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Linear polarization optimized Stokes polarimeter based on four-quadrant detector CHAO HE,1,2 JINTAO CHANG,1,3 YONG WANG,1,3 RAN LIAO,1 HONGHUI HE,1 NAN ZENG,1
AND
HUI MA1,3,*
1
Shenzhen Key Laboratory for Minimal Invasive Medical Technologies, Institute of Optical Imaging and Sensing, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China 2 Department of Biomedical Engineering, Tsinghua University, Beijing 100084, China 3 Department of Physics, Tsinghua University, Beijing 100084, China *Corresponding author: [email protected] Received 19 January 2015; accepted 9 April 2015; posted 16 April 2015 (Doc. ID 232790); published 7 May 2015
A four-quadrant detector (4QD) consists of four well-balanced detectors. We report on a Stokes polarimeter with optimal linear polarization measurements based on a 4QD. We turned the four intensity-detection channels into four polarization-analyzing channels by placing four polarizers and one quarter-wave plate in front of the individual detectors. An optimization method for the four polarization-analyzing channels is proposed to improve measurement accuracy. Considering applications in favor of linear polarization measurements instead of global optimization for all the possible states of polarization (SOP), we optimize the polarimeter first for the linear polarization components and then for the circular polarization component. The polarimeter is capable of simultaneous measurements of fast varying SOP with improved performance for the linear polarizations. © 2015 Optical Society of America OCIS codes: (120.5410) Polarimetry; (130.5440) Polarization-selective devices; (120.2130) Ellipsometry and polarimetry. http://dx.doi.org/10.1364/AO.54.004458
1. INTRODUCTION Polarimetric techniques are widely used in many research fields such as medical diagnosis, material characterization, astronomy, and so on [1–5]. For applications involving nonstatic subjects, such as moving objects, living organisms, and particles in fluids, simultaneous polarization detections have to be used, since polarization components recorded at different times may lead to significant errors in the derived possible states of polarization (SOP) [6]. Nowadays, there are already many techniques for simultaneous SOP measurements, such as multiple-detector polarimeters [7–9], spatial modulation polarimeters [10–13], and spectral modulation polarimeters [14]. Among all those simultaneous Stokes polarimeters, most are designed for optimal performances for all the Stokes components. However, in many occasions, such as remote sensing [15], accurate measurements of linear polarization components are particularly important for retrieving polarization characteristics because natural illuminations outdoors and underwater, scattered and reflected are partially linear polarized [16]. For example, a previous study has shown that, for isotropic turbid media such as some biological tissues, it would be useful and convenient if the polarization parameters can be quantified with linear polarization measurements alone [17]. Simulations based on the sphere-cylinder scattering model also have shown that linear polarization features are more prominent in Mueller matrix element patterns [18]. 1559-128X/15/144458-06$15/0$15.00 © 2015 Optical Society of America
In this paper, we propose a simultaneous Stokes polarimeter, named the four-quadrant polarmeter (4QP) with optimal linear polarization measurements. The polarimeter is based on a 4QD, which includes four well-balanced high-speed photon detectors at the four quadrants. 4QD has been widely used in many areas, such as beam collimation and target tracking. In this paper, the four detectors of the 4QD are turned into the four polarimetric analyzing channels by placing four linear polarizers of different orientations in front of each detector and a wave plate in one of the detectors. As in many applications, the accuracy of linear polarization measurements is more important; thus, we first optimize the performance of the polarimeter for the linear polarization components and then for the circular polarization component. We also present a Stokes polarimeter calibration method and the experimental results. 2. INSTRUMENT STRUCTURE OF 4QP As shown in Fig. 1, the 4QP consists of four polarization analyzers A (including R and P) and a 4QD. The 4QD system is a commercial product (Daheng Optics, GCI-1301) including four packaged photon detectors, each of which is connected to a preamplifier and a 12 bit A/D convertor with 3 μs sampling time. The 4QD system is stable and easy to operate, since the detectors and their electronics are well balanced with matching parameters. Each of the four detectors can be converted to a
Research Article polarization-analyzing channel by using polarizers and wave plates. Optimization of the polarimeter can be achieved by carefully choosing the orientations of the polarizers as well as the orientation and retardance of the wave plates. Since the optical properties of the wave plates are more sensitive to the experimental conditions (such as ambient temperature, the angle and the wavelength of the incident light, than the polarizer), we use four film polarizers (Thorlabs, extinction ratio > 1000∶1 for 500–700 nm) for all four channels and a single achromatic film wave plate (Edmund, 1∕4λ achromatic retarder film for 350–800 nm) for Channel 1. Following the optimization process described in Section 4, the transmission axes of the polarizers are set to 45° (Channel 1), 0° (Channel 2), 60° (Channel 3), and 120° (Channel 4) measured from the horizontal direction. The wave plate in Channel 1 is set as a quarter-wave plate with its fast axis direction at 0°.
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S out M p1 · M r · S in ;
(1)
where S in is the Stokes vector of the incident beam, S in s0 ; s1 ; s 2 ; s3 T . S out is the Stokes vector of the detected light. M p1 and M r are the Mueller matrices of the polarizer and the quarter-wave plate. For the polarization-analyzing Channels 2, 3, and 4, each of which contains only a polarizer, the SOP transformation can be expressed as Eq. (2): S out M pn · S in ;
n 2; 3; 4;
(2)
where M pn denotes the polarizer for the nth channel. The Mueller matrix of a polarizer with arbitrary transmission axis angle is expressed as Eq. (3):
3 p2x − p2y cos 2θp ; p2x − p2y sin 2θp ; 0 p2x p2y ; 6 p2x − p2y cos 2θp ; p2x − p2y cos2 2θp 2px py sin2 2θp ; p2x − p2y cos 2θp sin 2θp − 2px py cos 2θp sin 2θp ; 0 7 7; Mp 6 4 p2 − p2 sin 2θp ; p2 − p2 cos 2θp sin 2θp − 2p p cos 2θp sin 2θp ; 2px py cos2 2θp p2x − p2y sin2 2θp 0 5 x y x y x y 0 0 0 2px py 2
(3) where px and py are the transmission ratios in the transmission axis and extinction axis, and θp is the angle of the direction of the transmission axis. The Mueller matrix of a wave plate is expressed as Eq. (4): 2 1; 3 0; 0; 0 2 2 0.5 sin 4θr 1 − cos δ; − sin 2θr sin δ 7 6 0; cos 2θr sin 2θr cos δ; (4) Mr 4 5; 0; 0.5 sin 4θr 1 − cos δ; sin2 2θr cos2 2θr cos δ; cos 2θr sin δ − cos 2θr sin δ; cos δ 0; sin 2θr sin δ; where δ and θr represent the retardance and fast axis direction, respectively. Figure 1 also shows the schematic diagram of the polarization state generator (PSG). The PSG is used to generate different known SOP to calibrate the 4QP and to evaluate its performance. The PSG consists of a laser (Changchun New Industries, MGL-III-532 nm), a polarizer P1 (Thorlabs, extinction ratio > 5000∶1), and quarter-wave plates R1 and R2 (Thorlabs, 532 mm, WPQ10E-532). R1 is set at a fixed angle to transform the linear polarization of the laser to circular polarization. P1 and R2 are mounted on two high-precision motorized rotation stages (Thorlabs, PRM1Z8E). For each incident SOP generated by the PSG, four specific analyzing SOP are detected by the detectors; then, the four intensity values are used to calculate the Stokes vector of the incident light beam.
