Abstract

An exact analysis of the working parameters of a modified Wollaston prism is presented. Parameters include the output splitting angle, the retardation, and the location of the plane of the interference fringes (plane of apparent splitting). Results are presented for the entire range of optical axis inclinations and wedge angles. Approximate expressions from the literature are evaluated. An angle of incidence that causes the plane of the interference fringes to be perpendicular to the axis of the optical system is found for each configuration analyzed. This is then applied to the design of modified Wollaston prisms for Nomarski differential interference contrast microscopy.

© 1999 Optical Society of America

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References

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  1. G. Nomarski, “Microinterféromètre différentiel à ondes polarisées,” J. Phys. Radium 16, 9S–11S (1955), in French.
  2. D. L. Lessor, J. S. Hartman, R. L. Gordon, “Quantitative surface topography determination by Nomarski reflection microscopy. I. Theory,” J. Opt. Soc. Am. 69, 357–366 (1979).
    [CrossRef]
  3. M. Pluta, Advanced Light Microscopy (Elsevier, Amsterdam, 1989), Vol. 2.
  4. J. Padawer, “The Nomarski interference microscope; an experimental basis for image interpretation,” J. R. Microsc. Soc. 88, 305–349 (1968).
    [CrossRef]
  5. E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
    [CrossRef]
  6. M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 212–221 (1992).
    [CrossRef]
  7. M. Sochacka, L. R. Staronski, “Phase-stepping DIC technique for reflecting surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 222–232 (1992).
    [CrossRef]
  8. M. Sochacka, “Optical fiber profiling by phase-stepping transverse interferometry,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 160–175 (1992).
    [CrossRef]
  9. J. Courtial, B. A. Patterson, W. Hirst, A. R. Harvey, A. J. Duncan, W. Sibbett, M. J. Padgett, “Static Fourier-transform ultraviolet spectrometer for gas detection,” Appl. Opt. 36, 2813–2817 (1997).
    [CrossRef] [PubMed]
  10. J. Courtial, B. A. Patterson, A. R. Harvey, W. Sibbett, M. J. Padgett, “Design of a static Fourier-transform spectrometer with increased field of view,” Appl. Opt. 35, 6698–6702 (1996).
    [CrossRef] [PubMed]
  11. M. J. Padgett, A. R. Harvey, “A static Fourier-transform spectrometer based on Wollaston prisms,” Rev. Sci. Instrum. 66, 2807–2811 (1995).
    [CrossRef]
  12. D. Steers, W. Sibbett, M. J. Padgett, “Dual-purpose, compact spectrometer and fiber-coupled laser wavemeter based on Wollaston prism,” Appl. Opt. 37, 5777–5781 (1998).
    [CrossRef]
  13. M. J. Padgett, A. R. Harvey, A. J. Duncan, W. Sibbett, “Single-pulse, Fourier-transform spectrometer having no moving parts,” Appl. Opt. 33, 6035–6040 (1994).
    [CrossRef] [PubMed]
  14. B. A. Patterson, M. Antoni, J. Courtial, A. J. Duncan, W. Sibbett, M. J. Padgett, “An ultra-compact static Fourier-transform spectrometer based on a single birefringent component,” Opt. Commun. 130, 1–6 (1996).
    [CrossRef]
  15. J. Wilk, “Localization of the point of beam shearing in birefringent prisms,” Optyka 11, 187–192 (1976), in Polish.
  16. M. Françon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971).
  17. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354–360 (1983).
    [CrossRef] [PubMed]
  18. M. C. Simon, “Wollaston prism with large split angle,” Appl. Opt. 25, 369–376 (1986).
    [CrossRef] [PubMed]
  19. E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
    [CrossRef]
  20. E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
    [CrossRef]
  21. E. Oliva, “Wedged double Wollaston, a device for a single shot polarimetric measurements,” Astron. Astrophys. Suppl. Ser. 123, 589–592 (1997).
    [CrossRef]
  22. R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).
  23. T. W. Ng, “Simultaneous recording of Hv and Vv small-angle light-scattering patterns for crystallization kinetics studies,” J. Polym. Sci. Part B 35, 199–201 (1997).
    [CrossRef]
  24. W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
    [CrossRef]
  25. J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
    [CrossRef]
  26. H. Yu, S. Meng, “Wollaston prism design and working parameters in the Nomarski polarized light interferometer,” Opt. Eng. 35, 2310–2312 (1996).
    [CrossRef]
  27. E. Wood, Crystals and Light (Dover, New York, 1977).
  28. T. K. Gaylord, Optical Modulation, Class Notes (School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Ga., 1997).
  29. F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Philadelphia, Pa., 1961).
  30. M. P. Curie, “Sur la symétrie dans les phénomènes physiques, symétrie d’un champ èlectrique et d’un champ magnétique,” J. Phys. (Paris) 3, 393–415 (1894), in French.

