OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 13 — Jun. 30, 2003
  • pp: 1510–1519
« Show journal navigation

Compact complete imaging polarimeter using birefringent wedge prisms

Kazuhiko Oka and Toshiaki Kaneko  »View Author Affiliations


Optics Express, Vol. 11, Issue 13, pp. 1510-1519 (2003)
http://dx.doi.org/10.1364/OE.11.001510


View Full Text Article

Acrobat PDF (432 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

This paper describes a method for the mapping of spatially-distributed states-of-polarization of light with a simple and compact configuration. A tiny block of polarization-analyzing optics, consisting of four thin birefringent wedge prisms and a sheet analyzer, are incorporated into an imaging polarimeter, such that mesh-like multiple fringes are generated over a CCD image sensor of a video camera. Fourier analysis of the obtained fringes provides information for determining the two-dimensional distribution of the state-of-polarization. No mechanical or active elements for analyzing polarization are used, and all the parameters related to the spatially-dependent monochromatic Stokes parameters corresponding to azimuth and ellipticity angles can be determined from a single frame. The effectiveness of this method is demonstrated by a prototype incorporating calcite wedge prisms.

© 2003 Optical Society of America

1. Introduction

Two-dimensional state-of-polarization (SOP) mapping of light plays an important role in many fields, such as photoelasticity [1

1. J.F.S. Gomes, “Photoelasticity” in Optical Metrology ed. by O. D. D. Soares, Martinus Nijhoff Publishers, Dordrecht, 1987.

], polarimetric testing of optoelectronic devices [2

2. Y. Otani, T. Shimada, T. Yoshizawa, and N. Umeda,“Two-dimensional BirefringenceMeasurement using the Phase Shifting Technique,” Opt. Eng. 33, 1604–1609 (1994). [CrossRef]

], machine vision [3

3. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A , 19, 687–694 (2002). [CrossRef]

], and remote sensing [4

4. D. J. Sanchez, S. A. Gregory, S. Storm, T. E. Payne, and C. K. Davis, “Photopolarimetric Measurements of Geosynchronous Satellites,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, Proc. SPIE4490, 221–236(2001). [CrossRef]

]. In conventional imaging polarimetry, polarization-analyzing optics, consisting of polarizers, rotators, and retarders, is usually used together with a CCD video camera. The SOP distribution is determined from several images, each of which is acquired at various settings of the polarization-analyzing optics. Unfortunately, the optical component that controls the polarization-analyzing optics, such as a rotating analyzer or a liquid-crystal retarder, generally includes either a mechanical element or an active element. These components often bring with them various problems such as vibration, heat generation, and spatial co-registration, and hence require considerable care be taken for precise operation. In addition, the incorporation of such polarization-controlling components into the imaging optics almost always results in a considerable increase in volume and complexity of the measurement system.

While recent progress in the design of elements for polarization control has remarkably enhanced the performance of imaging polarimeters, the development of imaging polarimeters without the need for mechanical or active elements for polarization control remains a beneficial objective. One approach to this objective is to use either micropolarizers or microretarders [5

5. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16, 1168–1174 (1999). [CrossRef]

, 6

6. H. Kikuta, K. Numata, H. Arimitsu, K. Iwata, and N. Kato, “Imaging polarimetry with an micro-retarder array,” in Proceedings of the SICE Annual Conference 2002, (Society of Instrument and Control Engineers, Tokyo, 2002), pp. 2583–2584.

]. With the rapid progress in microlithography, it is now possible to fabricate out of semiconductor materials subwavelength periodic structures that work as wire-grid polarizers or retarders. Imaging polarimetry can be realized with the arrayed micropolarizers or microretarders placed just in front of the CCD image sensor of a video camera. Most of the current research is confined to the infrared region because of the difficulties in the fabrication of the subwavelength structures, but some trials have been made for shorter wavelengths, namely the visible region.

