Channeled Mueller matrix spectropolarimeters (CMMSPs) have gained increasing popularity in recent years due to no moving parts. However, in order to obtain more accurate measurements, thorough studies on the influence and correction of their systematic errors are still needed. This paper presents a novel perspective for CMMSPs based on a signal processing technique, and propose a coherence demodulation method to extract channel signals in the modulated intensity. From theoretical analysis, the influence of phase deviation resulting from the imperfection of retarders is pinpointed. Meanwhile, the mechanism of phase deviation is described in theory and visually displayed by simulation. To mitigate the interference of retarder phase deviation, this work proposes a way for correction utilizing a vacuum and polarizer as determinant samples. Noticeably, the phase deviations are treated as a whole and represented by polynomials during correction. The reverse process of error mechanism is used to correct the influence. Finally, this means is proved by a series of simulation validations with a detector noise of 30 dB and retarder misalignment errors of 0.5°.
1. Introduction
The polarization properties of a medium have proven to be fundamental importance in many applications such as biomedical diagnosis [1,2] or material classification [3–7]. However, the imperfection of polarizing optics can introduce systematic errors into the resulting Mueller matrices. And the recent studies have indicated that resultant Mueller parameters are highly sensitive to these errors. For better measurement accuracy, it is important to understand how these defects affect measurement results and how to correct them.
In the past decades, the analysis and calibration of systematic errors in polarimeters have received a lot of attention. Hauge developed a method that can calibrate diattenuation errors in retarders to correct systematic errors in the dual-rotating-retarder Mueller matrix polarimeter (RRMP) [8]. Chipman proposed a system error model related to misalignment of the polarization elements and the thickness deviations of the retarders in RRMP [9]. Broch researched deeply the systematic errors at the second order and proposed a new calibration method based on four-zone averaging measurement [10]. Cheng proposed a calibration algorithm with reduced errors in magnification to improve the robustness of the previous calibration process [11]. The mechanism and calibration of misalignment errors of retarders and polarizers in polarimeters were systematically studied by Dai et al. [12–14]. Gu and his group tried to improve the calibration accuracy of six-channel Stokes polarimeter by characterization of beam splitters [15]. However, most research works reported in the past are not directed at CMMSPs. In fact, since it was proposed by Dubreuil and Hagen [16,17], CMMSP has been playing an increasingly important role because of faster measurement speed and no moving parts. Although Dubreuil et al. have made a very good progress in the research of systematic errors specific to snapshot Mueller matrix polarimeter [18], there are still many issues that have not been completely and thoroughly addressed, such as the influence and precise characteristic of phase deviations of all four retarders in CMMSPs, especially the one resulted from dispersion. Dispersion may be a major factor limiting broadband CMMSPs.
To clarify the effect mechanism of phase deviations of retarders in CMMSPs, a channeled polarimetry theory based on signal processing technique is proposed. And the method of coherence demodulation is introduced to extract the modulated channel signals in Section 2. In Section 3, the influence and characteristic of the phase deviations of retarders are investigated. A series of simulations are carried out to prove the performance of the proposed method. Finally, the conclusion is made in Section 4.
2. Channeled polarimetry theory
Although the theory of generalized channeled polarimetry has been described by Alenin [19], we still consider that it is necessary to systematically expound it with the aid of signal processing theory because of ease of use in error investigation. In signal processing theory, a baseband signal, also named the initial signal, often needs to be modulated into a high frequency carrier signal to generate a modulation signal for transmission. Let x(t) be the baseband signal, and the carrier signal be a cosine signal denoted as cos(ω0t), so the modulated signal can be expressed by:
(1)$${s_m}(t )= x(t )\cdot \cos ({{\omega_0}t} )$$
If the Fourier transform of the baseband signal is recorded as X(ω), that of the modulated signal sm(t) can be represented by Eq. (2). The fact that a baseband signal is modulated by a sinusoidal signal means that its spectrum is shifted by ω0 in the frequency domain.