The intensity signals from the four polarization-analyzing channels correspond to the first component of the four Stokes vectors on the corresponding detectors. These intensities can be expressed as an intensity vector I i 1 ; i 2 ; i 3 ; i 4 T . The relationship between I and S in can be expressed in a matrix form: I A · S in ;
(5)
where A is a 4 × 4 matrix, named the instrument matrix, which can be calculated from Eqs. (3) and (4): A1; ∶ p2x p2y ; cos 2θp1 · p2x − p2y · cos2 2θr cos δ · sin2 2θr sin 2θp1 · p2x − p2y · cos 2θr · sin 2θr − cos 2θr · sin 2θr · cos δ; sin 2θp1
3. MEASUREMENT PRINCIPLE OF 4QP
· p2x − p2y · cos δ · cos2 2θr sin2 2θr cos 2θp1
We use the Mueller calculus to describe the measurement principle of 4QP [19]. The Stokes vector can represent all possible SOP of light, including polarized, partially polarized, and unpolarized light. The Mueller matrix provides a complete description of the polarization transforming properties of a sample. For an incident light beam entering the polarizationanalyzing Channel 1 which contains a polarizer and a wave plate, the SOP transformation can be expressed as Eq. (1):
· p2x − p2y · cos 2θr · sin 2θr − cos 2θr · sin 2θr · cos δ; cos 2θr · sin 2θp1 · sin δ · p2x − p2y − cos 2θp1 · sin 2θr · sin δ · p2x − p2y ; A2; ∶
p2x
p2y ;
cos 2θp2 ·
· p2x − p2y ; 0;
(6) p2x
−
p2y ;
sin 2θp2 (7)
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I 0 A 0 · S in0 ;
(12)
0
where A is a 3 × 3 instrument matrix for the linear polarization measurements, which can be calculated from Eqs. (3) and (4): A 0 1; ∶ p2x p2y ; cos 2θp2 · p2x − p2y ; sin 2θp2 · p2x − p2y ; A 0 2; ∶ p2x p2y ; cos 2θp3 · p2x − p2y ; sin 2θp3
(13)
· p2x − p2y ; A 3; ∶ p2x p2y ; cos 2θp4 · p2x − p2y ; sin 2θp4
(14)
0
· p2x − p2y : Fig. 1. Schematic diagram of the PSG, beam expander, and 4QP. R1, angle-fixed quarter-wave plate. P1 and R2, rotatable polarizer and quarter-wave plate. L, beam expander. A (including R and P): Four polarimetric analyzing channels, including one quarter-wave plate R and four film polarizers P. D, four-quadrant detector. Red arrow on the film wave plate R represents the fast axis; grids on the film polarizers P represent the transmission axes.
A3; ∶ p2x p2y ; cos 2θp3 · p2x − p2y ; sin 2θp3 · p2x − p2y ; 0;
(8)
A4; ∶ p2x p2y ; cos 2θp4 · p2x − p2y ; sin 2θp4 · p2x − p2y ; 0:
(9)
In Eqs. (6–9), θp1 , θp2 , θp3 , and θp4 denote the transmission axes of the polarizers in Channels 1, 2, 3, and 4, respectively. Then S in is calculated through the matrix inversion, as shown in Eq. (10): S in A−1 · I;
(10)
A−1
denotes the matrix inverse of A. where However, real measurements contain noises δI in the intensity vector I. The instrument matrix A may also be associated with an error matrix δA. Then, we recast Eq. (5), as shown in Eq. (11): I δI A δA · S δS;
(11)
where δS represents the error of S. S δS can be calculated by S δS invA δA • I δI. To obtain high-precision measurements of S, both δI and δA should be minimized. The well-balanced four detectors and electronics of the 4QD help to minimize δI . For the time-sequential modulation polarimeters, errors in the motorized rotators present the main contribution to δA. For 4QP, however, the measurements of different polarization components are taken simultaneously; A is a known matrix and δA contains uncertainties due to the polarization optics, which should stay small after the initial calibration. We can reduce δS by the optimization of A, which will be discussed in Section 4. For applications in which the linear polarization components S in0 s0 ; s 1 ; s2 T are more important, the intensity signals from the 2, 3, and 4 polarization-analyzing channels can be expressed as an intensity vector I 0 i 2 ; i 3 ; i 4 T , and the relationship between I 0 and S in0 can also be expressed in a matrix form:
(15)
In Eqs. (13–15), θp2 , θp3 , and θp4 denote the transmission axes of the polarizers in Channels 2, 3, and 4, respectively. Then, S in is calculated through the matrix inversion, as shown in Eq. (16): S in0 A 0−1 · I 0 ; 0−1
(16) 0
where A denotes the matrix inverse of A . Finally, the overall measurement procedure is to first determine the linear polarization components s1 and s2 using Eq. (16) and then obtain the circular polarization component s3 . 4. OPTIMIZATION OF 4QP In this Section, we propose an optimization method for the instrument matrix A and A 0 based on minimizing the condition number (CN) of them. The CN of a matrix is widely used to estimate whether the instrument matrix is good or not in Stokes polarimetry [20–22]. We use the 2-norm CN of a matrix, which is the ratio of the largest singular value of the matrix to the smallest. Most of those optimization methods based on minimizing CN are targeted to the overall performance of the polarimeter for all four Stokes components [20–22]. However, for many applications, the measurement accuracies in the linear polarization components are more important for the performances of a Stokes polarimeter. Therefore, we adopt an optimization method, which gives higher priority to linear polarization measurements. First, we optimize the linear polarization measurements based on Channels 2, 3, and 4. We need at least three channels to determine the linear polarization, i.e., the Stokes parameters s0 , s 1 , and s 2 . In order to optimize the linear polarization measurements, we calculate the CN of A 0 . For convenience, we set the transmission axis of the polarizer in Channel 2 (θp2 ) at 0°; then we calculate the CN of A 0 as a function of the transmission axis angles of Channels 3 and 4 (θp3 and θp4 ). The CN distribution map is shown in Fig. 2. The results show that, when θp3 and θp4 are 60° and 120°, CN reaches 21∕2 , which is the minimal CN value for a linear polarization measurements system [21]. Actually, CN can reach 21∕2 as long as the three transmission axes of the polarizers in Channels 2, 3, and 4 are separated by 60°. The minimum CN of A 0 can guarantee measurement accuracy of the linear polarization components s1 and s2 . When the polarization directions of Channels 2, 3, and 4 are determined, we optimize Channel 1 by minimizing the CN of matrix A. For convenience, we set the transmission axis of the polarizer in Channel 1 (θp1 ) at 45°; then we calculate the CN of A as a function of the retardance δ and fast axis direction θr of
Research Article
Fig. 2. CN map of A 0 . θp2 is set to 0°. Horizontal and vertical coordinates represent the θp3 and θp4 , respectively. Color bar represents the common logarithm of the CN for better data visualization.
