1. Introduction

Birefringence and polarization-dependent loss or gain (PDL/G) are two fundamental polarization effects. In some optical systems, they may exist simultaneously. An example of such a system in the infrared band is an optical fiber communication link composed of single-mode optical fibers (SMF), passive and active optical components [1]. In the terahertz (THz) band, both polarization effects have also been found in certain materials. These polarization effects could be used as potential fingerprints in THz spectroscopy to characterize these materials [2]. In principle, an optical system, having both birefringence and PDL/G, can be completely depicted by a $4\times 4$ real Mueller-Jones matrix M, or equivalently, a $2\times 2$ complex Jones matrix J. Polarization properties of an optical system, such as PDL/G vector, polarization mode dispersion (PMD) vector and so on, can be extracted from M or J using appropriate algorithms [3–6]. Hence, the measurement of Mueller matrix or Jones matrix is of great importance in both basic and applied research. In this paper, we only consider the measurement of the Mueller matrix.

To date, several Mueller matrix measurement methods, such as dual rotating retarders polarimetry [7], null ellipsometry [8] and the Stokes methods [9], have been proposed. All these methods realize the measurement by setting some input states of polarization (SOP) and measuring the corresponding outputs. A potential technique for the fast infrared polarization modulation was also reported based on the high birefringence and low linear diattenuation of ferroelectric liquid crystals in some spectral bands [10]. In some applications based on Mueller matrix measurement, the smallest number of input SOPs should be used to take the measurements because a series of Mueller matrices are required to be measured in finite time. Two examples of such applications are PMD vector measurement [4] and polarization measurement in terahertz time-domain spectroscopy (THz-TDS) [11]. However, most of the above-mentioned methods are not good candidates in such applications.

Actually, it has been demonstrated that at least three input SOPs must be used to realize the Mueller matrix measurement when the system under test has both birefringence and PDL/G [12,13]. In 1947, R. C. Jones proposed a Jones matrix measurement approach using the three SOPs ${\left(1,1,0,0\right)}^{\text{T}}$, ${\left(1,-1,0,0\right)}^{\text{T}}$ and ${\left(1,0,1,0\right)}^{\text{T}}$(“T” denotes the matrix transpose) [12]. We had also proposed a Mueller matrix measurement approach using the three input SOPs ${\left(1,1,0,0\right)}^{\text{T}}$, ${\left(1,-0.5,0.866,0\right)}^{\text{T}}$ and ${\left(1,-0.5,-0.866,0\right)}^{\text{T}}$ [13]. Theoretically, if the input and output SOPs are set and measured without any errors, the error-free Mueller matrix can be measured using three arbitrary input SOPs [13]. Unfortunately, errors arising from environmental perturbations, imperfect components and alignments, do exist in practice. Hence, the measured input and output SOPs definitely have errors. As a result, these errors will be transferred to the calculated Mueller matrix. The optimizations of polarimeters and noise influences on the SOP measurement have been reported in many papers [14–17]. Analysis of the Mueller matrix measurement error, induced by imperfect components and alignments, has also been presented [18,19]. On the other hand, for a pure birefringent system, we have demonstrated that the error of the Mueller matrix measurement is a function of two input SOPs and the statistically minimum error can be achieved when two input SOPs are orthogonal in Stokes space [20]. However, to the best of our knowledge, for a system with both birefringence and PDL/G, the relationship between the error of the calculated Mueller matrix and three input SOPs has not been presented until now.

A general analysis of this relationship, for an optical system with arbitrary PDL/G, is very complicated. In this paper, we present the theoretical and simulation results of this relationship in a simple case, where the system under test has a small PDL/G and the measurement errors of the output SOPs are far larger than those of the input SOPs. Some optical systems satisfy the first precondition. For example, an optical fiber system, composed of SMF, optical isolators and optical couplers, usually has a small PDL. Some measurement setups may approximately satisfy the second precondition. For example, when the input SOPs are generated by precisely rotating a high-quality polarizer and the output SOPs are measured by a fiber-type polarimeter, the measurement errors of the input SOPs must be less than those of the output SOPs.

