We consider the typical Stokes polarimetry system, which performs four intensity measurements to estimate a Stokes vector. We show that if the total integration time of intensity measurements is fixed, the variance of the Stokes vector estimator depends on the distribution of the integration time at four intensity measurements. Therefore, by optimizing the distribution of integration time, the variance of the Stokes vector estimator can be decreased. In this paper, we obtain the closed-form solution of the optimal distribution of integration time by employing Lagrange multiplier method. According to the theoretical analysis and real-world experiment, it is shown that the total variance of the Stokes vector estimator can be significantly decreased about 40% in the case discussed in this paper. The method proposed in this paper can effectively decrease the measurement variance and thus statistically improves the measurement accuracy of the polarimetric system.
© 2015 Optical Society of America
Stokes polarimetry has many applications in physical instrumentation and imagery [1–4 ]. It consists of performing a series of intensity measurements that are linear combinations of the elements of the Stokes vector, and inverting this linear relation in order to retrieve the Stokes vector. In practice, Stokes vector measurements are always perturbed by noise, which causes the retrieved Stokes vector to deviate from the true value. For Stokes polarimetry, the improvement of the measurement accuracy of Stokes vector is of course a key issue.
Statistically speaking, the measurement accuracy of Stokes vector is determined by the variance of Stokes vector estimator. Up to now, various methods for reducing the variance of the estimator have been proposed [5–12 ] including optimizing the number of intensity measurements  and optimizing the measurement matrix of a polarization state analyzer (PSA) [6–8 ]. However, the total integration time is usually distributed equally to all the intensity measurements in the polarimetry, while the distribution of integration time of intensity measurements has not been considered yet, which could also considerably influence the variance of the Stokes vector estimator.
In this paper, we consider the typical configuration of passive Stokes polarimeters with four intensity measurements to estimate the Stokes vector [6, 7, 13], which are disturbed by additive Gaussian noise. For the constant total integration time, we investigate the dependence of the variance of Stokes vector estimator on the distribution of integration time of four intensity measurements in the Stokes polarimetry, and we obtain the analytical solution of the optimal distribution of integration times by employing Lagrange multiplier method. In addition, we perform the real world experiment to verify the effect of optimizing the distribution of integration time on decreasing the variance of Stokes vector estimator.
2. Relation between the variance and integration time
We consider Stokes polarimeters that perform four intensity measurements at four different states of PSA to estimate the Stokes vector. Let us denote S = (S 0, S 1, S 2, S 3)T the four-dimensional Stokes vector to estimate and I = (I 1, I 2, I 3, I 4)T the intensity vector consisted by four measured intensities. In the presence of additive noise, one has the relation:Eq. (1) refers to the zero mean additive Gaussian noise with standard deviation of σ. The noise leads to the variation of the measured intensity Ii, i ∈ [1, 4] with standard deviation same to σ in the statistical sense. The Stokes vector is retrieved from the measurements I by inverting Eq. (1): Eq. (2) that the measured intensity Ii is divided by the integration time ti, and then be transformed to the Stokes vector estimator Ŝ. Therefore, the integration time multiplicatively modulates the noise variation, and then the noise variation modulated by integration time is transformed to the variation of the Stokes estimator through W −1.
The mean of Ŝ is <Ŝ>= (TW)−1 <I>= (TW)−1 TW S = S, and therefore, Ŝ is an unbiased estimator. For the unbiased estimator, the measurement accuracy is determined by the variance. The variance of Ŝ with multiple dimensions can be characterized by its covariance matrix. According to Eq. (2), the covariance matrix of Ŝ can be deduced as:Eq. (1) that the variance of I is equal to the variance of the noise N, and thus ΓI is given by: Eq. (3), we can get the covariance matrix of Stokes vector as:
It can be seen from Eq. (6) that the total variance of Stokes estimator depends not only on the measurement matrix W, but also on the integration time of the intensity measurements. In particular, it is noticed in Eq. (6) that the total variance of the Stokes vector estimator is composed of four variance terms corresponding to ti, which refer to the influences of the noise in the intensity measurements of Ii on the total variance of Stokes estimator respectively. In addition, it is obvious that the noise in the intensity measurements of Ii influence the total variance of Stokes estimator with different magnitudes, which can be characterized by the weight determined by W and ti.
