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Single-snapshot full-Stokes imaging polarimetry is a powerful tool for the acquisition of the spatial polarization information in real time. According to the general linear model of a polarimeter, to recover full Stokes parameters at least four polarimetric intensities should be measured. In this paper, four types of single-snapshot full-Stokes division-of-aperture imaging polarimeter with four subapertures are presented and compared, with maximum spatial resolution for each polarimetric image on a single area-array detector. By using the error propagation theories for different incident states of polarization, the performance of four polarimeters are evaluated for several main sources of error, including retardance error, alignment error of retarders, and noise perturbation. The results show that the configuration of four 132° retarders with angular positions of ( ± 51.7°, ± 15.1°) is an optimal choice for the configuration of four subaperture single-snapshot full-Stokes imaging polarimeter. The tolerance and uncertainty of this configuration are analyzed.
© 2015 Optical Society of America
1. Introduction
The spatial variance of the state of polarization (SOP) is an important indicator for the identification of object’s properties. Any SOP can be represented by four parameters (S0, S1, S2, and S3) of a well-known Stokes vector. S0 is the total power of the light, S1 denotes the preference for 0° over 90° linearly polarized components, S2 indicates the difference between 45° and 135° linearly polarized components, and S3 for right over left circularly polarized components. Other important polarization parameters such as the angle of linear polarization, the degree of linear polarization, and the degree of circular polarization can also be deduced from four Stokes parameters [1].To acquire the SOP of a scene and thus enhance the contrast of captured object, various imaging polarimeters have been developed [2]. It has aroused wide interests in fields of space exploration, earth remote sensing, machine vision, and biomedical diagnosis [3–5].
In terms of temporal resolution for measuring full Stokes parameters across a scene, current imaging polarimetry can be divided into two categories: division-of-time, and single-snapshot [2]. Division-of-time polarimetry usually employs electrically tunable elements or mechanically rotatable components to demodulate the incident light field. However, any variation in polarization elements, light field, or ambient environment would introduce misregistration and false polarization signal. Meanwhile, single-snapshot full-Stokes imaging polarimeter (SSFSIP) is proven to be a powerful tool for mapping SOP across most of scenarios (stable and variable), owing to its typical capability of real-time parallel acquisition. There are several typical architectures for snapshot imaging polarimetry, such as division-of-focal-plane polarimetry, division-of-amplitude polarimetry, division-of-aperture polarimetry (DoAP), and channeled-imaging polarimetry. The comprehensive comparison of their characteristics can be found from Tyo’s well-known review [2]. Therein, DoAP is one of feasible schema for SSFSIP with a single area-array detector [6–11]. It has the advantages of compactness, stabilization, simple and efficient recovery algorithm. According to the self-consistency of Mueller-Stokes formalism, at least four polarimetric intensities should be measured to get full Stokes parameters [1]. Therefore, to maximize the available spatial resolution of the single area-array detector in SSFSIP, the configuration of four subapertures would be an optimal solution for DoAP.
In the last decade, the optimizations of four-channel full-Stokes polarimeters have been largely addressed to suppress three main error sources [12–27]: (1) the random fluctuations in the detected intensity, (2) the misalignment in the azimuthal angle of retarder, and (3) the non-ideal retardation of retarder. The first type of four-channel full-Stokes polarimeters is that employs a rotatable retarder and a fixed polarizer (RRFP). By minimizing the condition numbers of the system measurement matrix, Ambirajan and Look [12] proposed four optimal rotation angles (−45°, 0°, 30°, 60°) for the RRFP with a standard quarter-wave plate. In this case, the estimated Stokes vector has a minimum sensitivity to the fluctuations in the detected flux and errors in the azimuthal angle of retarder. However, with the retardance of the retarder as a variable for optimization, Sabatke et al [14] found that the RRFP with a 132° retarder and a principal axis set (−51.7°, −15.1°, 15.1°, 51.7°) is more robust to the signal-independent Gaussian additive noise. This arrangement can minimize and equalize the noise power on the last three Stokes parameters. Furthermore, Goudail [22] verified that this configuration is also partial optimum for minimizing the signal-dependent Poisson shot noise. The second type of four-channel full-Stokes polarimeters uses two variable retarders and a fixed linear polarizer (VRFP). Tyo [15] proposed a noise-equalization VRFP system that uses two liquid crystal variable retarders with fixed angular positions and variable retardance. The angular positions of the two variable retarders are fixed at 22.5° and 45°, and a set of optimum retardance is (−158°, 50.6°), (127°, −178°), (47°, −16.9°), (0.66°, 126°). Other configurations are also feasible, different angular positions introduce different optimal retardance [19]. Recently, Zallat et al [21] found that, for a VRFP with two standard quarter-wave plates and a set of optimum angular positions (−20.3°, −41.14°), (−20.3°, 41.14°), (20.3°, −41.14°), (20.3°, 41.14°), the noises in raw image data also can be reduced and propagated equally to the Stokes channels. The above four optimized configurations of DoTP are summarized in Table 1.