the wave plate in Channel 1. The map of CN in Fig. 3 shows that when the retardance and fast axis direction of the wave plate in Channel 1 are 90° and 0°, respectively, CN reaches the minimum value 2.482. Actually, our calculations show that CN can reach the minimum as long as a quarter-wave plate (δ 90°) is used and there is a 45° difference between θr and θp1 . After the optimal measurements of the linear polarization components s 1 and s2 are completed, the minimum CN of A can guarantee the measurement accuracy of the circular polarization component s3 . In addition, as previously mentioned in Section 2, since the commercially available wave plate is usually more sensitive to the experimental conditions (such as ambient temperature, angle, and wavelength of the incident light) than the polarizer, the above design using only one wave plate should be more accurate than the global optimization design using four wave plates when measuring the linear polarization components. 5. CALIBRATION METHOD AND EXPERIMENTAL RESULTS The Stokes vector S (or S 0 ) is calculated from the intensity vector I (or I 0 ) and the instrument matrix A (or A 0 ) using Eq. (10) [or Eq. (16)]. For 4QP, the vector I (or I 0 ) can be read from the detectors. To determine the instrument matrix A (or A 0 ), we
Fig. 3. CN map of A. θp1 , θp2 , θp3 , and θp4 are set at 45°, 0°, 60°, and 120°. Horizontal and vertical coordinates represent the retardance δ and fast axis direction θr of the wave plate in Channel 1, respectively. Color bar represents the common logarithm of CN for better data visualization.
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can measure all the Mueller matrix elements of the polarizers and the quarter-wave plate in the four channels and extract the first row of each channel to establish the instrument matrix. However, we can also use a simpler but more practical calibration method using the PSG, as shown in Fig. 1, to determine the instrument matrix. To calibrate a Stokes polarimeter, we need to introduce light with known SOP and measure the polarimeter responses. When calibrating the instrument matrix A, the detected light intensity vector I and Stokes vector S in of the incident light in Eq. (5) are known and the instrument matrix A needs to be calculated. For incident beams of N N ≥ 4 different SOP, 4 × N different intensities are measured (4 is the number of detectors), giving a 4 × N calibration Stokes matrix S and a 4 × N intensity matrix I. The relation can be expressed as Eq. (17): I A · S:
(17)
The expanded expression of Eq. (17) can be expressed as Eq. (18): 2
i 2 i 3 · i N i 1 1 1 1 1
3
2
A11 A12 A13 A14
7 6 6 1 2 3 N 7 6i A21 6 2 i2 i2 · i2 7 6 76 6 1 2 3 6 N 7 6i 4 3 i 3 i 3 · i 3 5 4 A31 A41 i 1 i 2 i 3 · i N 4 4 4 4 2 1 s0 6 1 6s 6 1 · 6 1 6s 4 2 s 1 3
3
7 A22 A23 A24 7 7 A32 A33 A34 7 5 A42 A43 A44 3 s2 s3 · sN 0 0 0 7 7 s2 s3 · sN 1 1 1 7 7; 7 s2 s3 · sN 2 2 2 5 s2 s3 · sN 3 3 3
(18)
where the superscript of i denotes the intensity of the N th calibration step and the subscript of i denotes different detectors. The superscript of s denotes the Stokes vector of the N th calibration step and the subscript of s denotes the individual parameters of a Stokes vector. Then the instrument matrix A can be calculated by A I • S−1 , where S−1 represents the inverse of the matrix S when N 4 or the pseudo inverse of the matrix S when N > 4. Similarly, when calibrating the instrument matrix A 0 , the detected light intensity vector I 0 0 and Stokes vector S in0 of the incident light in Eq. (12) are known and the instrument matrix A 0 needs to be calculated. For incident beams of N 0 N 0 ≥ 3 different SOP, 3 × N 0 different intensities are measured (3 is the number of detectors), giving a 3 × N 0 calibration Stokes matrix S 0 and a 3 × N 0 intensity matrix I 0 . The relation can be expressed as Eq. (19): I 0 A 0 · S 0 :
(19)
The expanded expression of Eq. (19) can be expressed as Eq. (20):
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2
i 1 1
6 1 6i 4 2 i 1 3
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i 2 1 i 2 2 i 2 3
3
2 0 0 0 3 A11 A12 A13 7 0 0 0 0 7 76 · i N 5 4 A21 A22 A23 5 2 0 0 0 0 A31 A32 A33 · i N 3 2 1 2 0 3 s0 s0 · sN 0 6 1 2 0 7 7; ·6 s1 · sN 4 s1 5 1 · i N 1
0
s 1 2
s 2 2
·
(20)
0 sN 2
where the superscript of i denotes the intensity of the N 0 th calibration step and the subscript of i denotes different detectors. The superscript of s denotes the Stokes vector of the N 0 th calibration step and the subscript of s denotes the individual parameters of a Stokes vector. Then the instrument matrix A 0 can be calculated by A 0 I 0 • S 0 −1 , where S 0 −1 represents the inverse of the matrix S 0 when N 0 3 or the pseudo inverse of the matrix S 0 when N 0 > 3. The PSG, as shown in Fig. 1, can generate different full polarized SOP to calibrate the 4QP. In this paper, N is set to 18 and N 0 is set to 9. A simple but practical calibration procedure is as follows: first, we fix the transmission axis of the polarizer P1 at 0° and rotate the fast axis of the quarter-wave plate R2 from 0° to 180° in nine equal steps. The corresponding SOP are all on the blue solid path on the Poincare sphere, as shown in Fig. 4. Second, the transmission axis of the polarizer P1 and the fast axis of the quarter-wave plate R2 are rotated from 0° to 180° in nine equal steps simultaneously. The corresponding SOP are on the red-dashed path shown in Fig. 4. All 18 SOP are used to calculate A; the nine SOP on the reddashed path shown in Fig. 4 are used to calculate A 0 . The horizontal linear polarized light from the laser is turned into circular polarization by the angle-fixed quarter-wave plate R1; then it passes through the rotating polarizer P1 and the rotating quarter-wave plate R2. For different angles of P1 and R2, the light intensities entering the 4QP should remain the same, which is required in the calculation of the instrument matrix A using Eq. (18). In actual experiments, however, we predetermine the polarization angular intensity distribution of the light passing through R1 and normalize the intensities to minimize errors due to deviations from the ideal circular polarization.
