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Digital holographic three-dimensional Mueller matrix imaging TOSHITAKA KOBATA1
AND
TAKANORI NOMURA2,*
1
Graduate School of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan *Corresponding author: [email protected]‑u.ac.jp
2
Received 6 March 2015; accepted 18 May 2015; posted 26 May 2015 (Doc. ID 235770); published 10 June 2015
A digital holographic Mueller matrix imaging method is proposed. A Mueller matrix is obtained from Stokes vectors that are calculated from the complex amplitude of the two orthogonal polarized lights. The complex amplitude is obtained by digital holography. In the proposed method, three-dimensional Mueller matrix imaging is achieved by 12 recordings. Optical experimental results confirm the feasibility of the proposed method by measuring the Mueller matrices of four linear polarizers, a quarter-wave plate, and two linear polarizers placed at different positions. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (110.5405) Polarimetric imaging; (120.5410) Polarimetry. http://dx.doi.org/10.1364/AO.54.005591
1. INTRODUCTION Polarization imaging is a useful technique for investigating the internal structure or the stress of optical elements, biomedical tissues, and materials [1–8]. Several methods have been proposed to obtain the polarization property of a specimen. In general, methods based on rotating polarizers or retarders have been used [9–12]. In these methods, the polarization property is obtained with mechanical operations for controlling polarization states. In addition, mechanical scanning is required to measure a twodimensional polarization property. To remove the mechanical operations for controlling polarization states, spectroscopic polarimetry with a channeled spectrum [13] or a Wollaston prism [14] has been proposed. These methods are capable of measuring in a short time, although mechanical scanning is also used to measure a two-dimensional polarization property. Methods using a polarization-imaging camera [7], a polarization grating [15], ghost imaging [16], or compressive sensing [17] have been proposed to measure two-dimensional polarization properties without mechanical scanning. The method that uses a polarization-imaging camera consisting of a polarizer and/or a retarder array on an image sensor. However, the alignment and calibration of this device is quite difficult. The method that uses a polarization grating cannot be realized without a special device, e.g., a polarization grating. The ghost imaging or compressive sensing approaches can obtain two-dimensional polarization properties by using a single pixel photodetector, although upward of one thousand recordings are required. Three-dimensional polarization imaging has been realized by using integral imaging [18] and digital holography [19], which is capable of measuring polarization properties at each 1559-128X/15/175591-06$15/0$15.00 © 2015 Optical Society of America
pixel and at arbitrary depth without mechanical scanning. In the integral imaging approach, the spatial resolution of a reconstructed image is limited by a lens array. On the other hand, the spatial resolution obtained by the digital holographic approach is higher than that of the integral imaging approach. Furthermore, imaging systems and special elements, such as a polarizer array, a retarder array, or a polarization grating, are not required. Digital holography can record and reconstruct the complex amplitude of a lightwave at arbitrary depth. Thus, a polarization characteristic can be obtained at arbitrary depth. Digital holographic Jones matrix imaging has been proposed [6,20–22]. This method can measure the polarization state distribution of specimens with a Jones matrix. However, it is not applicable to unpolarized and partially polarized light. A Stokes vector is applicable to all polarization states, such as natural, totally, or partially polarized light, and a Mueller matrix that represents the polarization characteristics of a specimen can be obtained from the Stokes vectors. Stokes vector imaging with digital holography has been reported [19,23], although Mueller matrix imaging has not been applied. Therefore, we propose a digital holographic threedimensional Mueller matrix imaging method. In the proposed method, the Stokes vectors of the transmitted light, which are mandatory to calculate a Mueller matrix, corresponding to six kinds of polarized incident light are obtained by digital holography from the complex amplitude of the two orthogonal polarized lights. Mueller matrix imaging is achieved by twelve recordings. In this paper, we demonstrate the proposed method by using three type specimens which are four linear polarizers, a
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quarter-wave plate, and two linear polarizers placed at different positions.
