Error analysis of single-snapshot full-Stokes division-of-aperture imaging polarimeters Tingkui Mu,1,2,* Chunmin Zhang,1 Qiwei Li,1 and Rongguang Liang2 1
Institute of Space Optics, Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Science, Xi’an Jiaotong University, Xi’an 710049, China 2 College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA * [email protected]
Abstract: 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 OCIS codes: (120.5410) Polarimetry; (110.5405) Polarimetric imaging; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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18. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). 19. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). 20. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006). 21. J. Zallat, A. S, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006). 22. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009). 23. D. Lara and C. Paterson, “Stokes polarimeter optimization in the presence of shot and Gaussian noise,” Opt. Express 17(23), 21240–21249 (2009). 24. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimization and performance criteria of a Stokes polarimeter based on two variable retarders,” Opt. Express 18(10), 9815–9830 (2010). 25. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimized Stokes polarimeters based on a single twisted nematic liquid-crystal device for the minimization of noise propagation,” Appl. Opt. 50(28), 5437–5445 (2011). 26. A. Peinado, A. Lizana, and J. Campos, “Optimization and tolerance analysis of a polarimeter with ferroelectric liquid crystals,” Appl. Opt. 52(23), 5748–5757 (2013). 27. E. Chironi and C. Iemmi, “Bounding the relative errors associated with a complete Stokes polarimeter,” J. Opt. Soc. Am. A 31(1), 75–80 (2014). 28. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, Third Edition (McGraw-Hill, 2003). 29. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999). 30. J. L. Pezzaniti and D. B. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. SPIE 5888, 58880V (2005). 31. H. Dai and C. Yan, “Measurement errors resulted from misalignment errors of the retarder in a rotating-retarder complete Stokes polarimeter,” Opt. Express 22(10), 11869–11883 (2014). 32. H. Dong, M. Tang, and Y. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013). 33. W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014).
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 singlesnapshot [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-offocal-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-
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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 nonideal 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. Table 1. Four optimized configurations in [12], [14], [15], and [21] with different retardance and angular positions of retarders Configuration
Retardance
Azimuth
(I) [12]
90°
−45°, 0°, 30°, 60°
(II) [14]
132°
(III) [21]
(90°, 90°)
−51.7°, −15.1°, 15.1°, 51.7° (−41.14°, −20.3°), (41.14°, −20.3°) (−41.14°, 20.3°), (41.14°, 20.3°)
(IV) [15]
(158°, 50.6°), (127°, −178°) (47°, −16.9°), (0.66°, 126°)
(22.5°, 45°)
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
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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, M = M P (0) ⋅ M R (δ , θ i ),
(1)
M = M P (0) ⋅ M R 2 (δ 2i , θ 2i ) ⋅ M R1 (δ1i , θ1i ),
(2)
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, S out = M ⋅ Sin . (3) 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 I = A ⋅ Sin ,
(4)
where A is the 4 × 4 ideal measurement matrix. For the system in Fig. 1 (a), the measurement matrix A is 1 1 1 A(δ ,θ i ) = 2 1 1
cos 2 2θ1 + sin 2 2θ1 cos δ cos 2 2θ 2 + sin 2 2θ 2 cos δ cos 2 2θ 3 + sin 2 2θ3 cos δ cos 2 2θ 4 + sin 2 2θ 4 cos δ
sin 4θ1 sin 2 (δ / 2) sin 4θ 2 sin 2 (δ / 2) sin 4θ3 sin 2 (δ / 2) sin 4θ 4 sin 2 (δ / 2)
− sin 2θ1 sin δ − sin 2θ 2 sin δ . (5) − sin 2θ 3 sin δ − sin 2θ 4 sin δ
The row of the measurement matrix A the system in Fig. 1(b) is
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T