Another approach for the development of imaging polarimetry without mechanical or active elements is the use of mesh-like multiple fringes. In this approach, multiple interference patterns are generated at the CCD image sensor such that each interference fringe carries the information of a different polarized component of the light under measurement. The spatial frequency filtering of each fringe allows us to demodulate the spatial distribution of the SOP. Based on this concept, one of the present authors has developed an interferometric imaging polarimeter using a reference beam consisting of two orthogonal linearly-polarized components [7

7. K. Oka, J. Ikeda, and Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993). [CrossRef]

9

9. Y. Ohtsuka and K. Oka, “Contour mapping of the spatiotemporal state of polarization of light,” Appl. Opt. 33, 2633–2636 (1994). [CrossRef] [PubMed]

]. A similar idea has been adopted by Colomb et al. for their holographic imaging polarimeter [10

10. 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). [CrossRef] [PubMed]

]. Developing the idea even further, another type of imaging polarimeter with multiple fringes was also developed using a couple of Savart plates [11

11. N. Saito and K. Oka, “Two-dimensional measurement of polarization using spatial carrier,” in Extended Abstracts of the 47th Spring Meeting of the Japan Society of Applied Physics and Related Societies (Japan Society of Applied Physics, Tokyo, 2000), p. 1101 (in Japanese).

, 12

12. N. Saito and K. Oka, “Spatiotemporal polarimeter using spatial carrier fringes,” in Proceedings of the Optics Japan 2000, (Optical Society of Japan, Tokyo, 2000), p. 345–346 (in Japanese).

].

Fig. 1. Schematic of the imaging polarimeter using birefringent wedge prisms.
Fig. 2. Configuration of the block of the polarimetric devices.

2. Performance principle

A schematic of the developed imaging polarimeter is illustrated in Fig. 1. A beam of light from a He-Ne laser source is expanded and spatially filtered by means of a lens pair and a pinhole. It then passes through a polarizer. This expanded linearly-polarized beam is launched into a birefringent sample plate with spatial irreguralities. The transmitted wave turns into elliptically-polarized light, the SOP of which varies with position over the sample. Our objective is to determine the spatial distribution of the SOP as a function of two-dimensional x and y coordinates.

The light emerging from the sample passes through an imaging lens so that the output face of the sample is imaged over a CCD image sensor. A block of polarimetric devices, consisting of four birefringent wedge prisms PR1, PR2, PR3, and PR4 and an analyzer A, is placed just in front of the CCD image sensor. The fast axes of PR1, PR2, PR3, and PR4 are oriented at 0°, 90°, 45°, and -45° respectively relative to the x-axis, and the transmission axis of A is aligned with the x-axis. The planes of contact between PR1 and PR2, and between PR3 and PR4, are inclined with respect to the y and x axes with an angle α, whereas the other contact planes as well as the input face of PR1 are parallel to the xy plane. For now, we assume that the wedge angle α is small enough that the refraction occurring at the inclined contact surfaces is negligible.

We now formulate the analytic expression of the polarimeter. It is useful to think of the block of the polarimetric devices as composed of three elements, namely the pair of PR1 and PR2, the pair of PR3 and PR4, and the analyzer A, as illustrated in the blue, yellow, and gray parts of Fig. 2. The first and second elements act as retarders whose retardations linearly vary with y and x coordinates, respectively. The orthogonal principal axes of the first element are coincident with the x and y axes, whereas those of the second element are oriented at ±45° to the x-axes. Let S 0(x,y), S 1(x,y), S 2(x,y), and S 3(x,y) be the spatially-dependent Stokes parameters of the light emerging from the birefringent sample. Mueller calculus of the optical system allows us to derive the analytic expression for the intensity pattern formed over the CCD image sensor as

I(x,y)=12S0(x,y)+12S1(x,y)cos2πUx
+14S23(x,y)cos{2πU(xy)+arg[S23(x,y)]}
14S23(x,y)cos{2πU(x+y)arg[S23(x,y)]},
(1)

with

S23(x,y)=S2(x,y)+iS3(x,y),
(2)
U=2Bλtanα,
(3)

and where B and λ respectively denote the birefringence of the prisms and the wavelength of the incident light. This implies that the obtained interference pattern consists of one low-frequency component and three quasi-cosinusoidal components which can be seen as fringes in the spatial pattern. One fringe has the period 1/U, and the other two fringes have the period 1/(√2U). The low-frequency component is proportional to S 0(x,y), whereas each fringe carries the information of either S 1(x,y) or S 23(x,y).

