1. Introduction

Measurement of the state of polarization (SOP) of an electromagnetic (EM) wave is attracting more and more attention because it plays an important role in many regions, such as biomedicine [1], solar astronomy [2] and remote sensing [3]. The Stokes vector is a widely accepted means to describe the SOP of an incident EM wave. The simplest complete Stokes polarimeter is a rotatable retarder fixed polarizer (RRFP) Stokes polarimeter with a rotatable retarder, a fixed polarizer and a photodetector, which is shown in Fig. 1. Often, a quarter-wave retarder is chosen in a commercial polarimeter. Its structure is simple, thus a RRFP Stokes polarimeter can be easily integrated in many application systems, such as imaging polarimeters [4], Mueller matrix polarimeters [5] and solar polarimetric telescopes [6].

 

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

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A RRFP Stokes polarimeter determines the four Stokes parameters of an incident EM wave by rotating the retarder to N (N ≥4) different angular orientations and measuring the corresponding optical intensities of the EM wave. There are three potential error sources in such measurements: first, the random noise in the measured intensity; second, deviations in nominal values of retardance and diattenuation of the polarization optics; and third, the relative angular orientation errors away from the nominal angular orientations. These three error sources introduce Stokes parameter errors because they all induce errors in the measured intensity of the incident EM wave.

The first two error sources have been systematically investigated. The noise properties of RRFP Stokes polarimeters in the presence of Gaussian noise were theoretically investigated by Sabatke and Tyo [7, 8]. They found that a 132° retarder and an angle set of (−51.7°, −15.1°, 15.1°, 51.7°) can result in the largest SNR for the system. Moreover, Sabatke found that uniformly spaced angles over 360° are a good choice, and the practical difficulty and time expenditure of aligning the retarder at many different non-uniformly spaced angles is unnecessary [7]. Goudail looked at signal-dependent shot noise, and found that in a certain circumstance, the polarimeters optimal for Gaussian noise are also optimal for signal-dependent shot noise [9]. H. Dong (2013) investigated the measurement noises of Stokes parameters resulted from photodetector noises in RRFP Stokes polarimeters employing uniformly spaced angles over 180° or 360° by theoretical analysis and simulation [10]. For the second source of error, such as retarder non-uniformity and low extinction ratio of a linear polarizer, Tyo and Wei had determined that a low extinction ratio of a linear polarizer did not depress the signal-to-noise ratio [11]. H. Dong (2012) studied measurement errors induced by retardance deviation and derived theoretically the relationship between the retarance error and the measurement errors in RRFP Stokes polarimeters [12].

However, the above mentioned studies all assumed that the optical intensities were measured at the ideal retarder orientation angles. In practice, there will be angular orientation errors of the retarder during the procedure of measurement. Further, angular orientation error is far more important in a RRFP imaging polarimeter [11]. Tyo studied the sensitivity of polarimeters to this kind of error source analytically, and discussed the relationship between system condition and error minimization by means of numerical analysis [13]. But the analytical result presented in [13] can only be used for the case that the incident SOPs are uniformly distributed over the Poincaré sphere.

In this paper, we address the third error source of RRFP Stokes polarimeters measuring intensities in N (N≥5) uniformly spaced angles by theoretical analysis and simulation. The retarder misalignment errors are treated as statistic errors, and the covariance matrices of measurement errors induced by the retarder misalignment errors are derived analytically. Based on the covariance matrices, it can be concluded that 1) the measurement errors from the retarder misalignment errors are independent of the incident intensity s₀, but being dependent on Stokes parameters s₁, s₂, s₃ under test; 2) for any incident SOP, the optimal configuration of a RRFP polarimeter including the initial angle and the retardance can be achieved; 3) for the case of N = 5, 10, 12, the effect of the retarder misalignment error has been remarkably weakened when an initial angle leading to the smallest WS is used; 4) the covariance matrix for N uniformly spaced angles over 180° has the same form as that for 2N uniformly spaced angles over 360°. In the end of the paper, simulations are performed to verify that these theoretical results agree well with the simulation results.