1, 2 2 2 2 (cos 2θ 2i + sin 2θ 2i cos δ 2i )(cos 2θ1i + sin 2θ1i cos δ1i ) + (sin δ 2i / 2) 2 sin 4θ 2i (sin δ1i / 2) 2 sin 4θ1i − sin 2θ 2i sin δ 2i sin 2θ1i sin δ1i , 1 . (cos 2 2θ 2i + sin 2 2θ 2i cos δ 2i )(sin δ1i / 2)2 sin 4θ1i + 2 2 2 2 (sin δ 2i / 2) sin 4θ 2i (sin 2θ1i + cos 2θ1i cos δ1i ) + sin 2θ 2i sin δ 2i cos 2θ1i sin δ1i , 2 2 −(cos 2θ 2i + sin 2θ 2i cos δ 2i ) sin 2θ1i sin δ1i + 2 (sin δ 2i / 2) sin 4θ 2i cos 2θ1i sin δ1i − sin 2θ 2i sin δ 2i cos δ1i (6)
The incident Stokes vector can be evaluated from Eq. (4) with a least-square algorithm, Sin = B ⋅ I,
(7)
where B is the inverse matrix of the ideal measurement matrix, named as synthesis matrix, B01 B02 B03 B04 B B12 B13 B14 B = 11 . (8) B21 B22 B23 B24 B31 B32 B33 B34 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 S e = B ⋅ A′ ⋅ Sin ,
(9)
where A' is the measurement matrix that accounts for the imperfect retarders. Then the error in reconstructed Stokes vector will become
ε S = S e − Sin = B ⋅ ΔA ⋅ Sin ,
(10)
where 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 A(θ i + ξi ) − A (θi ) A(δ + ς i ) − A(ς i ) ΔA a = ξi + ςi ξi ςi ∂A(θ ) ∂A(δ ) = ξi + ςi , ∂θ θi ∂δ ς i
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(11)
Received 13 Jan 2015; revised 9 Apr 2015; accepted 15 Apr 2015; published 17 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.010822 | OPTICS EXPRESS 10826
A(θ1i + ξ1i ) − A(θ1i ) A(θ 2i + ξ 2i ) − A(θ 2i ) ΔA b = ξ1i + ξ 2i ξ1i ξ 2i A(δ1i + ς 1i ) − A(ς 1i ) A(δ 2i + ς 2i ) − A(ς 2i ) + ς 1i + ς 2i ς 1i ς 2i = ξ1i
(12)
∂A(θ1 ) ∂A(θ 2 ) ∂A(δ1 ) ∂A(δ 2 ) + ξ 2i + ς 1i + ς 2i , ∂θ1 θ ∂θ 2 θ ∂δ1 ς ∂δ 2 ς 1i
2i
1i
2i
where ξi is the alignment error and ς i 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
σ I = I i +σ G2 ,
(13)
i
where σ G2 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]
σS
2
∂( Sk ) 2 = σ Ii , i =1 ∂ ( I i ) 4
k
(14)
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 as σ S2 B 2 20 012 σ S1 B11 2 = 2 σ S2 B21 σ 2 B312 S3
B022
B032
B122
B132
B222
B232
B322
B332
2 B042 σ I1 2 B142 σ I 2 . B242 σ I23 B342 σ I2 4
(15)
Then the standard deviation of the noise in the estimated Stokes parameters can be written as
σS = k
4
B i =1
2 ki
⋅ σ I2i .
(16)
As 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, σ I21 = σ I22 = σ I23 = σ I24 . 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].
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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] 1 cos 2ψ cos 2 χ , 0 ≤ψ < π , − π ≤ χ ≤ π , S= sin 2ψ cos 2 χ 4 4 sin 2 χ
(17)
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. 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
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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.
Fig. 3. (a) Uniform sampling of SOPs along a spiral locus around the Poincare sphere, and (b) the normalized intensity for each sampling point. Table 2. Figures of merit for the optimized configurations. Configuration
к1
к2
к∞
кF
EWV
(I)
8.3316
3.6268
5.8868
6.5800
21.6480
(II)
6.0346
1.7378
4.6298
4.4722
10.0002
(III)
6.1833
1.7509
4.7022
4.4729
10.0034
(IV)
5.6377
1.7367
4.1932
4.4722
10.0003
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.
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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.
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.
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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. 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. Configuration With 1° retardance error With 0.5° alignment error
(I)
(II)
(III)
(IV)
1.75% 4.03%
1.57% 3.69%
3.31% 2.28%
1.57% 1.58%
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
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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], S = SU + SP ,
(18)
where S = [ S0
S1
S2
S3 ] , T
(19a)
S (U) = (1 − P) S0 [1 0 0 0] , T
S (P) = PS0 [1 S1 PS0
S2 PS0
(19b)
S3 PS0 ] ,
P = S12 + S 22 + S32 S0 , 0 ≤ P ≤ 1,
T
(19c) (19d)
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.
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.
