\sigma )}}{{n_o^2(\sigma )}}{\cos ^2}\Phi + {\sin ^2}\Phi ] \approx 1$, and an extremely small difference between $\sqrt {n_e^2(\sigma ) - {{\sin }^2}\beta } $ and $\sqrt {n_o^2(\sigma ) - {{\sin }^2}\beta } $.
Fig. 3. Variation of the phase retardation of R2 for different materials when the incident angle changes from 0° to 5°. The selected wavelength is 589.3 nm, and the thickness of R2 is 2 mm.
Simulations were conducted to demonstrate the effectiveness of the proposed method for polarimetric calibration and reconstruction. Figure 4 presents the schematic layout of the fore-optics that we designed. The maximum incident angle on R1 and R2 is 4.77° in the working waveband, which meets the requirement of being less than 5°. The half FOV is 18.6° and the entrance pupil diameter is 4.75 mm. The lenses are coated with custom-designed anti-reflective films. The input diattenuation and phase retardations were obtained through a polarization ray trace of the fore-optics. The wavenumber range is 12000–17143 cm-1 (583–833 nm), and the thicknesses of R1 and R2 are 6 mm and 2 mm, respectively.
The input values and calibration results of the polarization effects, including the transmittance, diattenuation, and retardance of the fore-optics at different view angles, are illustrated in Fig. 5. Consistent with our predictions, the polarization effects become more pronounced as the FOV increases. At the maximum FOV, the magnitudes of the diattenuation and retardation are approximately 0.011 and 0.105 rad, respectively, which will significantly affect the measurement accuracy of the fieldable CSP if they are overlooked in the reconstruction, a point we will elucidate in the following discussion. The polarization effects at the central FOV are zero because a single ray trace was used, and the rays are normally incident. The relative errors of the calibration results of $A(\sigma ,\theta )$, ${D_{fore}}(\sigma ,\theta )$ and ${\delta _{fore}}(\sigma ,\theta )$ are less than 1.7 × 10−4, 1.7 × 10−4, and 8.9 × 10−4 rad, respectively, for each corresponding view angle. These results demonstrate that the polarization effects of fore-optics can be accurately determined. Based on these calibration results, we can reconstruct the Stokes parameters from the obtained spectra using our presented reconstruction method.
Fig. 5. Input and calibration values of the polarization effects of the fore-optics (a) 2Afore(σ, θ), (b) Dfore(σ, θ) and (c) δfore(σ, θ) in the central wavenumber at different view angles.
To compare the reconstruction results with conventional methods, specifically the “reference beam calibration technique” [19], the orientations of the two retarders were set to 0° and 45°, respectively, and linearly polarized light at 30° was used as the input. Figure 6 presents the reconstruction results of the Stokes parameters. Using the traditional method, the largest RMSEs for S1/S0, S2/S0, and S3/S0 across the working wavenumber range are 4.3 × 10−3, 1.9 × 10−3 and 4.9 × 10−4, respectively, corresponding to the errors at the maximum FOV. Using our proposed methods, these values decreased to 9.7 × 10−5, 6.1 × 10−4, and 6.5 × 10−4, respectively. The RMSE of the DOP is reduced from 3.8 × 10−3 to 5.3 × 10−4. The reconstruction accuracy of the Stokes parameters is improved by approximately one order of magnitude. The errors of the conventional method tend to increase further if there are alignment errors. For example, if R1 and R2 are misaligned by 0.5° and -0.5° respectively, the RMSEs for S1/S0, S2/S0, and S3/S0 increase to 1.1 × 10−2, 8.5 × 10−3 and 8.4 × 10−4, respectively. Furthermore, the RMSE of the DOP increases to 1.3 × 10−2. However, when using the proposed methods, the RMSEs of the Stokes parameters and the DOP remain at an order of magnitude 1e-4.
Fig. 6. Reconstruction results of the Stokes parameters. The dotted lines denote the results using the traditional method, while the solid lines denote the results using the presented method. Theoretical values are S1/S0 = 0.5, S2/S0 = 0.866, and S3/S0 = 0.
Simulation results suggest that polarization effects from fore-optics significantly impact Stokes parameter reconstruction and should be considered. The proposed methods effectively mitigate such effects, thereby demonstrating their validity.