an incident Stokes vector S in into the transmitted Stokes vector S out : S out MS in :
2. PRINCIPLE OF DIGITAL HOLOGRAPHIC MUELLER MATRIX IMAGING A Mueller matrix is obtained from a Stokes vector, which can be calculated from both the complex amplitude distributions of the horizontal and vertical components. The complex amplitude distributions are obtained by digital holography. The complex amplitude distributions of the horizontal and vertical polarization components of an object wave are given by Aoh X ; Y aoh X ; Y expfiϕoh X ; Y g;
(1)
Aov X ; Y aov X ; Y expfiϕov X ; Y g;
(2)
and respectively, where aoh X ; Y and ϕoh X ; Y denote the amplitude and the phase distributions of the horizontal polarization component of the object wave, respectively. aov X ; Y and ϕov X ; Y denote the amplitude and the phase distributions of the vertical polarization component of the object wave, respectively. The Stokes vector is given by 3 2 2 3 jAoh X ; Y j2 jAov X ; Y j2 S 0 X ; Y 6 S 1 X ; Y 7 6 jAoh X ; Y j2 − jAov X ; Y j2 7 7 6 7 SX ; Y 6 4 S 2 X ; Y 5 4 2 ReAov X ; Y A X ; Y 5; oh 2 ImAov X ; Y Aoh X ; Y S 3 X ; Y (3) where Re and Im denote the real and imaginary parts of the complex amplitude, respectively. The notation denotes a complex conjugate. Equation (3) shows that the Stokes vector can be obtained from only two complex amplitude distributions. M is assumed as the Mueller matrix of a specimen. The Mueller matrix M is defined as the matrix that transforms
The matrix M represents all of the polarization characteristics of a specimen. Figure 1 shows that all of the components of M can be calculated with six different types of polarized incident light, which are horizontally, vertically, 45 deg linearly, and right-handed and left-handed circularly polarized light. The first and second columns of M are calculated from S out of horizontally and vertically polarized incident light. Similarly, the first and third columns of M are calculated from S out of 45 deg linearly polarized incident light, and the first and fourth columns of M are calculated from S out of right-handed and left-handed circularly polarized incident light. The first column of M calculated from the three pairs are used to calibrate M . All of the components of M can be calculated in order to obtain S out corresponding to six different types of polarized incident light. 3. EXPERIMENTAL PROCEDURE The optical experiment of Mueller matrix imaging was performed for three types of specimen, which are four linear polarizers, a quarter-wave plate, and two linear polarizers at different depths. To confirm the feasibility of the proposed method, simple objects are used for experiments. Figure 2 shows an off-axis digital holography setup for Mueller matrix imaging. An He–Ne laser with wavelength of 632.8 nm was used as a vertically polarized light source. A CCD camera with 1024 × 768 pixels and a 4.65 μm × 4.65 μm pixel was used to record holograms. To avoid the effects of the half-mirror and the mirror on the polarization state of the beams, the same polarization devices are used in both arms. Figure 3 shows
Fig. 2. Optical setup for Mueller matrix imaging.
Fig. 1. Acquisition of all components of Mueller matrix M .
(4)
Fig. 3. Specimens used in the experiments.
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Fig. 4. Two polarizers located at different distances between the CCD camera and the polarizer.
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the two specimens used, which have different polarization properties. The left side is linear polarizers with transmission axes of −45, 0, 45, and 90 deg, of which the extinction ratio is 600:1. The right side is the quarter-wave plate designed for a wavelength of 632.8 nm, for which the fast axis is set at an angle of 90 deg against the horizontal axis. Figure 4 shows the two linear polarizers located at different distances between the CCD camera and the polarizers to measure the polarization properties at different depths. Polarizers 1 and 2 with extinction ratio of 600:1 were located at 410 and 250 mm from the CCD camera, respectively. They were set to transmission axes of 90 and 0 deg, respectively. In the six different kinds of polarized incident light, holograms of the horizontal and vertical polarization components were recorded with an analyzer at the transmitted axes of 0 and 90 deg, respectively. Twelve (six kinds of incidentlight × two polarization components) reconstructed images were obtained by a numerical Fresnel diffraction integral. The Mueller matrix M was obtained from the Stokes vector S out corresponding to the six different types of incident light obtained from the reconstructed images by using Eq. (3). The time for data acquisition is about 10 min for twelve holograms. 4. EXPERIMENTAL RESULTS
Fig. 5. Mueller matrix images of polarizers with four different kinds of transmission axis.
