1. Introduction

Channeled spectropolarimetry (CSP) is a snapshot technique for measuring spectrally resolved Stokes vector of the light, which has aroused wide interest in various fields like remote sensing [1–3], thin-film analysis [4], material characterization [5–7], biomedical diagnosis [8,9], etc. It utilizes two high-order retarders and a polarizer to introduce spectral modulation, then the four components of the Stokes vector are encoded into a single modulated spectrum, achieving the snapshot nature [10–12]. Note that, after spectral modulation, the spectral modulation transfer function (SMTF) of the spectrometer significantly lowers the contrast of the output spectrum and suppresses the high-frequency channel amplitudes in the frequency domain (optical path difference domain in CSP) [13,14]. This will reduce the accuracy of the reconstructed Stokes spectra if not compensated. Moreover, the SMTF effect has been observed long before but its mechanism and theory in CSP have not been fully explained and derived.

SMTF was first proposed to determine the resolution power of the spectroscopic systems in analogy with the MTF of the imaging system [15–17]. The SMTF of the spectrometer is the ratio between the contrast of the output spectrum and the input spectrum [18]. Under most conditions, to compensate for SMTF in CSP, additional algorithms or instruments are required. Zhou et al. [19,20] used a well-calibrated dual-rotating-retarder spectropolarimeter (DRRSP) to compensate for SMTF. However, including the calibration process of DRRSP, 80 independent data acquisitions are required, which can be time-consuming and restrict its application. Besides, compensating for SMTF algorithmically is an alternative. Several methods have already been proposed, of which accurate simulations [21], well-trained neural networks [22], and inverse filters [3] are required. These approaches are sophisticated and applicable for actual use but require complex programming. The direct measurement of SMTF itself is another potential option for compensating for SMTF, of which white light interferometry is a well-known method [23–25]. It generates a sinusoidally modulated spectrum by the Michelson interferometer (MI). By varying the optical path difference (OPD), the SMTF can be recorded. The interferometric method is sophisticated, but it focuses on the SMTF measurement, not compensation. Hagen et al. [26,27] discuss the possibility of measuring SMTF using high-order retarders but do not present a specific compensation method in CSP. Moreover, although the compensation methods of SMTF mentioned above have achieved better results, there is still a lack of rigorous derivation for SMTF.

Note, the existing literature does not clearly state that, in the traditional method (or reference beam calibration method called in other literature), the SMTF is automatically compensated for by taking the ratio between the reference and measurement data. Generally, in this method, a reference beam linearly polarized at 22.5° is input into the CSP [28], calibrating the phase factors and SMTF. However, alignment and phase errors in the CSP system are not considered in this method. Accurate alignment can be challenging even in a laboratory, let alone in practice. Therefore, the calibration of alignment and phase errors has been studied extensively in the past [29–39], but the compensation of SMTF is rarely mentioned. To this end, it is of considerable research value to develop a method for calibrating both SMTF and errors.

This paper rigorously derives the SMTF theory, which can give insight into spectro-polarimeter design and data analysis. Further, an efficient method that can calibrate SMTF effects, alignment, and phase errors together in situ, is proposed. By simply rotating the retarder twice, the SMTF value can be obtained through channel shifting and the result is consistent with the rigorous interferometric measurement result in [23–25]. Furthermore, a reference beam is utilized to calibrate the alignment errors; and phase nonlinearity is compensated algorithmically via channel shifting. It is easy to implement the proposed method since no additional instruments or algorithms are required other than a rotator and a polarizer.

2. Channeled spectropolarimetry

2.1 Modulation theory of the channeled spectropolarimeter

A typical CSP is shown in Fig. 1. The fast axes of ${{R}_{1}}$ and ${{R}_{2}}$ are set at 0° and 45° about the x-axis, and their thickness ratio is 1:2.

 

Fig. 1. Sketch of a typical CSP. The red arrows represent the fast axes, black arrows represent the slow axes of the ${{R}_{1}}$, ${{R}_{2}}$, or the transmission axis of P.