I˜(fx,fy)=12A0(fx,fy)+14A1(fxU,fy)+14A1*(fxU,fy)
+18A23(fxU,fy+U)+18A23*(fxU,fy+U)
18A23*(fxU,fyU)18A23(fxU,fyU),
(4)

where

A0(fx,fy)=[S0(x,y)],
(5)
A1(fx,fy)=[S1(x,y)],
(6)
A23(fx,fy)=[S23(x,y)].
(7)

Here fx and fy denote the spatial frequencies and stands for the operator of the two-dimensional Fourier transformation. The seven components included in Ĩ(fx, fy), which are centered at (fx, fy)=(0,0), (±U,0), (±U,∓U), and (±UU), are satisfactorily separable from one another over the spatial frequency domain if the inclination angle α of the birefringent wedge prisms are properly selected. It follows that any one of the components can be extracted by spatial-frequency filtering. The Fourier inversion of the first, second, and fourth terms of Eq. (4) respectively give

F0(x,y)=12S0(x,y),
(8)
F1(x,y)=14S1(x,y)exp[i2πUx],
(9)
F23(x,y)=18S23(x,y)exp[i2πU(xy)].
(10)

θ(x,y)=12tan1[S2(x,y)S1(x,y)],
(11)
ε(x,y)=12tan1[S3(x,y)S1(x,y)2+S2(x,y)2].
(12)

3. Experimental procedure

To demonstrate the performance principle, a prototype of the imaging polarimeter was fabricated. Figure 3 is the photograph of the fabricated imaging polarimeter. The left-hand side of the photograph is the imaging lens, Micro Nikkor (f=50 mm) with an extension ring, whose magnification is set to be unity. The right-hand side of the photograph shows the CCD video camera incorporating the birefringent wedge prisms and the analyzer. It should be noted that the polarization-analyzing devices are wholly included within the housing of the CCD video camera, and cannot be seen in this photograph. The CCD video camera, Hamamatsu C3077, possesses a 2/3 inch-size CCD image sensor, and provides the NTSC-formatted analog video signal. This signal is sent to a computer to be digitized into the image consisting of 512 pixel × 512 pixel, each pixel corresponding to the area of 12.9 µm square over the image sensor of the CCD video camera. The block of the four birefringent wedge prisms made of calcite and one polarcor analyzer is placed just in front of the image sensor. Its height and width are 12 mm × 12 mm, and the total thickness is less than 2 mm. The wedge angle α is set to be 1.5°, which determines the periods of the fringes formed over the image sensor as 1/U=72.5 µm (5.62 pixel) and 1/(√2U)=51.3 µm (4.07 pixel). It should be noted that the wedge angle α is selected so that the beam displacement at the image sensor due to the double refraction at the inclined interfaces do not exceed the pixel size.

Fig. 3. Photograph of the fabricated imaging polarimeter.

A 90° twisted nematic liquid crystal cell was supplied for the experiment as a birefringent sample. Part of the cell is covered with patterned transparent electrodes so that electric fields can be applied across the cell. When light passes through the regions without the electrodes, the azimuth of the polarization ellipse rotates by 90°. On the other hand, the light passing through the regions with electrodes having sufficient electric fields experiences no change in its SOP. Since only part of the liquid crystal cell is covered with the transparent electrodes, the SOP of the light transmitted through the cell is spatially-dependent. The SOP distribution was measured with the fabricated imaging polarimeter.

4. Results and discussion

Figure 4 shows the intensity pattern I(x,y) obtained when the sample was illuminated with a linearly-polarized collimated wave whose polarization azimuth angle is -67.5°. In this figure, a 208 pixel × 190 pixel portion of the intensity pattern is extracted and enlarged for the convenience of seeing the details of the intensity pattern. The mesh-like fringes carry the information of all the spatially-dependent Stokes parameters. This figure implies that the SOP of the region inside the transparent electrode of the shape ‘A’ is different from that of the region outside the electrode. This intensity pattern is Fourier transformed to give the spectrum Ĩ(fx, fy) by means of the fast Fourier transform (FFT) algorithm. The power spectrum |Ĩ(fx, fy)|2 is shown in Fig. 5. This spectrum consists of seven peaks. The peak around (fx, fy)=(0,0) possesses the information of S 0(x,y), as designated by Eq. (4). Two peaks centered at (fx, fy)=(+(512/5.62),0)=(+91.1,0) and (fx, fy)=(-91.1,0) carry the information of S 1(x,y), and the rest of the peaks at (fx, fy)=(+91.1,+91.1), (+91.1,-91.1), (-91.1,+91.1), and (-91.1,-91.1) are determined by S 23(x,y). The seven peaks included in the spectrum Ĩ(fx, fy) are successfully separated from one another in the spatial frequency domain, and hence can easily be extracted from the spectrum.