Fig. 5. Two experimental results representing the Stokes vector measurement accuracy of the 4QP. Result (a) P1 in Fig. 1 is fixed to 0° and the fast axis of R2 is rotated from 0° to 180°. Result (b) P1 in Fig. 1 is fixed to 90° and the fast axis of R2 is rotated from 0° to 180°.
After optimization and calibration, the performance of the 4QP is shown in Fig. 5. When the polarizer P1 in Fig. 1 is fixed to 0° and the fast axis of the quarter-wave plate R2 is rotated from 0° to 180°, the measured Stokes parameters are shown in Fig. 5(a). The mean measurement errors of the linear polarization components s 1 and s 2 are 1.02% and 1.28%, and the mean measurement error of the circular polarization component s 3 is 1.95%. When the polarizer P1 in Fig. 1 is fixed to 90° and the fast axis of the quarter-wave plate R2 is rotated from 0° to 180°, the measured Stokes parameters are shown in Fig. 5(b). The mean measurement errors of the linear polarization components s 1 and s 2 are 1.33% and 1.00%, and the mean measurement error of the circular polarization component s 3 is 2.28%. From Figs. 5(a) and 5(b), we can see that the measurements for all the Stokes parameters are accurate, especially for the linear polarization components. The possible sources for the remaining errors are (1) the intensity instability and the SOP fluctuation of the laser; (2) the beam wanders due to rotations of the polarizer P1 and quarter-wave plate R2 in the PSG; and (3) the unequal and temporal polarization variations of wave front division among the four detectors. In addition to the uniform polarization, the incident beam of 4QP should also have the uniform amplitude, as with other division of wavefront polarimeters. 6. CONCLUSIONS
Fig. 4. Calibration paths on the Poincare sphere. SOP points are generated by the PSG, as shown in Fig. 1.
In this paper, we present a linear polarization optimized Stokes polarimeter. The polarimeter, named 4QP, is based on a 4QD whose four well-balanced detectors are turned into four polarization-analyzing channels by four polarizers at each channel and a wave plate at one channel. Considering applications in favor of linear polarization measurements, instead of optimization for all the Stokes parameters, we optimize the polarimeter first for the linear polarization components and then for the circular polarization component. Following such an optimization process, the transmission axes of the polarizers are set to 45°, 0°, 60°, and 120°, respectively, for Channels 1 to 4. The optimal retardance of the wave plate on Channel 1 is 90° (a quarter-wave plate) and its fast axis direction is 0°. We can calibrate the 4QP to improve its accuracy using known SOP generated by a rotating polarizer and a rotating quarter-wave
Research Article plate. The 4QP can measure the Stokes vector simultaneously and accurately, especially in the part of linear polarization. The 4QP is simple, stable, low cost, and can find wide applications in ellipsometry, Mueller matrix polarimetry, and other polarization studies. National Natural Science Foundation of China (NSFC) (11174178, 11374179, 41106034, 61205199); Science and Technology Project of Shenzhen (CXZZ20140509172959978). REFERENCES 1. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001). 2. E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19, 076013 (2014). 3. N. Ghosh and A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16, 110801 (2011). 4. F. Snik, J. H. H. Rietjens, A. Apituley, H. Volten, B. Mijling, A. D. Noia, S. Heikamp, R. C. Heinbroek, O. P. Hasekamp, J. M. Smit, J. Vonk, D. M. Stam, G. V. Harten, J. D. Boer, and C. U. Keller, “Mapping atmospheric aerosols with a citizen science network of smartphone spectropolarimeters,” Geophys. Res. Lett. 41, 7351–7358 (2014). 5. F. Snik, J. C. Jones, M. Escuti, S. Fineschi, D. Harrington, A. D. Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014). 6. R. Liao and H. Ma, “A study on errors of non-simultaneous polarizedlight scattering measurement of suspended rod-shaped particles,” Appl. Opt. 54, 418–424 (2015). 7. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four Stokes parameters of light,” Int. J. Opt. 29, 685–689 (1982). 8. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
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9. A. Peinado, A. Turpin, A. Lizana, E. Fernandez, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38, 4100–4103 (2013). 10. T. Wakayama, Y. Otani, and T. Yoshizawa, “Axisymmetrical Mueller matrix polarimeter,” Proc. SPIE 7461, 74610M (2009). 11. V. Gruev, A. Ortu, N. Lazarus, J. V. Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express 15, 4994–5007 (2007). 12. X. Zhao, X. Pan, X. Fan, P. Xu, A. Bermak, and V. G. Chigrinov, “Patterned dual-layer achromatic micro-quarter-wave-retarder array for active polarization imaging,” Opt. Express 22, 8024–8034 (2014). 13. J. Chang, N. Zeng, H. He, Y. He, and H. Ma, “Single-shot spatially modulated Stokes polarimeter based on a GRIN lens,” Opt. Lett. 39, 2656–2659 (2014). 14. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999). 15. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006). 16. L. B. Wolff, “Polarization vision a new sensory approach to image understanding,” Image Vis. Comp. 15, 81–93 (1997). 17. M. K. Swami, S. Manhas, P. Buddhiwant, N. Ghosh, A. Uppal, and P. K. Gupta, “Polar decomposition of 3 × 3 Mueller matrix: a tool for quantitative tissue polarimetry,” Opt. Express 14, 9324–9337 (2006). 18. H. He, N. Zeng, R. Liao, T. Yun, W. Li, Y. He, and H. Ma, “Application of sphere-cylinder scattering model to skeletal muscle,” Opt. Express 18, 15104–15112 (2010). 19. R. A. Chipman, Handbook of Optics, 3rd ed. (McGraw-Hill, 2009), vol. I. 20. A. Ambirajan and D. C. Look, “Optimum angles for a polarimeter part I,” Opt. Eng. 34, 1651–1655 (1995). 21. J. S. Tyo, “Design of optimal polarimeters: maximization of signal-tonoise ratio and minimization of systematic error,” Appl. Opt. 41, 619–630 (2002). 22. D. Layden, M. F. G. Wood, and I. A. Vitkin, “Optimum selection of input polarization states in determining the sample Mueller matrix: a dual photoelastic polarimeter approach,” Opt. Express 20, 20466–20481 (2012).