(2)$${S_m}(\omega )= \frac{1}{2}[{X({\omega + {\omega_0}} )+ X({\omega - {\omega_0}} )} ]$$
Usually, all spectroscopic elements of Mueller matrix of a sample overlap together along the frequency domain, so it is nearly impossible to measure them separately before all elements are separated in the frequency domain. It happens that sinusoidal modulation can help to achieve it. If these sixteen Mueller parameters mi or their linear combinations are modulated by sinusoidal signals with different carrier frequencies ωi, every Mueller element can be retrieved by signal demodulation. Assume that the modulated intensity is expressed as:
(3)$$I(\delta )= {a_0} + \sum\limits_{i = 1}^N {\{{{a_i}(\delta )\cos ({{\omega_i}\delta } )+ {b_i}(\delta )\textrm{sin}({{\omega_i}\delta } )} \}} $$
where N denotes the number of modulation frequencies, and δ represents the modulation domain, which may be the time, space or wavenumber domain. a0 is the constant term, while ai and bi are Mueller elements or their linear combinations and called the modulated channel signals. The Fourier transform of Eq. (3) is (4)$$I(\Delta )= {\alpha _0} + \frac{1}{2}\left\{ {\sum\limits_{i = 1}^N {[{{\alpha_i}({\Delta + {\omega_i}} )+ {\alpha_i}({\Delta - {\omega_i}} )- j{\beta_i}({\Delta + {\omega_i}} )+ j{\beta_i}({\Delta - {\omega_i}} )} ]} } \right\}$$
where αi and βi are the Fourier transforms of ai and bi, respectively. j is the imaginary unit. Δ is the corresponding Fourier domain, which depends on the modulation domain and could be the temporal frequency, spatial frequency or optical path difference (OPD) domain. In fact, these three types of channeled Mueller matrix polarimeters have been implemented multiple times in the past. Temporal modulation has been the most used method in the past, and it is usually achieved by using rotating polarization devices or electrically tuned liquid-crystal retarders. Wavenumber modulation is realized by means of four thick retarders. Spatial modulation often is achieved by a set of Savart plates or polarization gratings [20,21]. If the carrier frequencies are different and the interval between two adjacent ones is not less than double the bandwidth of mi, all Mueller elements could be completely separated in the frequency domain. According to the Nyquist law, the sampling frequency of detector in the modulation domain must be not less than twice the highest frequency of the modulated intensity I(δ). In other words, the resolution of the intensity in the modulation domain determines whether the high frequency parts of Mueller elements overlap together. For CMMSPs, the spectral resolution decides the upper bound of the detectable information capacity, which is equal to the sum of the allotted bandwidths of all Mueller elements in OPD domain. Thus, the two important prerequisites for channel Mueller matrix polarimeters are a reasonable selection of carrier frequencies and a high enough sampling resolution. The reasonable carrier frequency settings for CMMSPs have been suggested by Lemaillet [22], and these modulation frequencies were distributed as evenly spaced as possible in OPD domain to guarantee the bandwidth of every channel signal. It is a good choice to evenly distribute the bandwidth to all channels before the deeper understanding of spectroscopic polarization feature for the target is achieved.In the past, extracting Mueller matrix by windowing a channel in the Fourier domain and performing an inverse Fourier Transform was almost the only used method of obtaining the channel signal. A new method based on coherent demodulation without Fourier transform is proposed to make it more convenient and efficient to investigate the influence and characteristic of phase deviations. Coherent demodulation is usually achieved by a multiplying modulator and a low-pass filter. For the channel signal ai modulated in the ith channel, a cosine demodulation signal cos(ωiδ) is multiplied by the modulated intensity signal I(δ). The resultant intensity could be expressed by Eq. (5).
(5)$$\begin{aligned} {{\hat{I}}_i}(\delta ) & = \frac{1}{2}{a_i}(\delta ) + {a_0}\cos ({\omega _i}\delta ) + \frac{1}{2}{a_i}(\delta )\cos (2{\omega _i}\delta ) + \frac{1}{2}{b_i}(\delta )\textrm{sin}(2{\omega _i}\delta )\\ & + \sum\limits_{k = 1}^N {\{ {a_k}(\delta )\cos ({\omega _k}\delta )\cos ({\omega _i}\delta ) + {b_k}(\delta )\textrm{sin}({\omega _k}\delta )\cos ({\omega _i}\delta )\} } \;,\;k \ne i \end{aligned}$$
It is found out that ai is the only low-frequency part and could be extracted by low-pass filtering. The suitable cutoff frequency of the low-pass filter should be half of the modulation frequency interval. Note that due to the inherent characteristics of the low-pass filter, the filtered channel signal is shifted backwards by about half of the filter order so that the end signal is slightly lost. By optimizing the filter order, the amount of loss is negligible. In addition, the strength of the extracted channel information needs to be doubled to restore its original strength. Similarly, bi could be obtained by multiplying a sine demodulation signal sin(ωiδ). Ideally, the relationship between the demodulated channel signals by coherence demodulation and the modulated channel signals are given by Eq. (6). Finally, a non-singular system of equations connecting channel signals with Mueller matrix is constructed to obtain all of spectrally resolved Mueller elements.
(6)$$\left\{ \begin{array}{l} {A_i} = \frac{1}{2}{a_i}{\kern 1pt} {\kern 1pt} \\ {B_i} = \frac{1}{2}{b_i} \end{array} \right.$$
where Ai and Bi are the demodulated channel signals by coherence demodulation.3. Retarder phase deviation in CMMSPs
In this section, the influence and characteristic of phase deviations in CMMSPs are represented. The configuration of the investigated CMMSP is shown in Fig. 1, which is proposed by Dubreuil [16]. The polarization modulating system is composed of four high-order retarders Ri (i=1, 2, 3, 4) with thickness di, and two polarizers P1 oriented at 90°, P2 oriented at 0°. The thickness ratio of these four retarders d1:d2:d3:d4 is 1:1:5:5, and their fast axes are at 45°, 0°, 0° and 45° in turn. In this case, the channel number N is 12, and all of modulated channel signals ai and bi in Eq. (3) are listed by Eq. (11) in Appendix. The corresponding carrier frequency ωi is given by Eq. (7). All channel signals could be extracted by coherence demodulation. In this situation, every channel of CMMSPs has a bandwidth of Δn0d1/2 in OPD domain that is called single-channel bandwidth. It can be considered as the upper limit of the bandwidth of the Mueller matrix under test.