The Mueller matrices of the specimens shown in Figs. 3 and 4 were obtained. First, Fig. 5 shows the obtained Mueller matrix image of linear polarizers that have the four kinds of transmission axis. These images are normalized by M 00 X ; Y . At the top left of Fig. 5, the directions of the arrows indicate the transmission axes of the linear polarizers. The obtained Mueller matrix image was different in each area, corresponding to four polarizers. The experimental and theoretical Mueller matrices and their errors are shown in Table 1. The error indicates the difference between the experimental value and the theoretical value. The following errors are obtained in the same way. The experimental values were the averages in rectangular areas that are of 30 × 30 pixels each, as shown in the top left of Fig. 5. The areas were chosen to avoid a ringing from the edges of the
Table 1. Experimental and Theoretical Mueller Matrices of Linear Polarizers with Four Kinds of Transmission Axis Area (deg) 0
90
45
–45
Experimental Value 2 1.00 0.09 −0.09 0.15 3 6 0.87 0.91 −0.12 0.15 7 5 4 0.11 0.08 0.01 0.00 0.00 0.02 −0.01 0.03 2 1.00 −0.79 0.13 0.05 3 −0.08 0.94 −0.13 −0.03 7 6 5 4 −0.02 0.03 0.01 0.01 −0.02 0.02 0.01 −0.01 2 1.00 0.07 0.77 0.14 3 0.03 7 6 0.09 0.04 −0.11 5 4 0.86 0.01 0.74 0.13 0.06 0.08 −0.09 −0.20 2 1.00 0.00 −0.88 0.09 3 0.04 −0.13 0.02 7 6 0.04 5 4 −0.82 0.06 0.84 −0.14 −0.24 −0.05 −0.19 0.17
Theoretical Value 21 1 0 03 61 1 0 07 5 4 0 0 0 0 0 0 0 0 2 1 −1 0 0 3 6 −1 1 0 0 7 5 4 0 0 0 0 0 0 0 0 21 0 1 03 60 0 0 07 5 4 1 0 1 0 0 0 0 0 2 1 0 −1 0 3 6 0 0 0 07 5 4 −1 0 1 0 0 0 0 0
Error 2 0.00 0.10 0.09 6 0.13 0.09 0.12 4 0.11 0.08 0.01 0.00 0.02 0.01 2 0.00 0.21 0.13 6 0.20 0.06 0.13 4 0.02 0.03 0.01 0.02 0.02 0.01 2 0.00 0.07 0.23 6 0.09 0.04 0.11 4 0.14 0.01 0.26 0.06 0.08 0.09 2 0.00 0.00 0.12 6 0.04 0.04 0.13 4 0.18 0.06 0.16 0.24 0.05 0.19
0.15 3 0.15 7 5 0.00 0.03 0.05 3 0.03 7 5 0.01 0.01 0.14 3 0.03 7 5 0.13 0.20 0.09 3 0.02 7 5 0.14 0.17
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Fig. 7. Mueller matrix images of two linear polarizers obtained from reconstructed images at 410 mm.
Fig. 6. Mueller matrix image of a quarter-wave plate.
polarizers. The experimental values obtained were approximately close to the theoretical values. Next, Fig. 6 shows the obtained Mueller matrix image of a quarter-wave plate, which is also normalized by M 00 X ; Y . The experimental and, theoretical Mueller matrices and their errors are shown in Table 2. The experimental values were the averages in rectangular areas that are of 50 × 50 pixels, as shown in the top left of Fig. 6. In some components, in particular M 22 , M 23 , M 32 , and M 33 , large errors were caused. The maximum error was 45% and its average was 13%. Next, Figs. 7 and 8 show the obtained Mueller matrix images that are calculated from the reconstructed images at 410 and 250 mm from a CCD camera. These two Mueller matrix images were each normalized by M 00 X ; Y . The directions of the arrows in the top left of Figs. 7 and 8 indicate the transmission axes of 0 and 90 deg linear polarizers. Tables 3 and 4 show the experimental and theoretical Mueller matrices and their errors, which are calculated from the reconstructed images at 410 and 250 mm, respectively. The experimental values were averages in rectangular areas that have 20 × 20 pixels and 33 × 33 pixels, as shown in the top left of Figs. 7 and 8, respectively. The reason that the rectangular sizes are different is so that we can evaluate for the same scale because the spatial resolution depends on the propagation distance. In Tables 3 and 4, each experimental value obtained was approximately close to each theoretical value at different depths. Therefore, a three-dimensional Mueller matrix can be obtained by the proposed method.
Fig. 8. Mueller matrix images of two linear polarizers obtained from reconstructed images at 250 mm.