2. Theoretical analysis

RRFP Stokes polarimeters determine the Stokes vector $\overrightarrow{S}$ of an incident EM wave by obtaining N (N≥4) intensity measurements $\overrightarrow{I}={(I_0,I_1,\cdots ,I_N)}^T$ corresponding to N angular orientations of the retarder. Without any errors, the two vectors are related by

When there is only the misalignment error of the retarder, the real measurement matrix of the Stokes polarimeter is denoted by $W\text{'}$. Supposed that$W\text{'}=W+\text{Δ}W$, then the measurement error of the intensity induced by the misalignment error of the retarder is governed by

To obtain the relationship between the intensity error $\Delta \overrightarrow{I}$and the angular orientation error$\overrightarrow{\xi }$, Eq. (3) is transformed as

When a RRFP Stokes polarimeter is operating, the retarder is rotated harmonically to modulate the incident EM wave. The angular orientation errors of the retarder arise from the following factors: 1) the initial angle deviation from the nominal value θ₁; 2) non-harmonic rotation of the retarder; 3) non-uniform time intervals of intensity measurements. The sources of these kinds of errors are so varied that the angular orientation errors of the retarder are assumed to be independent of each other and obey a uniform normal distribution to simplify the analysis, and the standard deviation of the distribution is σ. Based on this assumption, the measurement error of the Stokes vector $\overrightarrow{\in }$ consists of four random variables, and their variances are usually used to assess the measurement errors of Stokes parameters [7, 14]. From Eq. (8), the covariance matrix on Stokes parameters can be derived as

During the procedure of deriving Eq. (14) and Eq. (15), lots of trigonometric identities are used, which all have been validated by MATLAB programs. From Eq. (11), Eq. (14) and Eq. (15), when N uniformly spaced angles θ₁, θ₁ + 360°/N, …, θ₁ + (N-1)360°/N (N = 9, 11, 13, 15, 17, 18, 19, …,∞) are used, the matrix Γ can be given by

From Eq. (11), the matrix Γ is independent of the alignment error of the retarder. Its diagonal elements represent gain factors in the propagation of variance from the measurements to the Stokes vector estimate. The off-diagonal elements indicate where correlations in errors in the components of the Stokes vector estimate arise [14], which have been derived, but are not shown here because they won’t be used in this paper. The diagonal elements of the covariance matrix Γ_∈ denote the corresponding measurement errors of the four Stokes parameters. So the sum of four measurement errors of Stokes parameters ${\displaystyle {\sum }_{i=0}^3〈{\in }_i{}^2〉}$, which is equal to the trace of the covariance matrix Γ_∈, can be used to evaluate the overall measurement error induced by angular orientation errors of the retarder. Based on Eq. (16), it is given as

From Eq. (17), it is concluded that the measurement error induced by the angular orientation of retarder is independent of the incident intensity s₀, but seriously depends on the incident Stokes parameters (s₁, s₂, s₃) and the retardance δ. The measurement error is also inversely proportional to N, which agrees well with the prediction in [8]. With the number of intensity measurement N increasing, the measurement error decreases rapidly. But when N exceeds some value, the change will not be obvious any more. Since the polarization degree of an EM wave is defined by Eq. (19), we can conclude that for the case of N = 9, 11, 13, 15, 17, 18, 19 …∞, the higher the polarization degree of an incident EM wave is, the larger the measurement error is.

The coefficients C₁, C₂, C₃, C₄, C₅ and C₆ can be seen as weights of contributions of Stokes parameters (s₁, s₂, s₃) to the measurement error. The coefficients C₁, C₂ and C₄ denote the dependence of the measurement error on the linear polarized component in the incident EM wave. The third coefficient C₃ represents the dependence on the circular polarized component. The coefficients C₅ and C₆ indicate the dependence on the elliptical polarized component. The weight factors C₁, C₂, C₃ for the case of N = 9, 11, 13, 15, 17, 18, 19 … ∞ are plotted in Fig. 2. Based on Fig. 2, it is found that when the incident SOPs are mostly distributed near the equator of the Poincaré sphere, in order to minimize the measurement error, the retardance of the retarder should not be larger than 130°. For example, the quarter-wave retarder is a good choice in this instance. When the incident SOPs are mostly distributed over the two poles of the Poincaré sphere, the retardance of the retarder should not be smaller than 85°.

 

Fig. 2 The relationships between C₁, C₂, C₃, WS₁, WS₂ and the retardance δ for the case of employing N (N = 9, 11, 13, 15, 17, 18, 19…∞) uniformly spaced angles over 360°.