It can be analyzed from Eq. (6) that for the polarimetric system with a given W, the total variance of Stokes vector estimator depends on ti, which makes it possible to modulate the variance of the Stokes vector estimator by adjusting the distribution of integration time, while keeping the total integration time constant. In this case, the problem of optimizing the distribution of integration time to achieve the minimum variance of Stokes vector estimator can be expressed as:
3. Optimizing integration time by Lagrange multiplier method
Lagrange multiplier methods can be applied for the problem of constrained optimization , and thus it is applied to solve optimization problem given by Eq. (7). In order to facilitate the discussion, we denoteEq. (7) can be expressed as Eq. (9) is that the total integration time of four intensity measurements is constant. If the total integration time is considered to be 4, then the constraint is given by: 14]: Eq. (9) could be replaced by the equivalent problem:
The solution of ti for the optimization problem in Eq. (12) should fulfill that the partial derivatives of L equal to zero, which are given by:Eq. (13) should follow:
According to Eq. (14) and Eq. (6), it can be seen that the optimal distribution of the integration time equalizes the influences of the noise in the four intensity measurements on the total variance of Stokes estimator to achieve the minimum variance of the estimator. However, the equalized integration time cannot always equalize the influences of the noise due to the asymmetry of the measurement matrix W in some cases. In these cases, changing the integration time from an equal one to different ones for different states of PSA can balance the inequality of the influences of the noise in intensity measurements induced by W, and thus can decrease the variance of estimator to improve the polarization measurement accuracy.
It can be seen from Eq. (15) that the optimal distribution of integration time is independent of the measured Stokes vector, and it only relates to Ci and thus to the measurement matrix W. According to Eq. (15), when the total integration time is 4, then the close-form expression of optimal integration time of each intensity measurement is given by:Eq. (6), we can obtain the total variance of Stokes vector estimator at optimal integration times ti as :
While for the case of equipartition time(t 1 = t 2 = t 3 = t 4 = 1), the total variance of Ŝ can be calculated from Eq. (6) as:
According to the analysis above, we address an approach that can reduce the variance of Stokes vector estimator, by optimizing the distribution of integration time while keeping the total integration time constant. In particular, it is noticed in Eq. (19) that the performance of variance reduction by optimizing the distribution of integration time fully depends on W, while it does not depend on the value of the measure Stokes vector S.
4. Validation on experimental measurements
Let us illustrate the effect of optimizing the distribution of integration time for a Stokes po-larimeter with a given W, and the noise is assumed to be ideal additive Gaussian noise. For illustration purposes, we consider the measurement matrix of
According to the measurement matrix W given by Eq. (20), it can be directly calculated that C 1 = 147.3, C 2 = 7.2, C 3 = 7.0, C 4 = 160.0 based on the Eq. (8). It needs to be clarified that the variance of noise σ 2 is assumed to be a constant in this paper, which does not affect the optimal solution given by Eq. (16), and thus σ 2 is neglected in the calculation of Ci. It can be seen that the values of Ci are different from each other, which makes it possible to optimize the distribution of integration time to decrease the variance of the estimator. According to Eq. (9), the optimization problem can be described as:
In particular, C 1 and C 4 are much greater than C 2 and C 3, which indicates that the influences of the noise in the intensity measurements of I 1 and I 4 on the total variance are dominant, according to Eq. (21). Therefore, by increasing t 1 and t 4, the decrease of the variance terms corresponding to t 1 and t 4 can overcome the consequent increase of the variance terms corresponding to t 2 and t 3, and thus the total variance of Stokes vector estimator can be decreased.
According to Eq. (16), we can obtain the optimal distribution of the integration time as t 1 : t 2 : t 3 : t 4 = 1.46 : 0.53 : 0.52 : 1.49. We assume the total integration time is 4s, and thus the corresponding optimal integration times for four intensity measurements are
The variances of the Stokes vector estimators with equal and with optimal integration times of intensity measurements are shown in Table 1. It can be seen from Table 1 that the total variance of the Stokes estimator can be decrease by 40.5% with optimal integration times. Besides, it is noticed in Table 1 that the variances of S 2 and S 3 at optimal integration times are greater than those at equal integration time, which indicates that the optical integration time for the global variance of Stokes vector estimator can not make sure the decrease of the variance for all elements in the Stokes vector.
In order to verify the theoretical results above, we perform the corresponding real world experiment with the Stokes polarimetry setup shown in Fig. 1. A He-Ne laser is employed as the light source. The laser passes through the beam expanding system and is reflected by the mirror. Polarizer P1 is employed to modulate the polarization state (Stokes vector) of the light to be measured. The output polarized light is filtered by PSA, which is composed of a quarter wave plate (QWP) and a linear polarizer P2, and finally reach the CCD camera. By adjusting the orientations of the quarter wave plate and the Polarizer P2 in PSA, we can achieve the measurement matrix W given by Eq. (20).