1998 (1)

1997 (6)

J. Courtial, B. A. Patterson, W. Hirst, A. R. Harvey, A. J. Duncan, W. Sibbett, M. J. Padgett, “Static Fourier-transform ultraviolet spectrometer for gas detection,” Appl. Opt. 36, 2813–2817 (1997).
[CrossRef] [PubMed]

E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
[CrossRef]

T. W. Ng, “Simultaneous recording of Hv and Vv small-angle light-scattering patterns for crystallization kinetics studies,” J. Polym. Sci. Part B 35, 199–201 (1997).
[CrossRef]

W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

E. Oliva, “Wedged double Wollaston, a device for a single shot polarimetric measurements,” Astron. Astrophys. Suppl. Ser. 123, 589–592 (1997).
[CrossRef]

1996 (4)

B. A. Patterson, M. Antoni, J. Courtial, A. J. Duncan, W. Sibbett, M. J. Padgett, “An ultra-compact static Fourier-transform spectrometer based on a single birefringent component,” Opt. Commun. 130, 1–6 (1996).
[CrossRef]

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

H. Yu, S. Meng, “Wollaston prism design and working parameters in the Nomarski polarized light interferometer,” Opt. Eng. 35, 2310–2312 (1996).
[CrossRef]

J. Courtial, B. A. Patterson, A. R. Harvey, W. Sibbett, M. J. Padgett, “Design of a static Fourier-transform spectrometer with increased field of view,” Appl. Opt. 35, 6698–6702 (1996).
[CrossRef] [PubMed]

1995 (1)

M. J. Padgett, A. R. Harvey, “A static Fourier-transform spectrometer based on Wollaston prisms,” Rev. Sci. Instrum. 66, 2807–2811 (1995).
[CrossRef]

1994 (1)

1989 (1)

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

1986 (1)

1983 (1)

1979 (1)

1976 (1)

J. Wilk, “Localization of the point of beam shearing in birefringent prisms,” Optyka 11, 187–192 (1976), in Polish.

1968 (1)

J. Padawer, “The Nomarski interference microscope; an experimental basis for image interpretation,” J. R. Microsc. Soc. 88, 305–349 (1968).
[CrossRef]

1955 (1)

G. Nomarski, “Microinterféromètre différentiel à ondes polarisées,” J. Phys. Radium 16, 9S–11S (1955), in French.

1894 (1)

M. P. Curie, “Sur la symétrie dans les phénomènes physiques, symétrie d’un champ èlectrique et d’un champ magnétique,” J. Phys. (Paris) 3, 393–415 (1894), in French.

Alighieri, S. D. S.

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

Antoni, M.

B. A. Patterson, M. Antoni, J. Courtial, A. J. Duncan, W. Sibbett, M. J. Padgett, “An ultra-compact static Fourier-transform spectrometer based on a single birefringent component,” Opt. Commun. 130, 1–6 (1996).
[CrossRef]

Aten, J. A.

E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
[CrossRef]

Bloss, F. D.

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Philadelphia, Pa., 1961).

Bock, W. J.

W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Caruso, A.

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

Chernova, G. P.

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Ciofini, M.

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

Courtial, J.

Curie, M. P.