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By using Stokes-Mueller formalism, the theoretical process can be derived.

The Stokes vector of incident light is shown as Eq. (1), $\sigma $ denotes the wavenumber, and T is the matrix transpose operator. The Mueller matrix of the system is

$\Delta {n^{(0 )}}$ is the zero-order birefringence of the retarders, ${L_1}$ and ${L_2}$ denote the optical path difference introduced by ${{R}_{1}}$ and ${{R}_{2}}$. ${\delta _1}$ and ${\delta _2}$ are composed of constants and high-order terms. Generally, they are small, so we often consider that the retardance ${\varphi _i}\; ({i = 1,2} )$ almost has a linear relation with wavenumber $\sigma $, and has the central frequency of ${L_i}$ [4].

The detector can only respond to the radiation intensity, only the ${S_0}$ component of the output spectrum is obtained. The obtained spectrum is shown as Eq. (4), where ${S_{23}} = {S_2} + i{S_3}$, ${\ast} $ denotes the complex conjugate. Dispersive spectrometers obtain spectral information in wavelength distribution. The method described in [26] converts uniform sampling from the wavelength domain to the wavenumber domain.

Equation (4) shows that the Stokes parameters are modulated by three different OPD frequencies. By doing the inverse Fourier transform, the Stokes parameters are distributed to seven frequency domain regions called “channels”. The autocorrelation of $B(\sigma )$ is given by

2.2 Traditional method

Generally, ${{C}_{0}}$, ${{C}_{2}}$ and ${{C}_{3}}$ channels contain all the Stokes parameters. The traditional method usually calibrates phase factors by a 22.5° linearly polarized reference beam [28]. Then the phase factors are given by

The Stokes spectra of incident light can be reconstructed from Eqs. (12)-(15), where Re and Im are operators to extract the real and imaginary parts, respectively. The SMTF reduces the high-frequency channel amplitudes proportionally. From Eqs. (13)-(15), the SMTF is already eliminated by taking the ratio of the measurement and reference data.

3. Derivation and analysis of spectral modulation transfer function

3.1 Effect of SMTF

The spectrometer can be considered a linear filter of spectral modulation frequencies, and the Fourier transform of its point spread function is the spectral modulation transfer function (SMTF). Similar to the MTF of an imaging system, the SMTF of the spectrometer decreases as the frequency increases. The spectrum in CSP is modulated by high frequencies, therefore, will be considerably affected by SMTF.

To analyze the SMTF in the spectral (wavenumber) and OPD (frequency) domains, we start from the convolution of the point spread function. Assuming the input is a single-frequency-modulated spectrum with wavenumber distribution, which is given by

$M(h )$ is the Fourier transform of $m(\sigma )$, which is the SMTF of the system, and is shown as Eq. (19).

The footmark s and a denote the Fourier transform of the symmetric and antisymmetric parts. Assuming the point spread function to be symmetric, then the phase $\phi $ is zero [15,18]. From Eq. (18), the SMTF is the contrast of the obtained spectrum.

Meanwhile, in the frequency domain, the Fourier-transform spectrum (or OPD spectrum) can be given by

Since $M(h )< 1$, the amplitude of the sideband has been suppressed, and the baseband is nearly unaffected. So far, the decreasing contrast of the spectrum and the reducing amplitudes of the high-frequency channels have been elucidated as the result of the SMTF.

3.2 SMTF derivation for a Czerny-Turner spectrometer

The finite slit, CCD pixel, and grating width determine the SMTF [25,40,41]. Essentially, the SMTF results from diffraction. For instance, we derive the point spread function (PSF) and SMTF of a Czerny-Turner spectrometer in this section. A typical Czerny-Turner spectrometer is schematically represented in Fig. 2. Under isoplanatic and linear superposition conditions, we derive the point spread function (PSF) employing diffraction theory.