In the case of low light intensity, the dominant source of noise can be considered additive Gaussian noise with a variance independent of the intensity of illumination . Therefore, we perform the experiment at low level of light intensity, and we check that the variance of the measured intensity is almost independent of the illumination intensity and integration time. In addition, it is also checked from the histograms of the measured intensities that the PDF (Probability Density Function) of the noise is Gaussian shape. Therefore, the statistical performance of the noise is close to that of the additive Gaussian noise in our experiment.
We assume the Stokes vector to be measured is S = (1, −1, 0, 0)T, and thus we adjust the orientation of the linear polarizer P1 in Fig. 1 to generate the linearly polarized light with the Stokes vector of S = (1, −1, 0, 0)T. In the experiment, we set the total integration time to be 400ms. Therefore, in the case of equipartition integration time, t 1 = t 2 = t 3 = t 4 = 100ms. While in the case of optimal integration times, it can be calculated based on Eq. (16) that t 1 = 146ms, t 2 = 53ms, t 3 = 52ms, t 4 = 149ms.
In order to analysis the statistical characteristic of the Stokes vector estimator, we perform 4×104 realizations, and we obtain the histogram of each element in Stokes vector estimator (see Fig. 2). It can be seen that the measured elements of Stokes vector follow Gaussian distribution. By comparing the histograms in Fig. 2(a) with the corresponding ones in Fig. 2(b), we clearly note that the profiles for S 0 and S 1 become more narrow after optimizing the integration times, while the profiles for S 2 and S 3 become broader after optimizing the integration time. Since the scales of the horizontal axis for S 2 and S 3 are much less than those of S 0 and S 1, the global variance of the Stokes vector estimator should be decreased after optimizing the integration time.
We can quantify these contributions with reference to Table 2, which quantitatively shows the variance before-and-after the optimization. It can be seen from Table 2 that the performances for the elements S 0 and S 1 dominate the degree of optimization. In particular, it can be seen that by optimizing the distribution of integration time, the reduction of the variances for the S 0 and S 1 can overcome the increase of the variance for the S 2 and S 3, and thus the total variance of Stokes vector is decreased about 38.5%.
By comparing Table 1 and Table 2, it can be seen that the experiment result in Table 2 consists well with the theoretical results in Table 1, and thus the possibility of decreasing the variance of the Stokes vector estimator by optimizing the integration time of the intensity measurements is theoretically and experimentally demonstrated.
In addition, we also perform the Stokes polarimetry experiments for other different Stokes vectors by changing the orientations of polarizer P1, and the results are almost same to those in Table 2, which experimentally demonstrates that performance of optimization does not depend on the measured Stokes vector, as analyzed in Section 3.
It needs to be clarified that the degree of optimization ψ is determined by Ci and thus by the measurement matrix W, as indicated in Eq. (19). Therefore, different W could lead to different ratios of variance reduction through optimizing the distribution of integration time. In particular, the degree of optimization will increase when the difference between the four parameters Ci is increased.
In conclusion, we consider the Stokes polarimetric measurements perturbed by the additive Gaussian noise, and we have shown that the variance of Stokes vector estimator depends on the distribution of the integration time of intensity measurements in the Stokes polarimetry. In particular, the equal distribution of the integration time may not lead to the minimum estimation variance in some cases. In this paper, we obtain the closed-form solution of the optimal distribution of the integration time at a given measurement matrix, by employing Lagrange multiplier method. In addition, we perform the corresponding theoretical deduction and real world experiment, and it is found that by optimizing the integration times of the intensity measurements, the variance of Stokes vector estimator can be decreased about 40% for the measurement matrix discussed in this paper. The experiment result agrees well with the theoretical deduction, which demonstrates the feasibility of the method proposed in this paper.
This work has many perspectives. For example, it will be interesting to consider other types of noise, such as Poisson shot noise and speckle noise. In addition, the idea of decreasing the variance by optimizing the integration time proposed in this paper can be extended to other types of polarimetric systems, such as Muller polarimetric system.
This work is supported by the National Natural Science Foundation of China (No. 61405140 and No. 61379014), the National Instrumentation Program (No. 2013YQ030915), and the Natural Science Foundation of Tianjin (No. 15JCQNJC02000). Haofeng Hu acknowledges the Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA) and the China Scholarship Council (CSC).
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