M. P. Curie, “Sur la symétrie dans les phénomènes physiques, symétrie d’un champ èlectrique et d’un champ magnétique,” J. Phys. (Paris) 3, 393–415 (1894), in French.

Duncan, A. J.

Fontaine, M.

W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Fosbury, R. A. E.

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

Françon, M.

M. Françon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971).

Gaylord, T. K.

T. K. Gaylord, Optical Modulation, Class Notes (School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Ga., 1997).

Gennari, S.

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

Geyer, E. H.

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Gordon, R. L.

Grattan, K. T. V.

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

Hartman, J. S.

Harvey, A. R.

Hirst, W.

Jiang, X. Q.

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

Jockers, K.

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Kemp, J.

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

Kiselev, N. N.

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

Lessor, D. L.

Mallick, S.

M. Françon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971).

Meng, S.

H. Yu, S. Meng, “Wollaston prism design and working parameters in the Nomarski polarized light interferometer,” Opt. Eng. 35, 2310–2312 (1996).
[CrossRef]

Munster, E. B. V.

E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
[CrossRef]

Ng, T. W.

T. W. Ng, “Simultaneous recording of Hv and Vv small-angle light-scattering patterns for crystallization kinetics studies,” J. Polym. Sci. Part B 35, 199–201 (1997).
[CrossRef]

Ning, Y. N.

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

Nomarski, G.

G. Nomarski, “Microinterféromètre différentiel à ondes polarisées,” J. Phys. Radium 16, 9S–11S (1955), in French.

Oliva, E.

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

E. Oliva, “Wedged double Wollaston, a device for a single shot polarimetric measurements,” Astron. Astrophys. Suppl. Ser. 123, 589–592 (1997).
[CrossRef]

Padawer, J.

J. Padawer, “The Nomarski interference microscope; an experimental basis for image interpretation,” J. R. Microsc. Soc. 88, 305–349 (1968).
[CrossRef]

Padgett, M. J.

Palmer, A. W.

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

Patterson, B. A.

Pluta, M.

M. Pluta, Advanced Light Microscopy (Elsevier, Amsterdam, 1989), Vol. 2.

Provost, F. L.

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 212–221 (1992).
[CrossRef]

Quinn, P. J.

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

Sibbett, W.

Simon, M. C.

Sochacka, M.

M. Sochacka, L. R. Staronski, “Phase-stepping DIC technique for reflecting surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 222–232 (1992).
[CrossRef]

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 212–221 (1992).
[CrossRef]

M. Sochacka, “Optical fiber profiling by phase-stepping transverse interferometry,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 160–175 (1992).
[CrossRef]

Staronski, L. R.

M. Sochacka, L. R. Staronski, “Phase-stepping DIC technique for reflecting surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 222–232 (1992).
[CrossRef]

Steers, D.

Tadhunter, C. N.

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

Urbanczyk, W.

W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

Vanzi, L.

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

Vliet, L. J. V.

E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
[CrossRef]

Wilk, J.

J. Wilk, “Localization of the point of beam shearing in birefringent prisms,” Optyka 11, 187–192 (1976), in Polish.

Wood, E.

E. Wood, Crystals and Light (Dover, New York, 1977).

Yu, H.

H. Yu, S. Meng, “Wollaston prism design and working parameters in the Nomarski polarized light interferometer,” Opt. Eng. 35, 2310–2312 (1996).
[CrossRef]

Appl. Opt. (6)

Astron. Astrophys. Suppl. Ser. (2)

E. Oliva, S. Gennari, L. Vanzi, A. Caruso, M. Ciofini, “Optical materials for near infrared Wollaston prisms,” Astron. Astrophys. Suppl. Ser. 123, 179–182 (1997).
[CrossRef]

E. Oliva, “Wedged double Wollaston, a device for a single shot polarimetric measurements,” Astron. Astrophys. Suppl. Ser. 123, 589–592 (1997).
[CrossRef]

Astrophys. Space Sci. (1)

E. H. Geyer, K. Jockers, N. N. Kiselev, G. P. Chernova, “A novel quadruple beam imaging polarimeter and its application to comet Tanaka-Machholz 1992 X,” Astrophys. Space Sci. 239, 259–274 (1996).
[CrossRef]

ESO Messenger (1)

R. A. E. Fosbury, S. D. S. Alighieri, C. N. Tadhunter, P. J. Quinn, “Imaging polarimetry of high-redshift radio galaxies with EFOSC,” ESO Messenger 57, 49–53 (1989).