5. Experimental results
The configuration of the demonstration experiment is shown in Fig. 7. The light source consisted of a stabilized tungsten halogen lamp, a collimator, and a rotatable polarizer P that generates the beams required for the polarimetric calibration and reconstruction. These components are mounted on a precision turntable and guide rail, allowing beam adjustment to illuminate the system at different view angles. The system being calibrated includes fore-optics, a PSIM module, relay optics, and a grating-based spectrometer (FieldSpec 3, Analytical Spectral Devices). The retarder thickness and wavenumber range are consistent with those in the simulations.
First, we set the orientations of R1 and R2 to 0° and 45°, respectively, to compare the proposed method with the traditional method. We set P to 30° as the target light for measurement and then input linearly polarized beam at 45° and 22.5°, respectively, as the reference beam for the proposed and the traditional methods. The reconstruction results for this case are shown in Fig. 8. In line with our expectations, the traditional method, which overlooks the polarization effects of the optical system, leads to deviations in the Stokes parameters. The RMSEs of S1/S0, S2/S0, and S3/S0 are less than 8.5 × 10−3, 1.3 × 10−2, and 3.6 × 10−3, respectively, and the RMSE of the DOP is less than 8.2 × 10−3. It should be noted that the polarization effects at the central FOV are not zero because the light at the edge of the aperture cannot be assumed to be normally incident [11]. However, when using the proposed method, the reconstruction accuracy remains favorable. The RMSEs of S1/S0, S2/S0, and S3/S0 are less than 4.2 × 10−3, 4.7 × 10−3, and 3.1 × 10−3, respectively, while the RMSEs of the DOP are less than 4.4 × 10−3.
Fig. 8. Reconstructed Stokes parameters from experimental measurements. The dotted lines denote the results using the traditional method, while the solid lines denote the results using the presented method. Theoretical values are S1/S0 = 0.5, S2/S0 = 0.866, and S3/S0 = 0.
We then changed the orientations of R1 and R2 to verify the calibration and demodulation methods for other retarder orientations. In the experiment, we first rotated P to 45° relative to A to generate a reference beam for determining parameters $A(\sigma ,\theta )$, ${D_{fore}}(\sigma ,\theta )$ and ${\delta _{fore}}(\sigma ,\theta )$. The Stokes parameters of the target light were then reconstructed using the determined parameter values. Subsequently, we set the polarizer P to 30° as the input. As the polarization effects are more evident at larger view angles, we allowed the light source to illuminate at the maximum view angle. The reconstruction results for this case are shown in Fig. 9, concurrently with the theoretically calculated values. The RMSEs of S1/S0, S2/S0, and S3/S0 are less than 4.5 × 10−3, 3.5 × 10−3, and 4.7 × 10−3, respectively; while the RMSEs of the DOP are less than 2.5 × 10−3, which meets the requirements of practical applications [22]. These results demonstrate that by using our proposed polarimetric calibration and reconstruction methods, we effectively reduced the influences of polarization effects of the optical system, variations in phase factors, and misalignment of the PSIM module.
Fig. 9. Reconstructed Stokes parameters for other retarder orientations with theoretical values of S1/S0 = 0.5, S2/S0 = 0.866, and S3/S0 = 0.
The experimental results indicate that our proposed method effectively improves the reconstruction accuracy of the Stokes parameters. For systems with a wider FOV or smaller F-number, which amplify the impact of polarization effects and variable phase factors, our proposed methods become increasingly crucial for ensuring accurate measurements.
6. Conclusion
We have proposed a simple method for calibrating the polarization effects of the optical system and compensating for the reconstruction of Stokes parameters of a fieldable CSP. Our primary focus is on the fore-optics, as the aft-optics only affect the overall transmittance and typically do not require calibration. We first derive a theoretical model for the polarimetric calibration, which is simple to implement and only requires a single reference beam of known SOP. Additionally, the misalignment of retarders is also considered, and the proposed method is still feasible. Simulation results validate that the polarization effects can be accurately determined, and our proposed method outperforms the conventional method, resulting in an order of magnitude improvement in measurement accuracy despite the complexity of the structure and coating of fore-optics. The validity of our proposed model has been experimentally verified and found to be in good agreement with corresponding theoretical values. Therefore, the presented method is helpful for accurate measurement with CSP.
Funding
National Natural Science Foundation of China (62105331, 62275114); High-altitude large UAV airborne visible infrared spectral imager procurement project (E13361X3IZ); HY-Multi-angle polarization spectral imaging remote sensing system (E21021S6IZ); K. C. Wong Education Foundation; Shandong Province Key Agricultural Project for Application Technology Innovation (SD2019ZZ007); Supported by the Taishan Industrial Experts Program.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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