5. DISCUSSION In all of the measurement results, the errors of the third and fourth rows of the Mueller matrix were larger than the others, except the light was not transmitted by the analyzer or specimens. Therefore, we consider that these errors are caused by the
Table 2. Experimental and Theoretical Mueller Matrices of a Quarter-Wave Plate Experimental Value 2 1.00 0.11 0.01 6 0.10 4 −0.04 −0.07
0.09 3 0.99 −0.20 0.06 7 5 0.00 0.45 −0.75 −0.01 0.71 0.33
Theoretical Value 21 0 0 03 07 60 1 0 4 5 0 0 0 −1 0 0 1 0
Error 2 0.00 0.11 0.01 6 0.10 0.01 0.20 4 0.04 0.00 0.45 0.07 0.01 0.29
0.09 3 0.06 7 5 0.25 0.33
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Table 3. Experimental and Theoretical Mueller Matrices Obtained from Reconstructed Images at 410 mm Experimental Value 2 1.00 −0.96 0.01 −0.05 3 0.97 −0.01 0.04 7 6 −0.96 5 4 −0.07 0.06 0.01 0.07 −0.07 0.06 0.09 0.01
Theoretical Value 2 1 −1 0 0 3 6 −1 1 0 0 7 5 4 0 0 0 0 0 0 0 0
Error 2 0.00 0.04 0.01 6 0.04 0.03 0.01 4 0.07 0.06 0.01 0.07 0.06 0.09
0.05 3 0.04 7 5 0.07 0.01
Table 4. Experimental and Theoretical Mueller Matrices Obtained from Reconstructed Images at 250 mm Experimental Value 2 1.00 0.89 0.07 −0.01 3 0.90 0.06 −0.01 7 6 0.88 5 4 −0.14 −0.13 0.15 −0.06 −0.04 0.01 −0.06 −0.09
Theoretical Value 21 1 0 03 61 1 0 07 5 4 0 0 0 0 0 0 0 0
phase difference distribution for calculating the Stokes vectors. A Stokes vector to calculate a Mueller matrix is obtained by Eq. (3). Equation (3) can be rewritten as 2 S X ; Y 3 0
6 S X ; Y 7 SX ; Y 4 1 5 S 2 X ; Y S 3 X ; Y 3 2 a2oh X ; Y a2ov X ; Y 2 2 aoh X ; Y − aov X ; Y 7 6 4 5; 2aoh X ; Y aov X ; Y cosfϕX ; Y g 2aoh X ; Y aov X ; Y sinfϕX ; Y g
0.01 3 0.01 7 5 0.06 0.09
be applicable to the industrial and the biomedical fields for measuring three-dimensional polarization properties of specimens, such as birefringence and diattenuation distribution, in a short measurement time. Japan Society for the Promotion of Science (JSPS) (KAKENHI 26390085). REFERENCES
(5)
where ϕX ; Y is the phase difference distribution between the vertical and horizontal polarization components: ϕX ; Y ϕov X ; Y − ϕoh X ; Y :
Error 2 0.00 0.11 0.07 6 0.12 0.10 0.06 4 0.14 0.13 0.15 0.04 0.01 0.06
(6)
As shown in Eq. (5), S 2 X ; Y and S 3 X ; Y are calculated from the amplitude and the phase distributions, whereas S 0 X ; Y and S 1 X ; Y are calculated from only the amplitude distributions. The phase distribution is more sensitive to disturbance and measurement error caused by the alignment of the setup and the profile irregularity of the optical elements. This error can be reduced by using a known polarized area in the object plane to calibrate the phase [6]. The long acquisition time is one of the candidates for the error. In these experiments, the actual degree of the polarization states in both the reference and object arms and the real Mueller matrices of the specimens were unknown. If these states are measured by some other method, the reason for the error will be eliminated. 6. CONCLUSION A digital holographic Mueller matrix imaging method was proposed. This proposed method can measure the threedimensional Mueller matrix of a specimen from twelve recordings with digital holography. The proposed method was demonstrated by using three types of specimen, which were linear polarizers with four kinds of transmission axis, a quarterwave plate, and two linear polarizers at different depths. The last experimental results confirmed the three-dimensional Mueller matrix imaging. The proposed method is expected to
1. J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997). 2. S. Jiao and V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 101–103 (2002). 3. M. Yamanari, Y. Lim, S. Makita, and Y. Yasuno, “Visualization of phase retardation of deep posterior eye by polarization-sensitive swept-source optical coherence tomography with 1-μm probe,” Opt. Express 17, 12385–12396 (2009). 4. Y. Yasuno, S. Makita, Y. Sutoh, M. Itoh, and T. Yatagai, “Birefringence imaging of human skin by polarization-sensitive spectral interferometric optical coherence tomography,” Opt. Lett. 27, 1803–1805 (2002). 5. C.-W. Sun, L.-S. Lu, C. C. Yang, Y.-W. Kiang, and M.-J. Su, “Myocardial tissue characterization based on the time-resolved Stokes-Mueller formalism,” Opt. Express 10, 1347–1353 (2002). 6. T. Colomb, P. Dahlgren, D. Beghuin, E. Cuche, P. Marquet, and C. Depeursinge, “Polarization imaging by use of digital holography,” Appl. Opt. 41, 27–37 (2002). 7. S. Yoneyama, H. Kikuta, and K. Moriwaki, “Simultaneous observation of phase-stepped photoelastic fringes using a pixelated microretarder array,” Opt. Eng. 45, 083604 (2006). 8. T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146–2148 (2007). 9. D. E. Aspnes, “Optimizing precision of rotating-analyzer ellipsometers,” J. Opt. Soc. Am. 64, 639–646 (1974). 10. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. 2, 148–150 (1978). 11. D. H. Goldstein, “Mueller matrix dual-rotating retarder polarimeter,” Appl. Opt. 31, 6676–6683 (1992). 12. M. H. Smith, “Optimization of a dual-rotating-retarder Mueller matrix polarimeter,” Appl. Opt. 41, 2488–2493 (2002). 13. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999). 14. J. D. Perreault, “Triple Wollaston-prism complete-Stokes imaging polarimeter,” Opt. Lett. 38, 3874–3877 (2013).