IEEE Trans. Instrum. Meas. (1)

W. J. Bock, W. Urbanczyk, M. Fontaine, “Characterization of highly birefringent optical fibers using interferometry techniques,” IEEE Trans. Instrum. Meas. 46, 903–907 (1997).
[CrossRef]

J. Microsc. (Oxford) (1)

E. B. V. Munster, L. J. V. Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope,” J. Microsc. (Oxford) 188, 149–157 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. (Paris) (1)

M. P. Curie, “Sur la symétrie dans les phénomènes physiques, symétrie d’un champ èlectrique et d’un champ magnétique,” J. Phys. (Paris) 3, 393–415 (1894), in French.

J. Phys. Radium (1)

G. Nomarski, “Microinterféromètre différentiel à ondes polarisées,” J. Phys. Radium 16, 9S–11S (1955), in French.

J. Polym. Sci. Part B (1)

T. W. Ng, “Simultaneous recording of Hv and Vv small-angle light-scattering patterns for crystallization kinetics studies,” J. Polym. Sci. Part B 35, 199–201 (1997).
[CrossRef]

J. R. Microsc. Soc. (1)

J. Padawer, “The Nomarski interference microscope; an experimental basis for image interpretation,” J. R. Microsc. Soc. 88, 305–349 (1968).
[CrossRef]

Opt. Commun. (1)

B. A. Patterson, M. Antoni, J. Courtial, A. J. Duncan, W. Sibbett, M. J. Padgett, “An ultra-compact static Fourier-transform spectrometer based on a single birefringent component,” Opt. Commun. 130, 1–6 (1996).
[CrossRef]

Opt. Eng. (1)

H. Yu, S. Meng, “Wollaston prism design and working parameters in the Nomarski polarized light interferometer,” Opt. Eng. 35, 2310–2312 (1996).
[CrossRef]

Optyka (1)

J. Wilk, “Localization of the point of beam shearing in birefringent prisms,” Optyka 11, 187–192 (1976), in Polish.

Rev. Sci. Instrum. (1)

M. J. Padgett, A. R. Harvey, “A static Fourier-transform spectrometer based on Wollaston prisms,” Rev. Sci. Instrum. 66, 2807–2811 (1995).
[CrossRef]

Other (9)

M. Françon, S. Mallick, Polarization Interferometers (Wiley-Interscience, New York, 1971).

M. Pluta, Advanced Light Microscopy (Elsevier, Amsterdam, 1989), Vol. 2.

M. Sochacka, F. L. Provost, “Implementation of phase-stepping interferometry to transmitted-light DIC microscopy for dielectric surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 212–221 (1992).
[CrossRef]

M. Sochacka, L. R. Staronski, “Phase-stepping DIC technique for reflecting surface evaluation,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 222–232 (1992).
[CrossRef]

M. Sochacka, “Optical fiber profiling by phase-stepping transverse interferometry,” in Phase Contrast and Differential Interference Contrast Imaging Techniques and Applications, M. Pluta, ed., Proc. SPIE1846, 160–175 (1992).
[CrossRef]

E. Wood, Crystals and Light (Dover, New York, 1977).

T. K. Gaylord, Optical Modulation, Class Notes (School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Ga., 1997).

F. D. Bloss, An Introduction to the Methods of Optical Crystallography (Holt, Philadelphia, Pa., 1961).

J. Kemp, X. Q. Jiang, Y. N. Ning, A. W. Palmer, K. T. V. Grattan, “Dual interferometric displacement measurement system, incorporating a Wollaston interferometer,” in Laser Diode and LED Applications III, K. J. Linden, ed., Proc. SPIE3000, 82–89 (1997).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

