1. Introduction

It is well known that the state of polarization (SOP) of an electromagnetic (EM) wave is completely depicted by a set of four Stokes parameters [1]. Stokes polarimeters are optoelectronic instruments to measure Stokes parameters of an EM wave under test; among them, complete Stokes polarimeters are those that can measure all of four Stokes parameters. Rotatable retarder fixed polarizer (RRFP) Stokes polarimeters are probably the simplest complete Stokes polarimeters, which are constructed with a rotatable retarder (also termed waveplate), a fixed polarizer and a photodetector. A schematic illustration of a RRFP Stokes polarimeter is shown in Fig. 1. Benefiting from such a simple and reliable structure, RRFP Stokes polarimeters have been widely used in many applications and can be easily integrated in polarization measurement systems,such as polarimetric optical time domain reflectometry (POTDR) systems or Terahertz frequency domain spectroscopy (THz-FDS) systems [2,3].

 

Fig. 1 The schematic structure of a RRFP Stokes polarimeter.

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RRFP Stokes polarimeters determine the four Stokes parameters of an incident EM wave by rotating the retarder to $N\left(N\ge 4\right)$angular orientations and measuring the corresponding optical intensities with the photodetector. In practice, two types of angular orientations of the retarder are mostly used in RRFP Stokes polarimeters. The first type of angular orientations is $N\left(N\ge 5\right)$angles uniformly spaced over 180° or 360° [4]. Most of commercialized RRFP Stokes polarimeters, Thorlabs PAX5720 for instance, adopt such a scheme by rotating a quarter-wave plate (90° retarder) at a constant angular velocity and performing a discrete Fourier transform of data [5]. The second type of angular orientations includes some sets of non-uniformly spaced angles that are chosen by optimizing various figures of merit, for example (−51.7°, −15.1°, 15.1°, 51.7°), (−45°, 0°, 30°, 60°) or (−59.6°, −36.8°, −10.1°, 10.1°, 36.8°, 59.6°) [6,7].

All photodetectors have intrinsic noises, which induce the random fluctuations in the intensity measurements. In practice, two kinds of noises are dominant in the commonly-used photodetectors, which are the signal-independent Gaussian thermal noise and the signal-dependent Poisson shot noise [8,9]. In RRFP Stokes polarimeters, these photodetector noises are important error sources and lead to the measurement noises of four Stokes parameters [6]. D. S. Sabatke et al (2000) and F. Goudail (2009) theoretically investigated the noise properties of RRFP Stokes polarimeters in the presence of Gaussian noise and Poisson noise, respectively [7,10]. They found, for the RRFP Stokes polarimeters employing the second type of angular orientations (four optimized angles), that 1) a 132° retarder and an angle set (−51.7°, −15.1°, 15.1°, 51.7°) are optimum to achieve the smallest measurement noises of Stokes parameters by using the equally weighted variance (EWV) as the figure of merit [7,10]; 2) for the above-mentioned polarimeter structure, a diagonal noise covariance matrix on Stokes parameters can be obtained and noise equalization on the last three Stokes parameters can be achieved for Gaussian noise [7]. These results are very useful in optimizing the RRFP Stokes polarimeters employing the second type of angular orientations, achieving noise equalization in Stokes parameter images and analyzing the Mueller matrix measurement systems employing such Stokes polarimeters.

It has been demonstrated that the RRFP Stokes polarimeters employing $N\left(N\ge 5\right)$uniformly spaced angles are more robust than those employing the second type of angular orientations in reducing the measurement errors induced by the imperfections of the retarder [11]. However, the noise properties of the RRFP Stokes polarimeters employing the first type of angular orientations ($N\ge 5$uniformly spaced angles over 180° or 360°) have not been investigated sufficiently to date. For Gaussian noise, one numerical result presented in [7] shows that uniformly spaced angles lead to a slightly larger EWV than four optimized angles (−51.7°, −15.1°, 15.1°, 51.7°) when 90° or 132° retardance is adopted. But there was no analytical result presented to show the relationship between the covariance matrices and the retardance. Hence, the retardances leading to the minimum EWV and noise equlization cannot be found out. Further, there is no result reported for Poisson noise at all, to the best of our knowledge.