These existing studies show that no one optimal configuration is better than the other, because the system with the minimum condition number (that is maximum signal-to-noise ratio) would not be insensitive to systematic error [18,20], and the noise equalizations are implemented only at special SOPs for signal-dependent noise. In this paper, the above optimized four-channel full-Stokes polarimeters will be converted to optimal DoAPs first, then the systematic errors and noise variance on the estimated Stokes vector at different incident SOPs will be evaluated comprehensively.
2. Configuration and principle
The first type of DoAP in Fig. 1(a) comprises a four-quadrant retarder array (R) with the same optimal retardance and four different optimal azimuths of fast axis, a uniform polarizer (P) with the principle axis along x axis, and an area-array detector. The second type in Fig. 1(b) consists of two four-quadrant retarder arrays (R1 and R2) with different optimal retardance and eight optimal angular positions of fast axis. Since these configurations originate from the optimized four-channel full-Stokes polarimeters, the noise immunity for each of the recovered Stokes parameters will be enhanced. Since the principal axis of the polarizer is fixed, we don’t need to consider the polarization sensitivities on the four portions of the detector.
Fig. 1 Scheme of DoAP using optimal four-quadrant polarization array. (a) A retarder array R plus a polarizer, and (b) two retarder arrays R1 and R2 plus a polarizer.
The system matrix for each subaperture formed by four-quadrant polarization array in Fig. 1(a) and (b) can be expressed as, respectively,
where δ is the retardance of the retarder, θ is the fast axis with respect to the x axis, and subscripts i = 1,2,3,4 represents measurement channel. The exiting Stokes vector of each polarimetric subaperture is derived with Stokes-Muller formalism,Since the detector only responses to intensity, the first row of system matrix depicted in Eq. (3) should be used to describe the measured intensity [1]. To get full Stokes parameters, four intensities corresponding to different polarization modulations should be measured. These measured intensities can be expressed as a matrix form
where A is the 4 × 4 ideal measurement matrix. For the system in Fig. 1 (a), the measurement matrix A isThe row of the measurement matrix A the system in Fig. 1(b) isThe incident Stokes vector can be evaluated from Eq. (4) with a least-square algorithm,where B is the inverse matrix of the ideal measurement matrix, named as synthesis matrix,Since the measurement matrix is determined by the retardance and azimuths of the retarders, the retardance errors and alignment error of retarders would propagate into the estimated Stokes vector. In the meantime, the detected intensities would be influenced by the noise of the detector, reducing the signal-to-noise ratio. Therefore, the measurement matrix should be calibrated rigorously to suppress these errors. To achieve this goal, the first important step is to build the theoretical models of the systematic error and noise perturbation.
3. Error theoretical models
3.1 Systematic error
Taking errors of retardance and angular position into consideration, the reconstructed Stokes vector will become
where A' is the measurement matrix that accounts for the imperfect retarders. Then the error in reconstructed Stokes vector will becomewhere ΔA = A'−A is the residual error matrix due to the imperfect retarders, named as error matrix. Obviously, the error of Stokes vector depends on the incident SOP as well as on the systematic error. If the retardance error and alignment error are relatively small, the error matrixes for the systems in Figs. 1(a) and 1(b) can be approximated by the first term of Taylor-series expansion, respectively where is the alignment error and is the retardance error of the retarders. Analytical expressions for the elements in Eqs. (11) and (12) are very complex and are not presented here.3.2 Noise variance
There are two types of noises in the detection system: the signal-independent Gaussian additive noise and the signal-dependent Poisson shot noise. Generally, they are inherent in the commonly-used photodetectors. The summation of the two types of noises in quadrature will give an estimation of the total noise. For a pixel in the array with a mean photon signal Ii, the standard deviation of the signal can be given as
where denotes the variance of additive noise. Assuming the noise in each detected signal is statistically independent from the others, the standard deviation of the estimated Stokes parameters can be deduced with standard error propagation equation as [28]where subscripts k = 0, 1, 2, 3 denote four Stokes parameters. Carrying out the differentiations in Eq. (14) with respect to Eq. (8), the noise variance can be derived asThen the standard deviation of the noise in the estimated Stokes parameters can be written asAs can be seen, the noise variance of Stokes vector is determined by the noise perturbation of the four detected intensities. If the dominant noise is Gaussian distribution, the noise perturbations will be equal to each other, . Therefore, the noise variance on the estimated Stokes is independent on the incident SOP, and can be equalized and minimized by optimizing the measurement matrix [14]. In contrast, if the dominant noise is Poisson distribution, the noise perturbations will be different. The noise variance will depend on the incident SOPs and the noise minimization and equalization will become difficult [22].4. Simulation and analysis
4.1 Sampling states of polarization
Since the errors in the estimated Stokes vector originated from the systematic error and noise perturbation are related to the incident Stokes vector, it is necessary to analyze the error tolerance in different incident SOPs. Typically, Stokes vector can be defined with the ellipse polarization parameters across the Poincare sphere [1]
where ψ is azimuth angle, χ is ellipticity. As shown in Fig. 2(a), to ensure the uniform sampling of the incident SOP, N = 1000 points along a spiral locus around the Poincare sphere are selected, where 20 circles with different ellipticities go from the south pole to the north pole, and 50 azimuth angles in each circle goes clockwise. Figure 2(b) represents the normalized intensity for each sampling point. The incident SOPs are linearly polarized around the equator and circularly polarized around the south and north poles. The sampling method of SOPs is analogous to those presented in [23] and [26].Fig. 2 (a) Uniform sampling of SOPs along a spiral locus around the Poincare sphere, and (b) the normalized intensity for each sampling point.
4.2 Measurement matrix and figure of merits
The last three elements on each row of the measurement matrix are coordinates of eigenvectors of the polarization system. These vectors (red points), normalized to unit magnitude, inscribe a tetrahedron inside a Poincaré sphere of unit radius as shown in Fig. 3. As hypothesized by Ambirajan and Look [12], a regular tetrahedron really represents an optimal configuration. However, they were not able to optimize the RRFP configuration, because the tetrahedron in Fig. 3(a) is not regular as they expected. In contrast, the configurations (II)-(IV) are optimized, because all of tetrahedrons in Figs. 3(b)-3(d) are approximately regular.
Fig. 3 (a) Uniform sampling of SOPs along a spiral locus around the Poincare sphere, and (b) the normalized intensity for each sampling point.
To optimize different polarimeters, some figures of merit have been introduced. Firstly, four condition numbers к are defined and used to determine whether the measurement matrix is well-conditioned or not [12,15,21]. Optimal configurations are usually achieved by minimizing condition number. Then the equally weighted variance (EWV) of the measurement matrix is developed to account for the propagation of noise from the measurement intensities to Stokes parameters [14]. In Table 2, the figures of merit for the optimized configurations are calculated with the math functions in MATLAB Toolbox. As can be seen, the condition numbers and EWV of the configuration (I) are much larger than that of others. Configurations (II) and (IV) are slightly better than the configuration (III) due to their lower condition number к2 and EWV. However, the tolerance of the systematic error and noise variance at different incident SOPs cannot be determined from the tetrahedrons and figures of merit.
Table 2. Figures of merit for the optimized configurations.
4.3 Stokes vector errors introduced by systematic errors
As pointed out in Eq. (10), the error of estimated Stokes vector depends on the incident SOPs and the error matrix ΔA. Substituting the sampled SOPs in Fig. (2) into Eq. (10) sequentially, we can get the error in the estimated Stokes vector when each retarder has an retardance error of 1°, as depicted in Fig. 4. Although larger retardance error may be introduced for a nonideal retarder due to the wave band and incident angle. As can be seen from Fig. 4, the error in the estimated Stokes vector obviously varies with incident SOPs. For the configuration (I) in Fig. 4(a), there is almost no error in the estimated Stokes parameter S3. For the other three configurations, the error in S3 is more sensitive to the retardance error. Table 3 shows the maximum errors of the estimated Stokes vector introduced by the retardance error.
Fig. 4 The errors in the estimated Stokes vector due to the retardance errors are plotted with the sampled incident SOPs along the spiral locus around the Poincare sphere for four configurations in (a)-(d). The retardance error of each retarder is 1°, no alignment error and noise are considered in the simulation.
Table 3. The maximum errors of the estimated Stokes vector introduced by the 1° retardance error and 0.5° alignment error respectively in the four configurations.