(7)$${\omega _i} = i \cdot 2\pi \cdot \Delta n \cdot {d_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,2, \cdots 12$$
where Δn is the birefringence of retarders.3.1 Influence of retarder phase deviation
The phase deviation of retarder is so important that it is worth discussing its consequence. Limited to the accuracy of the crystal cut, manufacturing errors in thickness for retarders is inevitable. Then, the real thickness scheme of retarders becomes (d1+Δd1, d2+Δd2, d3+Δd3, d4+Δd4), for which Δdi is the thickness deviation of the retarder Ri. Besides, crystal dispersion is also an important factor which has to be taken into consideration. In light of this, the true phase Θi of the retarder Ri is given by Eq. (8) instead of ωiδ.
(8)$${\Theta _i} = 2\pi [{\Delta {n_0}{d_i} + \Delta {n_0}\Delta {d_i} + o(\delta )({{d_i} + \Delta {d_i}} )} ]\delta$$
where Δn0 is the constant term of birefringence, and o(δ) is its dispersion terms dependent on wave-number. In fact, the dispersion for birefringence can be expressed as a polynomial in a limited bandwidth. According to these features, we assume that the true phase of retarder Θi is a polynomial function of wave-number δ. In general, the magnitude of 2πΔn0diδ is treated as the nominal phase of retarder. Therefore, the phase deviation of the ith retarder θi could be denoted by Eq. (9). (9)$${\theta _i} = 2\pi [{\Delta {n_0}\Delta {d_i} + o(\delta )({{d_i} + \Delta {d_i}} )} ]\delta$$
This assumption allows that the phase deviation could be investigated as a whole, no matter what it is resulted from. It means that the phase deviation θi expressed by a polynomial can include not only the dispersion and thickness deviation of retarders, but also the deflection of the beam, stress birefringence, frequency deviation of the demodulation carrier, etc. This assumption holds at least when the influence of dispersion and thickness deviation dominates. One should note that the function form can vary with the dominant error source.
Ideally, the retarder thickness decides the location of every channel in OPD domain. However, the real position of central frequency of all channel components must not be the nominal one, and their shifts are determined by the first-order component of θi . The second-order and higher-order components result in non-linear phase modulation, which will cause the spectrum of the signal ai and bi to widen in OPD domain. From Eq. (9), the non-linear phase deviations are proportional to the thickness di+Δdi, which means the higher the central frequency of the channel, the greater spectrum broadening it suffers.
Moreover, a contradiction between dispersion and thickness of retarder must be faced during design process. It is that the retarder thickness di needs to be increased to provide a broader bandwidth in OPD domain because of the influence of dispersion, while the effect of dispersion also increases with thickness. The combination of birefringent crystals with positive and negative dispersion seems to be a nice way to suppress dispersion in high-frequency channels. For example, the retarders R1 and R3 is with positive dispersion, while the retarders R2 and R4 is with negative dispersion.
Employing coherent demodulation, the demodulated channel signals with phase deviations will be obtained. After derivation, it is found out that the demodulation error mainly comes from two aspects. On the one hand, the demodulated signal will be modulated by a sinusoidal wave, whose phase is a simple linear combination of phase deviations θi. On the other hand, the crosstalk signal from the respective symmetrical channel is included in the demodulated channel signal. The mathematically expression of this process is given by Eq. (10).