The spatial-distribution of the Stokes parameters are demodulated from the respective peaks in the spectrum Ĩ(fx, fy), and then the azimuth and ellipticity angles are computed by use of Eqs. (11) and (12). Figure 6 represents the three dimensional plots of the obtained azimuth angle θ(x,y) and ellipticity angle ε(x,y), and Figure 7 shows their cross section, cut at the red dotted planes in Fig. 6. It should be noted that these paired angles are suited for the assessment of the light transmitted through the twisted nematic liquid crystal cell. That is because the twisted nematic liquid crystal does not affect the ellipticity angle of the transmitting light; only the azimuth angle changes while passing through the liquid crystal cell. As can be seen from Fig. 6, an apparent difference exists between the azimuth angles of the light which does and does not pass the electrode of the ‘A’ shape. On the other hand, the ellipticity angle is almost constant over the observation area except for the edge of the electrode. The cross section shows that the azimuth step is almost 90°, and the ellipticity angle is everywhere almost 0°. This implies that the light which has passed the electrode is a linearly-polarized wave with an azimuth of -67.5°, which is almost the same as that of the incident light, whereas the light which has not passed the electrode experiences 90° rotation in its polarization azimuth. It should be noted that some artifacts are found around the edge of the electrode; the width of the artifact is around 10 pixels. This determines the spatial resolution of the fabricated system. Aside from the artifacts, the result agrees well with the function of the 90° twisted nematic liquid crystal, and hence proves the performance principle of the present method.

Fig. 4. Intensity pattern with mesh-like fringes.
Fig. 5. Power spectrum of the intensity pattern.
Fig. 6. Spatial distribution of (a) azimuth angle θ(x, y) and (b) ellipticity angle ε(x, y).
Fig. 7. Cross sections of Fig. 6.

It should be noted, however, that the current configuration of the calcite wedge prisms is not suited for the application areas which require higher spatial resolution. In the method presented here, the four spatially-dependent Stokes parameters are modulated over the fringes in the single intensity pattern. This implies that the spatial variation of the SOP of the light that is being measured should vary gradually compared to the period of the fringes. If the SOP abruptly changes at a certain point in the observation area, this method cannot offer correct results around the point. That is why some artifacts are found around the edge of the electrode in the experimental results. To increase the spatial resolution, the fringe period should be minimized. However the present configuration of the calcite prisms is not feasible to offer such finely spaced fringes. Although the fringe periods can be decreased with an increase in the wedge angle α, this increase also causes the significant beam splitting effect due to the double refraction at the inclined interface of the prisms, much like the function of Wollaston prisms. For applications which require higher spatial resolution, another configuration using wedge prisms with smaller beam displacement should be selected [17

17. T. Tsuruta, Applied Optics II, Sec.5.1, (Baifukan, Tokyo, 1990).

]. Detailed discussion about the optimization of the configuration and the parameters for higher spatial resolution will be presented elsewhere.

Finally, some discussion of the applicabilities of this method to time-varying objects is in order. It is obvious that the present method is useful for stationary objects, and may also have some time-varying applications, because no rapidly controllable mechanical or active components for polarization-analyzing are required. The present method can be conveniently applied to time-variation measurements with the above-mentioned features of simplicity and compactness remaining intact. Each frame taken from the video camera offers an SOP distribution at a different time, and hence the temporal resolution to follow a change in SOP is determined only with the frame rate of the CCD video camera. It should be noted, however, that this fact does not directly suggest the superiority of the measurement speed of the present method. In general, many high-speed video cameras increase frame rate by sacrificing pixel numbers. The present method requires a video camera with many pixels to acquire the fine structure of the mesh-like intensity pattern, and is therefore not compatible with many high-speed cameras. When this method is used to follow a rapid change in SOP distribution, it is necessary to consider not only the frame rate but also the pixel numbers of the video camera.