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15. M. W. Kudenov, M. J. Escuti, N. Hagen, E. L. Dereniak, and K. Oka, “Snapshot imaging Mueller matrix polarimeter using polarization gratings,” Opt. Lett. 37, 1367–1369 (2012). 16. D. Shi, S. Hu, and Y. Wang, “Polarimetric ghost imaging,” Opt. Lett. 39, 1231–1234 (2014). 17. V. Duran, P. Clemente, M. Fernadez-Alonso, E. Tajahuerce, and J. Lancis, “Single-pixel polarimetric imaging,” Opt. Lett. 37, 824–826 (2012). 18. O. Matoba and B. Javidi, “Three-dimensional polarimetric integral imaging,” Opt. Lett. 29, 2375–2377 (2004). 19. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481–483 (2007).
Research Article 20. T. Colomb, E. Cuche, F. Montfort, P. Marquet, and C. Depeursinge, “Jones vector imaging by use of digital holography: simulation and experimentation,” Opt. Commun. 231, 137–147 (2004). 21. M. Yokota, Y. Terui, and I. Yamaguchi, “Analysis of polarization state by digital holography with polarization modulation,” Opt. Rev. 13, 405–409 (2006). 22. M. Yokota, “Polarization analysis by off-axis digital holography with an improved optical system and an evaluation of its performance by simulation,” Appl. Opt. 47, 6325–6333 (2008). 23. T. Tahara, Y. Awatuji, Y. Shimozato, T. Kakue, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Single-shot polarization-imaging digital holography based on simultaneous phase-shifting interferometry,” Opt. Lett. 36, 3254–3256 (2011).
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Digital holographic three-dimensional Mueller matrix imaging TOSHITAKA KOBATA1
AND
TAKANORI NOMURA2,*
1
Graduate School of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan Faculty of Systems Engineering, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan *Corresponding author: [email protected]‑u.ac.jp
2
Received 6 March 2015; accepted 18 May 2015; posted 26 May 2015 (Doc. ID 235770); published 10 June 2015
A digital holographic Mueller matrix imaging method is proposed. A Mueller matrix is obtained from Stokes vectors that are calculated from the complex amplitude of the two orthogonal polarized lights. The complex amplitude is obtained by digital holography. In the proposed method, three-dimensional Mueller matrix imaging is achieved by 12 recordings. Optical experimental results confirm the feasibility of the proposed method by measuring the Mueller matrices of four linear polarizers, a quarter-wave plate, and two linear polarizers placed at different positions. © 2015 Optical Society of America OCIS codes: (090.1995) Digital holography; (110.5405) Polarimetric imaging; (120.5410) Polarimetry. http://dx.doi.org/10.1364/AO.54.005591
1. INTRODUCTION Polarization imaging is a useful technique for investigating the internal structure or the stress of optical elements, biomedical tissues, and materials [1–8]. Several methods have been proposed to obtain the polarization property of a specimen. In general, methods based on rotating polarizers or retarders have been used [9–12]. In these methods, the polarization property is obtained with mechanical operations for controlling polarization states. In addition, mechanical scanning is required to measure a twodimensional polarization property. To remove the mechanical operations for controlling polarization states, spectroscopic polarimetry with a channeled spectrum [13] or a Wollaston prism [14] has been proposed. These methods are capable of measuring in a short time, although mechanical scanning is also used to measure a two-dimensional polarization property. Methods using a polarization-imaging camera [7], a polarization grating [15], ghost imaging [16], or compressive sensing [17] have been proposed to measure two-dimensional polarization properties without mechanical scanning. The method that uses a polarization-imaging camera consisting of a polarizer and/or a retarder array on an image sensor. However, the alignment and calibration of this device is quite difficult. The method that uses a polarization grating cannot be realized without a special device, e.g., a polarization grating. The ghost imaging or compressive sensing approaches can obtain two-dimensional polarization properties by using a single pixel photodetector, although upward of one thousand recordings are required. Three-dimensional polarization imaging has been realized by using integral imaging [18] and digital holography [19], which is capable of measuring polarization properties at each 1559-128X/15/175591-06$15/0$15.00 © 2015 Optical Society of America
pixel and at arbitrary depth without mechanical scanning. In the integral imaging approach, the spatial resolution of a reconstructed image is limited by a lens array. On the other hand, the spatial resolution obtained by the digital holographic approach is higher than that of the integral imaging approach. Furthermore, imaging systems and special elements, such as a polarizer array, a retarder array, or a polarization grating, are not required. Digital holography can record and reconstruct the complex amplitude of a lightwave at arbitrary depth. Thus, a polarization characteristic can be obtained at arbitrary depth. Digital holographic Jones matrix imaging has been proposed [6,20–22]. This method can measure the polarization state distribution of specimens with a Jones matrix. However, it is not applicable to unpolarized and partially polarized light. A Stokes vector is applicable to all polarization states, such as natural, totally, or partially polarized light, and a Mueller matrix that represents the polarization characteristics of a specimen can be obtained from the Stokes vectors. Stokes vector imaging with digital holography has been reported [19,23], although Mueller matrix imaging has not been applied. Therefore, we propose a digital holographic threedimensional Mueller matrix imaging method. In the proposed method, the Stokes vectors of the transmitted light, which are mandatory to calculate a Mueller matrix, corresponding to six kinds of polarized incident light are obtained by digital holography from the complex amplitude of the two orthogonal polarized lights. Mueller matrix imaging is achieved by twelve recordings. In this paper, we demonstrate the proposed method by using three type specimens which are four linear polarizers, a
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quarter-wave plate, and two linear polarizers placed at different positions.