Figure 5 shows the error in the estimated Stokes vector when each retarder has alignment error of 0.5°. It is found that, for the polarimeter with a single retarder array, the error in S3 is robust to any incident SOP as shown in Figs. 5(a) and 5(b). In contrast, for the polarimeters with two retarder arrays, the errors in S0, S1 and S2 are relatively small, less than 1.6%, in Figs. 5(c) and 5(d). Table 3 also shows the maximum errors of the estimated Stokes vector introduced by the alignment error. It can be concluded that the alignment error dominates the retardance error for the polarimeter with a single retarder array. The configuration (III) is more subject to the retardance error. The configuration (IV) is less sensitive to both the retardance and alignment errors.
Fig. 5 The errors in the estimated Stokes vector due to the alignment errors are plotted with the sampled incident SOPs along the spiral locus around the Poincare sphere for four configurations in (a)-(d). The alignment error of each retarder is 0.5°, no alignment error and noise are considered in the simulation.
4.4 Noise Variance on estimated Stokes vector
Assuming there are no retardance and alignment errors, and the Gaussian noise signal only accounts for 1% mean photon signal. By combining Eq. (4) with Eqs. (13)-(16), the propagations of the noise perturbation in the measured intensities into the reconstructed Stokes vector can be depicted in Fig. 6.
Fig. 6 The standard deviations of noise in the estimated Stokes vector due to the noise perturbation are plotted with the sampled incident SOPs along the spiral locus around the Poincare sphere for four configurations in (a)-(d). Assuming the retarder arrays are perfect.
Notably, the noise levels for the configuration (I) in Fig. 6(a) are different in each of the four Stokes parameters. For the optimized configurations (II) and (III) in Figs. 6(b) and 6(c), the noise equalization and minimization for the last three Stokes parameters can be achieved in some special SOPs. These incident SOPs become sparse for the optimized configurations (IV) in Fig. 6(d). The noise deviation on S0 component is almost a constant at any incident SOP. The noise variance is less dependent on the incident SOPs with the increase of Gaussian noise.
4.5 For partially polarized light
In the above analysis, the sampling SOPs are assumed to be fully polarized. If the incident SOPs are partially polarized, the results will depend on the degree of polarization. The Stokes vector S of the partially polarized light can be represented by a superposition of the Stokes vector S(U) of the unpolarized component and the Stokes vector S(P) of the fully polarized component [1],
where T denotes the transpose operation, P is the degree of polarization, P = 0 denotes the unpolarized light, and P = 1 denotes the fully polarized light. By combining Eq. (18) with Eqs. (4), (10) and (16), two special cases will be produced.(1) If P << 1, that is the unpolarized component is much stronger than the fully polarized component, the incident SOPs will be dominated by the S0 component. The fraction of the last three components S1-S3 are reduced relative to the S0 component. Therefore, the effect of the incident SOPs could be neglected. Assuming the fully polarized components are also the sampling points depicted in Fig. 2, the errors in the estimated Stokes vector for the configuration (II) are shown in Figs. 7(a)-7(c) when P = 0.1.
Fig. 7 The errors in the estimated Stokes vector for the configuration (II). The top row is for P = 0.1 and the bottom row is for P = 0.5. In (a) and (d), the retardance error of each retarder is 1°. In (b) and (e), the alignment error of each retarder is 0.5°. (c) and (f) are the standard deviations of noise on the estimated Stokes vector.
(2) If the unpolarized component approaches to the fully polarized component, the effects of the incident SOPs should be considered. Figures 7(d)-7(f) show the errors in the estimated Stokes vector in the configuration (II) when P = 0.5.
5. Discussion
The above simulation results show that the configuration (IV) is an optimal choice when the noise is the signal-independent additive noise or S0 component of the incident SOP is much larger than others, because of its immunity to the retardance and alignment errors. In contrast, the configurations (II) and (III) are preferable to others due to their advantages of noise immunization when signal-dependent noise is dominant. Meanwhile, the configuration (II) is robust to the retardance and the configuration (III) is relative robust to the alignment error. However, the system alignment of the configuration (III) is more difficult because two retarders in tandem are needed. Therefore, the configuration (II) is the better choice for the implementation of the DoAP.