Research Article
Vol. 54, No. 14 / May 10 2015 / Applied Optics
Linear polarization optimized Stokes polarimeter based on four-quadrant detector CHAO HE,1,2 JINTAO CHANG,1,3 YONG WANG,1,3 RAN LIAO,1 HONGHUI HE,1 NAN ZENG,1
AND
HUI MA1,3,*
1
Shenzhen Key Laboratory for Minimal Invasive Medical Technologies, Institute of Optical Imaging and Sensing, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China 2 Department of Biomedical Engineering, Tsinghua University, Beijing 100084, China 3 Department of Physics, Tsinghua University, Beijing 100084, China *Corresponding author: [email protected] Received 19 January 2015; accepted 9 April 2015; posted 16 April 2015 (Doc. ID 232790); published 7 May 2015
A four-quadrant detector (4QD) consists of four well-balanced detectors. We report on a Stokes polarimeter with optimal linear polarization measurements based on a 4QD. We turned the four intensity-detection channels into four polarization-analyzing channels by placing four polarizers and one quarter-wave plate in front of the individual detectors. An optimization method for the four polarization-analyzing channels is proposed to improve measurement accuracy. Considering applications in favor of linear polarization measurements instead of global optimization for all the possible states of polarization (SOP), we optimize the polarimeter first for the linear polarization components and then for the circular polarization component. The polarimeter is capable of simultaneous measurements of fast varying SOP with improved performance for the linear polarizations. © 2015 Optical Society of America OCIS codes: (120.5410) Polarimetry; (130.5440) Polarization-selective devices; (120.2130) Ellipsometry and polarimetry. http://dx.doi.org/10.1364/AO.54.004458
1. INTRODUCTION Polarimetric techniques are widely used in many research fields such as medical diagnosis, material characterization, astronomy, and so on [1–5]. For applications involving nonstatic subjects, such as moving objects, living organisms, and particles in fluids, simultaneous polarization detections have to be used, since polarization components recorded at different times may lead to significant errors in the derived possible states of polarization (SOP) [6]. Nowadays, there are already many techniques for simultaneous SOP measurements, such as multiple-detector polarimeters [7–9], spatial modulation polarimeters [10–13], and spectral modulation polarimeters [14]. Among all those simultaneous Stokes polarimeters, most are designed for optimal performances for all the Stokes components. However, in many occasions, such as remote sensing [15], accurate measurements of linear polarization components are particularly important for retrieving polarization characteristics because natural illuminations outdoors and underwater, scattered and reflected are partially linear polarized [16]. For example, a previous study has shown that, for isotropic turbid media such as some biological tissues, it would be useful and convenient if the polarization parameters can be quantified with linear polarization measurements alone [17]. Simulations based on the sphere-cylinder scattering model also have shown that linear polarization features are more prominent in Mueller matrix element patterns [18]. 1559-128X/15/144458-06$15/0$15.00 © 2015 Optical Society of America
In this paper, we propose a simultaneous Stokes polarimeter, named the four-quadrant polarmeter (4QP) with optimal linear polarization measurements. The polarimeter is based on a 4QD, which includes four well-balanced high-speed photon detectors at the four quadrants. 4QD has been widely used in many areas, such as beam collimation and target tracking. In this paper, the four detectors of the 4QD are turned into the four polarimetric analyzing channels by placing four linear polarizers of different orientations in front of each detector and a wave plate in one of the detectors. As in many applications, the accuracy of linear polarization measurements is more important; thus, we first optimize the performance of the polarimeter for the linear polarization components and then for the circular polarization component. We also present a Stokes polarimeter calibration method and the experimental results. 2. INSTRUMENT STRUCTURE OF 4QP As shown in Fig. 1, the 4QP consists of four polarization analyzers A (including R and P) and a 4QD. The 4QD system is a commercial product (Daheng Optics, GCI-1301) including four packaged photon detectors, each of which is connected to a preamplifier and a 12 bit A/D convertor with 3 μs sampling time. The 4QD system is stable and easy to operate, since the detectors and their electronics are well balanced with matching parameters. Each of the four detectors can be converted to a
Research Article polarization-analyzing channel by using polarizers and wave plates. Optimization of the polarimeter can be achieved by carefully choosing the orientations of the polarizers as well as the orientation and retardance of the wave plates. Since the optical properties of the wave plates are more sensitive to the experimental conditions (such as ambient temperature, the angle and the wavelength of the incident light, than the polarizer), we use four film polarizers (Thorlabs, extinction ratio > 1000∶1 for 500–700 nm) for all four channels and a single achromatic film wave plate (Edmund, 1∕4λ achromatic retarder film for 350–800 nm) for Channel 1. Following the optimization process described in Section 4, the transmission axes of the polarizers are set to 45° (Channel 1), 0° (Channel 2), 60° (Channel 3), and 120° (Channel 4) measured from the horizontal direction. The wave plate in Channel 1 is set as a quarter-wave plate with its fast axis direction at 0°.