5. Conclusion

A revolutionary method has been developed for the measurement of the two-dimensional SOP distribution. Introducing a tiny block of polarization-analyzing optics, consisting of four thin birefringent wedge prisms and a sheet analyzer, into the CCD video camera makes it possible to measure the SOP distribution without using mechanical or active optics for the polarization analysis. The performance principle was verified by the demonstrative experiment for the 90° twisted nematic liquid crystal cell.

Acknowledgements

The authors are grateful to Noshi Baba and Tsuyoshi Ishigaki, for many helpful discussions. They are also grateful to Hiroyuki Kowa of Uniopt and Shinji Yasuda ofMiyuki Kogaku for the help in fabricating the polarimeter. They thank Euctace L. Dereniak, Nathan Hagen, and Tod C. Picone for their useful comments for the manuscript.

References and links

1.

J.F.S. Gomes, “Photoelasticity” in Optical Metrology ed. by O. D. D. Soares, Martinus Nijhoff Publishers, Dordrecht, 1987.

2.

Y. Otani, T. Shimada, T. Yoshizawa, and N. Umeda,“Two-dimensional BirefringenceMeasurement using the Phase Shifting Technique,” Opt. Eng. 33, 1604–1609 (1994). [CrossRef]

3.

D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, “Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,” J. Opt. Soc. Am. A , 19, 687–694 (2002). [CrossRef]

4.

D. J. Sanchez, S. A. Gregory, S. Storm, T. E. Payne, and C. K. Davis, “Photopolarimetric Measurements of Geosynchronous Satellites,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, Proc. SPIE4490, 221–236(2001). [CrossRef]

5.

G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, “Micropolarizer array for infrared imaging polarimetry,” J. Opt. Soc. Am. A 16, 1168–1174 (1999). [CrossRef]

6.

H. Kikuta, K. Numata, H. Arimitsu, K. Iwata, and N. Kato, “Imaging polarimetry with an micro-retarder array,” in Proceedings of the SICE Annual Conference 2002, (Society of Instrument and Control Engineers, Tokyo, 2002), pp. 2583–2584.

7.

K. Oka, J. Ikeda, and Y. Ohtsuka, “Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,” J. Mod. Opt. 40, 1713–1723 (1993). [CrossRef]

8.

K. Oka and Y. Ohtsuka, “Polarimetry for spatiotemporal photoelastic analysis,” Exp. Mech. 33, 44–48 (1993). [CrossRef]

9.

Y. Ohtsuka and K. Oka, “Contour mapping of the spatiotemporal state of polarization of light,” Appl. Opt. 33, 2633–2636 (1994). [CrossRef] [PubMed]

10.

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). [CrossRef] [PubMed]

11.

N. Saito and K. Oka, “Two-dimensional measurement of polarization using spatial carrier,” in Extended Abstracts of the 47th Spring Meeting of the Japan Society of Applied Physics and Related Societies (Japan Society of Applied Physics, Tokyo, 2000), p. 1101 (in Japanese).

12.

N. Saito and K. Oka, “Spatiotemporal polarimeter using spatial carrier fringes,” in Proceedings of the Optics Japan 2000, (Optical Society of Japan, Tokyo, 2000), p. 345–346 (in Japanese).

13.

T. Kaneko and K. Oka, “Two-Dimensional Mapping of Polarization Using Birefringent Wedges,” in Extended Abstracts of the 49th Spring Meeting of the Japan Society of Applied Physics and Related Societies, (Japan Society of Applied Physics, Tokyo, 2002), p. 977 (in Japanese).

14.

K. Oka and T. Kaneko, “Polarization Mapping Using Birefringent Prism,” in Proceedings of the SICE Annual Conference 2002 (Society of Instrument and Control Engineers, Tokyo, 2002), pp. 2581–2582.

15.

K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24, 1475–1477 (1999). [CrossRef]

16.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1988) p 51.

17.