 

Fig. 2. The schematic configuration of a Czerny-Turner spectrometer. Dashed and solid lines denote light at different wavelengths.

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In ideal conditions, the slit and CCD pixel have infinitesimal width and appear as Dirac functions. Then the PSF is determined only by the grating. The grating function is composed of single-slit diffraction and multi-slit interference factors. In general, the first diffraction order is used in spectrometers. Therefore, the single-slit diffraction factor is near-constant in this range and is omitted. Apart from the constant terms, the PSF of the grating is given by

Since the entrance slit and CCD pixel in a real spectrometer have finite widths, the point spread function is the convolution of the slit, pixel, and grating function [41]. In the absence of aberrations, nonlinear response, and noise, the PSF at a fixed wavelength is given by

P is the angular dispersion at wavenumber ${\sigma _0}$, describing the relation between the exiting diffraction angle from the grating and wavenumber. Then the relation between the image plane coordinate x and wavenumber can be obtained by

Then the PSF can be rewritten as a function of wavenumber, the PSF and SMTF are given by

In Eq. (30), the first two sinc(.) functions result from the limited width of the slit and CCD pixel, while the third tri(.) function, is the consequence of the limited grating width. From Eq. (30), SMTF is a function of optical path difference at a particular wavenumber. In experimental conditions, Eq. (30) can be simplified. According to [24], the authors measured SMTF while the slit or grating dominated the SMTF and verified that the shape of the measurement result corresponded to that of the sinc(.) and tri(.) functions.

Broadband spectrum is employed in CSP, from Eq. (30), the SMTF varies with both OPD and wavenumber. Then, the shift-invariant condition actually is not satisfied in the CSP system, because the wavenumber resolution varies across the spectrum [27]. Therefore, the mechanism of the SMTF in CSP system is more complicated than the MTF in other shift-invariant systems. MTF is expressed as multiplication form in the frequency domain, but in CSP, this is not rigorously applicable for SMTF.

According to [18], the SMTF varies by only 10% in the region from 380 to 750 nm. Additionally, Martinez-Matos et al. [24] utilized a Michaelson interferometer to introduce varying OPD and measured the SMTF at different wavelengths. The measured SMTF at various wavelengths shows significant differences only when the OPD exceeds approximately 400 µm. In our experimental setup, the maximum OPD is about 150 µm, which is considerably smaller. Furthermore, if the OPD introduced by the retarders exceeds 400 µm, it often surpasses the resolving power of grating spectrometers [27]. In such cases, a Fourier transform spectrometer [34] is recommended.

Therefore, for CSPs employing grating spectrometers, an approximation can be made that the SMTF is primarily a function of OPD and remains relatively consistent across wavenumbers. Consequently, the MTF concept can be employed to analyze the SMTF in the CSP system. This concept is expressed as a multiplication factor in the frequency domain (OPD domain in CSP).

As the optical path difference increases, the SMTF decreases, and at zero OPD, the SMTF is equal to 1. In CSP, ${{S}_{0}}$ component remains unmodulated and has a narrow bandwidth in the OPD domain. As a result, the baseband channel is nearly immune to the SMTF. Building upon this nature, we proposed a method via channel shifting to measure the SMTF and correct the high-frequency channels to compensate for it. Additionally, if the spectrometer specifications are known, the computation result of Eq. (30) can serve as a reference value for measuring the SMTF.

4. SMTF and errors calibration method

4.1 SMTF measurement and calibration

The SMTF can be easily determined from Eq. (18) using the contrast and compensated accordingly. Many prior works [23–25] have measured the SMTF from contrast by interferometric methods. However, this method is limited to single-frequency-modulated spectra. Even if we obtain the SMTF with the wavenumber distribution as mentioned in [27], it is difficult to compensate for the SMTF in the spectral (wavenumber) domain. That is because the output spectrum is modulated by different heterodyne OPD frequencies, and the contrast is thus affected by different SMTF values. However, in the OPD domain, different channels are affected by specific SMTF values. Consequently, correcting the amplitudes of high-frequency channels after obtaining the SMTF values is straightforward.