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If the statistic distribution of the incident SOPs under test can be theoretically predicted or be known from the previous data, a figure of merit, which is based on the weights on the importance of the respective Stokes components to the polarimeter’s intended application, can be designed by composing a weight sum of the six coefficients C₁, C₂, C₃, C₄, C₅ and C₆ to evaluate the measurement error. The figure of merit is defined as the Weight Sum (WS) by Eq. (20).

Again, when incident SOPs are uniformly distributed over the Poincaré sphere, it means that the mean incident SOP is ${\left(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3}\right)}^T$. The figure of merit WS₂, which is still shown in Fig. 2, is given by

By similar methods to the above, it can be obtained that when N = 5, 7, 10, 12, 14, 16, the matrix Λ is a function matrix of the initial angle θ₁. So the covariance matrix Γ_∈ is also a function matrix of the initial angle θ₁. These analytical results for N = 5, 7, 10, 12, 14, 16 will be presented in what follows.

When N (N = 5, 10) uniformly spaced angles θ₁, θ₁ + 360°/N, …, θ₁ + (N-1)360°/N are used, the corresponding matrix Γ becomes

We find that the covariance matrix for N = 5, 7, 10, 12, 14, 16 is a trigonometric function matrix of the initial angle θ₁, and the period is 360°/N. Though the analytical results for the case of N = 5, 7, 10, 12, 14, 16 are relatively complex, a RRFP Stokes polarimeter, which employs N = 5, 7, 10, 12, 14, 16 intensity measurements, is recommended. That is because that when N = 5, 7, 10, 12, 14, 16, the added degree of freedom by θ₁ can be used to optimize the configuration of a RRFP Stokes polarimeter.

From Eq. (23), Eq. (24), Eq. (25), and Eq. (26), the total measurement error induced by the angular orientation error for the case of N = 5, 7, 10, 12, 14, 16 can also be denoted by Eq. (17). Hence, for N = 5, 7, 10, 12, 14, 16, the weight sum (WS) of these six corresponding coefficients C₁, C₂, C₃, C₄, C₅, and C₆ can still be used to evaluate the measurement error induced by the misalignment error of the retarder. When the mean incident SOP is assumed as ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$, the figure of merit WSs for the case of N = 5, 7, 10, 12, 14, 16 are plotted in Fig. 3. It is obvious that the measurement error is a cosine function of the initial angle. As illustrated in Fig. 3, the amplitude for N = 7, 14, 16 is too small but that for N = 5, 10, 12 is large enough to optimize θ₁. RRFP Stokes polarimeters which use N (N = 5, 10, 12) uniformly spaced angles are recommended because an additional parameter of the initial angle θ₁ can be utilized to optimize their configuration. The optimal initial angle for the case of N = 5 or 10 is 12.89° + M·36°, while that for the case of N = 12 is 21.74° + M·30°. M is an arbitrary positive integer.

 

Fig. 3 The relationship between WS and the initial orientation angle θ₁ when N = 5 or 10, 7 or 14, 12, 16.

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When the mean incident SOP is supposed as ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$, all of these coefficients and WSs for the case N = 5 or 10, 7 or 14, 12, 16 are plotted in Fig. 4. It is not difficult to find that there must be an optimal retardance resulting in the smallest measurement error for all of N. The optimal initial angles and retardances for all of N discussed above are listed in Table 1.

 

Fig. 4 The coefficients and WSs for N = 5 or 10, 7 or 14, 12, 16 at the respective optimal initial angle when the mean incident SOP is ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$.

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Table 1. Optimal Initial Angle and Retardance for the Incident SOP ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$

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In addition, when employing N (N = 4, 6, 8) uniformly spaced angles over 360°, the measurement matrix W is a singular matrix. In this case, the RRFP polarimeter cannot determine all of Stokes parameters, which has no practical significance for a generalized RRFP Stokes polarimeter because a RRFP polarimeter is usually used as a complete polarimeter. So the cases of N = 4, 6, 8 are not covered in our discussion.