an incident Stokes vector S in into the transmitted Stokes vector S out : S out MS in :
2. PRINCIPLE OF DIGITAL HOLOGRAPHIC MUELLER MATRIX IMAGING A Mueller matrix is obtained from a Stokes vector, which can be calculated from both the complex amplitude distributions of the horizontal and vertical components. The complex amplitude distributions are obtained by digital holography. The complex amplitude distributions of the horizontal and vertical polarization components of an object wave are given by Aoh X ; Y aoh X ; Y expfiϕoh X ; Y g;
(1)
Aov X ; Y aov X ; Y expfiϕov X ; Y g;
(2)
and respectively, where aoh X ; Y and ϕoh X ; Y denote the amplitude and the phase distributions of the horizontal polarization component of the object wave, respectively. aov X ; Y and ϕov X ; Y denote the amplitude and the phase distributions of the vertical polarization component of the object wave, respectively. The Stokes vector is given by 3 2 2 3 jAoh X ; Y j2 jAov X ; Y j2 S 0 X ; Y 6 S 1 X ; Y 7 6 jAoh X ; Y j2 − jAov X ; Y j2 7 7 6 7 SX ; Y 6 4 S 2 X ; Y 5 4 2 ReAov X ; Y A X ; Y 5; oh 2 ImAov X ; Y Aoh X ; Y S 3 X ; Y (3) where Re and Im denote the real and imaginary parts of the complex amplitude, respectively. The notation denotes a complex conjugate. Equation (3) shows that the Stokes vector can be obtained from only two complex amplitude distributions. M is assumed as the Mueller matrix of a specimen. The Mueller matrix M is defined as the matrix that transforms
The matrix M represents all of the polarization characteristics of a specimen. Figure 1 shows that all of the components of M can be calculated with six different types of polarized incident light, which are horizontally, vertically, 45 deg linearly, and right-handed and left-handed circularly polarized light. The first and second columns of M are calculated from S out of horizontally and vertically polarized incident light. Similarly, the first and third columns of M are calculated from S out of 45 deg linearly polarized incident light, and the first and fourth columns of M are calculated from S out of right-handed and left-handed circularly polarized incident light. The first column of M calculated from the three pairs are used to calibrate M . All of the components of M can be calculated in order to obtain S out corresponding to six different types of polarized incident light. 3. EXPERIMENTAL PROCEDURE The optical experiment of Mueller matrix imaging was performed for three types of specimen, which are four linear polarizers, a quarter-wave plate, and two linear polarizers at different depths. To confirm the feasibility of the proposed method, simple objects are used for experiments. Figure 2 shows an off-axis digital holography setup for Mueller matrix imaging. An He–Ne laser with wavelength of 632.8 nm was used as a vertically polarized light source. A CCD camera with 1024 × 768 pixels and a 4.65 μm × 4.65 μm pixel was used to record holograms. To avoid the effects of the half-mirror and the mirror on the polarization state of the beams, the same polarization devices are used in both arms. Figure 3 shows
Fig. 2. Optical setup for Mueller matrix imaging.
Fig. 1. Acquisition of all components of Mueller matrix M .
(4)
Fig. 3. Specimens used in the experiments.
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Fig. 4. Two polarizers located at different distances between the CCD camera and the polarizer.