In CSP, only a few OPD frequencies are employed and appear as channels in the OPD domain. Due to this nature, we can successively estimate the SMTF at the high-frequency channels, namely ${{C}_{1}}$, ${{C}_{2}}$, ${{C}_{3}}$. The experimental setup is shown in Fig. 3, generating the spectra modulated by only one of the three heterodyne frequencies.

 

Fig. 3. Experimental setup to calculate the SMTF. ${{P}_{1}}$ is a linear polarizer whose transmission axis is fixed at 0°. The three times measurement need different fast axis angles of ${{R}_{1}}$, respectively -45°, 0° and 45°. The numbers in the brackets represent the sequence of measurement.

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While constructing the CSP system, the transmission axis of the last polarizer P is the fiducial direction and set as x-axis. Other components need to be aligned to P, due to the detector’s electronic noise and the insufficient accuracy of the mechanic rotation, there will be angle errors while aligning. Also, the vibrations caused by the insertion and removal in the repeated experiment will make the errors become prominent. The alignment method can be referred to in [45], but it is tedious to repeat accurate alignment before each time use. Therefore, calibration through an algorithm is simple and practical for actual use. From the perspective of experimental operation, the sufficiently accurate alignment between ${{P}_{1}}$ and P can be achieved and maintained. But the accurate alignment between thick retarder and P is somewhat difficult because its transmission varies with wavelengths. Therefore, we considered that alignment angle errors existed in retarders. Moreover, there will be thickness errors during manufacture.

Hence, in Fig. 3, the reason for choosing 0° linearly polarized light as an input is to provide a quantitative relationship between the baseband and sideband channels and meanwhile avoid errors. It is considered that the angle errors of ${{R}_{1}}$ and ${{R}_{2}}$ are ${\varepsilon _1}$ and ${\varepsilon _2}$, and the actual retardance with thickness errors are $\varphi _1^{\prime}(\sigma )$ and $\varphi _2^{\prime}(\sigma )$ respectively. The obtained spectra are given by

Since the baseband of the autocorrelations of ${I_1}(\sigma )$, ${I_2}(\sigma )$, ${I_3}(\sigma )$ are same and is given by

Considering the SMTF effects, from the autocorrelation of ${I_1}(\sigma )$, ${I_2}(\sigma )$, ${I_3}(\sigma )$, we can get

Since ${SMT}{{F}_{i}}{(h)}$ is an even function, the values in ${{C}_{i}}$ and ${{C}_{{ - i}}}$ channels are the same. Therefore, we can obtain the shifted ${{C}_{i}}$ channel and ${SMT}{{F}_{i}}{(h)}$ by calculation, and they are given by

Moreover, the experimental obtained SMTF value in a single channel bandpass looks like the same (which is shown in Fig. 5 in section 5), but after doing the Fourier transform of it, the result does slightly vary with wavenumber, as mentioned in section 3.2 and Eq. (30). By multiplying the reciprocal of the obtained SMTF in the OPD domain and correcting the high-frequency channels’ amplitudes, the SMTF effect can be well compensated.

It is noteworthy that, in Eq. (38) and (39), from $exp[{i2{\Phi _i}(\sigma )} ]$ to obtain ${\Phi _i}(\sigma )$, we take the argument of the former phase, unwrap it to avoid the $\pi $ jumping, and divide it by two afterward. There will be phase bias ambiguity during this process. However, unlike the self-calibration method, the phase factor is obtained through three measurements rather than one. Therefore, the phase factors do not rely on the thickness ratio of the retarders, and there will be no bias phase like $3\pi /2$ as mentioned in [44]. In the obtained phase factors, the bias phases are expressed as $exp({im\pi } )$, where m is an integer. These bias phases solely impact the sign of the obtained phase factors, namely, introducing an additional coefficient of ${\pm} 1$. Therefore, it is no need to find the exact value of m like [44], obtaining the additional coefficient is enough. Through channel shifting from Eq. (39), we can obtain the coefficient by dividing the shifted channel by the original channel. In our experiment, the additional coefficients of $exp[\varphi _2^{\prime}(\mathrm{\sigma } )- \varphi _1^{\prime}(\mathrm{\sigma } )]$, $exp[\varphi _2^{\prime}(\mathrm{\sigma } )]$, and $exp[{\varphi_2^{\prime}(\sigma )+ \varphi_1^{\prime}(\sigma )} ]$ are 1,1 and -1, respectively.