In this paper, we address the noise properties of the RRFP Stokes polarimeters employing $N\left(N\ge 5\right)$uniformly spaced angles by theoretical analysis and simulations. The noise covariance matrices on Stokes parameters are derived analytically for both Gaussian noise and Poisson noise. These noise covariance matrices show that the measurement noises of Stokes parameters depend seriously on the retardance of the retarder. For Gaussian noise dominated systems, it can be concluded that 1) the retardance 130.48° can lead to the smallest EWV; 2) the retardance in the range from 126.06° to 134.72° can result in a nearly-minimum EWV by only 1% larger than the minimum; 3) the retardance 126.87° can result in the equalized measurement noises of the last three Stokes parameters. For Poisson noise dominated systems, it can be found that 1) the measurement noises of Stokes parameters depend on the SOP under test; 2) the retardance in the range from 126.06° to 134.72° can still result in nearly-minimum measurement noise for Poisson noise; 3) $N=5,10,12$uniformly spaced angles over 360° have slightly different covariance matrices that depend on the initial angle. To verify these theoretical findings, simulations are performed to show the good agreements between the simulation results and theoretical results.

2. Covariance matrix of the measurement noises of Stokes parameters

Four Stokes parameters are usually written in a form of Stokes vector $\overrightarrow{S}={\left(s_0,s_1,s_2,s_3\right)}^T$, where the superscript “T” denotes the transposition of a vector or a matrix. RRFP Stokes polarimeters determine a Stokes vector $\overrightarrow{S}$ by performing $N\left(N\ge 4\right)$ intensity measurements $\overrightarrow{I}={\left(I_0,I_1,\cdots ,I_N\right)}^T$corresponding to $N$ angular orientations of the retarder. In the absence of any errors, two vectors are related by [10]

In a photodetector, the intensity measurement error $\Delta I$ follows a stationary random process. For a well-calibrated photodetector, $\Delta I$ should have a zero mean $〈\Delta I〉=0$. Hence, the measurement noise of a photodetector is depicted by the variance ${\sigma }^2=〈\Delta I^2〉$. In a photodetector, Gaussian thermal noise and Poisson shot noise are two fundamental noise sources. Gaussian thermal noise is generated by the load resistor, which is described by Gaussian statistics. The variance of Gaussian noise ${\sigma }_G^2=〈\Delta I_G^2〉$ is independent on the intensity $I$ [8,9]. Poisson shot noise is a manifestation of the fact that an electric current consists of a stream of electrons that are generated at random times. Mathematically, it is a stationary random process with Poisson statistics. Hence, the variance of Poisson shot noise ${\sigma }_P^2=〈\Delta I_P^2〉=kI$($k$is constant for a given photodetector and explained in [8] and [9]) is proportional to the intensity $I$ [8,9]. From the properties of Gaussian noise and Poisson noise, the fluctuations $\Delta I_i,i=1,2,\cdots ,N$ are statistically independent from one intensity measurement to the other [10]. Hence, it has $〈\Delta I_i\Delta I_j〉=0,i\ne j$. Therefore, from Eq. (5), it has

3. Gaussian noise

When Gaussian noise dominates in a RRFP Stokes polarimeter, the variances of the intensity fluctuations are independent on the intensities. Then, it has [8,9]

To obtain an overall evaluation of the measurement noises, the sum of four measurement noises of Stokes parameters ${\displaystyle \sum _{i=0}^3〈\Delta s_i^2〉}$ is used, which is equal to the trace of the covariance matrix ${\Gamma }_G$. From Eq. (8) and Eq. (9), it is calculated as

The relationship between the EWV and the retardance $\delta$of the retarder, governed by Eq. (13), is illustrated in Fig. 2. It is obvious that the EWV reaches its minimum value 10.43 × (4/N) when the retardance is 130.48°. As a comparison, the EWV has a value of 21 × (4/N) when a commonly-used quarter-wave plate (the retardance is 90°) is used, which is 2 times larger than the minimum. Further, it can be easily observed from the inset of Fig. 2 that the EWV only increases by 1% from the minimum in the retardance range from 126.06° to 134.72°. It means the retarder in this retardance range can result in nearly minimum overall measurement noises by using the EWV as the criterion.