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the two specimens used, which have different polarization properties. The left side is linear polarizers with transmission axes of −45, 0, 45, and 90 deg, of which the extinction ratio is 600:1. The right side is the quarter-wave plate designed for a wavelength of 632.8 nm, for which the fast axis is set at an angle of 90 deg against the horizontal axis. Figure 4 shows the two linear polarizers located at different distances between the CCD camera and the polarizers to measure the polarization properties at different depths. Polarizers 1 and 2 with extinction ratio of 600:1 were located at 410 and 250 mm from the CCD camera, respectively. They were set to transmission axes of 90 and 0 deg, respectively. In the six different kinds of polarized incident light, holograms of the horizontal and vertical polarization components were recorded with an analyzer at the transmitted axes of 0 and 90 deg, respectively. Twelve (six kinds of incidentlight × two polarization components) reconstructed images were obtained by a numerical Fresnel diffraction integral. The Mueller matrix M was obtained from the Stokes vector S out corresponding to the six different types of incident light obtained from the reconstructed images by using Eq. (3). The time for data acquisition is about 10 min for twelve holograms. 4. EXPERIMENTAL RESULTS
Fig. 5. Mueller matrix images of polarizers with four different kinds of transmission axis.
The Mueller matrices of the specimens shown in Figs. 3 and 4 were obtained. First, Fig. 5 shows the obtained Mueller matrix image of linear polarizers that have the four kinds of transmission axis. These images are normalized by M 00 X ; Y . At the top left of Fig. 5, the directions of the arrows indicate the transmission axes of the linear polarizers. The obtained Mueller matrix image was different in each area, corresponding to four polarizers. The experimental and theoretical Mueller matrices and their errors are shown in Table 1. The error indicates the difference between the experimental value and the theoretical value. The following errors are obtained in the same way. The experimental values were the averages in rectangular areas that are of 30 × 30 pixels each, as shown in the top left of Fig. 5. The areas were chosen to avoid a ringing from the edges of the
Table 1. Experimental and Theoretical Mueller Matrices of Linear Polarizers with Four Kinds of Transmission Axis Area (deg) 0
90
45
–45
Experimental Value 2 1.00 0.09 −0.09 0.15 3 6 0.87 0.91 −0.12 0.15 7 5 4 0.11 0.08 0.01 0.00 0.00 0.02 −0.01 0.03 2 1.00 −0.79 0.13 0.05 3 −0.08 0.94 −0.13 −0.03 7 6 5 4 −0.02 0.03 0.01 0.01 −0.02 0.02 0.01 −0.01 2 1.00 0.07 0.77 0.14 3 0.03 7 6 0.09 0.04 −0.11 5 4 0.86 0.01 0.74 0.13 0.06 0.08 −0.09 −0.20 2 1.00 0.00 −0.88 0.09 3 0.04 −0.13 0.02 7 6 0.04 5 4 −0.82 0.06 0.84 −0.14 −0.24 −0.05 −0.19 0.17
Theoretical Value 21 1 0 03 61 1 0 07 5 4 0 0 0 0 0 0 0 0 2 1 −1 0 0 3 6 −1 1 0 0 7 5 4 0 0 0 0 0 0 0 0 21 0 1 03 60 0 0 07 5 4 1 0 1 0 0 0 0 0 2 1 0 −1 0 3 6 0 0 0 07 5 4 −1 0 1 0 0 0 0 0
Error 2 0.00 0.10 0.09 6 0.13 0.09 0.12 4 0.11 0.08 0.01 0.00 0.02 0.01 2 0.00 0.21 0.13 6 0.20 0.06 0.13 4 0.02 0.03 0.01 0.02 0.02 0.01 2 0.00 0.07 0.23 6 0.09 0.04 0.11 4 0.14 0.01 0.26 0.06 0.08 0.09 2 0.00 0.00 0.12 6 0.04 0.04 0.13 4 0.18 0.06 0.16 0.24 0.05 0.19
0.15 3 0.15 7 5 0.00 0.03 0.05 3 0.03 7 5 0.01 0.01 0.14 3 0.03 7 5 0.13 0.20 0.09 3 0.02 7 5 0.14 0.17
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Fig. 7. Mueller matrix images of two linear polarizers obtained from reconstructed images at 410 mm.
Fig. 6. Mueller matrix image of a quarter-wave plate.
polarizers. The experimental values obtained were approximately close to the theoretical values. Next, Fig. 6 shows the obtained Mueller matrix image of a quarter-wave plate, which is also normalized by M 00 X ; Y . The experimental and, theoretical Mueller matrices and their errors are shown in Table 2. The experimental values were the averages in rectangular areas that are of 50 × 50 pixels, as shown in the top left of Fig. 6. In some components, in particular M 22 , M 23 , M 32 , and M 33 , large errors were caused. The maximum error was 45% and its average was 13%. Next, Figs. 7 and 8 show the obtained Mueller matrix images that are calculated from the reconstructed images at 410 and 250 mm from a CCD camera. These two Mueller matrix images were each normalized by M 00 X ; Y . The directions of the arrows in the top left of Figs. 7 and 8 indicate the transmission axes of 0 and 90 deg linear polarizers. Tables 3 and 4 show the experimental and theoretical Mueller matrices and their errors, which are calculated from the reconstructed images at 410 and 250 mm, respectively. The experimental values were averages in rectangular areas that have 20 × 20 pixels and 33 × 33 pixels, as shown in the top left of Figs. 7 and 8, respectively. The reason that the rectangular sizes are different is so that we can evaluate for the same scale because the spatial resolution depends on the propagation distance. In Tables 3 and 4, each experimental value obtained was approximately close to each theoretical value at different depths. Therefore, a three-dimensional Mueller matrix can be obtained by the proposed method.