(a) CWP made from a positive uniaxial birefringent material. Two triangular pieces of birefringent material are cemented together with their optical axes orthogonal to each other. In one piece, the optical axis (o.a.) is perpendicular to the plane of incidence, whereas in the other piece the optical axis lies in the plane of incidence and is parallel to the entrance and exit faces of the prism. A linearly polarized beam is incident on the entrance face and is split into two orthogonally polarized beams as shown. Two cases are considered. Case 1 is defined when the incident beam goes first through the region whose optical axis is perpendicular to the plane of incidence. In case 2, the incident beam goes first through the region whose optical axis lies in the plane of incidence. (b) MWP made from positive uniaxial birefringent material. The two optical axes are again perpendicular to each other. However, the optical axis that lies in the plane of incidence is inclined with respect to the entrance and the exit faces of the prism. Rays are shown in each case.

Fig. 2
Fig. 2

(a) Definition of the parameters used in the ray-tracing analysis inside a MWP in the case 1 configuration. A linearly polarized beam is incident on the prism at an angle α relative to the normal to the entrance face and at point (0, y 0) in the x, y system. The incident beam is split into two orthogonally polarized beams inside the prism: the TE (dot) and the TM (double arrow) polarized beams. In region 2, the allowed wave-vector surface cross section in the plane of incidence is two circles. The wave-vector directions are given by α2 TE, β2 TE, α2 TM, β2 TM, for TE and TM, respectively. In region 3, the allowed wave-vector surface cross section in the plane of incidence is a circle and an ellipse for TE and TM, respectively. The wave-vector directions are given by β3 TE, α3 TE, β3k TM, α3k TM. The ray (Poynting vector) direction for TM is given by β3S TM, α3S TM. The beams are then refracted into region 4. The splitting angle is defined in region 4 by Δβ = β4 TE - β4 TM. The point of intersection of the two beams is (x PAS, y PAS). Also shown are the thickness of the prism w, the optical axis (o.a.) inclination angle δ relative to the x axis, and the wedge angle of the prism γ. (b) Allowed wave-vector surface cross section of region 3. The quantities θ e and ψ are, respectively, the angle between the TM wave-vector direction and the optical axis in region 3 and the angle between the normal to the middle interface and the TM wave-vector direction in region 3. The quantity ϕ e is the angle between the TM Poynting vector direction and the optical axis in region 3. The quantity θ kS is the angle between the TM wave vector and the TM Poynting vector in region 3.

Fig. 3
Fig. 3

Angular directions β4 TE and β4 TM of TE and TM beams in region 4 as a function of the optical axis inclination angle δ of a MWP made of quartz in the case 1 configuration. Each curve is labeled with the value in degrees of the wedge angle γ to which it corresponds. The wavelength is λ = 555 nm, and the angle of incidence is α = 0. The vertical dashed line indicates the case of the CWP. Also shown is the case of the Rochon prism.

Fig. 4
Fig. 4

Calculation of the splitting angle Δβ as a function of the optical axis inclination δ of a MWP made of quartz in the case 1 configuration. Each curve is labeled with the value in degrees of the wedge angle γ to which it corresponds. The wavelength is λ = 555 nm and the angle of incidence is α = 0. The vertical dashed line indicates the case of the CWP. The horizontal dashed lines are the values of the splitting angle computed with the approximate relation Δβ = 2(n E - n O ) tan(90° - γ) at γ = 45° (upper dashed curve) and γ = 89.25° (lower dashed curve). Also shown is the case of the Rochon prism.

Fig. 5
Fig. 5

Locations of the PAS’s of a MWP made of quartz in the case 1 configuration for various values of the optical axis inclination angle δ. The PAS’s are calculated for beams at normal incidence all along the entrance face. The PAS’s are labeled with the corresponding values of δ. The prism outline is shown.

Fig. 6
Fig. 6

Locations of the PAS’s of a MWP made of quartz in the case 2 configuration for various values of the optical axis inclination angle δ. The PAS’s are calculated for beams at normal incidence all along the entrance face. The PAS’s are labeled with the various values in degrees of δ to which they correspond. The prism outline is shown.

