1. Introduction

For an optical system having birefringence and small polarization-dependent loss or gain (PDL/G), we have demonstrated that, by theoretical analysis and simulations, the Mueller matrix measurement error (M³E) seriously depends on the choice of the three input states of polarization (SOP) [1]. When we use the preconditions that 1) the measurement errors of output SOPs are far larger than those of input SOPs and 2) SOP measurement errors independently and identically follow the Gaussian distribution $N\left(0,{\sigma }^2\right)$, it can be demonstrated that the minimum M³E is statistically achieved when the three input SOPs are coplanar with an angle of 120° between any two of them in Stokes space [1]. Three standard input SOPs ${\left(1,\hspace{0.17em}1,\hspace{0.17em}0,\hspace{0.17em}0\right)}^{\mathrm{T}}$, ${\left(1,\hspace{0.17em}-0.5,\hspace{0.17em}0.866,\hspace{0.17em}0\right)}^{\mathrm{T}}$ and ${\left(1,\hspace{0.17em}-0.5,\hspace{0.17em}-0.866,\hspace{0.17em}0\right)}^{\mathrm{T}}$has been suggested as they obviously have a larger probability to result in a smaller M³E than other input SOPs [1].

However, the above conclusion is valid only when the PDL/G is less than 0.35dB. Some optical systems may have a large PDL/G value from several dB to tens of dB, for example, a long-distance optical fiber communication system [2]. In such a system, PDL/G has adverse effect on both analog and digital optical signals [3, 4]. Combined effect of PDL/G and polarization mode dispersion (PMD) gives rise to anomalous pulse broadening and deteriorates the bit error rate [2, 5]. To monitor the PDL/G and PMD values in such a system, its Mueller matrix is required to be accurately measured [6, 7].

When the PDL/G has a finite value, such as 5 or 10 dB, the optimum input SOPs will definitely rely on the PDL/G of the system under test, including both the modulus and the direction of PDL/G vector. In this paper, we will present the detailed theoretical and simulation results of this problem. Firstly, in Section 2, the condition number (CN) of the matrix $F_{\text{ou}\mathrm{t}}$($F_{\text{ou}\mathrm{t}}$has been defined in Ref [1].) is used to evaluate M³E. By calculating the minimum of this CN, the relationship among the three optimum input SOPs, which can lead to a smaller M³E in a single test, is clearly presented. Secondly, in Sections 3 and 4, the statistical relationship between M³E and the three input SOPs is investigated under the same two preconditions mentioned above. Finally, some simulation results are used to verify the theoretical findings.

2. Optimization using CN as the criterion

In this section, the measurement errors of both input and output SOPs will be considered. From Eq. (10) of Ref [1], we have

When the Frobenius matrix norm is adopted, the detailed expression of the CN is calculated as

Due to ${\overrightarrow{S}}_{\text{ou}\mathrm{t}}=\tilde{M}{\overrightarrow{S}}_{\mathrm{i}\mathrm{n}}$, considering the input lights are partially polarized, we can obtain the generalized form of Eq. (7) in Re [1]. Then, an equation group, relating the input and output powers, can be derived as

Fortunately, most of the light sources used in modern polarization measurement systems, such as the tunable laser source in a polarization-mode dispersion measurement system [11] and the photoconductive switch emitter in a terahertz time-domain spectroscopy system [12], are completely polarized. Moreover, the input power can remain unchanged by using some well-designed polarization state generation approaches [7, 13]. Therefore, in the rest of this paper, we consider that $s_{\mathrm{i}\mathrm{n}0}=t_{\mathrm{i}\mathrm{n}0}=u_{\mathrm{i}\mathrm{n}0}$ and ${\rho }_{\mathrm{i}\mathrm{n}}^{\mathrm{s}}={\rho }_{\mathrm{i}\mathrm{n}}^{\mathrm{t}}={\rho }_{\mathrm{i}\mathrm{n}}^{\mathrm{u}}=1$ are always satisfied for both theoretical analysis and simulations. Further, since all input and output Stokes vectors can be normalized by the input power, $s_{\mathrm{i}\mathrm{n}0}=t_{\mathrm{i}\mathrm{n}0}=u_{\mathrm{i}\mathrm{n}0}=1$ can be adopted without any influence to the results.

Under the above conditions, Eqs. (3) and (4) can be simplified as

 

Fig. 1 The relationships between the optimum angles of inputs and the values of PDL/G when the CN takes the global minimum.