Fig. 8. Mueller matrix images of two linear polarizers obtained from reconstructed images at 250 mm.
5. DISCUSSION In all of the measurement results, the errors of the third and fourth rows of the Mueller matrix were larger than the others, except the light was not transmitted by the analyzer or specimens. Therefore, we consider that these errors are caused by the
Table 2. Experimental and Theoretical Mueller Matrices of a Quarter-Wave Plate Experimental Value 2 1.00 0.11 0.01 6 0.10 4 −0.04 −0.07
0.09 3 0.99 −0.20 0.06 7 5 0.00 0.45 −0.75 −0.01 0.71 0.33
Theoretical Value 21 0 0 03 07 60 1 0 4 5 0 0 0 −1 0 0 1 0
Error 2 0.00 0.11 0.01 6 0.10 0.01 0.20 4 0.04 0.00 0.45 0.07 0.01 0.29
0.09 3 0.06 7 5 0.25 0.33
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Table 3. Experimental and Theoretical Mueller Matrices Obtained from Reconstructed Images at 410 mm Experimental Value 2 1.00 −0.96 0.01 −0.05 3 0.97 −0.01 0.04 7 6 −0.96 5 4 −0.07 0.06 0.01 0.07 −0.07 0.06 0.09 0.01
Theoretical Value 2 1 −1 0 0 3 6 −1 1 0 0 7 5 4 0 0 0 0 0 0 0 0
Error 2 0.00 0.04 0.01 6 0.04 0.03 0.01 4 0.07 0.06 0.01 0.07 0.06 0.09
0.05 3 0.04 7 5 0.07 0.01
Table 4. Experimental and Theoretical Mueller Matrices Obtained from Reconstructed Images at 250 mm Experimental Value 2 1.00 0.89 0.07 −0.01 3 0.90 0.06 −0.01 7 6 0.88 5 4 −0.14 −0.13 0.15 −0.06 −0.04 0.01 −0.06 −0.09
Theoretical Value 21 1 0 03 61 1 0 07 5 4 0 0 0 0 0 0 0 0
phase difference distribution for calculating the Stokes vectors. A Stokes vector to calculate a Mueller matrix is obtained by Eq. (3). Equation (3) can be rewritten as 2 S X ; Y 3 0
6 S X ; Y 7 SX ; Y 4 1 5 S 2 X ; Y S 3 X ; Y 3 2 a2oh X ; Y a2ov X ; Y 2 2 aoh X ; Y − aov X ; Y 7 6 4 5; 2aoh X ; Y aov X ; Y cosfϕX ; Y g 2aoh X ; Y aov X ; Y sinfϕX ; Y g
0.01 3 0.01 7 5 0.06 0.09
be applicable to the industrial and the biomedical fields for measuring three-dimensional polarization properties of specimens, such as birefringence and diattenuation distribution, in a short measurement time. Japan Society for the Promotion of Science (JSPS) (KAKENHI 26390085). REFERENCES
(5)
where ϕX ; Y is the phase difference distribution between the vertical and horizontal polarization components: ϕX ; Y ϕov X ; Y − ϕoh X ; Y :
Error 2 0.00 0.11 0.07 6 0.12 0.10 0.06 4 0.14 0.13 0.15 0.04 0.01 0.06
(6)
As shown in Eq. (5), S 2 X ; Y and S 3 X ; Y are calculated from the amplitude and the phase distributions, whereas S 0 X ; Y and S 1 X ; Y are calculated from only the amplitude distributions. The phase distribution is more sensitive to disturbance and measurement error caused by the alignment of the setup and the profile irregularity of the optical elements. This error can be reduced by using a known polarized area in the object plane to calibrate the phase [6]. The long acquisition time is one of the candidates for the error. In these experiments, the actual degree of the polarization states in both the reference and object arms and the real Mueller matrices of the specimens were unknown. If these states are measured by some other method, the reason for the error will be eliminated. 6. CONCLUSION A digital holographic Mueller matrix imaging method was proposed. This proposed method can measure the threedimensional Mueller matrix of a specimen from twelve recordings with digital holography. The proposed method was demonstrated by using three types of specimen, which were linear polarizers with four kinds of transmission axis, a quarterwave plate, and two linear polarizers at different depths. The last experimental results confirmed the three-dimensional Mueller matrix imaging. The proposed method is expected to
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