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S out M p1 · M r · S in ;
(1)
where S in is the Stokes vector of the incident beam, S in s0 ; s1 ; s 2 ; s3 T . S out is the Stokes vector of the detected light. M p1 and M r are the Mueller matrices of the polarizer and the quarter-wave plate. For the polarization-analyzing Channels 2, 3, and 4, each of which contains only a polarizer, the SOP transformation can be expressed as Eq. (2): S out M pn · S in ;
n 2; 3; 4;
(2)
where M pn denotes the polarizer for the nth channel. The Mueller matrix of a polarizer with arbitrary transmission axis angle is expressed as Eq. (3):
3 p2x − p2y cos 2θp ; p2x − p2y sin 2θp ; 0 p2x p2y ; 6 p2x − p2y cos 2θp ; p2x − p2y cos2 2θp 2px py sin2 2θp ; p2x − p2y cos 2θp sin 2θp − 2px py cos 2θp sin 2θp ; 0 7 7; Mp 6 4 p2 − p2 sin 2θp ; p2 − p2 cos 2θp sin 2θp − 2p p cos 2θp sin 2θp ; 2px py cos2 2θp p2x − p2y sin2 2θp 0 5 x y x y x y 0 0 0 2px py 2
(3) where px and py are the transmission ratios in the transmission axis and extinction axis, and θp is the angle of the direction of the transmission axis. The Mueller matrix of a wave plate is expressed as Eq. (4): 2 1; 3 0; 0; 0 2 2 0.5 sin 4θr 1 − cos δ; − sin 2θr sin δ 7 6 0; cos 2θr sin 2θr cos δ; (4) Mr 4 5; 0; 0.5 sin 4θr 1 − cos δ; sin2 2θr cos2 2θr cos δ; cos 2θr sin δ − cos 2θr sin δ; cos δ 0; sin 2θr sin δ; where δ and θr represent the retardance and fast axis direction, respectively. Figure 1 also shows the schematic diagram of the polarization state generator (PSG). The PSG is used to generate different known SOP to calibrate the 4QP and to evaluate its performance. The PSG consists of a laser (Changchun New Industries, MGL-III-532 nm), a polarizer P1 (Thorlabs, extinction ratio > 5000∶1), and quarter-wave plates R1 and R2 (Thorlabs, 532 mm, WPQ10E-532). R1 is set at a fixed angle to transform the linear polarization of the laser to circular polarization. P1 and R2 are mounted on two high-precision motorized rotation stages (Thorlabs, PRM1Z8E). For each incident SOP generated by the PSG, four specific analyzing SOP are detected by the detectors; then, the four intensity values are used to calculate the Stokes vector of the incident light beam.
The intensity signals from the four polarization-analyzing channels correspond to the first component of the four Stokes vectors on the corresponding detectors. These intensities can be expressed as an intensity vector I i 1 ; i 2 ; i 3 ; i 4 T . The relationship between I and S in can be expressed in a matrix form: I A · S in ;
(5)
where A is a 4 × 4 matrix, named the instrument matrix, which can be calculated from Eqs. (3) and (4): A1; ∶ p2x p2y ; cos 2θp1 · p2x − p2y · cos2 2θr cos δ · sin2 2θr sin 2θp1 · p2x − p2y · cos 2θr · sin 2θr − cos 2θr · sin 2θr · cos δ; sin 2θp1
3. MEASUREMENT PRINCIPLE OF 4QP
· p2x − p2y · cos δ · cos2 2θr sin2 2θr cos 2θp1
We use the Mueller calculus to describe the measurement principle of 4QP [19]. The Stokes vector can represent all possible SOP of light, including polarized, partially polarized, and unpolarized light. The Mueller matrix provides a complete description of the polarization transforming properties of a sample. For an incident light beam entering the polarizationanalyzing Channel 1 which contains a polarizer and a wave plate, the SOP transformation can be expressed as Eq. (1):
· p2x − p2y · cos 2θr · sin 2θr − cos 2θr · sin 2θr · cos δ; cos 2θr · sin 2θp1 · sin δ · p2x − p2y − cos 2θp1 · sin 2θr · sin δ · p2x − p2y ; A2; ∶
p2x
p2y ;
cos 2θp2 ·
· p2x − p2y ; 0;
(6) p2x
−
p2y ;
sin 2θp2 (7)
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I 0 A 0 · S in0 ;
(12)
0
where A is a 3 × 3 instrument matrix for the linear polarization measurements, which can be calculated from Eqs. (3) and (4): A 0 1; ∶ p2x p2y ; cos 2θp2 · p2x − p2y ; sin 2θp2 · p2x − p2y ; A 0 2; ∶ p2x p2y ; cos 2θp3 · p2x − p2y ; sin 2θp3
(13)
· p2x − p2y ; A 3; ∶ p2x p2y ; cos 2θp4 · p2x − p2y ; sin 2θp4
(14)
0
· p2x − p2y : Fig. 1. Schematic diagram of the PSG, beam expander, and 4QP. R1, angle-fixed quarter-wave plate. P1 and R2, rotatable polarizer and quarter-wave plate. L, beam expander. A (including R and P): Four polarimetric analyzing channels, including one quarter-wave plate R and four film polarizers P. D, four-quadrant detector. Red arrow on the film wave plate R represents the fast axis; grids on the film polarizers P represent the transmission axes.
A3; ∶ p2x p2y ; cos 2θp3 · p2x − p2y ; sin 2θp3 · p2x − p2y ; 0;
(8)
A4; ∶ p2x p2y ; cos 2θp4 · p2x − p2y ; sin 2θp4 · p2x − p2y ; 0:
(9)
In Eqs. (6–9), θp1 , θp2 , θp3 , and θp4 denote the transmission axes of the polarizers in Channels 1, 2, 3, and 4, respectively. Then S in is calculated through the matrix inversion, as shown in Eq. (10): S in A−1 · I;
(10)
A−1
denotes the matrix inverse of A. where However, real measurements contain noises δI in the intensity vector I. The instrument matrix A may also be associated with an error matrix δA. Then, we recast Eq. (5), as shown in Eq. (11): I δI A δA · S δS;
(11)
where δS represents the error of S. S δS can be calculated by S δS invA δA • I δI. To obtain high-precision measurements of S, both δI and δA should be minimized. The well-balanced four detectors and electronics of the 4QD help to minimize δI . For the time-sequential modulation polarimeters, errors in the motorized rotators present the main contribution to δA. For 4QP, however, the measurements of different polarization components are taken simultaneously; A is a known matrix and δA contains uncertainties due to the polarization optics, which should stay small after the initial calibration. We can reduce δS by the optimization of A, which will be discussed in Section 4. For applications in which the linear polarization components S in0 s0 ; s 1 ; s2 T are more important, the intensity signals from the 2, 3, and 4 polarization-analyzing channels can be expressed as an intensity vector I 0 i 2 ; i 3 ; i 4 T , and the relationship between I 0 and S in0 can also be expressed in a matrix form:
(15)
In Eqs. (13–15), θp2 , θp3 , and θp4 denote the transmission axes of the polarizers in Channels 2, 3, and 4, respectively. Then, S in is calculated through the matrix inversion, as shown in Eq. (16): S in0 A 0−1 · I 0 ; 0−1
(16) 0
where A denotes the matrix inverse of A . Finally, the overall measurement procedure is to first determine the linear polarization components s1 and s2 using Eq. (16) and then obtain the circular polarization component s3 . 4. OPTIMIZATION OF 4QP In this Section, we propose an optimization method for the instrument matrix A and A 0 based on minimizing the condition number (CN) of them. The CN of a matrix is widely used to estimate whether the instrument matrix is good or not in Stokes polarimetry [20–22]. We use the 2-norm CN of a matrix, which is the ratio of the largest singular value of the matrix to the smallest. Most of those optimization methods based on minimizing CN are targeted to the overall performance of the polarimeter for all four Stokes components [20–22]. However, for many applications, the measurement accuracies in the linear polarization components are more important for the performances of a Stokes polarimeter. Therefore, we adopt an optimization method, which gives higher priority to linear polarization measurements. First, we optimize the linear polarization measurements based on Channels 2, 3, and 4. We need at least three channels to determine the linear polarization, i.e., the Stokes parameters s0 , s 1 , and s 2 . In order to optimize the linear polarization measurements, we calculate the CN of A 0 . For convenience, we set the transmission axis of the polarizer in Channel 2 (θp2 ) at 0°; then we calculate the CN of A 0 as a function of the transmission axis angles of Channels 3 and 4 (θp3 and θp4 ). The CN distribution map is shown in Fig. 2. The results show that, when θp3 and θp4 are 60° and 120°, CN reaches 21∕2 , which is the minimal CN value for a linear polarization measurements system [21]. Actually, CN can reach 21∕2 as long as the three transmission axes of the polarizers in Channels 2, 3, and 4 are separated by 60°. The minimum CN of A 0 can guarantee measurement accuracy of the linear polarization components s1 and s2 . When the polarization directions of Channels 2, 3, and 4 are determined, we optimize Channel 1 by minimizing the CN of matrix A. For convenience, we set the transmission axis of the polarizer in Channel 1 (θp1 ) at 45°; then we calculate the CN of A as a function of the retardance δ and fast axis direction θr of
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Fig. 2. CN map of A 0 . θp2 is set to 0°. Horizontal and vertical coordinates represent the θp3 and θp4 , respectively. Color bar represents the common logarithm of the CN for better data visualization.