 

Fig. 2 The relationship between the EWV and the retardance when Gaussian noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

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On the other hand, the noise equalization$〈\Delta s_1^2〉=〈\Delta s_2^2〉=〈\Delta s_3^2〉$ is expected in some applications, such as Stokes parameter imaging [10,13,14]. From Eq. (9) and Eq. (10), it can be easily observed that $〈\Delta s_1^2〉=〈\Delta s_2^2〉$ is always valid for any retardance. Further, to find where $〈\Delta s_1^2〉=〈\Delta s_3^2〉$ is valid, two curves in Fig. 3 are plotted to show the relationships between $〈\Delta s_1^2〉$, $〈\Delta s_3^2〉$ and the retardance $\delta$, respectively.

 

Fig. 3 The relationships between the meansurement noises of Stokes parameters $〈\Delta s_i^2〉,i=0,1,3$ and the retardance. The inset shows the “zoom in” views of the same data.

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It is obvious that when the retardance is 126.87°, the noise equalization $〈\Delta s_1^2〉=〈\Delta s_2^2〉=〈\Delta s_3^2〉$ can be achieved. Actually, the retardance of 126.87° can lead to a special covariance matrix as

4. Poisson noise

When Poisson noise dominates, the variances of the intensity fluctuations are proportional to the intensities [8–10]. Then, it has

Then, from Eq. (6), Eq. (17) and Eq. (18), the covariance matrix can be derived a

Similarly, to evaluate the measurement noises using an overall factor, the sum of the four measurment noises, ${\displaystyle \sum _{i=0}^3〈\Delta s_i^2〉}$, can be calculated from Eq. (21) as

 

Fig. 4 The relationship between C₀, C₁ defined in Eq. (23) and the retardance when Poisson noise dominates in a RRFP Stokes polarimeter employing N uniformly spaced angles. The inset shows the “zoom in” views of the same data.

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Last but not least, in the derivations of the Eqs in Section 3 and Section 4, $N$uniformly spaced angles over 360°, viz. ${\theta }_1,{\theta }_1+{360}^{\circ }/N,\cdots ,{\theta }_1+\left(N-1\right){360}^{\circ }/N$, have been used. From the measurement matrix shown in Eq. (2), it is easy to understand that the same results can be achieved for $N$uniformly spaced angles over 180°, viz. ${\theta }_1,{\theta }_1+{180}^{\circ }/N,\cdots ,{\theta }_1+\left(N-1\right){180}^{\circ }/N$ [11]. In the case of 180°, $N$ can be any integer larger than 4. In this case, for Poisson noise, when $N=5$, the covariacne matrix is described by Eq. (30) in Appendix I; when $N=6$, the covariance matrix is described by Eq. (36) in Appendix II; however, when $N=10$ or $N=12$, the covariacne matrix is still depicted by Eq. (21).

5. Simulation results

Simulations are performed to verify the theoretical findings in Section 3, Section 4, Appendix I and Appendix II. Simulations are carried out for 100 different incident SOPs, which uniformly span the whole Poincaré Sphere. Due to the page limit, only the results of the SOP ${\left(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3}\right)}^T$ are shown in this paper. In the first step, Gaussian noise is considered and the variance is taken as ${\sigma }_G^2=0.01$. $N=5$uniformly-spaced angles over 180°, viz. (−90°, −54°, −18°, 18°, 54°) are used in the simulations. The simulations are performed in the retardance range from 40° to 170° with 1° step with the following procedures: 1) for a given retardance $\delta$, calculate the measurement matrix $W$; 2) calculate the error-free intensity vector using $\overrightarrow{I}=W\overrightarrow{S}$; 3) generate 360000 Gaussian noise error vectors $\Delta \overrightarrow{I}$with the above-given variance; 4) calculate the mesurement errors with $\Delta \overrightarrow{S}=W^{-1}\left(\overrightarrow{I}+\Delta \overrightarrow{I}\right)-\overrightarrow{S}$ for each noise realization; 5) calculate the variance of $\Delta \overrightarrow{S}$. Both the simulation results and the theoretical results calculated using Eq. (8) ~Eq. (10) are plotted in Fig. 5 for comparison. It is obvious that, for the measurement noises of all of four Stokes parameters, good agreements between the simulation results and the theoretical results are achieved.