(10)$$\left\{ {\begin{array}{{c}} \begin{array}{l} A_i^{\prime} = filte{r_{low - pass}}\{{[{{a_i}\cos ({{\omega_i}\delta + \Delta {\Phi _i}} )+ {b_i}\sin ({{\omega_i}\delta + \Delta {\Phi _i}} )} ]\cdot \cos {\omega_i}\delta } \}\\ = \frac{1}{2}({{a_i}\cos \Delta {\Phi _i} + {b_i}\sin \Delta {\Phi _i}} )\end{array}\\ \begin{array}{l} {\kern 1pt} B_i^{\prime} = filte{r_{low - pass}}\{{[{{a_i}\cos ({{\omega_i}\delta + \Delta {\Phi _i}} )+ {b_i}\sin ({{\omega_i}\delta + \Delta {\Phi _i}} )} ]\cdot \sin {\omega_i}\delta } \}\\ ={-} \frac{1}{2}({{a_i}\sin \Delta {\Phi _i} - {b_i}\cos \Delta {\Phi _i}} )\end{array} \end{array}} \right.$$
where Ai’ and Bi’ are the obtained demodulated channel signals with phase deviations. ΔΦi is the accumulated phase deviation in the ith channel. For example, ΔΦi is θ1 + θ2−θ4 in the third channel. From Eq. (10), the retarder phase deviation ΔΦi is separated from the total phase of retarder Θi by coherence demodulation. It means that retarder phase deviation could be studied independently instead of being superimposed on a large base. That will greatly improve the accuracy and efficiency of phase error correction.To visually display the influence of phase deviations, a MATLAB simulation is performed. It is carried out among 5300–7600 cm−1 in near-infrared band, whose central wavelength is 1550 nm. Only the dispersion and thickness deviation of retarders are taken into account. Calcite is used as the birefringent material, whose birefringence with dispersion is from Ghosh’s work [23]. The retarder thickness d1 is set as 10 mm, and the thickness errors (Δd1, Δd2, Δd3, Δd4) are set to (0.0075, -0.0053, 0.0025, -0.0089) mm. The measured sample is a polarizer oriented at 30°. A polarizer is chosen as the sample because its constant elements are beneficial to displaying the modulation effect of phase deviations. The modulated intensity with phase deviation is drawn in Fig. 2(a) to compare with that without phase deviation. The channel shift and spectrum widening can been seen and are more obvious in high-frequency channels. The high-order phase error derived from dispersion will introduce extremely serious measurement errors and largely limit the spectral width.
As a result of coherence demodulation, six typical channel signals are shown in Fig. 2(b) It shows that channel signals Ai and Bi are indeed modulated by sinusoidal signals whose phase varies non-linearly with wave-number. Since the dispersion of calcite is almost linear in the operation band, phase deviations could be treated as a quadratic polynomial. The position where the signal changes the slowest is related to the first-order component of phase deviation, while the change rate of the signal is determined by the second-order one.
3.2 Characteristic of retarder phase deviation
According to the above analysis, the essential cause of measurement errors is that phase deviations change the mapping of Mueller matrix to demodulated channel signals. Fortunately, the analytical form of this mapping is obtained after derivation. The analytical result, which is important for characteristic and correction of phase deviations, is given by Eq. (12) in Appendix. In order to correct these errors, it is necessary to precisely characterize all the phase deviations (θ1, θ2, θ3, θ4). One should note that there is no need to acquire their absolute phases, and it is enough to calibrate their initial ones. Based on Eq. (12), once they were determined, all Mueller elements could be accurately inverted. In our opinion, to retrieve all phase deviations, it is worth proceeding the two steps as follows.
Firstly, vacuum is measured as a sample. Vacuum means that only Mueller elements m00, m11, m22 and m33 are equal to 1, while the other are zero. According to Eq. (12), the analytical expressions of all corresponding demodulated channel signals Ai’ and Bi’ for vacuum are listed in Eq. (13). It shows that every non-zero channel signal is a sinusoidal wave with a non-linear phase, which is a linear accumulation of retarder phase deviations. And, Ai’ and Bi’ have the same phase. By constructing a complex number Ai’+jBi’, their phase ΔΦi could be obtained by a Matlab function called ‘phase’. Then, according to Eq. (13), a simple system of linear equations connecting retarder phase deviations (θ1, θ2, θ3, θ4) with the obtained phases ΔΦi could be constructed to calculate (θ1, θ2, θ3, θ4). Finally, polynomial fitting is used to fit the relationship between retarder phase deviations and wave-number. Piecewise fitting could be used to improve fitting accuracy. One should note that the role of polynomial fitting is quite crucial because of random errors and bad points. These channel signal A2’, B2’, A4’, B4’, A6’, B6’, A10’, B10’, A12’ and B12’ are chosen to construct the desired system of equations. The other non-zero signals are discarded because they include two or more sinusoidal waves with discrepant phases.
Note that these linear equations constructed must be non-singular. In order to obtain satisfactory equations, a polarizer oriented at 0° is also measured as a sample. It means m00, m01, m10, and m11 are equal to 1, while the other are zero. In this case, the analytical expression of Ai’ and Bi’ are listed by Eq. (14) in Appendix, among which the channel signal A1’, B1’, A4’, B4’, A5’, B5’, A6’, B6’, A8’, B8’, A10’, B10’, A12’ and B12’ are adopted.
Then, we performed a series of simulation experiments to verify the performance of this characteristic method. In order to mimic the real experiment, an additional noise of 30 dB was introduced into the intensity signal. Besides, the azimuths of the four retarders also were artificially added by a misalignment error of 0.5°. A polarizer oriented at 30° is treated as the measured sample. After solving equations and fitting, the retarder phase deviations (θ1, θ2, θ3, θ4) are acquired. According to the retrieved phase deviations and Eq. (12), the corrected Mueller matrix is obtained and plotted in Fig. 3 to compare with the true one. To clearly show the detail of figure, only four Mueller elements m00, m11, m22, and m33 are drawn. They are roughly consisted, and the inverted Mueller fluctuates around the truth one. The deviation between them is less than 10%. This deviation mainly comes from high order dispersion and fitting errors.