4.2 Alignment errors calibration method

Retarders’ errors in CSP consist of alignment and phase errors. Numerous calibration methods have been proposed, such as calibration using reference beams or additional components [28–35], self-calibration [36–38], phase rearrangement calibration [39], and total polarimetric errors calibration [19,20]. In section 4.1, we have already calibrated the phase with error, as indicated by Eq. (38). To simply calibrate the alignment errors, we utilize a 22.5° linearly polarized reference beam [31]. Considering that the angle errors of retarders are typically small, the small-angle approximation is employed. The obtained spectrum in the OPD domain is given by

As shown in Eqs. (42)-(45), the SMTF suppresses the sideband amplitude. Its effect can be compensated by correcting the amplitudes of high-frequency channels with the obtained SMTF values. Because the thickness ratio of retarders adopted is 1:2, generally, $\varphi _1^{\prime}(\sigma )$ is equal to $\varphi _2^{\prime}(\sigma )- \varphi _1^{\prime}(\sigma )$. Nevertheless, the thickness error could affect this equality relationship, resulting in an imperfect overlap between ${C}_{1}^{\prime}$ and ${C}_{4}^{\prime}$. Therefore, these channels will be abandoned in the follow-up process. By employing a 22.5° linearly polarized beam, the angle errors are given by

The actual phase factors have already been calculated in section 4.1, and all the unknown terms in Eqs. (41)-(45) are now available. Therefore, we can reconstruct the incident Stokes spectra by this modified theory.

5. Analysis of experimental results

5.1 Experimental setup

To verify the proposed method, an experimental configuration is set up according to Fig. 1. The customized high-order retarders ${{R}_{1}}$ and ${{R}_{2}}$ are made of quartz, and the thickness are nominal 5 mm and 10 mm. ${{R}_{1}}$ is mounted on a rotator (DDR100/M, Thorlabs, Inc., Newton, NJ, USA). A linear polarizer (10LP-VIS-B, Newport Corp., Irvine, CA, USA) is mounted after ${{R}_{2}}$. The light source is a warm white LED (MWWHLP1, Thorlabs), and its output is coupled into a fiber by a doublet lens, then coupled to another doublet lens to collimate the output beam. Finally, the modulated spectrum is obtained by a Czerny-Turner grating spectrometer (CCS100, Thorlabs). We combine a linear polarizer and an achromatic quarter-wave plate (AQWP, 10RP54-1B, Newport) to introduce the light at different polarized states.

The actual experimental setup is shown in Fig. 4. Firstly, we remove the AQWP and fix the transmission axis of ${{P}_{1}}$ at 0°, rotate ${{R}_{1}}$ to -45°, 0°, and 45°. Secondly, we fix ${{R}_{1}}$ at 0° to constitute a typical CSP system and rotate ${{P}_{1}}$ to 22.5°. After these two steps, all the data for calibrating CSP are acquired.

 

Fig. 4. Photograph of the experimental setup

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Figure 5 provides an illustration of both the LED spectrum and the modulated spectrum when the input is linearly polarized at 0°. However, it is evident that the contrast has been reduced due to the presence of SMTF, where the theoretical expectation that the peaks should align with the LED spectrum and the valleys should reach zero. It is worth noting that the acquisition of the LED spectrum occurs when the polarization components ${{P}_{1}}$, ${{R}_{1}}$, ${{R}_{2}}$, and P are aligned to 0°, to eliminate the effect of the transmittance of these polarization components.

 

Fig. 5. LED spectrum and the modulated spectrum while the input is 0° linearly polarized

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5.2 SMTF compensation

After calibration, the SMTF values at different channels are shown in Fig. 6. In our experimental data, in the OPD domain, only a few data points in the channel bandpass have the value relatively greater than zero. Therefore, the SMTF calculation results will distort at near-zero data points due to program processing. In Fig. 6, we eliminate them and choose the central points of each channel to illustrate the SMTF. It is noteworthy that in the SMTF compensation process, the whole bandpass SMTF value is used, rather than only the central points. We choose the several points at the center only to better show the tendency and value of the SMTF. Furthermore, the SMTF values in each channel appear to remain constant. This is attributed to the use of an LED spectrum with a narrow OPD bandwidth, which leads to tiny variations in the SMTF at each channel. In the proposed method, the obtained SMTF bandwidth aligns with the bandwidth of the source to be measured. Consequently, if the light source is changed, it is imperative to conduct a re-calibration.

 

Fig. 6. Measured SMTF in different channels.

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As shown in Fig. 6, the SMTF in different channels are red-marked. In ${{C}_{3}}$ channel, the SMTF is approximately 0.85, resulting in a reduction about 15% in the channel amplitude, which will lead to reconstruction errors beyond remedy.

While the incident light is 22.5° linearly polarized, Fig. 7 compares the measured data and the high-frequency channel amplitudes corrected data. The corrected data is the multiplication of the measured data and the reciprocal of the SMTF values. As mentioned in section 3, the SMTF will reduce the amplitudes of the OPD domain channels and lower the modulated spectrum's contrast in the wavenumber domain. Correspondingly, by doing the Fourier transform of the corrected data, the spectrum in the wavenumber domain is shown in Fig. 8. The calculated spectrum contrast is enhanced significantly, which corresponds to the nature of SMTF mentioned above.

 

Fig. 7. Absolute value of the autocorrelation of the modulated spectrum.

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Fig. 8. Comparison between the measured and contrast-enhanced spectrum when input is 22.5° linearly polarized light.

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5.3 Reconstruction results

We input the linearly and elliptically polarized light, respectively, to test the efficiency of the proposed method. First, we remove the AQWP and mount the linear polarizer on the rotator, rotating it from 0° to 180° in 15° steps. Second, we fix the polarizer's transmission axis at 0° and set the AQWP on the rotator, rotating like the first step.

For instance, the reconstructed spectra of 60° linearly polarized light are shown in Fig. 9. We use the theoretical values to be the reference truth and compare the results of the traditional method (mentioned in section 2.2), the angle errors calibration method, and the proposed method. In this paper, the angle errors calibration method denotes the method mentioned in section 4.2, but without the SMTF compensation in [31].

 

Fig. 9. Reconstructed results of different methods while input is 60° linearly polarized light

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Considering that the theoretical value of ${S_3}$ component of the linearly polarized light is zero, all the results seem close to the reference truth. The reconstruction results of ${S_1}/{S_0}$ and ${S_2}/{S_0}$ demonstrate the improvement of the proposed method.

It is noteworthy that much existing literature did not mention the SMTF effects when establishing the theoretical model, causing misunderstandings and leading to such phenomenon: the angle errors calibration method seems more precise than the traditional method (since alignment error is considered) and should yield better results, but the experiment results often indicate otherwise. Although the SMTF effect is not clearly mentioned in the traditional method as well, the reconstruction method inherently eliminates them since SMTF are both the denominator and the numerator in Eqs. (13)-(15), this is where the traditional method is clever. Therefore, the two chosen methods can serve as suitable control groups, since only errors or SMTF is taken into account, while the proposed method actually considers them both.

To verify that it is SMTF that makes the angle errors calibration method shows such enormous error, we have done simulations. The simulation condition is the same as the experimental setup. The angle errors of ${{R}_{1}}$ and ${{R}_{2}}$ are set as ${\varepsilon _1} = 0.3^\circ $ and ${\varepsilon _2} = 0.5^\circ $, and the reconstruction results with or without SMTF effects are compared in Fig. 10. It is noteworthy that, since some specific parameters of our spectrometer are unknown, the SMTF in the simulation is not the actual SMTF in the experiment.

 

Fig. 10. Simulation#results of angle errors calibration method while input is 60° linearly polarized light

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From Fig. 10, the reconstructed Stokes spectra without SMTF effects are highly consistent with the reference truth. This verified that the theoretical model of the angle errors calibration method is accurate, it just did not take SMTF into account. Therefore, when combined with SMTF effects, there are enormous errors just like Fig. 9 shows.

Therefore, we considered that the SMTF compensation step is primary if we need to calibrate the retarders’ errors in follow-up steps. An easy way to reduce the influence of SMTF is to employ thinner retarders. From our experimental results, the SMTF of the employed spectrometer is about 0.95 at the OPD introduced by 5 mm quartz crystal retarder. Therefore, employing thinner retarders when the corresponding SMTF is higher enough to ignore this influence can be an alternative, such as 125 µm and 250 µm mentioned in [36]. However, meanwhile, the channel width will decrease and cause severe crosstalk, the gains and losses need to be evaluated.

To quantify the reconstruction errors, we defined root-mean-square error (RMSE) to evaluate the proximity of the reconstruction results and the reference truth.

As Eq. (48) shows, N is the number of data points, and the footmark R and RT denote the reconstruction result and the reference truth, respectively.

RMSEs of the ${S_1}/{S_0}$ and ${S_2}/{S_0}$ at various input linearly polarized states are shown in Fig. 11. The RMSE of the proposed method is less than 0.01, which shows its efficiency. Further, we found that the results of 15° and 30° linearly polarized light have the lowest RMSE (the proposed method achieved about $2 \times {10^{ - 4}}$) and have the minimum difference among the methods we employed. Therefore, we considered that the Fourier demodulation methods might perform better while the azimuth of the input linearly polarized light is in this range.

 

Fig. 11. RMSEs of the reconstruction results of different methods at various input linearly polarized states.

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Surprisingly, many existing papers in the field of Channeled Spectropolarimetry [30–39] demonstrate only the experimental reconstruction result of linearly polarized light, while the elliptically polarized light’s counterpart lacks. There are difficulties to generate an ideal broad-band elliptically polarized light since the alignment error and the uneven retardance of AQWP. Therefore, it is difficult to quantify the accuracy of the reconstructed Stokes spectra as linearly polarized light does, since the ground truth of the input is unknown, which may be the reason. Even so, we consider it necessary to supplement the experiment while the input is elliptically polarized. Since the ${S_3}$ component is zero in linearly polarized light, and thus all the results are close to the zero, it is insufficient to verify the effectiveness in reconstructing ${S_3}$. In the elliptically polarized light experiment, the reference truth is not calculated by theory but experimentally measured by rotating retarder spectropolarimeter (namely PSA) after calibrating by the eigenvalue calibration method [46]. We fixed the transmission axis of the linear polarizer at 0° and varied the fast axis of AQWP, namely, to constitute a polarization state generator (PSG). The reconstruction results are shown in Fig. 12.