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It is evident that ${\alpha }_{\text{op}\mathrm{t}}$is close to 120° when PDL/G is small. This is consistent with the conclusion we have obtained in Ref [1]. When PDL/G increases, the optimum angle ${\alpha }_{\text{op}\mathrm{t}}$ decreases. For a given PDL/G vector, the three optimum input Stokes vectors should be equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector as shown in Fig. 2 .

 

Fig. 2 The relative relationship of three input Stokes vectors and the PDL/G vector on the Poincaré sphere to achieve the minimum of the CN.

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Figure 3 shows that, for different values of PDL/G, the minimum of the CN is also different. It is obvious that M³E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, its Mueller matrix cannot be accurately measured by using only three inputs in a single test.

 

Fig. 3 The relationship between the minimum of the CN and the values of PDL/G.

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Since the CN in Eq. (9) is a function of six angles ${\theta }_{\mathrm{s}},\hspace{0.17em}{\theta }_{\mathrm{t}},\hspace{0.17em}{\theta }_{\mathrm{u}}$,${\alpha }_{\mathrm{i}\mathrm{n}},\hspace{0.17em}{\beta }_{\mathrm{i}\mathrm{n}},\hspace{0.17em}{\gamma }_{\mathrm{i}\mathrm{n}}$, it is impossible to illustrate the whole function. In Fig. 4 , we only present a curve to partially show this function, where $D=0.5195$ (5 dB), ${\theta }_{\mathrm{s}}={\theta }_{\mathrm{t}}={\theta }_{\mathrm{u}}=\theta$and${\alpha }_{\mathrm{i}\mathrm{n}}={\beta }_{\mathrm{i}\mathrm{n}}={\gamma }_{\mathrm{i}\mathrm{n}}=\alpha$. Please note, in Fig. 4(a), α is related to θ by Eq. (11) with the minus sign.

 

Fig. 4 The relationships between the CN and (a) the angle αand (b) the angle θ when the value of PDL/G is 5 dB. The insets show the “zoom in” views of the same data.

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In this section, we use the CN as the criterion to find out the appropriate input SOPs. The results show that the minimum CN is achieved when the three input SOPs are equally-spaced on the Poincaré sphere and centered on the reversed PDL/G vector. The larger the PDL/G, the closer the three input SOPs is to the reversed PDL/G vector.

From Eq. (2), M³E depends on not only the CN, but also on the noise realization. When many tests can be performed, the mean of M³E should be investigated. To carry out such an investigation, we must know the statistical properties of the Stokes parameter measurement errors in advance. In this paper, we assume that all Stokes parameter measurement errors independently and identically follow the Gaussian distribution $N\left(0,{\sigma }^2\right)$, which has been used in Ref [1]. This means that we assume that an ideal polarimeter, which has such statistical properties, is used in the measurement system.

3. Statistical properties of $\Delta \sqrt{\left|\tilde{M}\right|}$

In the following, we still consider that the measurement errors of the input SOPs can be neglected as we have done in Ref [1]. Under the conditions $\Delta s_{\text{out}\mathrm{j}},\hspace{0.17em}\Delta t_{\text{out}\mathrm{j}},\Delta u_{\text{out}\mathrm{j}}\hspace{0.17em}\left(\mathrm{j}=1,2,3,4\right)\sim N\left(0,{\sigma }^2\right)$, we know that $〈\Delta \sqrt{\left|\tilde{M}\right|}〉=0$. Hence, the variance of $\Delta \sqrt{\left|\tilde{M}\right|}$ is needed to evaluate the measurement uncertainty. It can be derived as

Obviously, $K_{\Delta \left|M\right|}$ is also a symmetric function of the three input SOPs. Based on the Purkiss Principle and numerical calculations, its global minimum is also achieved when ${\theta }_{\mathrm{s}}={\theta }_{\mathrm{t}}={\theta }_{\mathrm{u}}=\theta$, ${\alpha }_{\mathrm{i}\mathrm{n}}={\beta }_{\mathrm{i}\mathrm{n}}={\gamma }_{\mathrm{i}\mathrm{n}}=\alpha$. When $K_{\Delta \left|M\right|}$ takes the global minimum, the relationships between the optimum angles and the PDL/G is shown in Fig. 5 using solid lines. As a comparison, the optimum angles, corresponding to the CN, are also plotted in Fig. 5 using dashed lines. It is clear that they are different corresponding to the same value of PDL/G.

 

Fig. 5 The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $K_{\Delta \left|M\right|}$ and dash lines are based on the minimum of the CN.

