## Abstract

Magneto-optic (MO) based Mueller matrix polarimetry (MMP) has several advantages of compact size, no-mechanical movement and high speed. Inaccuracies of components in the polarization state generator (PSG) optical parameters will influence the measurement accuracy of MMP. In this paper, we present a PSG self-calibration method in the compact MMP based on binary MO polarization rotators. Since PSG can generate enough numbers of non-degenerate polarization states, the optical parameters in PSG and the Mueller matrix of the sample can totally be numerically solved, which realizes a self-calibration in the PSG. Combining the previous self-calibration method in polarization state analyzer (PSA), we realize a complete self-calibration compact MO based MMP. Based on the numerical simulation results, the errors of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying the PSG self-calibration method. In experimental results of a variable retarder as a sample, the Euclidean distance of retardance between the measurement and reference curves comparing PSG self-calibration with no PSG self-calibration can be reduced from 0.035 rad to 0.033 rad and the Euclidean distance of optical axis can be reduced from 3.39° to 1.51°. Compared with the experimental results, the numerical simulation results more accurately verify the performance of the presented PSG self-calibration method without being influenced by other errors because the Mueller matrix of the sample is known and the error source only comes from these components in PSG.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Mueller matrix polarimetry (MMP) can reflect all the polarization information of the sample. It has many unique advantages such as a non-contact and in-situ technique for detecting tissue microstructure, which is currently one of the most widely used methods [1–4]. By decomposing Mueller matrix, the birefringence, attenuation and depolarization information of the sample can be obtained [5–7]. This makes polarization measurement great significance in biomedical applications [8,9], remote sensing [10], astronomy [11], plasmonic nanoscience [12], microstructure measurement [13] and other fields. However, most of these current high-precision MMPs are desktop devices. They require relatively large free space and complex optical components [14,15], which are not suitable for in-situ measurement application such as optical manufacturing.

A compact MMP can realize an in-situ measurement for samples without sacrificing too much accuracy, which has characteristics of high degree of freedom, low cost and easy operation. To deal with the problem like suppression of perfusion in the diagnosis of skin diseases, Fricke et al. developed a compact non-contact dermatoscope [16]. Manhas et al. presented a novel technique through an optical fiber, opening the way for endoscopic applications of Mueller polarimetry for biomedical diagnosis [1]. López-Morales et al. presented a compact complete Stokes polarimetric imager that can work for multiple wavelength bands with a frame-rate suitable for real-time applications [17]. We find that there are few reports about compact MMP. Magneto-optic (MO) crystals for compact in-situ polarimeters are raised concern for in-situ measurement as a result of their compact size, no-mechanical movement and high speed [18–20]. We presented a MMP for optical fiber polarization measurement using binary polarization rotators consisting of magneto-optic (MO) crystals based polarization state generator (PSG) and polarization state analyzer (PSA) [21–23]. This binary PSG is capable of generating five or six particular polarization states on the Poincaré Sphere with a repeatability of better than 0.1° using binary MO polarization rotators [22]. Based on the similar principle, we also designed a compact in-situ binary polarization rotators based MMP for on-machine bulk optical manufacture monitoring [15]. In these previous MMPs based on MO rotators, PSG and PSA contain optical components including MO rotators, quarter wave plate (QWP) and polarizer.

Inaccuracies in PSG and PSA caused by manufacture imperfections, temperature and wavelength dependences of these components will produce polarization measurement errors for MMP. The calibration for MMP is a necessary process to ensure the measurement accuracy of MMP. Several methods have been attempted to realize a calibration for MMP including eigenvalue calibration method [24], model calibration method [25] and beam drift calibration method [26], etc. However, these previous calibration methods need the use of previously characterized samples or the rotation of these samples. On one hand, these samples and operations for calibration will add error for MMP measurement. On the other hand, this calibration is not effective when the optical parameters in MMP is time-varying due to environmental disturbances such as temperature and vibration and so on [27]. Whereas, the self-calibration can solve this issue, which can realize to complete calibration and measurement process at the same time. Namely, dynamical calibration is applied for each measurement. In addition, self-calibration does not need extra operation for sample and previously characterized samples. Some self-calibration methods have been applied to MMP. For example, Shibata et al. presented a three-shot algorithm, which realize the retardance of the rotating retarder in the polarization imaging system be calibrated dynamically, at the same time as the Stokes parameter measurements [28]. We applied a PSA self-calibration method to reduce the inaccuracies of PSA effect on the measurement accuracy [23]. In this self-calibration method, the optical parameters of PSA can also be calculated by redundant non-degenerate logic states generated by PSA. However, the issue of inaccuracies in PSG impacting on the measurement accuracy has not been solved, which still could cause polarization measurement errors.

In this paper, we present a PSG self-calibration method in the compact Mueller-matrix polarimetry based on binary MO polarization rotators. Combining PSA self-calibration, we realize a complete self-calibration compact MO based MMP. In the presented PSG self-calibration method, we construct the equations including elements of sample’s Mueller matrix and the optical parameters in PSG as unknown variables. The PSG can generate enough number of non-degenerate polarization states, namely enough number of equations. The optical parameters in PSG and the Mueller matrix of the sample can totally be calculated by numerically solving these equations. We apply a numerical simulation using an ideal waveplate as sample to verify the effectiveness of the PSG self-calibration method. Based on the numerical simulation results, the errors of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying the PSG self-calibration method. From experimental results of a variable retarder as a sample, the Euclidean distance of retardance between the measurement curve and reference curve comparing PSG self-calibration with no PSG self-calibration can reduce from 0.035 rad to 0.033 rad and the Euclidean distance of optical axis can reduce from 3.39° to 1.51°.

## 2. Principle

#### 2.1 Compact Mueller matrix polarimetry system

The configuration of MMP system based on binary polarization rotators has been illustrated in [15] shown in Fig. 1. Here we briefly introduce our MMP system. It consists of a laser source with a polarization-maintaining (PM) output, PSG, PSA, a photo-detector, a data acquisition (DAQ) board and a computer. The probe containing PSG and PSA has a small size [71 mm (L) × 28 mm (W) × 25 mm (H)], which is very suitable to in-situ measurements for optical machining. The laser, photo-detector are connected with the probe with optical fiber, which can be far away from optical machining equipment. The PSG and PSA module consists of six binary MO rotators, a quarter-wave plate and a polarizer. In our MMP system, we use the fast axis of QWP as the reference axis, namely *x*-axis. The orientation of the polarizer is set 90° to the *x*-axis. When a positive magnetic field above the saturation level is applied, the MO rotator rotates the polarization state by precisely +22.5°. When a negative magnetic field above the saturation level is applied, the MO rotator rotates the polarization state by precisely −22.5°. By applying different magnetic fields to the six MO rotators to rotate them at different angles, the PSG can generate six non-degenerate polarization states including 0°, ±45°, and 90° and right-hand circular (RHC) and left-hand circular (LHC) [22]. The polarized light passes through the optical beam splitter and then projects on sample by lens. The light reflected by the sample enters the PSA through the beam splitter. Here MO rotators based PSA has an opposite structure with MO rotators based PSG. The Stokes vector of the reflected light can be obtained by the light intensities detected by the photo-detector when we apply different voltages to the MO rotators in the PSA. By utilizing redundant non-degenerate logic states generated by PSA, a PSA self-calibration method can reduce the inaccuracies of optical components in PSA effect on the measurement accuracy of Stokes vector under test [23]. At least four non-degenerate polarization states must be generated by the PSG and analyzed by the PSA to completely determine the Mueller Matrix of the sample. Since PSG can generate six non-degenerate polarization states, these redundant non-degenerate states give us the opportunity to design a PSG self-calibration method to reduce the error of polarization states generated by PSG due to inaccuracies of optical components in PSG.

#### 2.2 Principle of PSG self-calibration

In the presented MMP shown in Fig. 1, changing the rotation combination of MO rotators in PSG can produce different polarization states. After the output Stokes vectors generated by PSG ${\boldsymbol S}_j^{PSG} = {(S_{0j}^{PSG}\textrm{ }S_{1j}^{PSG}\textrm{ }S_{2j}^{PSG}\textrm{ }S_{3j}^{PSG})^T}$ pass through the sample *M _{sample}*, the Stokes vectors before entering the PSA ${\boldsymbol S}_j^{PSA} = {(S_{0j}^{PSA}\textrm{ }S_{1j}^{PSA}\textrm{ }S_{2j}^{PSA}\textrm{ }S_{3j}^{PSA})^T}$ can be expressed as:

*j*= 1,2,…,6 is the number of polarization states generated by PSG. To design a PSG self-calibration method, we need to analyze the output Stokes vectors when polarized light passes through optical components in PSG, which have been deduced in [15]. The output Stokes vectors of PSG can be expressed as:

*μ*is the angle between polarizer and

*x*-axis,

*δ*is the retardance of the QWP,

*ξ*is the rotating angle of the first two MO rotators before the QWP, and

_{j}*φ*is the rotating angle of the last four MO rotators in PSG shown in Fig. 1. In our MMP system, we use the fast axis of QWP as the reference axis, namely

_{j}*x*-axis [22]. Submit Eq. (2) to (1):

*m*are elements of

_{ij}*M*. When a positive magnetic field above the saturation level is applied, the MO rotator rotates the polarization state by a precise angle

_{sample}*θ*. When a negative magnetic field above the saturation level is applied, the MO rotator rotates the polarization state by precisely −

*θ*. Therefore, when two MO rotators rotate the polarization state in the same direction, the net rotation is 2

*θ*. Conversely, if the two MO rotators rotate the polarization state in opposite directions, the net rotation is zero. Here

*θ*is about 22.5°. Here we assume that

*θ*of the first two MO rotators before QWP are identical, which is represented by

*θ*. We assume that

_{ξ}*θ*of the last four MO rotators after QWP are identical, which is represented by

*θ*. In Eq. (3), the output polarization states generated by PSG are determined by

_{φ}*ξ*and

_{j}*φ*, which are the combination of

_{j}*θ*and

_{ξ}*θ*. Here we use six non-degenerate polarization states including 0°, ±45°, and 90° and RHC and LHC corresponding to six different combinations of

_{φ}*ξ*and

_{j}*φ*as:

_{j}*μ*,

*δ*,

*θ*into Eq. (3) and (4) for calculation, where

*μ*=90°,

*δ*=1/4π,

*θ*=22.5°. From Eq. (3), at least 4 groups of Eq. (2) are required to get 4 groups of Stokes vectors. The PSG output matrix with ideal values of

*μ*,

*δ*,

*θ*can be expressed as:

*G*. Whereas, we still submit

_{PSG}*G*with ideal values to calculate

_{PSG}*M*based on Eq. (1), which will produce measurement errors of

_{sample}*M*. Since PSG can generate six non-degenerate polarization states, redundant non-degenerate states give us an opportunity to calculate

_{sample}*m*and these parameters of components in PSG including

_{ij}*μ*,

*δ*,

*θ*and

_{ξ}*θ*at the same time. We construct the equations including sixteen

_{φ}*m*and

_{ij}*μ*,

*δ*,

*θ*and

_{ξ}*θ*as 20 unknown variables and the Stokes vectors analyzed by the PSA as known quantities. Substitute Eq. (4) into (3) and rewrite Eq. (3) to build the equations

_{φ}*i*= 0,1,2,3 represents the number of rows of Stokes vectors and Mueller matrix. It is enough to solve 20 unknown variables in Eq. (6). We use the advance-retreat method [29] to numerically solve Eq. (6).

The flow chart of numerical solving Eq. (6) by the advance-retreat method is shown in Fig. 2. All unknown variables *m _{ij}*,

*μ*,

*θ*,

_{ξ}*θ*,

_{φ}*δ*in their respective cycles will finally satisfy this equation as:

*ε*is the difference between the value of Eq. (6)’s right side

*f*and the actual measurement of Stokes vectors $S_{ij}^{PSA}$. In the algorithm, we firstly set the initial values of unknown variables. Here $m_{ij}^0\textrm{ = }0$, ${\mu ^0}\textrm{ = }0^\circ ,\theta _\xi ^0\textrm{ = }22.5^\circ ,\theta _\varphi ^0\textrm{ = }22.5^\circ ,{\delta ^0}\textrm{ = }\pi \textrm{/}2$, where superscript 0 represents the initial value before the algorithm enters the cycles. Then we set initial step size

_{ij}*h*= 1. Secondly, we submit all the initial values of $m_{ij}^0,{\mu ^0},\theta _\xi ^0,\theta _\varphi ^0,{\delta ^0}$ into Eq. (6) to calculate ${f_{ij}}^0$. And then we calculate the initial value of ${\varepsilon ^0} = \{ \sum\nolimits_i {[{{\sum\nolimits_j {(S_{ij}^{PSA} - f_{ij}^0)} }^2}]} \}$. Then we let $\varepsilon={\varepsilon ^0}$.

*ε*is a variable that changes continuously with the cycle process, which is corresponding to Eq. (7). Then we choose the first unknown variable and start algorithm. In the algorithm, we construct four sets of cycles. Here we use unknown variable

*m*

_{00}as an example to show how to numerically search the value of

*m*

_{00}. In Cycle 1, we firstly let $m_{00}^1\textrm{ = }m_{00}^0 + {( - 1)^r} \cdot h$, where

*r*= 0,1,2,… and (−1)

*represents the stepping direction. The superscript 1 represents that the algorithm enters the first loop of Cycle 1. For the*

^{r}*n-*th Cycle 1, we use $m_{00}^{n + 1}\textrm{ = }m_{00}^n + {( - 1)^r} \cdot h$, where

*n*= 0,1,2,… represents the current number of Cycle 1, Then we put $m_{00}^{n + 1}$ and other unknown variables into Eq. (6) and calculate ${\varepsilon ^{n + 1}} = \{ \sum\nolimits_i {[{{\sum\nolimits_j {(S_{ij}^{PSA} - f_{ij}^{n + 1})} }^2}]} \}$. Then we compare the value of

*ε*

^{n}^{+1}and

*ε*. If

*ε*

^{n}^{+1}<

*ε*, we let

*ε = ε*

^{n}^{+1}and continue Cycle 1. If

*ε*

^{n}^{+1}>

*ε*, the algorithm will go to Cycle 2. In Cycle 2, we let $m_{00}^{n + 1}\textrm{ = }m_{00}^n - {( - 1)^r} \cdot h$,

*h = h*/ 2,

*r*=

*r*+ 1 and

*r*represents the number of Cycle 2. Then we compare the value of

*h*and its lower limit

*h*

_{min}. If

*h > h*

_{min}, the algorithm will go back to Cycle 1. If

*h < h*

_{min}, we finish searching the unknown variable

*m*

_{00}and the algorithm go to Cycle 3. In Cycle 3, the algorithm will change the unknown variable. In our example, we change the unknown variable from

*m*

_{00}to

*m*

_{01}. After re-initializing

*h*,

*r*and

*n*, the algorithm will go back to Cycle 1. The number of Cycle 3 is corresponding to the number of the unknown variables. When we finish to search all the unknown variables, the algorithm will go to Cycle 4. Here we consider Cycle 1 to 3 as a global cycle as Cycle 4. The number of Cycle 4 is represented by

*C*. In Cycle 4, since we already obtained the final value of

*ε*after the first Cycle 1 to 3, we compare the value of

*ε*and

*ε*

_{min}, where

*ε*

_{min}is a setting value that represents minimal difference between the numerical calculation

*f*and the actual measurement $S_{ij}^{PSA}$ of Stokes vectors. If

_{ij}*ε*>

*ε*

_{min}, we use all numerical solutions after Cycle 1 to 3 as unknown variables’ new initial values and initializing

*h*,

*r*and

*n*, then the algorithm will go back to Cycle 1. When

*ε*<

*ε*

_{min}, the algorithm stops and all final numerical solutions for unknown variables are obtained. In fact, if

*ε*

_{min}is set too small, the terminal condition of Cycle 4

*ε*<

*ε*

_{min}is difficult to be satisfied. Therefore, we need to set an upper limit of number of Cycle 4

*C*

_{max}to prevent the occurrence of infinite Cycle 4. If

*C*>

*C*

_{max}, the algorithm also stops. The setting of

*C*

_{max}is need to balance the precision of numerical solutions and the efficiency of the entire algorithm. If

*C*

_{max}is too small, the precision of numerical solutions will be deteriorated because the algorithm will be stopped without satisfying the terminal condition of

*ε*<

*ε*

_{min}. If

*C*

_{max}is too large, the algorithm will be got caught in an invalid cycle and the efficiency of algorithm will be decreased.

Here we will discuss the value of *h*_{min} and *ε*_{min}. Based on Eq. (7), the precision of the numerical solutions depends on the *ε*_{min}. In the experiment, the precision of *ε*_{min} is depended on the precision of Stokes vector analyzed by PSA *P*_{min}. Based on Eq. (7), *ε*_{min} is the square of *P*_{min}. *h* is the step value of the unknown variable in each numerical search process, which determines the calculation precision of every unknown variable. Here we set *h*_{min} to be less than *P*_{min} an order of magnitude, which can make *h*_{min} consistent with *P*_{min} in the final step when searching for each unknown parameter. For example, if *P*_{min} is 10^{−4}, we set *ε*_{min} = 1×10^{−8} and *h*_{min} = 1×10^{−5}. Under this circumstance, the final precision digit of *h* in Cycle 2 is 10^{−4}, which is consistent with *P*_{min}. In the numerical simulation, we found that *ε* < *ε*_{min} can be satisfied after 2∼5 cycles. Therefore, to ensure the integrity of algorithm, we set *C*_{max} = 10.

## 3. Numerical simulation and Experiment

#### 3.1 Numerical simulation

We firstly verify the PSG self-calibration method by a numerical simulation. In the numerical simulation, the Mueller matrix of the sample is known and the error source only comes from these components in PSG, which can accurately verify the performance of the presented PSG self-calibration method without influencing by other errors. We choose a normal wave-plate as a sample under test and set a certain error for parameters of components in PSG. The Mueller matrix of an ideal waveplate as a sample is

*ϕ*and

*σ*are the phase retardance and the optical axis of the sample, respectively. Here we calculate the average Euclidean distance $d$ between each measurement value and setting ideal value as the evaluation condition and

*d*can be expressed as: where

*n*is the number of measurement. (

*x*

_{n,}

*y*

_{n}) is the coordinate of measurement value and (

*x*

_{0n,}

*y*

_{0n}) is the coordinate of reference value namely the truth value. In the numerical simulation, we set the precision of the PSA can achieve the measurement of Stokes vectors at 10

^{−4}accuracy. Therefore, When

*P*

_{min }= 10

^{−4}, we set

*ε*

_{min}= 1×10

^{−8}and

*h*

_{min}= 1×10

^{−5}for numerical search. The algorithm shown in Fig. 2 needs about 0.2 seconds to complete numerical solution for Eq. (6) to acquire Mueller matrix of the sample. The CPU we used is Intel Core i7-8750H CPU @ 2.20GHz, RAM is 16GB.

The output Stokes vectors of PSG are determined by *μ, $\xi_j$*, *φ _{j}*,

*δ*based on Eq. (2). Here we use

*μ*as an example. In order to analysis the influence of each parameter error in the measurement of the sample’s phase retardance and optical axis, we set

*μ*=

*μ*

_{0}+ Δ

*μ*, where

*μ*

_{0}represents to its ideal value and

*μ*

_{0}= 90°, Δ

*μ*represents to the error we set and Δ

*μ*= -10°, -9°,…, 0°,…,10°. We compare the measured phase retardance and optical axis of the sample with and without using the PSG self-calibration under the influence of different Δ

*μ*. Figure 3(a1) to (a3) show the simulation results of the phase retardance from 0∼

*π*when Δ

*μ*= 0°, 5°, 10°. The greater Δ

*μ*will cause the greater measurement error of phase retardance and optical axis. We keep two significant digits after the decimal point for the simulation results here. When Δ

*μ*= 0°, no error occurs. When Δ

*μ*= 10°,

*d*is 0.090 rad without the PSG self-calibration. After the PSG self-calibration

*d*is only 6.6×10

^{−6}rad. Figure 3(b1) to (b3) show the simulation results of the optical axis from 0°∼180° when Δ

*μ*= 0°, 5°, 10°. When Δ

*μ*= 10°,

*d*is 4.22° without the PSG self-calibration. After the PSG self-calibration,

*d*is only 0.00046°. In order to observe differences between the calibrated and non-calibrated simulation results intuitively, we show the deviations of retardance and optical axis between with and without self-calibration.

Similarly, μ, *$\xi_j$*, *φ _{j}*,

*δ*will cause different errors from the simulation results. Figure 4(a1) to (a3) show the simulation results of the phase retardance from 0 ∼

*π*when Δ

*δ*= 0 rad, 0.087 rad, 0.17 rad. when Δ

*δ*= 0 rad, no error occurs. The results with and without PSG self-calibration are comparable. When Δ

*δ*=0.17 rad,

*d*is 0.082 rad without the PSG self-calibration. After PSG self-calibration

*d*is only 7.33×10

^{−6}rad. Figure 4(b1) to (b3) show the simulation result of the optical axis from 0°∼180° when Δ

*δ*= 0 rad, 0.087 rad, 0.17 rad. When Δ

*δ*= 0.17 rad,

*d*is 1.52° without the PSG self-calibration. After the PSG self-calibration,

*d*is only 0.00042°.

Figure 5(a1) to (a3) show the simulation results of the phase retardance from 0 ∼ *π* when Δ*θ _{ξ}* = 0°, 2.5°, 5°. When Δ

*θ*= 0°, no error occurs. The results with and without PSG self-calibration are comparable. When Δ

_{ξ}*θ*= 5°,

_{ξ}*d*is 0.071rad without the PSG self-calibration. After the PSG self-calibration

*d*is only 6.63×10

^{−6}rad. Figure 5(b1) to (b3) show the simulation result of the optical axis from 0°∼180° when Δ

*θ*= 0°, 2.5°, 5°. When Δ

_{ξ}*θ*= 5°,

_{ξ}*d*is 3.36° without the PSG self-calibration. After the PSG self-calibration,

*d*is only 0.00042°.

Figure 6(a1) to (a3) show the simulation results of the phase retardance from 0 ∼ *π* when Δ*θ _{φ}* = 0°, 2.5°, 5°. When Δ

*θ*= 0°, no error occurs. The results with and without PSG self-calibration are comparable. When Δ

_{φ}*θ*= 5°,

_{φ}*d*is 0.12 rad without PSG self-calibration. After the PSG self-calibration

*d*is only 1.92×10

^{−6}rad. Figure 6(b1) to (b3) show the simulation result of the optical axis from 0°∼180° when Δ

*θ*= 0°, 2.5°, 5°. When Δ

_{φ}*θ*= 5°,

_{φ}*d*is 4.73° without the PSG self-calibration. After the PSG self-calibration

*d*is only 0.00063°. For above, based on the simulation results, the error of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying the PSG self-calibration method.

#### 3.2 Experiment results and discussion

In the experiment, we apply an adjustable liquid crystal variable retarder (LCVR, LCC1223-C from Thorlabs Inc.) to evaluate the performance of the presented complete self-calibration MMP. The voltage range applied on LCVR is 0∼25 Volts. We still measure the phase retardance of the LCVR by detecting reflected light from the mirror and the light transmits the LCVR twice. We measure every 0.1 Volts in the range of 0-1 Volts, every 0.02 Volts in the range of 1 to 3 Volts, every 0.1 Volts in the range of 3 to 10 Volts and every 0.2 Volts in the range of 10 to 25 Volts. A total of 256 phase retardance points is measured, which is consistent with the manufacturer’s measurement range. We use Benchtop LC Controller (LCC25 from Thorlabs Inc.) to realize voltage control. To reduce the random noises, we repeat the measurement of each measurement point 10 times and take average of 10 measurements as final measured results. In the experiment, due to the addition of random noises, the final choice of *ε* _{min} in the algorithm depends on the measurement precision of PSA. The PSA used can achieve the measurement of Stokes vectors at the measurement precision of 10^{−2}∼10^{−3} [23]. Therefore, When *P*_{min} = 10^{−2}, we set *ε* _{min} = 1×10^{−4} and *h*_{min} = 1×10^{−5} for numerical search. In addition, based on the experimental results, we find that a large amount of data satisfied *ε* <*ε* _{min} after 3∼7 cycles, so we set *C*_{max} = 10. The MMP with and without PSG self-calibration are applied with the same Stokes vectors data analyzed by PSA. In the MMP without PSG self-calibration, we still submit *G _{PSG}* with ideal values in Eq. (5) and these corresponding four Stokes vectors analyzed by PSA to calculate

*M*based on Eq. (1). In the MMP with the PSG self-calibration, six Stokes vectors analyzed by PSA are involved in calculation.

_{sample}We show the phase retardance curve with and without the PSG self-calibration and the manufacturer’s reference curve in Fig. 7(a). We calculate Euclidean distance *d* using the measured phase retardance with and without PSG calibration compared with the manufacturer’s reference values. The *d* is reduced from 0.350 rad to 0.326 rad after using PSG self-calibration. Some jumps on the curve are reduced after applying the PSG self-calibration shown in enlarged image of Fig. 7(a). The improvement is not obvious by the PSG self-calibration. The reason is that the output of PSG is closed to ideal values. The PSG output polarization state calculated by self-calibration is shown in Table 1. From Fig. 7(a), the deviation between the measured and reference phase retardance is not small even after the PSG self-calibration. There are several reasons as below: First, the light projected by the presented MMP is not completely vertical to the sample. Second, the phase retardance of LCVR is depend on the temperature. The room temperature in our experiment occurs deviation with that of the manufacturer’s test environment. Third, the phase retardance of LCVR will change along with the usage increments [30]. In addition, a large deviation occurs when the applied voltage is less than 1 Volt, which is the threshold voltage of LCVR. The reason is that when the driving voltage is below 1 Volt, the liquid crystal molecules in LCVR have not been polarized [15].

We also measure the optical axis of the LCVR. We keep the voltage on LCVR at a constant and change the optical axis direction of the LCVR by rotating the dial on LCVR. We measure 20 times optical axis of the LCVR at each interval of 10° with an overall range of 0° to 180°. The reference curve comes from the dial on LCVR. From the results shown in Fig. 7(b), the measured optical axis curve with the PSG self-calibration is much closer to the reference curve than the curve without the PSG calibration. We also calculate *d* using the measured optical axis with and without the PSG self-calibration compared with the reference values. The value of *d* is reduced from 3.39° to 1.51° after using the PSG self-calibration, which verifies that the presented PSG self-calibration method is effective. The other errors such as manual rotation accuracy will also impact the accuracy of the reference curve.

Stokes vectors generated by PSG can also be calibrated by PSA using a mirror as a sample. However, this standard PSG calibration method will add error that comes from the mirror for MMP measurement. In addition, this calibration is not effective when the optical parameters in MMP is time-varying due to environmental disturbances such as temperature and vibration and so on. To compare the presented PSG self-calibration method, we also show the experimental results of LCVR’s retardance and optical axis using this standard PSG calibration method. Four PSG Stokes vectors of ±45°, RHC and LHC measured by PSA using a mirror as a sample shown in Table 2. We submit *G _{PSG}* calibrated by mirror in Table 2 and these corresponding four Stokes vectors analyzed by PSA to calculate

*M*of LCVR based on Eq. (1). The measured retardance and optical axis of LCVR using the PSG calibration by mirror are shown in Fig. 7(a) and 7(b), respectively. The

_{sample}*d*with the PSG self- calibration is still lower than

*d*with the PSG calibration by mirror. The deviation between the PSG self-calibration and the PSG calibration by mirror is not large because the optical parameters in MMP are not varied significantly along with time. When the MMP is applied with environmental disturbances such as temperature and vibration, the advancement of the PSG self-calibration will be manifested.

Since the system error and random error occur in the experiment and the output of PSG are closed to ideal values, the advantage of the presented PSG self-calibration method is not obvious shown in the experiment results shown in Fig. 7. It is difficult to reflect the advantage of the presented PSG self-calibration method. Whereas, in the numerical simulation, the errors of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying PSG self-calibration method shown in Fig. 3 to 6. Compared with the experiment using LCVR, in the numerical simulation, the Mueller matrix of the sample is known and the error source only comes from these components in PSG. This can accurately verify the performance of the presented PSG self-calibration method without being influenced by other errors. Retardances of the LCVR are need to be calibrated by the other commercial MMP with high accuracy. Anyway, we will measure retardances of the LCVR by the other commercial MMP with high accuracy to evaluate the accuracy of the presented MO based MMP in the future.

In principle, the sample with arbitrary Mueller matrix can be measured based on Eq. (6), so these dichroic samples as polarizers can also be measured. We apply a numerical simulation when the sample is a polarizer. We assume an ideal linear polarizer as a sample under test. The Mueller matrix of the ideal polarizer *M _{p}* can be expressed as:

*θ*is the orientation angle of the polarizer. In order to verify the validity of this self-calibration for a polarizer as a sample, we add Δ

_{P}*μ*, Δ

*δ*, Δ

*θ*and Δ

_{ξ}*θ*four kinds of errors on PSG at the same time. Here Δ

_{φ}*μ*= 5° , Δ

*δ*= 0.087 rad, Δ

*θ*= 2.5° and Δ

_{ξ}*θ*= 2.5°. In the numerical simulation,

_{φ}*θ*is set from 0° ∼ 180°. Here we set

_{P}*θ*= 30° as an example. Table 3 shows our numerical simulation results of

_{P}*M*. Here reference matrix is the true value of

_{p}*M*. We can find that the measured

_{p}*M*with the PSG self-calibration is equal to the reference matrix. However, the measured

_{p}*M*without PSG self-calibration has deviations with the reference matrix. Other measured

_{p}*M*under different

_{p}*θ*has a similar result. The numerical simulation results verify this PSG self-calibration is valid for a polarizer as a sample. In the future, we will verify the validity of this self-calibration for a polarizer as a sample in the experiment.

_{P}## 4. Conclusion

We present a PSG self-calibration method in the compact MMP based on binary MO polarization rotators. Since PSG can generate enough number of non-degenerate polarization states, the optical parameters in PSG and the Mueller matrix of the sample can totally be calculated by numerically solving these equations, which realize a self-calibration process in PSG. Based on the simulation results, the errors of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying PSG self-calibration method. From experiment, the Euclidean distance of retardance between the measurement curve and reference curve comparing PSG self-calibration with no PSG self-calibration can reduce from 0.035 rad to 0.033 rad and the Euclidean distance of optical axis can reduce from 3.39° to 1.51°. Compared with the experimental results, the numerical simulation results more accurately verify the performance of the presented PSG self-calibration method without being influenced by other errors because the Mueller matrix of the sample is known and the error source only comes from these components in PSG. Combining previous PSA self-calibration method, we realize a complete self-calibration compact MMP.

## Funding

National Natural Science Foundation of China (61975147, 61635008, 61735011, 61505138); Key Technologies Research and Development Program (2019YFC0120701); Special Technical Support Project of China Market Supervision and Administration (2021YJ027).

## 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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