 

Fig. 5 Simulation and theoretical results of $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ for Gaussian noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

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In the second step, Poisson noise is taken into simulation and the coefficient $k$is taken as $k=0.01$. The simulation procedures are the same as the above except Poisson noises are applied here. Firstly, $N=5$uniformly-spaced angles over 180°, viz. (−90°, −54°, −18°, 18°, 54°) are still used in the simulation. Because $N=5$, the covariance matrix should be calculated from Eq. (30) that is derived in Appendix I. However, since ${\theta }_1=-{90}^{\circ },$then $e_1=\sin \left(10{\theta }_1\right)=0,$thus the theoretical results of the $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ can still be calculated from Eq. (21). Both the simulation results and the theoretical results are shown in Fig. 6. Once again, good consistences between the simulation results and the theoretical results can be observed from all of four figures in Fig. 6.

 

Fig. 6 Simulation and theoretical results of $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ for Poisson noise and the five uniformly spaced angles (−90°, −54°, −18°, 18°, 54°). The inset shows the “zoom in” views of the same data.

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Secondly, $N=5$uniformly-spaced angles over 180° are chosen as (−81°, −45°, −9°, 27°, 63°). Because ${\theta }_1=-{81}^{\circ },$ then $e_1=\sin \left(10{\theta }_1\right)=-1,$ hence the theoretical results of the $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ cannot be obtained from Eq. (21), but Eq. (30) that is derived in Appendix I. The simulation results, together with two sets of theoretical results calculated from Eq. (21) and Eq. (30) respectively, are shown in Fig. 7. From the results in Fig. 7, we can conclude that 1) for N = 5 uniformly-spaced angles, the covariance matrix is determined by Eq. (30) when Possion noise dominates; 2) the difference between Eq. (30) and Eq. (21) is fairly small, especially when the retardance is getting larger.

 

Fig. 7 Simulation and theoretical results of $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ for Poisson noise and the five uniformly spaced angles (−81°, −45°, −9°, 27°, 63°). The inset shows the “zoom in” views of the same data.

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Finally, $N=12$uniformly-spaced angles over 360° are chosen as (0°, 30°, 60°, …, 300°, 330°), the simulation results for Poisson noise, together with two sets of theoretical results calculated from Eq. (21) and Eq. (36) that is derived in Appendix II, respectively, are shown in Fig. 8. It is obvious that 1) for $N=12$uniformly-spaced angles, the covariance matrix is determined by Eq. (36) when Possion noise dominates; 2) the difference between Eq. (36) and Eq. (21) is fairly small, especially when the retardance is getting larger.

 

Fig. 8 Simulation and theoretical results of $〈\Delta s_{{}_i}^2〉,i=0,1,2,3$ for Poisson noise and the twelve uniformly spaced angles (0°, 30°, 60°, …, 300°, 330°). The inset shows the “zoom in” views of the same data.

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7. Conclusion

In this paper, we investigated, by theoretical analysis and simulation validation, the measurement noises of Stokes parameters induced by photodetector noises in RRFP Stokes polarimeters employing $N\left(N\ge 5\right)$uniformly spaced angles over 180° or 360°. The noise covariance matrices on Stokes parameters are derived analytically for both Gaussian noise and Poisson noise dominated situations. These noise covariance matrices show that the measurement noises of Stokes parameters depend heavily on the retardance of the retarder. For Gaussian noise dominated systems, the retardance 130.48° can lead to the smallest EWV and the retardance 126.87° can result in the equalized measurement noises of Stokes parameters. For Poisson noise, the measurement noises of Stokes parameters are dependent on the incident SOP under test. If using the sum of the measurement noises of four Stokes parameters as the figure of merit, the retardance in the range from 126.06° to 134.72° can result in nearly-minimum measurement noise for both Gaussian and Poisson noise. In the case of Poisson noise, $N=5,10,12$uniformly spaced angles over 360° have slightly different covariance matrices that depend on the initial angle.