(1) If P
Institute of Space Optics, Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Science, Xi’an Jiaotong University, Xi’an 710049, China 2 College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA * [email protected]
Abstract: 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 OCIS codes: (120.5410) Polarimetry; (110.5405) Polarimetric imaging; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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18. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-to-noise ratio and minimization of systematic error,” Appl. Opt. 41(4), 619–630 (2002). 19. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). 20. J. S. Tyo and H. Wei, “Optimizing imaging polarimeters constructed with imperfect optics,” Appl. Opt. 45(22), 5497–5503 (2006). 21. J. Zallat, A. S, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A, Pure Appl. Opt. 8(9), 807–814 (2006). 22. F. Goudail, “Noise minimization and equalization for Stokes polarimeters in the presence of signal-dependent Poisson shot noise,” Opt. Lett. 34(5), 647–649 (2009). 23. D. Lara and C. Paterson, “Stokes polarimeter optimization in the presence of shot and Gaussian noise,” Opt. Express 17(23), 21240–21249 (2009). 24. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimization and performance criteria of a Stokes polarimeter based on two variable retarders,” Opt. Express 18(10), 9815–9830 (2010). 25. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos, “Optimized Stokes polarimeters based on a single twisted nematic liquid-crystal device for the minimization of noise propagation,” Appl. Opt. 50(28), 5437–5445 (2011). 26. A. Peinado, A. Lizana, and J. Campos, “Optimization and tolerance analysis of a polarimeter with ferroelectric liquid crystals,” Appl. Opt. 52(23), 5748–5757 (2013). 27. E. Chironi and C. Iemmi, “Bounding the relative errors associated with a complete Stokes polarimeter,” J. Opt. Soc. Am. A 31(1), 75–80 (2014). 28. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, Third Edition (McGraw-Hill, 2003). 29. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. 38(16), 3490–3502 (1999). 30. J. L. Pezzaniti and D. B. Chenault, “A Division of Aperture MWIR Imaging Polarimeter,” Proc. SPIE 5888, 58880V (2005). 31. H. Dai and C. Yan, “Measurement errors resulted from misalignment errors of the retarder in a rotating-retarder complete Stokes polarimeter,” Opt. Express 22(10), 11869–11883 (2014). 32. H. Dong, M. Tang, and Y. Gong, “Noise properties of uniformly-rotating RRFP Stokes polarimeters,” Opt. Express 21(8), 9674–9690 (2013). 33. W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014).
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 singlesnapshot [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-offocal-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-
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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 nonideal 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. Table 1. Four optimized configurations in [12], [14], [15], and [21] with different retardance and angular positions of retarders Configuration
Retardance
Azimuth
(I) [12]
90°
−45°, 0°, 30°, 60°
(II) [14]
132°
(III) [21]
(90°, 90°)
−51.7°, −15.1°, 15.1°, 51.7° (−41.14°, −20.3°), (41.14°, −20.3°) (−41.14°, 20.3°), (41.14°, 20.3°)
(IV) [15]
(158°, 50.6°), (127°, −178°) (47°, −16.9°), (0.66°, 126°)
(22.5°, 45°)
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
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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, M = M P (0) ⋅ M R (δ , θ i ),
(1)
M = M P (0) ⋅ M R 2 (δ 2i , θ 2i ) ⋅ M R1 (δ1i , θ1i ),
(2)
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, S out = M ⋅ Sin . (3) 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 I = A ⋅ Sin ,
(4)
where A is the 4 × 4 ideal measurement matrix. For the system in Fig. 1 (a), the measurement matrix A is 1 1 1 A(δ ,θ i ) = 2 1 1
cos 2 2θ1 + sin 2 2θ1 cos δ cos 2 2θ 2 + sin 2 2θ 2 cos δ cos 2 2θ 3 + sin 2 2θ3 cos δ cos 2 2θ 4 + sin 2 2θ 4 cos δ
sin 4θ1 sin 2 (δ / 2) sin 4θ 2 sin 2 (δ / 2) sin 4θ3 sin 2 (δ / 2) sin 4θ 4 sin 2 (δ / 2)
− sin 2θ1 sin δ − sin 2θ 2 sin δ . (5) − sin 2θ 3 sin δ − sin 2θ 4 sin δ
The row of the measurement matrix A the system in Fig. 1(b) is