In this paper, under the two above-mentioned preconditions, an approximate upper limit of the mean of the Mueller matrix measurement error is derived. This upper limit is determined by three angles among three input SOPs in Stokes space. It shows that the three best input SOPs should be coplanar in Stokes space and the angles between any two of them should be 120°. In fact, this upper limit is strictly valid only when there is no PDL/G. Generally, the Mueller matrix measurement error should also be related to the PDL/G vector. Thus, the best input SOPs should also depend on the PDL/G in the system. However, as long as the PDL/G is small enough, the three best input SOPs are very close to the relationship mentioned above. Therefore, if the system under test has a small PDL/G, we can use the inputs${\left(1,1,0,0\right)}^{\text{T}}$, ${\left(1,-0.5,0.866,0\right)}^{\text{T}}$ and ${\left(1,-0.5,-0.866,0\right)}^{\text{T}}$as a substitute for the best inputs.

This paper is organized as follows: firstly, some useful equations are deduced based on the properties of the Mueller matrix in Section 2; secondly, the equations governing Mueller matrix measurement using three arbitrary input SOPs and error propagations are derived in Section 3; the statistical properties of Mueller matrix measurement error are investigated in Section 4 and Section 5; finally, some simulation results are used to verify the theoretical finding.

2. Some properties of Mueller-Jones matrix

It has been demonstrated that the Mueller matrix of a system having both birefringence and PDL/G satisfies the Lorentz transformation [21,22]. Thus, we can express such a Mueller matrix in the form of a $4\times 4$ complex matrix $\colorbox[rgb]{}{$\tilde{M}$}$

It has been demonstrated that such a Mueller matrix can be decomposed as [23]

Sub-matrix ${\colorbox[rgb]{}{$m$}}_{\text{R}}$ is a $3\times 3$ orthogonal matrix, which can be expressed as [23]

Sub-matrix ${\colorbox[rgb]{}{$m$}}_{\text{D}}$ is a $3\times 3$ symmetric matrix, which can be written as [23]

From Eqs. (3), (4) and (5), the determinant $\left|\colorbox[rgb]{}{$\tilde{M}$}\right|$ and the Frobenius matrix norm $\Vert \colorbox[rgb]{}{$\tilde{M}$}\Vert$ of $\colorbox[rgb]{}{$\tilde{M}$}$, which is defined as $\Vert \colorbox[rgb]{}{$\tilde{M}$}\Vert =\sqrt{\mathrm{Tr}\left({\colorbox[rgb]{}{$\tilde{M}$}}^{\text{H}}\colorbox[rgb]{}{$\tilde{M}$}\right)}$, can be calculated as

From Eq. (3), we can easily obtain

Another useful equation between input and output SOPs is ${\overrightarrow{S}}_{\text{out}}\cdot {\overrightarrow{T}}_{\text{out}}=\sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}{\overrightarrow{S}}_{\text{in}}\cdot {\overrightarrow{T}}_{\text{in}}$ [13]. Based on this equation, we have

3. Measurement approach and measurement error

In an experimental setup [13], three input SOPs ${\overrightarrow{S}}_{\text{in}}$, ${\overrightarrow{T}}_{\text{in}}$, ${\overrightarrow{U}}_{\text{in}}$ and three output SOPs ${\overrightarrow{S}}_{\text{out}}$, ${\overrightarrow{T}}_{\text{out}}$, ${\overrightarrow{U}}_{\text{out}}$ can be measured. Then, the complex Mueller matrix $\colorbox[rgb]{}{$\tilde{M}$}$ can be calculated using the following two equations [13]

When the measured output SOPs have errors $\Delta {\overrightarrow{S}}_{\text{out}}$, $\Delta {\overrightarrow{T}}_{\text{out}}$ and $\Delta {\overrightarrow{U}}_{\text{out}}$, the exact form of Eq. (9) becomes