the wave plate in Channel 1. The map of CN in Fig. 3 shows that when the retardance and fast axis direction of the wave plate in Channel 1 are 90° and 0°, respectively, CN reaches the minimum value 2.482. Actually, our calculations show that CN can reach the minimum as long as a quarter-wave plate (δ 90°) is used and there is a 45° difference between θr and θp1 . After the optimal measurements of the linear polarization components s 1 and s2 are completed, the minimum CN of A can guarantee the measurement accuracy of the circular polarization component s3 . In addition, as previously mentioned in Section 2, since the commercially available wave plate is usually more sensitive to the experimental conditions (such as ambient temperature, angle, and wavelength of the incident light) than the polarizer, the above design using only one wave plate should be more accurate than the global optimization design using four wave plates when measuring the linear polarization components. 5. CALIBRATION METHOD AND EXPERIMENTAL RESULTS The Stokes vector S (or S 0 ) is calculated from the intensity vector I (or I 0 ) and the instrument matrix A (or A 0 ) using Eq. (10) [or Eq. (16)]. For 4QP, the vector I (or I 0 ) can be read from the detectors. To determine the instrument matrix A (or A 0 ), we
Fig. 3. CN map of A. θp1 , θp2 , θp3 , and θp4 are set at 45°, 0°, 60°, and 120°. Horizontal and vertical coordinates represent the retardance δ and fast axis direction θr of the wave plate in Channel 1, respectively. Color bar represents the common logarithm of CN for better data visualization.
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can measure all the Mueller matrix elements of the polarizers and the quarter-wave plate in the four channels and extract the first row of each channel to establish the instrument matrix. However, we can also use a simpler but more practical calibration method using the PSG, as shown in Fig. 1, to determine the instrument matrix. To calibrate a Stokes polarimeter, we need to introduce light with known SOP and measure the polarimeter responses. When calibrating the instrument matrix A, the detected light intensity vector I and Stokes vector S in of the incident light in Eq. (5) are known and the instrument matrix A needs to be calculated. For incident beams of N N ≥ 4 different SOP, 4 × N different intensities are measured (4 is the number of detectors), giving a 4 × N calibration Stokes matrix S and a 4 × N intensity matrix I. The relation can be expressed as Eq. (17): I A · S:
(17)
The expanded expression of Eq. (17) can be expressed as Eq. (18): 2
i 2 i 3 · i N i 1 1 1 1 1
3
2
A11 A12 A13 A14
7 6 6 1 2 3 N 7 6i A21 6 2 i2 i2 · i2 7 6 76 6 1 2 3 6 N 7 6i 4 3 i 3 i 3 · i 3 5 4 A31 A41 i 1 i 2 i 3 · i N 4 4 4 4 2 1 s0 6 1 6s 6 1 · 6 1 6s 4 2 s 1 3
3
7 A22 A23 A24 7 7 A32 A33 A34 7 5 A42 A43 A44 3 s2 s3 · sN 0 0 0 7 7 s2 s3 · sN 1 1 1 7 7; 7 s2 s3 · sN 2 2 2 5 s2 s3 · sN 3 3 3
(18)
where the superscript of i denotes the intensity of the N th calibration step and the subscript of i denotes different detectors. The superscript of s denotes the Stokes vector of the N th calibration step and the subscript of s denotes the individual parameters of a Stokes vector. Then the instrument matrix A can be calculated by A I • S−1 , where S−1 represents the inverse of the matrix S when N 4 or the pseudo inverse of the matrix S when N > 4. Similarly, when calibrating the instrument matrix A 0 , the detected light intensity vector I 0 0 and Stokes vector S in0 of the incident light in Eq. (12) are known and the instrument matrix A 0 needs to be calculated. For incident beams of N 0 N 0 ≥ 3 different SOP, 3 × N 0 different intensities are measured (3 is the number of detectors), giving a 3 × N 0 calibration Stokes matrix S 0 and a 3 × N 0 intensity matrix I 0 . The relation can be expressed as Eq. (19): I 0 A 0 · S 0 :
(19)
The expanded expression of Eq. (19) can be expressed as Eq. (20):
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2
i 1 1
6 1 6i 4 2 i 1 3
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i 2 1 i 2 2 i 2 3
3
2 0 0 0 3 A11 A12 A13 7 0 0 0 0 7 76 · i N 5 4 A21 A22 A23 5 2 0 0 0 0 A31 A32 A33 · i N 3 2 1 2 0 3 s0 s0 · sN 0 6 1 2 0 7 7; ·6 s1 · sN 4 s1 5 1 · i N 1
0
s 1 2
s 2 2
·
(20)
0 sN 2
where the superscript of i denotes the intensity of the N 0 th calibration step and the subscript of i denotes different detectors. The superscript of s denotes the Stokes vector of the N 0 th calibration step and the subscript of s denotes the individual parameters of a Stokes vector. Then the instrument matrix A 0 can be calculated by A 0 I 0 • S 0 −1 , where S 0 −1 represents the inverse of the matrix S 0 when N 0 3 or the pseudo inverse of the matrix S 0 when N 0 > 3. The PSG, as shown in Fig. 1, can generate different full polarized SOP to calibrate the 4QP. In this paper, N is set to 18 and N 0 is set to 9. A simple but practical calibration procedure is as follows: first, we fix the transmission axis of the polarizer P1 at 0° and rotate the fast axis of the quarter-wave plate R2 from 0° to 180° in nine equal steps. The corresponding SOP are all on the blue solid path on the Poincare sphere, as shown in Fig. 4. Second, the transmission axis of the polarizer P1 and the fast axis of the quarter-wave plate R2 are rotated from 0° to 180° in nine equal steps simultaneously. The corresponding SOP are on the red-dashed path shown in Fig. 4. All 18 SOP are used to calculate A; the nine SOP on the reddashed path shown in Fig. 4 are used to calculate A 0 . The horizontal linear polarized light from the laser is turned into circular polarization by the angle-fixed quarter-wave plate R1; then it passes through the rotating polarizer P1 and the rotating quarter-wave plate R2. For different angles of P1 and R2, the light intensities entering the 4QP should remain the same, which is required in the calculation of the instrument matrix A using Eq. (18). In actual experiments, however, we predetermine the polarization angular intensity distribution of the light passing through R1 and normalize the intensities to minimize errors due to deviations from the ideal circular polarization.