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4. Upper bound of $〈\Vert \Delta \tilde{M}\Vert 〉$

When D has a finite value, from Eq. (14) of Ref [1], we have

Based on the definition of the Frobenius matrix norm, it is easy to know that $\Vert \Delta \tilde{M}\Vert =\Vert \Delta {\tilde{M}}^{\mathrm{T}}\Vert$. Then, by substituting Eq. (18) into Eq. (16), we have

Unfortunately, we cannot directly calculate $〈\Vert \Delta \tilde{M}\Vert 〉$ because of the difficulties in mathematics. As an alternative, we can calculate an upper bound as

 

Fig. 6 The relationships between the optimum angles and the PDL/G. Solid lines are based on the minimum of $K_{\Vert \Delta M\Vert }$ and dash lines are based on the minimum of the CN and$K_{\Delta \left|M\right|}$.

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For different values of PDL/G, the minimum of $K_{\Vert \Delta M\Vert }$ is also different. As shown in Fig. 7 , M³E will dramatically increase when the value of PDL/G is up to tens of dB. Therefore, when the system under test has such a big PDL/G, the mean of the Mueller matrix also cannot be accurately measured.

 

Fig. 7 The relationship between the minimum of $K_{\Vert \Delta M\Vert }$ and the values of PDL/G.

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When the value of PDL/G is 5 dB, the relationship between $K_{\Vert \Delta M\Vert }$ and the angle α is shown in Fig. 8 . The curve in Fig. 8 gives us a partial information of the function of $K_{\Vert \Delta M\Vert }$.

 

Fig. 8 The relationship between $K_{\Vert \Delta M\Vert }$and the angle αwhen the value of the PDL/G is 5 dB. The inset shows the “zoom in” view of the same data.

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5. Simulation results

To verify the theoretical finding in Section 4, simulations are performed. The parameters of the system under simulation are 1) Birefringence:$\varphi =5\pi /3$, $r=\left(0.66,\hspace{0.17em}0.74,\hspace{0.17em}-0.1296\right)$; 2) PDL/G: $0<D<1$, $\stackrel{⌢}{D}={\left(0,\hspace{0.17em}0,\hspace{0.17em}1\right)}^{\mathrm{T}}$and 3) PIDL/G: $T_{\mathrm{u}}=1$. In Section 4, the theoretical result shows that the upper bound does not depend on the PIDL/G $T_{\mathrm{u}}$. Then, we take $T_{\mathrm{u}}=1$ in the following simulations. In this paper, we only show the simulation results when ${\alpha }_{\mathrm{i}\mathrm{n}}={\beta }_{\mathrm{i}\mathrm{n}}={\gamma }_{\mathrm{i}\mathrm{n}}=\alpha$and ${\theta }_{\mathrm{s}}=\hspace{0.17em}{\theta }_{\mathrm{t}}=\hspace{0.17em}{\theta }_{\mathrm{u}}=\theta$. Then, the three input SOPs can be written as

 

Fig. 9 The relationships between the theoretical upper bound (in red), simulation results (in blue) of $〈\Vert \Delta \tilde{M}\Vert 〉$ and (a) the angle αand (b) the angle θ when the values of PDL/G are 5 dB, 10 dB and 13 dB, respectively.

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6. Conclusion

We presented the relationship between three input SOPs and M³E in a system having birefringence and a finite PDL/G. Firstly, by using the CN as the criterion, it has been demonstrated that the three optimum input SOPs should be equally-spaced and centred on the reversed PDL/G vector for achieving a smaller M³E in a single test. Secondly, the statistical relationship, which is expressed as an upper bound of the mean of M³E, has been derived when SOP measurement errors follow the same Gaussian distribution. This upper bound also tells us that the minimum M³E will be statistically achieved when the three input SOPs are equally-spaced and centred on the reversed PDL/G vector. Finally, the simulation results confirm the validity of the proposed conclusion.

This conclusion can be used in real measurements. If the PDL/G vector of the system under test can be known before the measurement, the three input SOPs, with the relative relationship shown in Section 2, should be adopted in a single test. If the PDL/G vector is unknown, the polarimeter has the statistical properties we have used in this paper and the measurement can be repeated for many times [14], the measurement can be performed using the following steps:

Moreover, if the measurement errors of input SOPs cannot be neglected, the three statistically optimum input SOPs will depend on not only the PDL/G vector, but also on the birefringence. A detailed analysis will be presented in another paper.

On the other hand, if the polarimeter used in the measurements does not have the statistical properties we adopted in this paper, the statistical relationship between M³E and the three input SOPs will not be as same as the one we obtained in this paper. It will be polarimeter-dependent. Further analysis will be presented in other papers.