Similarly, when errors exist, the exact form of Eq. (10) should be

For three input SOPs, α, β and γ are the angles between ${\overrightarrow{S}}_{\text{in}}$ and ${\overrightarrow{T}}_{\text{in}}$, ${\overrightarrow{S}}_{\text{in}}$ and ${\overrightarrow{U}}_{\text{in}}$, ${\overrightarrow{T}}_{\text{in}}$ and ${\overrightarrow{U}}_{\text{in}}$ in Stokes space, respectively. Please note that we have ignored the subscript “in” for the three angles. These angles are bounded by $0\le \alpha ,\beta ,\gamma \le {180}^{\circ }$ and $\alpha +\beta +\gamma \le {360}^{\circ }$. In fact, these angles are not independent. They should satisfy $\left|\alpha -\beta \right|\le \gamma \le \alpha +\beta$. Then we have the following relations

 

Fig. 1 $\left(\alpha ,\beta ,\gamma \right)$ can only take values on the surface and the inner space of the red tetrahedron.

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4. Statistical properties of $\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}$

From Eq. (12), the smallest $\left|\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right|$ may be achieved corresponding to different input SOPs in different tests because the measurement errors of the output SOPs vary. Therefore, the relationship between the statistical parameter of $\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}$ and input SOPs should be considered. Since a good polarimeter should have a completely random measurement error with zero mean, then $〈\Delta {\overrightarrow{S}}_{\text{out}}〉=〈\Delta {\overrightarrow{T}}_{\text{out}}〉=〈\Delta {\overrightarrow{U}}_{\text{out}}〉=0$. $〈\cdot 〉$ stands for the mean of a random variable. Obviously, $〈\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}〉=0$. Thus, we need to use the variance $\mathrm{Var}\left(\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right)$ to evaluate the uncertainty. A smaller variance $\mathrm{Var}\left(\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right)$ means the larger possibility of$\sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}$ of having a smaller measurement error. In this paper, we assume all SOP measurement errors follow the Gaussian distributions, that is, $\Delta s_{\text{outj}},\Delta t_{\text{outj}},\Delta u_{\text{outj}}\left(\text{j}=1,2,3\right)\sim N\left(0,{\sigma }^2\right)$. Then, $\mathrm{Var}\left(\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right)$ can be calculated as

Equation (17) means that$\mathrm{Var}{\left(\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right)}_{D<<1}$ is completely determined by α, β and γ if $T_{\text{u}}$ is not taken into consideration. From Eq. (17), it is easy to know that $\mathrm{Var}{\left(\Delta \sqrt{\left|\colorbox[rgb]{}{$\tilde{M}$}\right|}\right)}_{D<<1}$ is the smallest when $K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}$ takes its minimum value. Actually, all $\left(\alpha ,\beta ,\gamma \right)$, with the same $K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}$, form a curved surface as shown in Fig. 2 . From Fig. 2, $K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}$ has the minimum value when $\alpha =\beta =\gamma ={120}^{\circ }$, which means three input SOPs are coplanar and have angles of 120° between any two of them in Stokes space.

 

Fig. 2 Curved surfaces formed by $\left(\alpha ,\beta ,\gamma \right)$ with the same values of$K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}$.

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To illustrate more clearly, two special cases are plotted in Fig. 3 : (a) $\alpha =\beta =\gamma$ and (b) $\alpha =\beta \text{\hspace{0.17em} and}\gamma =\min \left(2\alpha ,2\pi -2\alpha \right)$. In Fig. 3(a), “orthogonal inputs” is equivalent to$\alpha =\beta =\gamma ={90}^{\circ }$. In Fig. 3(b), “Jones inputs” is equivalent to $\alpha =\beta ={90}^{\circ }\text{\hspace{0.17em} and}\gamma ={180}^{\circ }$.