Although the discussion above is performed for the case of N uniformly spaced angles over 360°, it is easy to prove that the covariance matrix for N uniformly spaced angles over 180° has the same form as that for 2N uniformly spaced angles over 360°. It means that in the case of 180°, N can be any integer larger than 4. Furthermore, the results are different between 180° and 360° only when N is 10, 12, 14 and 16, which are illustrated in Fig. 5. It is observed that when N = 14 and 16, N uniformly spaced angles over 180° are better to minimize the measurement error. When N = 12, 360° is a better choice. There is no quite obvious difference between 180° and 360° for N = 10.

 

Fig. 5 The comparison of WSs for N (N = 10, 12, 14, 16) uniformly spaced angles over between180 °and 360 °.

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3. Simulation verification

In order to verify the theoretical results in Eq. (16), Eq. (23), Eq. (24), Eq. (25) and Eq. (26), a series of simulations are performed in this section. These simulations are performed with the following procedures: 1) use MATLAB to generate 10⁴ random angular orientation errors, which obey the same normal distribution; 2) calculate the measurement errors with $\overrightarrow{\in }=W^{+}W\overrightarrow{S}-\overrightarrow{S}$ for each angular orientation error; 3) calculate the variance $〈{\in }_i{}^2〉$of the four components of $\overrightarrow{\in }$; 4) use Eq. (16), Eq. (23), Eq. (24), Eq. (25) and Eq. (26) to calculate the theoretical result of the corresponding measurement error components. These simulations are carried out in the retardance range from 20° to 160° for 112 different incident SOPs, which are uniformly distributed over the whole Poincaré sphere. Here, only the two typical SOPs ${\left(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3}\right)}^T$ and ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$ are shown in Fig. 6 and Fig. 7, respectively. The first SOP is measured using an initial angle θ₁ = 9° and σ = 0.05, while the second SOP is measured using an initial angle θ₁ = 13° and σ = 0.02. Both the simulation results and theoretical results are illustrated in the same figure to compare, where the real line “-” represents the simulation result while the star “*” denotes theoretical result. It can be clearly observed that, the theoretical results agree well with the simulation results for the measurement errors of all of the four Stokes parameters for all cases. The small deviation between the two results mostly derives from the truncation error induced by the approximation in Eq. (4).

 

Fig. 6 Simulation and theoretical results of $〈{\in }_i{}^2〉,i=0,1,2,3$ at the SOP ${\left(1,1/\sqrt{3},1/\sqrt{3},1/\sqrt{3}\right)}^T$and the initial angle ${\theta }_1=9^{\circ }$with σ = 0.05. The real line “-” represents the simulation result while the star “*” denotes theoretical result.

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Fig. 7 Simulation and theoretical results of $〈{\in }_i{}^2〉,i=0,1,2,3$ at the SOP ${\left(1,1/\sqrt{2},1/\sqrt{3},1/\sqrt{6}\right)}^T$and the initial angle ${\theta }_1={13}^{\circ }$ with σ = 0.02. The real line “-” represents the simulation result while the star “*” denotes theoretical result.

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4. Conclusion

For a RRFP Stokes polarimeter, which employs N (N≥5) uniformly spaced angles over 180° or 360°, the measurement error induced by the angular orientation error of the retarder is investigated by theoretical analysis and verified by simulation. We assume that the angular orientation errors are random variables which obey a uniform normal distribution. The covariance matrices of the measurement errors are derived analytically for the misalignment errors of the retarder. Based on the covariance matrices, we conclude that 1) the measurement errors of Stokes parameters are independent of the incident intensity s₀, but depend heavily on the incident Sstokes parameters (s₁,s₂,s₃) and the retardance of the retarder δ; 2) whatever the incident SOP is, the optimal initial angle and retardance both can be obtained by the analytical results presented above; 3) for the cases of N = 5, 7, 10, 12, 14, 16, the corresponding covariance matrices are different and heavily depend on the initial angle θ₁, and the measurement errors are cosine functions of the initial angle θ₁; 4) the covariance matrix for N uniformly spaced angles over 180° has the same form as that for 2N uniformly spaced angles over 360°. Except N = 10, 12, 14, 16, there is nearly no difference between 180° and 360° for the same N. Further, RRFP Stokes polarimeters, which employ N (N = 5 or 10) uniformly spaced angles over 360° or N (N = 5) uniformly spaced angles over 180°, have stronger immunity to the misalignment error of the retarder when the appropriate initial angle is chosen.