Fig. 7
Fig. 7

Retardation induced by a MWP as a function of the y coordinate of the incident beam y 0 and the optical axis inclination angle δ at normal incidence. Along the white curve, the retardation R is zero.

Fig. 8
Fig. 8

Approximate retardation induced by a MWP for small angles of incidence as a function of the y coordinate of the incident beam y 0 and the optical axis inclination angle δ. The approximation used is R aprx = Δβy 0, where y 0 is the y coordinate of the incident beam on the entrance face and Δβ is the output splitting angle.

Fig. 9
Fig. 9

Retardation induced by a MWP as a function of the thickness w and the optical axis inclination angle δ for a beam incident at y 0 = 10 mm.

Fig. 10
Fig. 10

Geometry of a MWP tilted with respect to the axis of the optical system showing the PAS. The angle of incidence is α. Shown are η, the angle to the PAS from the x axis; σ, the angle between the PAS and the axis of the optical system; and D, the distance from the exit face to the PAS. o.a., optical axis.

Fig. 11
Fig. 11

Angle of incidence α that is necessary to make the PAS perpendicular to the axis of the optical system as a function of the optical axis inclination angle δ. The prisms are quartz in the case 1 configuration. Each curve is labeled with the value in degrees of the wedge angle γ to which it corresponds.

Fig. 12
Fig. 12

Distance D from the exit face to the PAS as a function of the optical axis inclination angle δ when the PAS is perpendicular to the axis of the optical system. The prisms are quartz in the case 1 configuration. Each curve is labeled with the value in degrees of the wedge angle γ to which it corresponds.

Fig. 13
Fig. 13

Geometry of a reflection DIC microscope. The linearly polarized incident beam is split into two rays by the MWP. The dot represents the TE polarization, and the double arrow represents the TM polarization. These two beams intersect at point A in the PAS. The focal plane of the objective coincides with the PAS. The beams are then focused on the object plane and reflected by it. They intersect at point C in the PAS of the Wollaston prism. The beams are then recombined by the prism and interfere upon passing through the polarizer. o.a., optical axis.

Fig. 14
Fig. 14

Retardation R Bias induced by a MWP implemented in a reflection DIC system as a function of the y coordinate y 0 along the entrance face of the prism. The PAS of the prism coincides with the rear focal plane of the objective. The bias retardation is the sum of the retardation produced during the first pass through the prism and the retardation produced during the second pass.

Tables (4)

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Table 1 Extrema Angles of Incidence and the Corresponding Optical Axis Inclination Angles when the PAS is Perpendicular to the Incident Beama

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Table 2 Extrema Distances of the PAS from the Exit Face of the Prism and the Corresponding Optical Axis Inclination Angles when the PAS is Perpendicular to the Incident Beam for w = 3-mm-Thick Prisma

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Table 3 Extrema Distances of the PAS from the Exit Face of the Prism and the Corresponding Optical Axis Inclination Angles when the PAS is Perpendicular to the Incident Beam for w = 20-mm-Thick Prism

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Table 4 Bias Retardation when the PAS is Perpendicular to the Axis of the Optical System

Equations (57)