However, since Mueller matrix of a polarizer is constant, the above simulation cannot prove the ability to correct Mueller matrix varying with wavenumber. For this purpose, another simulation needs to be carried out. In this simulation, the measured Mueller matrix as a sample is a randomly generated 4×4 matrix. The simulation result and the measured Mueller matrix are drawn in Fig. 4. It shows that they mainly agree with each other.
Although the Mueller matrix used in the second simulation is not from a real sample, it does not affect the validation of this simulation. That is because this correction method according to Eq. (12) is independent of Mueller matrix and suitable for any 4×4 matrix whose bandwidth in OPD domain is less than the single-channel bandwidth Δn0d1/2. If the bandwidth of Mueller matrix exceeds it, an additional error resulted from overlap of high frequency signals will appear.
Therefore, this calibration method is feasible when the bandwidth of the Mueller matrix under test does not exceed the bandwidth of a single channel. The correction method for retarder phase deviation will help to break through the limit to spectral width from retarder dispersion and greatly improve the measurement accuracy of Mueller matrix.
4. Conclusion
In this paper, a theory explaining the operational principle of channeled Mueller matrix polarimeters was represented based on signal modulation/demodulation, which is borrowed from signal processing theory. At the same time, a demodulation method of extracting channel signals from the modulated intensity was proposed. Coherence demodulation can separate the phase deviation from the total phase of retarder, providing an opportunity to study and correct it separately. The proposed theory and method are also suitable for all kinds of channeled Stokes and Mueller matrix polarimeters.
The influence of phase deviations of retarders in CMMSPs was analytically investigated and verified by a simulation. The phase deviation could be investigated as a whole, no matter what it is resulted from. The high-frequency channels suffer greater influence. The combination of birefringent crystals with positive and negative dispersion could be utilized to suppress dispersion in high-frequency channels. In order to correct this kind of error, we used vacuum and a polarizer as determinant samples that have definite and simple Mueller matrix to characterize the phase deviations of four retarders. Two simulations were carried out to validate its performance. The results show that the residual error of the correction method is less than 10%. Similarly, this method could also be used to analysis and characterize channeled Stokes spectropolarimeters.
Appendix
When there exists no the phase deviation of retarder, all of modulated channel signals ai and bi in Eq. (3) are given by Eq. (11).
(11)$$\begin{array}{l} {a_0} = \frac{1}{2}{m_{00}} + \frac{1}{4}{m_{02}} - \frac{1}{4}{m_{20}} - \frac{1}{{16}}{m_{22}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_1} = \frac{1}{2}{m_{01}} - \frac{1}{8}{m_{21}}\\ {a_2} ={-} \frac{1}{4}{m_{02}} + \frac{1}{{16}}{m_{22}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 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{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_5} ={-} \frac{1}{2}{m_{10}} - \frac{1}{4}{m_{12}}\\ {a_6} ={-} \frac{1}{4}{m_{11}}; {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 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{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_7} = \frac{1}{8}{m_{12}}\\ {a_8} ={-} \frac{1}{{16}}{m_{22}} + \frac{1}{{16}}{m_{33}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_9} = \frac{1}{8}{m_{21}}\\ {a_{10}} = \frac{1}{4}{m_{20}} + \frac{1}{8}{m_{22}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_{11}} = \frac{1}{8}{m_{21}}\\ {a_{12}} ={-} \frac{1}{{16}}{m_{22}} - \frac{1}{{16}}{m_{33}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_1} = {b_4} = {b_5} = {b_6} = 0\\ {b_2} ={-} \frac{1}{4}{m_{03}} + \frac{1}{8}{m_{23}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_3} ={-} \frac{1}{8}{m_{13}}\\ {b_7} = \frac{1}{8}{m_{13}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_8} = \frac{1}{{16}}{m_{23}} + \frac{1}{{16}}{m_{32}}\\ {b_9} ={-} \frac{1}{8}{m_{31}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_{10}} ={-} \frac{1}{4}{m_{30}} - \frac{1}{8}{m_{32}}\\ {b_{11}} ={-} \frac{1}{8}{m_{31}};{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_{12}} ={-} \frac{1}{{16}}{m_{23}} + \frac{1}{{16}}{m_{32}} \end{array}$$
When there exists the phase deviation of retarder, the demodulated channel signals Ai’ and Bi’ by coherence demodulation will be interfered by retarder phase deviations. They become more complicated and are expressed by Eq. (12).