 

Fig. 12. Reconstructed results of different methods while the fast axis of AQWP is fixed at 60°

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However, while the input light is elliptically polarized, the reconstruction result does not show the same accuracy as the linearly polarized light does. Considering that the ${S_3}$ component is non-zero while the input is elliptically polarized light, from Eq. (44), the effect of the alignment errors will become more prominent, which might be part of the reason. To improve the accuracy, we introduced another reference light which is 45° linearly polarized, and further calibrated the system as Eqs. (42)-(45), to tolerate the effect of the alignment errors [44]. And also, considering that ${S_2}$ and ${S_3}$ are extracted as a whole channel in all the employed methods, there is crosstalk noise between symmetric channels [42] (the real and imaginary parts), where in linearly polarized light conditions, ${S_3}$ is zero, may avoid this kind of crosstalk. It can be alleviated by the coherence demodulation method [42], but the solution for the Fourier demodulation method still needs discussion.

Correspondingly, we give the RMSEs in Fig. 13. Due to the incident light being linearly polarized while the azimuth of AQWP is 0°, 90°, and 180°, we eliminate them in Fig. 13. Although the accuracy does not reach the same level as Fig. 11 shows, the accuracy improvement brought by the proposed method is still obvious.

 

Fig. 13. RMSEs of the reconstruction results of different methods at various elliptically polarized states.

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5.4 Discussion of nonlinearity compensation

In particular, this section briefly analyzes the nonlinearity effects and proposes a compensation method. The nonlinear phase of the retarder also suppresses the sideband amplitude, but the mechanism is different from the SMTF. Nonlinearity broadens the bandwidth and thus suppresses the sideband, then affects the reconstruction results by crosstalk. However, as shown in Fig. 6, the signal bandwidth is relatively small compared to the channel width. Therefore, even though nonlinearity causes blurring at the channel edge, the edge point is near zero, thus preventing severe crosstalk. That is the reason why nonlinearity is generally overlooked in CSP.

Inspired by [43], we convolute the OPD spectrum and the inverse Fourier transform of the phase factor, to shift the sideband to the baseband, and thus eliminate the whole phase factor to compensate for the nonlinearity. From our experimental results, with or without nonlinearity compensation makes the reconstructed Stokes spectra almost no difference. But it does offer an average improvement of about 0.001 in RMSE of ${S_3}/{S_0}$. RMSEs of ${S_1}/{S_0}$ and ${S_2}/{S_0}$ are almost the same. The improvement may be slight compared to the complexity of the data processing. Our experiment validates that nonlinearity compensation improves accuracy, despite being somewhat slight. Therefore, whether to adopt nonlinearity compensation requires evaluation of the experimental conditions, such as the nonlinear dispersion and thickness of the retarders, spectral range, light source bandwidth, etc.

6. Conclusion and discussion

This paper theoretically proposes and experimentally verifies an efficient calibration method for spectrometer response and systematic error in CSP. We derive the theory of the SMTF of a typical Czerny-Turner spectrometer and analyze its effect in the wavenumber and the OPD domain. The proposed method actually contains two steps and an optional step. Firstly, we derived a method to estimate the SMTF in the OPD domain and prove that the process is immune to the errors of retarders. By multiplying the reciprocal of SMTF values in the OPD domain and correcting the high-frequency channel amplitudes, we compensate for the SMTF. Secondly, we employed a reference beam to calibrate the angle errors of retarders. Finally, the nonlinearity can be compensated for through channel shifting.

Compared with other current methods, the proposed method shows high efficiency and stability under input linearly and elliptically polarized light conditions. It is noteworthy that, in our experiment, the reconstruction results of elliptically polarized light show lower accuracy compared to linearly polarized light. Therefore, it is worth further study, from our investigation, factors such as the uneven retardance of AQWP, and crosstalk between symmetric channels (real and imaginary channels) may affect the accuracy.

Moreover, the proposed method can be simplified based on different experimental conditions. For instance, if the retarders are very thin, it is possible to skip the SMTF compensation step; if the nonlinearity has little effect, it is unnecessary to compensate for it. However, the gains and losses associated with the simplification need to be evaluated dynamically.

The proposed method achieves in-situ SMTF and error calibration without requiring additional instruments or polarimetric components. Therefore, we believe it is feasible for most experimental conditions and will help improve the accuracy, reliability, and practicability of Stokes CSP.