 

Fig. 3 $K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}$ of Eq. (17) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.

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5. Upper limit of $〈\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert 〉$

When $D<<1$, the matrix ${\colorbox[rgb]{}{$m$}}_{\text{D}}$ in Eq. (5) is almost equal to I. Then, the complex Mueller matrix $\colorbox[rgb]{}{$\tilde{M}$}$ is close to $T_{\text{u}}\left(\begin{array}{l}\multicolumn{1}{c}{1{\overrightarrow{0}}^{\text{T}}}\\ \multicolumn{1}{c}{\overrightarrow{0}{\colorbox[rgb]{}{$m$}}_{\text{R}}}\end{array}\right)$. Since ${\colorbox[rgb]{}{$m$}}_{\text{R}}^{\text{H}}={\colorbox[rgb]{}{$m$}}_{\text{R}}^{\text{T}}={\colorbox[rgb]{}{$m$}}_{\text{R}}^{-1}$ [23], from Eq. (14), we have

where

(26)$$\{\begin{array}{l}\multicolumn{1}{c}{B_{\text{in1}}=\left|{\stackrel{⌢}{s}}_{\text{in}}\cdot \left({\stackrel{⌢}{t}}_{\text{in}}\times {\stackrel{⌢}{u}}_{\text{in}}\right)\right|=\sqrt{1-{\cos }^2\alpha -{\cos }^2\beta -{\cos }^2\gamma +2\cos \alpha \cos \beta \cos \gamma }}\\ \multicolumn{1}{c}{B_{\text{in2}}=\left|\left({\stackrel{⌢}{t}}_{\text{in}}-{\stackrel{⌢}{s}}_{\text{in}}\right)\times \left({\stackrel{⌢}{u}}_{\text{in}}-{\stackrel{⌢}{s}}_{\text{in}}\right)\right|=\sqrt{4\left(1-\cos \beta \right)\left(1-\cos \gamma \right)-{\left(1+\cos \alpha -\cos \beta -\cos \gamma \right)}^2}}\end{array}$$

In Eq. (23), the terms of ${\Omega }_{jk}$ are given by

(27)$$\{\begin{array}{l}\multicolumn{1}{c}{{\Omega }_{11}={\Omega }_{22}={\Omega }_{33}=2,{\Omega }_{44}=4\left(B_{\text{in1}}^2+B_{\text{in2}}^2\right)}\\ \multicolumn{1}{c}{{\Omega }_{12}={\Omega }_{21}=1+\cos \alpha ,{\Omega }_{13}={\Omega }_{31}=1+\cos \beta ,{\Omega }_{23}={\Omega }_{32}=1+\cos \gamma }\\ \multicolumn{1}{c}{{\Omega }_{14}={\Omega }_{24}={\Omega }_{34}=-{\Omega }_{41}=-{\Omega }_{42}=-{\Omega }_{43}=4i{B}_{\text{in1}}}\end{array}$$

In Eq. (24), the terms of${\Psi }_{jk}$ are given by

(28)$$\{\begin{array}{l}\multicolumn{1}{c}{{\Psi }_{11}=2-\cos \alpha -\cos \beta ,{\Psi }_{22}=2-\cos \alpha -\cos \gamma ,{\Psi }_{33}=2-\cos \beta -\cos \gamma }\\ \multicolumn{1}{c}{{\Psi }_{12}={\Psi }_{13}=1-\cos \gamma ,{\Psi }_{21}={\Psi }_{23}=1-\cos \beta ,{\Psi }_{31}={\Psi }_{32}=1-\cos \alpha }\end{array}$$