$$\begin{array}{l} A_0^{\prime} = \frac{{{m_{00}}}}{2} + \frac{{{m_{02}}}}{4}\cos ({\theta _1} - {\theta _2}) - \frac{{{m_{20}}}}{4}\cos ({\theta _3} - {\theta _4}) - \frac{{{m_{22}}}}{{16}}\left[ \begin{array}{l} cos({\theta_1} - {\theta_2} + {\theta_3} - {\theta_4})\\ + \cos ({\theta_1} - {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ + \frac{{{m_{33}}}}{{16}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} + {\theta_3} - {\theta_4})\\ - \cos ({\theta_1} - {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{03}}}}{4}\sin ({\theta _1} - {\theta _2}) - \frac{{{m_{23}}}}{{16}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} + {\theta_3} - {\theta_4})\\ + \sin ({\theta_1} - {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ - \frac{{{m_{30}}}}{4}\sin ({\theta _3} - {\theta _4}) + \frac{{{m_{32}}}}{{16}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} - {\theta_3} + {\theta_4})\\ - \sin ({\theta_1} - {\theta_2} + {\theta_3} - {\theta_4}) \end{array} \right]\\ A_1^{\prime} = \frac{{{m_{01}}}}{4}\cos {\theta _1} - \frac{{{m_{21}}}}{{16}}\left[ \begin{array}{l} \cos ({\theta_1} + {\theta_3} - {\theta_4})\\ + \cos ({\theta_1} - {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{31}}}}{{16}}\left[ \begin{array}{l} \sin ({{\theta_1} - {\theta_3} + {\theta_4}} )\\ - \sin ({{\theta_1} + {\theta_3} - {\theta_4}} )\end{array} \right]\\ A_2^{\prime} ={-} \frac{{{m_{02}}}}{8}\cos ({\theta _1} + {\theta _2}) + \frac{{{m_{22}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ + \cos ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{33}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ - \cos ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ \qquad + \frac{{{m_{03}}}}{8}\sin ({{\theta_1} + {\theta_2}} )\\ - \frac{{{m_{23}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ + \sin ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{32}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ - \sin ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ A_3^{\prime} = \frac{{{m_{12}}}}{{16}}\cos ({\theta _1} + {\theta _2} - {\theta _4}) - \frac{{{m_{13}}}}{{16}}\sin ({\theta _1} + {\theta _2} - {\theta _4})\\ A_4^{\prime} ={-} \frac{{m{}_{11}}}{8}\cos ({\theta _1} - {\theta _4})\\ A_5^{\prime} ={-} \frac{{{m_{10}}}}{4}\cos {\theta _4} - \frac{{{m_{12}}}}{{16}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} + {\theta_4})\\ + \cos ({\theta_1} - {\theta_2} - {\theta_4}) \end{array} \right] - \frac{{{m_{12}}}}{{16}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} + {\theta_4})\\ + \sin ({\theta_1} - {\theta_2} - {\theta_4}) \end{array} \right]\\ A_6^{\prime} ={-} \frac{{{m_{11}}}}{8}\cos ({\theta _1} + {\theta _4})\\ A_7^{\prime} = \frac{{{m_{12}}}}{{16}}\cos ({\theta _1} + {\theta _2} + {\theta _4}) - \frac{{{m_{13}}}}{{16}}\sin ({\theta _1} + {\theta _2} + {\theta _4}) \end{array}$$
(12)$$\begin{array}{l} A_8^{\prime} ={-} \frac{{{m_{22}}}}{{32}}\cos ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4}) + \frac{{{m_{33}}}}{{32}}\cos ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4}) + \frac{{{m_{23}}}}{{32}}\sin ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4})\\ + \frac{{{m_{32}}}}{{32}}\sin ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4})\\ A_9^{\prime} = \frac{{{m_{21}}}}{{16}}\cos ({\theta _1} - {\theta _3} - {\theta _4}) - \frac{{{m_{31}}}}{{16}}\sin ({\theta _1} - {\theta _3} - {\theta _4})\\ A_{10}^{\prime} = \frac{{{m_{20}}}}{8}\cos ({\theta _3} + {\theta _4}) + \frac{{{m_{22}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4})\\ + \cos ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}) \end{array} \right] + \frac{{{m_{33}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4})\\ - \cos ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}) \end{array} \right]\\ + \frac{{{m_{30}}}}{8}\sin ({\theta _3} + {\theta _4}) + \frac{{{m_{23}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4})\\ + \sin ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}) \end{array} \right] + \frac{{{m_{32}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4})\\ - \sin ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}) \end{array} \right]\\ A_{11}^{\prime} = \frac{{{m_{21}}}}{{16}}\cos ({\theta _1} + {\theta _3} + {\theta _4}) + \frac{{{m_{31}}}}{{16}}\sin ({\theta _1} + {\theta _3} + {\theta _4})\\ A_{12}^{\prime} ={-} \frac{{{m_{22}} + {m_{33}}}}{{32}}\cos ({\theta _1} + {\theta _2} + {\theta _3} + {\theta _4}) + \frac{{{m_{23}} - {m_{32}}}}{{32}}\sin ({\theta _1} + {\theta _2} + {\theta _3} + {\theta _4})\\ B_1^{\prime} ={-} \frac{{{m_{01}}}}{4}\sin {\theta _1} + \frac{{{m_{21}}}}{{16}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_3} + {\theta_4})\\ + \sin ({\theta_1} + {\theta_3} - {\theta_4}) \end{array} \right] + \frac{{{m_{31}}}}{{16}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_3} + {\theta_4})\\ - \cos ({\theta_1} + {\theta_3} - {\theta_4}) \end{array} \right]\\ B_2^{\prime} = \frac{{{m_{02}}}}{8}\sin ({\theta _1} + {\theta _2}) - \frac{{{m_{22}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ + \sin ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right] - \frac{{{m_{33}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ - \sin ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ + \frac{{{m_{03}}}}{8}\cos ({\theta _1} + {\theta _2}) - \frac{{{m_{23}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ + \cos ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{32}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} + {\theta_2} + {\theta_3} - {\theta_4})\\ - \cos ({\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}) \end{array} \right]\\ B_3^{\prime} = \frac{{{m_{12}}}}{{16}}\cos ({\theta _1} + {\theta _2} - {\theta _4}) - \frac{{{m_{13}}}}{{16}}\sin ({\theta _1} + {\theta _2} - {\theta _4})\\ B_4^{\prime} ={-} \frac{{m{}_{11}}}{8}\sin ({\theta _1} - {\theta _4})\\ B_5^{\prime} = \frac{{{m_{10}}}}{4}\sin {\theta _4} + \frac{{{m_{12}}}}{{16}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} + {\theta_4})\\ + \sin ({\theta_1} - {\theta_2} - {\theta_4}) \end{array} \right] - \frac{{{m_{13}}}}{{16}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} + {\theta_4})\\ - \cos ({\theta_1} - {\theta_2} - {\theta_4}) \end{array} \right]\\ B_6^{\prime} = \frac{{m{}_{11}}}{8}\sin ({\theta _1} + {\theta _4})\\ B_7^{\prime} ={-} \frac{{{m_{12}}}}{{16}}\sin ({\theta _1} + {\theta _2} + {\theta _4}) - \frac{{{m_{13}}}}{{16}}\cos ({\theta _1} + {\theta _2} + {\theta _4})\\ B_8^{\prime} ={-} \frac{{{m_{22}} - {m_{33}}}}{{32}}\sin ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4}) - \frac{{{m_{23}} + {m_{32}}}}{{32}}\cos ({\theta _1} + {\theta _2} - {\theta _3} - {\theta _4})\\ B_9^{\prime} = \frac{{{m_{21}}}}{{16}}\sin ({\theta _1} - {\theta _3} - {\theta _4}) + \frac{{{m_{31}}}}{{16}}\cos ({\theta _1} - {\theta _3} - {\theta _4})\\ B_{10}^{\prime} = \frac{{{m_{23}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4})\\ + \sin ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{30}}}}{4}\sin ({\theta _3} + {\theta _4}) + \frac{{{m_{32}}}}{{32}}\left[ \begin{array}{l} \sin ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4})\\ - \sin ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}) \end{array} \right]\\ + \frac{{{m_{22}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4})\\ + \cos ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}) \end{array} \right] + \frac{{{m_{20}}}}{8}\cos ({\theta _3} + {\theta _4}) + \frac{{{m_{33}}}}{{32}}\left[ \begin{array}{l} \cos ({\theta_1} - {\theta_2} + {\theta_3} + {\theta_4})\\ - \cos ({\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}) \end{array} \right]\\ B_{11}^{\prime} ={-} \frac{{{m_{21}}}}{{16}}\sin ({\theta _1} + {\theta _3} + {\theta _4}) + \frac{{{m_{31}}}}{{16}}\cos ({\theta _1} + {\theta _3} + {\theta _4})\\ B_{12}^{\prime} = \frac{{{m_{22}} + {m_{33}}}}{{32}}\sin ({\theta _1} + {\theta _2} + {\theta _3} + {\theta _4}) + \frac{{{m_{23}} - {m_{32}}}}{{32}}\cos ({\theta _1} + {\theta _2} + {\theta _3} + {\theta _4}) \end{array}$$
When vacuum is used as a sample, the demodulated channel signals by coherence demodulation Ai’ and Bi’ are given by Eq. (13).
(13)$$\begin{array}{l} {A_0}^{\prime} = \frac{1}{4} - \frac{1}{{16}}\cos ({{\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}} );\\ {A_1}^{\prime} = {A_3}^{\prime} = {A_5}^{\prime} = {A_7}^{\prime} = {A_8}^{\prime} = {A_9}^{\prime} = {A_{11}}^{\prime} = {B_1}^{\prime} = {B_3}^{\prime} = {B_5}^{\prime} = {B_7}^{\prime} = {B_8}^{\prime} = {B_9}^{\prime} = {B_{11}}^{\prime} = 0;\\ {A_2}^{\prime} = \frac{1}{{32}}\cos ({{\theta_1} + {\theta_2} + {\theta_3} - {\theta_4}} )\\ {B_2}^{\prime} ={-} \frac{1}{{32}}\sin ({{\theta_1} + {\theta_2} + {\theta_3} - {\theta_4}} )\\ {A_4}^{\prime} ={-} \frac{1}{{32}}\cos ({{\theta_1} - {\theta_4}} )\\ {B_4}^{\prime} ={-} \frac{1}{{32}}\sin ({{\theta_1} - {\theta_4}} )\\ {A_6}^{\prime} ={-} \frac{1}{{32}}\cos ({{\theta_1} + {\theta_4}} )\\ {B_6}^{\prime} = \frac{1}{{32}}\sin ({{\theta_1} + {\theta_4}} )\\ {A_{10}}^{\prime} = \frac{1}{{32}}\cos ({{\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}} )\\ {B_{10}}^{\prime} = \frac{1}{{32}}\sin ({{\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}} )\\ {A_{12}}^{\prime} ={-} \frac{1}{{32}}\cos ({{\theta_1} + {\theta_2} + {\theta_3} + {\theta_4}} )\\ {B_{12}}^{\prime} = \frac{1}{{32}}\sin ({{\theta_1} + {\theta_2} + {\theta_3} + {\theta_4}} )\end{array}$$
Additionally, in order to construct non-singular equations to invert all phase deviations of retarders, a polarizer oriented at 0° is also measured to produce enough equations. In this case, the demodulated channel signals Ai’ and Bi’ by coherence demodulation are analytically expressed by Eq. (14).
$$\begin{array}{l} {A_0}^{\prime} = \frac{1}{4} + \frac{1}{{32}}[{\cos ({{\theta_1} - {\theta_2} + {\theta_3} - {\theta_4}} )- \cos ({{\theta_1} - {\theta_2} - {\theta_3} + {\theta_4}} )} ]\\ {A_1}^{\prime} = \frac{1}{8}\cos {\theta _1}\\ {B_1}^{\prime} = \frac{1}{8}\sin {\theta _1}\\ {A_2}^{\prime} = \frac{1}{{64}}[{\cos ({{\theta_1} + {\theta_2} + {\theta_3} - {\theta_4}} )- \cos ({{\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}} )} ]\\ {B_2}^{\prime} ={-} \frac{1}{{64}}[{\sin ({{\theta_1} + {\theta_2} + {\theta_3} - {\theta_4}} )- \sin ({{\theta_1} + {\theta_2} - {\theta_3} + {\theta_4}} )} ]\\ {A_3}^{\prime} = {B_3} = {A_7} = {B_7} = {A_9} = {B_9} = {A_{11}} = {B_{11}} = 0\\ {A_4}^{\prime} ={-} \frac{1}{{16}}\cos ({{\theta_1} - {\theta_4}} )\\ {B_4}^{\prime} ={-} \frac{1}{{16}}\sin ({{\theta_1} - {\theta_4}} )\\ {A_5}^{\prime} ={-} \frac{1}{8}\cos {\theta _4}\\ {B_5}^{\prime} ={-} \frac{1}{8}\sin {\theta _4} \end{array}$$
(14)$$\begin{array}{l} {A_6}^{\prime} ={-} \frac{1}{{16}}\cos ({{\theta_1} + {\theta_4}} )\\ {B_6}^{\prime} = \frac{1}{{16}}\sin ({{\theta_1} + {\theta_4}} )\\ {A_8}^{\prime} = \frac{1}{{64}}\cos ({{\theta_1} + {\theta_2} - {\theta_3} - {\theta_4}} )\\ {B_8}^{\prime} = \frac{1}{{64}}\sin ({{\theta_1} + {\theta_2} - {\theta_3} - {\theta_4}} )\\ {A_{10}}^{\prime} = \frac{1}{{64}}[{\cos ({{\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}} )- \cos ({{\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}} )} ]\\ {A_{10}}^{\prime} = \frac{1}{{64}}[{\sin ({{\theta_1} - {\theta_2} - {\theta_3} - {\theta_4}} )+ \sin ({{\theta_1} - {\theta_2} + {\theta_3} + {\theta_4}} )} ]\\ {A_{12}}^{\prime} ={-} \frac{1}{{64}}\cos ({{\theta_1} + {\theta_2} + {\theta_3} + {\theta_4}} )\\ {B_{12}}^{\prime} = \frac{1}{{64}}\sin ({{\theta_1} + {\theta_2} + {\theta_3} + {\theta_4}} )\end{array}$$
Funding
National Key Laboratory of Science and Technology on Vacuum Technology and Physics (ZWK 1802); China Aerospace Science and Technology Corporation (no award number).
Acknowledgments
The authors thank Mr. Peng Zhu for helpful discussions and technical assistance.
Disclosures
The authors declare no conflicts of interest.
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