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n1 sin α=nE sin β2TE=nO sin β2TM,
α2TE=90°-γ+β2TE TE wave, α2TM=90°-γ+β2TM TM wave,
neθe=nOnEnO2 sin2 θe+nE2 cos2 θe.
ϕe=tan-1nO2nE2tan θe.
neSθe=neθe cos θkS,
α3TE=β3TE+γ-90° for the TE wave vector, TE Poynting vector,
α3kTM=β3kTM+γ-90° for the TM wave vector,
α3STM=β3STM+γ-90° for the TM Poynting vector.
β4TE=sin-1nO sin α3TEn1, β4TM=sin-1neθesin α3kTMn1.
Δβ=β4TE-β4TM.
y2TE=x2TE tan β2TE+y0, y2TM=x2TM tan β2TM+y0,
y3TE=x3TE-x2,3TE tan α3TE+y2,3TE, y3TM=x3TM-x2,3TM tan α3STM+y2,3TM.
y4TE=x4TE-wtan β4TE+y3,4TE, y4TM=x4TM-wtan β4TM+y3,4TM.
xPAS=y3,4TM-y3,4TEtan β4TE-tan β4TM+w, yPAS=y3,4TM-y3,4TEtan β4TEtan β4TE - tan β4TM+y3,4TE.
OPL+,-TE=nEx2,3TE2+y2,3TE-y021/2+nOw-x2,3TE2+y3,4TE-y2,3TE21/2±n1xPAS-w2+yPAS-y3,4TE21/2,
OPL+,-TM=nOx2,3TM2+y2,3TM-y021/2+neSθe×w-x2,3TM2+y3,4TM-y2,3TM21/2±n1xPAS-w2+yPAS-y3,4TM21/2.
R=OPL+,-TE-OPL+,-TM,
ΔΦ=2πλ R.
β4TE=-sin-1nO cossin-1nEnOcos γ+γ.
β4TM=-sin-1neθecossin-1nOneθecos γ+γ,
Δβ=2nE-nOtan90°-γ.
ΔΦaprx=2πλ Δβy0=2πλ 2nE-nOtan90°-γy0.
RL/2=nE-nOw,
R-L/2=nO-nEw,
η-α=90°.
I=sin2πλRBias±s dRobjdy,
tan γ=yPASy0=y1-yPASy0=y2xPASy0=y1-xPASy0=y2,
xPAS=y3,4TM-y3,4TEtan β4TE - tan β4TM+W, yPAS=y3,4TM tan β4TE-y3,4TE tan β4TMtan β4TE - tan β4TM.
η=tan-1A+B+CD+E,
A=tan β4TE tan γ - tan β2TE×tan γ + tan β2TM - 2 tan α3STM,
B=tan β4TM tan γ - tan β2TM×2 tan α3TE - tan γ - tan β2TE,
C=tan β4TE - tan β4TMtan γ - tan β2TM×tan γ - tan β2TE,
D=tan γ - tan β2TE×tan γ + tan β2TM - 2 tan α3STM,
E=tan γ - tan β2TM×2 tan α3TE - tan γ - tan β2TE.
β2TE=sin-1sin αnE,
β2TM=sin-1sin αnO,
α3TE=γ+sin-1nEnOcossin-1sin αnE-γ-90°,
α3STM=δ-tan-1nO2nE2tan θe,
β4TE=sin-1-nO cossin-1nEnO×cossin-1sin αnE-γ+γ,
β4TM=sin-1-neθecossin-1nOneθe×cossin-1sin αnO-γ+γ.
nO sin90°-γ+sin-1sin αnO=neθesin90°+δ-γ-θe,
neθe=nOnEnO2 sin2 θe+nE2 cos2 θe1/2.
A=tan β4TE tan γ - tan β2TE×tan γ + tan β2STM-2 tan α3TM,
B=tan β4TM tan γ - tan β2STM×2 tan α3TE - tan γ - tan β2TE,
C=tan β4TE - tan β4TMtan γ - tan β2STM×tan γ - tan β2TE,
D=tan γ - tan β2TE×tan γ + tan β2STM - 2 tan α3TM,
E=tan γ - tan β2STM×2 tan α3TE - tan γ - tan β2TE.
β2TE=sin-1sin αnO,
β2STM=δ-tan-1nO2nE2 tan θe,
α3TE=γ+sin-1nOnE×cossin-1sin αnO-γ-90°,
α3TM=γ+sin-1neθenO×cossin-1sin αneθe-γ-90°,
β4TE=sin-1-nE cossin-1nOnE×cossin-1sin αnO-γ+γ,
β4TM=sin-1-nO cossin-1neθenO×cossin-1sin αneθe-γ+γ.
sin α=neθesinδ-θe,
n-α=90°,
α=sin-1nO sinγ-cos-1neθenO×cosδ-γ-θe.
α=sin-1neθesinδ-θe.

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