Finally, we have

(29)$$〈\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert 〉\le \sqrt{〈{\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert }^2〉}=K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }\sigma$$

where

(30)$$K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }=\sqrt{2}{\left\{\begin{array}{l}\multicolumn{1}{c}{2{\displaystyle \colorbox[rgb]{}{$\sum _{j=1}^3{f}_{jj}$}}+f_{44}\left[\begin{array}{l}\multicolumn{1}{c}{4\left(3-\cos \alpha -\cos \beta -\cos \gamma \right)-}\\ \multicolumn{1}{c}{{\left(1-\cos \alpha \right)}^2-{\left(1-\cos \beta \right)}^2-{\left(1-\cos \gamma \right)}^2}\end{array}\right]+}\\ \multicolumn{1}{c}{K_{\Delta \left|\colorbox[rgb]{}{$M$}\right|}{\displaystyle \colorbox[rgb]{}{$\sum _{j=1}^4{\displaystyle \colorbox[rgb]{}{$\sum _{k=1}^4{f}_{jk}{\Omega }_{jk}$}}$}}-\genfrac{}{}{0.1ex}{}{{\displaystyle \colorbox[rgb]{}{$\sum _{j=1}^3{\displaystyle \colorbox[rgb]{}{$\sum _{k=1}^3{f}_{jk}{\Psi }_{jk}$}}$}}}{3-\cos \alpha -\cos \beta -\cos \gamma }}\end{array}\right\}}^{\genfrac{}{}{0.1ex}{}{1}{2}}$$

Apparently, this upper limit is completely determined by α, β and γ. From the calculation, we can find that $K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }$ has its minimum value also when $\alpha =\beta =\gamma ={120}^{\circ }$. All $\left(\alpha ,\beta ,\gamma \right)$, with the same $K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }$, form a curved surface as shown in Fig. 4 .

 

Fig. 4 Curved surfaces formed by $\left(\alpha ,\beta ,\gamma \right)$ with the same values of$K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }$.

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To illustrate more clearly, two special cases are plotted in Fig. 5 : (a) $\alpha =\beta =\gamma$ and (b) $\alpha =\beta \text{\hspace{0.17em} and}\gamma =\min \left(2\alpha ,2\pi -2\alpha \right)$. Strictly speaking, the upper limit shown in Eq. (29) is valid only when there is no PDL/G ($D=0$). However, when $D<<1$, it still approximately governs the statistical relationship between three input SOPs and the Mueller matrix measurement error $\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert$.

 

Fig. 5 $K_{\Vert \Delta \colorbox[rgb]{}{$M$}\Vert }$ of Eq. (29) as a function of angles between three inputs (a) three angles are identical and (b) three inputs are coplanar and two of them are identical. The insets show the “zoom in” views of the same data.

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6. Simulation results

To verify the theoretical finding in Section 5, simulations are performed. The parameters of the system under simulation are 1) Birefringence:$\varphi =5\pi /3$, $r=\left(0.66,0.74,-0.1296\right)$; 2) PDL/G: $0.02\le D\le 0.1$, $\stackrel{⌢}{D}$varies on the whole Poincaré sphere and 3) PIDL/G: $T_{\text{u}}=1$. In Section 5, theoretical result shows that the upper limit does not depend on the PIDL/G $T_{\text{u}}$. This is because we assume $D<<1$ and the variance ${\sigma }^2$is not a function of $T_{\text{u}}$. Then, we take $T_{\text{u}}=1$ in the following simulations. Further, we take the first two input SOPs as${\overrightarrow{S}}_{\text{in}}={\left(i,1,0,0\right)}^{\text{T}}$and ${\overrightarrow{T}}_{\text{in}}={\left(i,\cos \alpha ,\sin \alpha ,0\right)}^{\text{T}}$. In this paper, we only show the simulation results in the same two special cases as those in Section 4 and Section 5: (a) $\alpha =\beta =\gamma$and (b) $\alpha =\beta \text{\hspace{0.17em} and}\gamma =\min \left(2\alpha ,2\pi -2\alpha \right)$.

In (a),

(31)$${\overrightarrow{U}}_{\text{in}}={\left(i,\cos \alpha ,\genfrac{}{}{0.1ex}{}{\cos \alpha -{\cos }^2\alpha }{\sin \alpha },\genfrac{}{}{0.1ex}{}{\sqrt{{\sin }^4\alpha -{\left(\cos \alpha -{\cos }^2\alpha \right)}^2}}{\sin \alpha }\right)}^{\text{T}}$$

and in (b),

(32)$${\overrightarrow{U}}_{\text{in}}={\left(i,\cos \alpha ,-\sin \alpha ,0\right)}^{\text{T}}$$

In the simulations, $〈\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert 〉$ is calculated using 1000 independent noise realizations with $\sigma =0.03$. For three 4-dimensional Stokes vectors, this means that 12000 random values have been generated. In Fig. 6 , simulation results, with $\stackrel{⌢}{D}=\left(0,0,1\right)$ and different values of D, show that the theoretical upper limit (ULimit) is valid when $D\le 0.04$ (0.35dB). When D is less than 0.35 dB, $\alpha =\beta =\gamma ={120}^{\circ }$ can lead to a measurement error that is very close to the minimum. In fact, the real minimum measurement errors, corresponding to different values of D, have been shown in Fig. 6(a). However, to achieve these minimums, the PDL/G vectors must be known before the measurement. If the PDL/G vector is unknown, $\alpha =\beta =\gamma ={120}^{\circ }$ can be considered as the substitute for the best inputs when$D<<1$.

 

Fig. 6 Simulation results of $〈\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert 〉$as a function of angles between three inputs corresponding to different values of D (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.

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Next, we need to confirm the validity of the above conclusion for all $\stackrel{⌢}{D}$ when $D\le 0.04$. In Fig. 7 , $D=0.04$and $\stackrel{⌢}{D}$ varies on the whole Poincaré sphere. Results show that $\alpha =\beta =\gamma ={120}^{\circ }$ can be considered as the substitute for the best inputs whatever $\stackrel{⌢}{D}$ is. Based on these results, we can use${\left(1,1,0,0\right)}^{\text{T}}$, ${\left(1,-0.5,0.866,0\right)}^{\text{T}}$ and ${\left(1,-0.5,-0.866,0\right)}^{\text{T}}$as the standard input SOPs in the measurements. From the results shown in Fig. 6 and Fig. 7, these standard input SOPs can result in obviously better measurement accuracy than Jones inputs and orthogonal inputs.

 

Fig. 7 Simulation results of $〈\Vert \Delta \colorbox[rgb]{}{$\tilde{M}$}\Vert 〉$as a function of angles between three inputs corresponding to different values of $\stackrel{⌢}{D}$ (a) three angles are identical and (b) three inputs are coplanar and two of them are identical.

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7. Conclusion

We presented the statistical relationship between three input SOPs and the Mueller matrix measurement error in a system having small PDL/G. This statistical relationship is expressed as an upper limit of the measurement error, which is approximately valid when the PDL/G is small. Based on this upper limit, the minimum Mueller matrix measurement error will be statistically achieved when three input SOPs are coplanar with an angle of 120° between any two of them in Stokes space. From the simulations, these input SOPs are confirmed to lead to a measurement error that is very close to the minimum when the PDL/G of the system under test is less than 0.35 dB. The standard input SOPs ${\left(1,1,0,0\right)}^{\text{T}}$, ${\left(1,-0.5,0.866,0\right)}^{\text{T}}$ and ${\left(1,-0.5,-0.866,0\right)}^{\text{T}}$can be suggested since they have obviously larger probability to result in better Mueller matrix measurement accuracy than Jones inputs and orthogonal inputs.

Further, if the PDL/G is not very small, three optimum input SOPs will depends on the PDL/G vector. They will be not coplanar any more, but still equally separated. A detailed analysis will be presented in another paper.

Acknowledgements

This work is supported by Singapore A-star, Singapore Bioimaging Consortium, SBIC Grant Ref: SBIC RP C-014/2007.

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