Fig. 5. Two experimental results representing the Stokes vector measurement accuracy of the 4QP. Result (a) P1 in Fig. 1 is fixed to 0° and the fast axis of R2 is rotated from 0° to 180°. Result (b) P1 in Fig. 1 is fixed to 90° and the fast axis of R2 is rotated from 0° to 180°.
After optimization and calibration, the performance of the 4QP is shown in Fig. 5. When the polarizer P1 in Fig. 1 is fixed to 0° and the fast axis of the quarter-wave plate R2 is rotated from 0° to 180°, the measured Stokes parameters are shown in Fig. 5(a). The mean measurement errors of the linear polarization components s 1 and s 2 are 1.02% and 1.28%, and the mean measurement error of the circular polarization component s 3 is 1.95%. When the polarizer P1 in Fig. 1 is fixed to 90° and the fast axis of the quarter-wave plate R2 is rotated from 0° to 180°, the measured Stokes parameters are shown in Fig. 5(b). The mean measurement errors of the linear polarization components s 1 and s 2 are 1.33% and 1.00%, and the mean measurement error of the circular polarization component s 3 is 2.28%. From Figs. 5(a) and 5(b), we can see that the measurements for all the Stokes parameters are accurate, especially for the linear polarization components. The possible sources for the remaining errors are (1) the intensity instability and the SOP fluctuation of the laser; (2) the beam wanders due to rotations of the polarizer P1 and quarter-wave plate R2 in the PSG; and (3) the unequal and temporal polarization variations of wave front division among the four detectors. In addition to the uniform polarization, the incident beam of 4QP should also have the uniform amplitude, as with other division of wavefront polarimeters. 6. CONCLUSIONS
Fig. 4. Calibration paths on the Poincare sphere. SOP points are generated by the PSG, as shown in Fig. 1.
In this paper, we present a linear polarization optimized Stokes polarimeter. The polarimeter, named 4QP, is based on a 4QD whose four well-balanced detectors are turned into four polarization-analyzing channels by four polarizers at each channel and a wave plate at one channel. Considering applications in favor of linear polarization measurements, instead of optimization for all the Stokes parameters, we optimize the polarimeter first for the linear polarization components and then for the circular polarization component. Following such an optimization process, the transmission axes of the polarizers are set to 45°, 0°, 60°, and 120°, respectively, for Channels 1 to 4. The optimal retardance of the wave plate on Channel 1 is 90° (a quarter-wave plate) and its fast axis direction is 0°. We can calibrate the 4QP to improve its accuracy using known SOP generated by a rotating polarizer and a rotating quarter-wave
Research Article plate. The 4QP can measure the Stokes vector simultaneously and accurately, especially in the part of linear polarization. The 4QP is simple, stable, low cost, and can find wide applications in ellipsometry, Mueller matrix polarimetry, and other polarization studies. National Natural Science Foundation of China (NSFC) (11174178, 11374179, 41106034, 61205199); Science and Technology Project of Shenzhen (CXZZ20140509172959978). REFERENCES 1. R. S. Gurjar, V. Backman, L. T. Perelman, I. Georgakoudi, K. Badizadegan, I. Itzkan, R. R. Dasari, and M. S. Feld, “Imaging human epithelial properties with polarized light scattering spectroscopy,” Nat. Med. 7, 1245–1248 (2001). 2. E. Du, H. He, N. Zeng, M. Sun, Y. Guo, J. Wu, S. Liu, and H. Ma, “Mueller matrix polarimetry for differentiating characteristic features of cancerous tissues,” J. Biomed. Opt. 19, 076013 (2014). 3. N. Ghosh and A. Vitkin, “Tissue polarimetry: concepts, challenges, applications, and outlook,” J. Biomed. Opt. 16, 110801 (2011). 4. F. Snik, J. H. H. Rietjens, A. Apituley, H. Volten, B. Mijling, A. D. Noia, S. Heikamp, R. C. Heinbroek, O. P. Hasekamp, J. M. Smit, J. Vonk, D. M. Stam, G. V. Harten, J. D. Boer, and C. U. Keller, “Mapping atmospheric aerosols with a citizen science network of smartphone spectropolarimeters,” Geophys. Res. Lett. 41, 7351–7358 (2014). 5. F. Snik, J. C. Jones, M. Escuti, S. Fineschi, D. Harrington, A. D. Martino, D. Mawet, J. Riedi, and J. S. Tyo, “An overview of polarimetric sensing techniques and technology with applications to different research fields,” Proc. SPIE 9099, 90990B (2014). 6. R. Liao and H. Ma, “A study on errors of non-simultaneous polarizedlight scattering measurement of suspended rod-shaped particles,” Appl. Opt. 54, 418–424 (2015). 7. R. M. A. Azzam, “Division-of-amplitude photopolarimeter (doap) for the simultaneous measurement of all four Stokes parameters of light,” Int. J. Opt. 29, 685–689 (1982). 8. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10, 309–311 (1985).
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