5.1 Construction of the optimized SSFSIP
One of the methods to build a SSFSIP based on the optimized DoAP configuration is presented in Fig. 8. Light is firstly collimated by a front telescopic system (L1 and L2), and then reaches on a four-quadrant retarder array R and a uniform polarizer P. The light from four polarimetric subapertures are split by a pyramid prism PP. Lens L3 focuses each polarimetric beam on one of the four quadrants of a single area-array detector. Substituting the measured four intensities and measurement matrix into Eq. (4), the incident SOP could be estimated. In this configuration, the four sub-retarders can be tailored from a larger uniform 132° retarder to suppress the retardance diversity spatially. The retarder can be made from uniaxial birefringent crystal, and the retardance variation is then dominated by the incident angle. In contrast, the retardance of the liquid-crystal variable retarders varies with both spatial location and incident angle [24–26]. Components R, P and PP can be assembled together and placed in the aperture plane, so that all collimated rays will enter the retarder array at a given incident angle. Therefore, the retardance variation with incident angle will be similar on each sub-retarder and this variation could be calibrated on a pixel-by-pixel basis [29]. The influence of the retardance varies with aperture inhomogeneity, the corresponding optimization of retardance was addressed in detail in [20]. Alternatively, the pyramid prism PP and lens L3 can be replaced with the four mini-lens array, each mini lens forms an image of the object onto the area-array detector [30].
Fig. 8 The scheme of SSFSIP based on optimized DoAP. Lenses L1, L2 and L3, a retarder array R, a uniform polarizer P, a pyramid prism PP, and an area-array detector.
5.2 The tolerance of each retarder
The accuracy of the estimated Stokes parameters can be considered as a figure of merit in determining the tolerance of retardance and alignment in each sub-retarder. Figure 9 shows the absolute values of the accuracy corresponding to a set of specified tolerance (0.2°, 0.5°, 1°, 2°, 3°) in retardance and alignment. There is a linear dependence between the accuracy and tolerance. The accuracy is more subject to the alignment error than to the retardance error. With the linear interpolation method, the tolerance of each systematic parameter corresponding to the accuracy of 2% is calculated in Table 4.
Fig. 9 The dependence of the accuracy of estimated Stokes vector on the tolerance of each systematic parameter.
Table 4. The tolerance of each systematic parameter for the accuracy of 0.02 in the estimated Stokes parameters.
5.3 The uncertainty of the system
Usually, the standard deviation of each polarization parameter can be used to represent the uncertainty of the system. In the following, the standard deviations of Stokes parameters due to the variations of the retardance and angular position will be derived, if we assume the noise is small compared with the effect of the systematic error. When the alignment error (or retardance error ) varies randomly with a standard deviation , the covariance matrix in the estimated Stokes parameters can be expressed as [18, 20, 31, 32]
where C is the error propagation matrix, and the superscript “T” denotes transposition operation. If only retardance error exists in the system,When only alignment error exists,The four diagonal elements of the covariance matrix represent the variances of four Stokes parameters S0, S1, S2 and S3 respectively. Notably, the variances are directly dependent on the incident SOPs. Substituting δ = 132°, θi = (−51.7°, −15.1°, 15.1°, 51.7°) into Eqs. (8), and (20)-(22), the standard deviations of four Stokes parameters are calculated and shown in Fig. 10. The standard deviations σ of the retardance error and alignment error are 0.45° and 1° respectively. In this situation, the maximum standard deviation of four Stokes parameters is less than 2%. If the noise perturbation is included, the total deviation may be larger than 2%. Hsu et al [33] used this configuration to build a division-of-focal-plane polarimeter and showed the maximum measurement uncertainty was about 5%. This uncertainty could be further suppressed via the rigorous manufacture and calibration of measurement matrix.Fig. 10 The standard deviation of the estimated Stokes vector at different incident SOPs. (a) When the standard deviation of retardance error is 1°, and (b) when the standard deviation of alignment error is 0.45°
6. Conclusion
In this paper, four potential configurations for the implementation of DoAPs are compared in the aspect of systematic error and noise propagation. The comparisons take into consideration of the different incident SOPs around the Poincare sphere. We find that the noise minimization and equalization are unable to be achieved at any incident SOPs when the signal-dependent noise cannot be neglected. The optimized configuration with a minimum condition number cannot simultaneously has well immunity to the systematic error on the estimated Stokes vector. The configuration (II) is found to be an optimal choice for DoAP, because of its superior immunity to the systematic error and noise perturbation. In the future, we will build and calibrate a DoAP using this configuration to achieve accurate Stokes vector measurement.
Acknowledgments
The authors thank the anonymous reviewers for their helpful comments and constructive suggestions. The work was supported by the China Scholarship Council, the Fundamental Research Funds for the Central Universities of China (xjj2013044), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130201120047), the Natural Science Basic Research Plan in Shaanxi Province of China (2014JQ8362), the National Natural Science Foundation (61405153, 61275184) of China, and the 863 Program (2012AA121101) of China.
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