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T
1, 2 2 2 2 (cos 2θ 2i + sin 2θ 2i cos δ 2i )(cos 2θ1i + sin 2θ1i cos δ1i ) + (sin δ 2i / 2) 2 sin 4θ 2i (sin δ1i / 2) 2 sin 4θ1i − sin 2θ 2i sin δ 2i sin 2θ1i sin δ1i , 1 . (cos 2 2θ 2i + sin 2 2θ 2i cos δ 2i )(sin δ1i / 2)2 sin 4θ1i + 2 2 2 2 (sin δ 2i / 2) sin 4θ 2i (sin 2θ1i + cos 2θ1i cos δ1i ) + sin 2θ 2i sin δ 2i cos 2θ1i sin δ1i , 2 2 −(cos 2θ 2i + sin 2θ 2i cos δ 2i ) sin 2θ1i sin δ1i + 2 (sin δ 2i / 2) sin 4θ 2i cos 2θ1i sin δ1i − sin 2θ 2i sin δ 2i cos δ1i (6)
The incident Stokes vector can be evaluated from Eq. (4) with a least-square algorithm, Sin = B ⋅ I,
(7)
where B is the inverse matrix of the ideal measurement matrix, named as synthesis matrix, B01 B02 B03 B04 B B12 B13 B14 B = 11 . (8) B21 B22 B23 B24 B31 B32 B33 B34 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 S e = B ⋅ A′ ⋅ Sin ,
(9)
where A' is the measurement matrix that accounts for the imperfect retarders. Then the error in reconstructed Stokes vector will become
ε S = S e − Sin = B ⋅ ΔA ⋅ Sin ,
(10)
where 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 A(θ i + ξi ) − A (θi ) A(δ + ς i ) − A(ς i ) ΔA a = ξi + ςi ξi ςi ∂A(θ ) ∂A(δ ) = ξi + ςi , ∂θ θi ∂δ ς i
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(11)
Received 13 Jan 2015; revised 9 Apr 2015; accepted 15 Apr 2015; published 17 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.010822 | OPTICS EXPRESS 10826
A(θ1i + ξ1i ) − A(θ1i ) A(θ 2i + ξ 2i ) − A(θ 2i ) ΔA b = ξ1i + ξ 2i ξ1i ξ 2i A(δ1i + ς 1i ) − A(ς 1i ) A(δ 2i + ς 2i ) − A(ς 2i ) + ς 1i + ς 2i ς 1i ς 2i = ξ1i
(12)
∂A(θ1 ) ∂A(θ 2 ) ∂A(δ1 ) ∂A(δ 2 ) + ξ 2i + ς 1i + ς 2i , ∂θ1 θ ∂θ 2 θ ∂δ1 ς ∂δ 2 ς 1i
2i
1i
2i
where ξi is the alignment error and ς i 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
σ I = I i +σ G2 ,
(13)
i
where σ G2 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]
σS
2
∂( Sk ) 2 = σ Ii , i =1 ∂ ( I i ) 4
k
(14)
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 as σ S2 B 2 20 012 σ S1 B11 2 = 2 σ S2 B21 σ 2 B312 S3
B022
B032
B122
B132
B222
B232
B322
B332
2 B042 σ I1 2 B142 σ I 2 . B242 σ I23 B342 σ I2 4
(15)
Then the standard deviation of the noise in the estimated Stokes parameters can be written as
σS = k
4
B i =1
2 ki
⋅ σ I2i .
(16)
As 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, σ I21 = σ I22 = σ I23 = σ I24 . 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].
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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] 1 cos 2ψ cos 2 χ , 0 ≤ψ < π , − π ≤ χ ≤ π , S= sin 2ψ cos 2 χ 4 4 sin 2 χ
(17)
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. 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
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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.
Fig. 3. (a) Uniform sampling of SOPs along a spiral locus around the Poincare sphere, and (b) the normalized intensity for each sampling point. Table 2. Figures of merit for the optimized configurations. Configuration
к1
к2
к∞
кF
EWV
(I)
8.3316
3.6268
5.8868
6.5800
21.6480
(II)
6.0346
1.7378
4.6298
4.4722
10.0002
(III)
6.1833
1.7509
4.7022
4.4729
10.0034
(IV)
5.6377
1.7367
4.1932
4.4722
10.0003
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.
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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.
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.
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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. 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. Configuration With 1° retardance error With 0.5° alignment error
(I)
(II)
(III)
(IV)
1.75% 4.03%
1.57% 3.69%
3.31% 2.28%
1.57% 1.58%
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
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Received 13 Jan 2015; revised 9 Apr 2015; accepted 15 Apr 2015; published 17 Apr 2015 20 Apr 2015 | Vol. 23, No. 8 | DOI:10.1364/OE.23.010822 | OPTICS EXPRESS 10831
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], S = SU + SP ,
(18)
where S = [ S0
S1
S2
S3 ] , T
(19a)
S (U) = (1 − P) S0 [1 0 0 0] , T
S (P) = PS0 [1 S1 PS0
S2 PS0
(19b)
S3 PS0 ] ,
P = S12 + S 22 + S32 S0 , 0 ≤ P ≤ 1,
T
(19c) (19d)
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.
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.
(1) If P