T. Tsuruta, Applied Optics II, Sec.5.1, (Baifukan, Tokyo, 1990).

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.5410) Instrumentation, measurement, and metrology : Polarimetry
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: June 3, 2003
Revised Manuscript: June 20, 2003
Published: June 30, 2003

Citation
Kazuhiko Oka and Toshiaki Kaneko, "Compact complete imaging polarimeter using birefringent wedge prisms," Opt. Express 11, 1510-1519 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1510


Sort:  Journal  |  Reset  

References

  1. J.F.S.Gomes, �??Photoelasticity�?? in Optical Metrology ed. by O. D. D. Soares, Martinus Nijhoff Publishers, Dordrecht, 1987.
  2. Y.Otani, T.Shimada, T.Yoshizawa, and N.Umeda,�??Two-dimensional Birefringence Measurement using the Phase Shifting Technique,�?? Opt. Eng. 33, 1604�??1609 (1994). [CrossRef]
  3. D. Miyazaki, M. Saito, Y. Sato, and K. Ikeuchi, �??Determining surface orientations of transparent objects based on polarization degrees in visible and infrared wavelengths,�?? J. Opt. Soc. Am. A , 19, 687�??694 (2002). [CrossRef]
  4. D. J. Sanchez, S. A. Gregory, S. Storm, T. E. Payne, C. K. Davis, �??Photopolarimetric Measurements of Geosynchronous Satellites,�?? in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, Proc. SPIE 4490, 221�??236(2001). [CrossRef]
  5. G. P. Nordin, J. T. Meier, P. C. Deguzman, and M. W. Jones, �??Micropolarizer array for infrared imaging polarimetry,�?? J. Opt. Soc. Am. A 16, 1168�??1174 (1999). [CrossRef]
  6. H. Kikuta, K. Numata, H. Arimitsu, K. Iwata, and N. Kato, �??Imaging polarimetry with an micro-retarder array,�?? in Proceedings of the SICE Annual Conference 2002, (Society of Instrument and Control Engineers, Tokyo, 2002), pp. 2583�??2584.
  7. K. Oka, J. Ikeda, and Y. Ohtsuka, �??Novel polarimetric technique exploring spatiotemporal birefringent response of an anti-ferroelectric liquid crystal cell,�?? J. Mod. Opt. 40, 1713�??1723 (1993). [CrossRef]
  8. K. Oka and Y. Ohtsuka, �??Polarimetry for spatiotemporal photoelastic analysis,�?? Exp. Mech. 33, 44�??48 (1993). [CrossRef]
  9. Y. Ohtsuka and K. Oka, �??Contour mapping of the spatiotemporal state of polarization of light,�?? Appl. Opt. 33, 2633�??2636 (1994). [CrossRef] [PubMed]
  10. 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). [CrossRef] [PubMed]
  11. N. Saito and K. Oka, �??Two-dimensional measurement of polarization using spatial carrier,�?? in Extended Abstracts of the 47th Spring Meeting of the Japan Society of Applied Physics and Related Societies (Japan Society of Applied Physics, Tokyo, 2000), p. 1101 (in Japanese).
  12. N. Saito and K. Oka, �??Spatiotemporal polarimeter using spatial carrier fringes,�?? in Proceedings of the Optics Japan 2000, (Optical Society of Japan, Tokyo, 2000), p. 345�??346 (in Japanese).
  13. T. Kaneko and K. Oka, �??Two-Dimensional Mapping of Polarization Using Birefringent Wedges,�?? in Extended Abstracts of the 49th Spring Meeting of the Japan Society of Applied Physics and Related Societies, (Japan Society of Applied Physics, Tokyo, 2002), p. 977 (in Japanese).
  14. K. Oka and T. Kaneko, �??Polarization Mapping Using Birefringent Prism,�?? in Proceedings of the SICE Annual Conference 2002 (Society of Instrument and Control Engineers, Tokyo, 2002), pp. 2581�??2582.
  15. K. Oka and T. Kato, �??Spectroscopic polarimetry with a channeled spectrum,�?? Opt. Lett. 24, 1475�??1477 (1999). [CrossRef]
  16. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1988) p 51.
  17. T. Tsuruta, Applied Optics II, Sec.5.1, (Baifukan, Tokyo, 1990).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited