Abstract

We compare the phase measurements of a fused-silica witness sample made with a liquid-crystal point-diffraction interferometer (LCPDI) with measurements made with a Zygo Mark IV xp phase-shifting interferometer and find close agreement. Two phase-shift-error sources in the LCPDI that contribute to measurement discrepancies are frame-to-frame intensity changes caused by the dichroism of the dye and alignment distortions of the host liquid crystal. An empirical model of the phase-shift error caused by the host alignment distortions is presented and used to investigate the performance of two different phase-detection algorithms. It is suggested that by proper choice of LCPDI fabrication parameters and phase-acquisition methods, the device’s accuracy can be significantly improved.

© 2002 Optical Society of America

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  1. V. P. Linnik, “Simple interferometer for optical systems evaluation,” C.R. Acad. Sci. (USSR) 1, 208–210 (1933).
  2. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).
  3. D. Malacara, Optical Shop Testing, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley, New York, 1992).
  4. See, for example, several references in Ref. 6 and S. H. Lee, P. Naulleau, K. A. Goldberg, F. Piao, W. Oldham, J. Bokar, “Phase-shifting point-diffraction interferometry at 193 nm,” Appl. Opt. 39, 5768–5772 (2000), and references therein.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  9. Zygo Mark IV xp, Zygo Corporation, Middlefield, Conn. 06455.
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    [CrossRef]
  11. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. E. Novak, C. Ai, J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 114–121 (1997).
    [CrossRef]
  14. E. Novak, C. Ai, J. C. Wyant, “Errors caused by nearly parallel optical elements in a laser Fizeau interferometer utilizing strictly coherent imaging,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 456–460 (1997).
    [CrossRef]
  15. S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  20. R. Rao, “LCPDI modeling,” 1999Summer Research Program for High School Juniors at the University of Rochester’s Laboratory for Laser Energetics, Laboratory for Laser Energetics Report 311, NTIS document DOE/SF/19460-338 (Laboratory for Laser Energetics, Rochester, N.Y., 1999).
  21. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
    [CrossRef]
  22. The Orasol dyes were recommended by C. R. Mercer, NASA Glenn Research Center, 2100 Brookpark Road, Cleveland, Ohio 44135 (personal communication, 2000).
  23. E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
    [CrossRef]
  24. “Design and synthesis of near-infrared absorbing dyes for the liquid crystal point-diffraction interferometer (LCPDI),” Laboratory for Laser Energetics LLE Rev.81, 37–47 (1999), NTIS document DOE/SF/19460-335 (1999). Copies may be obtained from the National Technical Information Service, Springfield, Va. 22161.
  25. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14, 918–930 (1997).
    [CrossRef]
  26. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.
  27. H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
    [CrossRef]
  28. S. Garoff, R. B. Meyer, “Electroclinic effect of the A–C phase change in a chiral smectic liquid crystal,” Phys. Rev. A 19, 338–347 (1979).
    [CrossRef]
  29. A. Sneh, J. Y. Liu, K. M. Johnson, “High-speed analog refractive-index modulator that uses a chiral smectic liquid-crystal,” Opt. Lett. 19, 305–307 (1994).
    [CrossRef] [PubMed]
  30. K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).
  31. O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
    [CrossRef]
  32. J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. Ser. 1, 1–74 (1982).
  33. J. A. Castellano, “Alignment of liquid crystal molecules on various surfaces: myths, theories, facts,” in Proceedings of the American Chemical Society Symposium on Liquid Crystals and Ordered Fluids, A. C. Griffin, J. F. Johnson, eds. (Plenum, New York, 1984), pp. 763–780.
    [CrossRef]

2000 (2)

1999 (6)

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
[CrossRef]

E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
[CrossRef]

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

J. Schwider, T. Dresel, B. Manzke, “Some considerations of reduction of reference phase error in phase-stepping interferometry,” Appl. Opt. 38, 655–659 (1999).
[CrossRef]

1997 (1)

1996 (1)

1994 (2)

1991 (1)

1987 (2)

1983 (1)

1982 (1)

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. Ser. 1, 1–74 (1982).

1979 (1)

S. Garoff, R. B. Meyer, “Electroclinic effect of the A–C phase change in a chiral smectic liquid crystal,” Phys. Rev. A 19, 338–347 (1979).
[CrossRef]

1975 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

1933 (1)

V. P. Linnik, “Simple interferometer for optical systems evaluation,” C.R. Acad. Sci. (USSR) 1, 208–210 (1933).

Ai, C.

E. Novak, C. Ai, J. C. Wyant, “Errors caused by nearly parallel optical elements in a laser Fizeau interferometer utilizing strictly coherent imaging,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 456–460 (1997).
[CrossRef]

E. Novak, C. Ai, J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 114–121 (1997).
[CrossRef]

Arrasmith, S. R.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Bokar, J.

Brecker, R. P.

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Burow, R.

Burton, D. R.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Castellano, J. A.

J. A. Castellano, “Alignment of liquid crystal molecules on various surfaces: myths, theories, facts,” in Proceedings of the American Chemical Society Symposium on Liquid Crystals and Ordered Fluids, A. C. Griffin, J. F. Johnson, eds. (Plenum, New York, 1984), pp. 763–780.
[CrossRef]

Cognard, J.

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. Ser. 1, 1–74 (1982).

Corsello, S. M.

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Creath, K.

Dedieu, J. C.

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

Dresel, T.

Dumas, P.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Eiju, T.

Elssner, K.-E.

Farrant, D. I.

Frankena, H. J.

Garoff, S.

S. Garoff, R. B. Meyer, “Electroclinic effect of the A–C phase change in a chiral smectic liquid crystal,” Phys. Rev. A 19, 338–347 (1979).
[CrossRef]

Goldberg, K. A.

Golini, D.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Gregg, L. L.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Grzanna, J.

Guardalben, M. J.

M. J. Guardalben, N. Jain, “Phase shift error as a result of molecular alignment distortions in a liquid-crystal point-diffraction interferometer,” Opt. Lett. 25, 1171–1173 (2000).
[CrossRef]

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Gulik-Krzywicki, T.

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

Hariharan, P.

Hibino, K.

Hogan, S.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Jacobs, S. D.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Jain, N.

Kordonski, W. I.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Kozhinova, I. A.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Lalor, M. J.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Larkin, K. G.

Lee, S. H.

Linnik, V. P.

V. P. Linnik, “Simple interferometer for optical systems evaluation,” C.R. Acad. Sci. (USSR) 1, 208–210 (1933).

Lippa, I. A.

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Liu, J. Y.

M. Johnson, K.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley, New York, 1992).

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.

Manzke, B.

Marshall, K. L.

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Mercer, C. R.

Merkel, K.

Meyer, R. B.

S. Garoff, R. B. Meyer, “Electroclinic effect of the A–C phase change in a chiral smectic liquid crystal,” Phys. Rev. A 19, 338–347 (1979).
[CrossRef]

Mondain-Monval, O.

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

Moore, M. S.

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

Naulleau, P.

Novak, E.

E. Novak, C. Ai, J. C. Wyant, “Errors caused by nearly parallel optical elements in a laser Fizeau interferometer utilizing strictly coherent imaging,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 456–460 (1997).
[CrossRef]

E. Novak, C. Ai, J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 114–121 (1997).
[CrossRef]

Oldham, W.

Oreb, B. F.

Piao, F.

Plyusnina, T.

E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
[CrossRef]

Poulin, P.

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

Prudnikova, E.

E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
[CrossRef]

Rao, R.

R. Rao, “LCPDI modeling,” 1999Summer Research Program for High School Juniors at the University of Rochester’s Laboratory for Laser Energetics, Laboratory for Laser Energetics Report 311, NTIS document DOE/SF/19460-338 (Laboratory for Laser Energetics, Rochester, N.Y., 1999).

Romanofsky, H. J.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Schwider, J.

Servin, M.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.

Shorey, A. B.

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Smorenburg, C.

Sneh, A.

Spolaczyk, R.

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Thompson, B. J.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.

Turner, A. C.

A. C. Turner, “Ray tracing through the liquid crystal point diffraction interferometer,” in 1998 Summer Research Program for High School Juniors at the University of Rochester’s Laboratory for Laser Energetics, Laboratory for Laser Energetics Report 300, NTIS document DOE/SF/19460-299 (Laboratory for Laser Energetics, Rochester, N.Y., 1998).

Umanskii, B.

E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
[CrossRef]

van Wingerden, J.

Wyant, J. C.

E. Novak, C. Ai, J. C. Wyant, “Errors caused by nearly parallel optical elements in a laser Fizeau interferometer utilizing strictly coherent imaging,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 456–460 (1997).
[CrossRef]

E. Novak, C. Ai, J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 114–121 (1997).
[CrossRef]

Zhang, H.

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Am. Ceram. Soc. Bull. (1)

S. D. Jacobs, S. R. Arrasmith, I. A. Kozhinova, L. L. Gregg, A. B. Shorey, H. J. Romanofsky, D. Golini, W. I. Kordonski, P. Dumas, S. Hogan, “MRF: computer-controlled optics manufacturing,” Am. Ceram. Soc. Bull. 78, 42–48 (1999).

Appl. Opt. (7)

C.R. Acad. Sci. (USSR) (1)

V. P. Linnik, “Simple interferometer for optical systems evaluation,” C.R. Acad. Sci. (USSR) 1, 208–210 (1933).

Eur. J. Phys. B (1)

O. Mondain-Monval, J. C. Dedieu, T. Gulik-Krzywicki, P. Poulin, “Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions,” Eur. J. Phys. B 12, 167–170 (1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers (telescope testing),” Jpn. J. Appl. Phys. 14, 351–356 (1975).

Mol. Cryst. Liq. Cryst. (1)

E. Prudnikova, B. Umanskii, T. Plyusnina, “Synthesis of new dichroic dyes with negative dichroism for a black mixture,” Mol. Cryst. Liq. Cryst. 332, 37–41 (1999).
[CrossRef]

Mol. Cryst. Liq. Cryst. Suppl. Ser. (1)

J. Cognard, “Alignment of nematic liquid crystals and their mixtures,” Mol. Cryst. Liq. Cryst. Suppl. Ser. 1, 1–74 (1982).

Opt. Eng. (1)

H. Zhang, M. J. Lalor, D. R. Burton, “Robust, accurate seven-sample phase-shifting algorithm insensitive to nonlinear phase-shift error and second-harmonic distortion: a comparative study,” Opt. Eng. 38, 1524–1533 (1999).
[CrossRef]

Opt. Lett. (3)

Opt. Rev. (1)

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6, 529–538 (1999).
[CrossRef]

Phys. Rev. A (1)

S. Garoff, R. B. Meyer, “Electroclinic effect of the A–C phase change in a chiral smectic liquid crystal,” Phys. Rev. A 19, 338–347 (1979).
[CrossRef]

Other (13)

K. L. Marshall, M. J. Guardalben, S. M. Corsello, M. S. Moore, I. A. Lippa, R. P. Brecker, “Device applications of highly soluble near-infrared transition metal dyes in liquid crystal hosts,” presented at the OSA Annual Meeting, Providence, R.I., 22–26 October 2000 (Optical Society of America, Washington, D.C., 2000).

“Design and synthesis of near-infrared absorbing dyes for the liquid crystal point-diffraction interferometer (LCPDI),” Laboratory for Laser Energetics LLE Rev.81, 37–47 (1999), NTIS document DOE/SF/19460-335 (1999). Copies may be obtained from the National Technical Information Service, Springfield, Va. 22161.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, Vol. 61 of Optical Engineering Series, B. J. Thompson, ed. (Marcel Dekker, New York, 1998), pp. 196–199.

The Orasol dyes were recommended by C. R. Mercer, NASA Glenn Research Center, 2100 Brookpark Road, Cleveland, Ohio 44135 (personal communication, 2000).

A. C. Turner, “Ray tracing through the liquid crystal point diffraction interferometer,” in 1998 Summer Research Program for High School Juniors at the University of Rochester’s Laboratory for Laser Energetics, Laboratory for Laser Energetics Report 300, NTIS document DOE/SF/19460-299 (Laboratory for Laser Energetics, Rochester, N.Y., 1998).

R. Rao, “LCPDI modeling,” 1999Summer Research Program for High School Juniors at the University of Rochester’s Laboratory for Laser Energetics, Laboratory for Laser Energetics Report 311, NTIS document DOE/SF/19460-338 (Laboratory for Laser Energetics, Rochester, N.Y., 1999).

D. Malacara, Optical Shop Testing, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley, New York, 1992).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Chap. 5, pp. 351–393.

C. R. Mercer, N. Rashidnia, “Common-path phase-stepped interferometer for fluid measurements,” in Eighth International Symposium on Flow Visualization 1998, G. M. Carlomagno, I. Grant, eds. 1998), Vol. CD-ROM, pp. 256.1–256.9. CD-ROM available through http://www.ode-web.demon.co.uk/post-conf-web/flyer.html .

Zygo Mark IV xp, Zygo Corporation, Middlefield, Conn. 06455.

E. Novak, C. Ai, J. C. Wyant, “Transfer function characterization of laser Fizeau interferometer for high-spatial-frequency phase measurements,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 114–121 (1997).
[CrossRef]

E. Novak, C. Ai, J. C. Wyant, “Errors caused by nearly parallel optical elements in a laser Fizeau interferometer utilizing strictly coherent imaging,” in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE3134, 456–460 (1997).
[CrossRef]

J. A. Castellano, “Alignment of liquid crystal molecules on various surfaces: myths, theories, facts,” in Proceedings of the American Chemical Society Symposium on Liquid Crystals and Ordered Fluids, A. C. Griffin, J. F. Johnson, eds. (Plenum, New York, 1984), pp. 763–780.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic of the LCPDI. The laser beam is focused onto an area of the device containing a glass or plastic microsphere in the LC fluid gap that takes the place of the pinhole in the conventional PDI. The small central portion of the beam passing through the microsphere is diffracted/refracted by the microsphere to form the reference wave front of the interferometer, while light passing around the microsphere forms the object beam being tested. Phase shifting is accomplished through application of an electric field to the LCPDI, as described in Fig. 2.

Fig. 2
Fig. 2

Electric field applied to the LCPDI producing a controlled reorientation of the birefringent LC molecules, thereby shifting the phase of the object wave front relative to the reference wave front. Light that is polarized along the buff direction of the cell will first see an extraordinary refractive index n e followed by refractive-index values approaching the ordinary refractive index n o as voltage is applied. Attenuation of the object beam intensity by adding a guest dye to the LC fluid host allows high-contrast fringes to be obtained.

Fig. 3
Fig. 3

Absorbance at λ = 543 nm of the LCPDI with 1% wt./wt. concentration of Oil Red O dye in the nematic E7 host LC as a function of voltage applied to the device. The dichroism of the dye produces voltage-dependent changes in fringe intensity and contrast.

Fig. 4
Fig. 4

Experimental setup used for LCPDI measurements. The inset shows interference fringes from the test sample with a polished spot.

Fig. 5
Fig. 5

(a) LCPDI interference fringes obtained by phase shifting through 2π rad from 0.98 V (0) to 1.21 V (2π). (b) Two empty-cavity phase images ϕ1 and ϕ2 were subtracted to obtain the residual phase error Δϕ in the LCPDI. The phase-difference image reveals phase errors at one and two times the fringe frequency.

Fig. 6
Fig. 6

Phase measurements of a fused-silica window containing a magnetorheological finishing polishing spot comparing the LCPDI with a commercial interferometer (Zygo Mark IV xp). The LCPDI lineout matches that of the Zygo Mark IV in some areas and is ≈50 nm discrepant in other areas, most likely due to the absorption dichroism of the dye.

Fig. 7
Fig. 7

Absorbance (OD) of two different dye mixtures containing both positive and negative dichroic dye components in E7 showing very little change with applied voltage. Such mixtures can be used to reduce significantly a phase-shift error in the LCPDI caused by the absorption dichroism of a single dye: (a) 1.3% Orasol Red BL, 0.55% Orasol Black RLI, + Oil Red O; (b) 1.3% Orasol Red BL, 0.55% Orasol Black RLI + 0.2% Sudan III, 0.38% Sudan Black B. In each case the fluid path length was 22 µm.

Fig. 8
Fig. 8

(a) Gray-scale image showing the spatial form of f(x, y) defined in the text and used in Fig. 9. Black corresponds to f(x, y) = 0 with a maximum value of f(x, y) = 1. (b) Peak value of the phase-error function α0r [H exp(-Aα 0r ) f(x, y)] versus α0r for the set of constants AM in Table 5.

Fig. 9
Fig. 9

(a) Two interferograms from the six-frame series used to compare five- and six-frame algorithms. For the images shown the phase shift α0r = π/3 and 2π/3, corresponding, respectively, to r = 2, 3 for the six-frame algorithm. Focusing conditions were chosen so as to introduce only a moderate amount of LC host-induced phase-shift error to avoid possible phase unwrapping errors. (b) Gray-scale images of the residual phase error Δϕ = ϕperturbed - ϕideal for the five- and six-frame algorithms. For the experimental results shown ϕideal was determined by a five-term Zernike fit to the phase data. Table 5 gives p–v and rms errors.

Fig. 10
Fig. 10

Phase-shift values used to obtain the simulated results in Table 5 versus the unperturbed phase shift with f(x, y) = 1. Phase shift values were calculated by adding the appropriate constant Θ to each frame in Eq. (8) where, ○, Θ = -π for the five-frame algorithm and, ■, Θ = -5π/6 for the six-frame algorithm. If no phase error were present, phase shift values would lie on the diagonal line. The phase errors shown here are taken from the plot in Fig. 8. Note that the first frame of each algorithm has zero phase error, corresponding to α01 = 0 and representing experimental observations near the Frederiks transition of the LCPDI.

Fig. 11
Fig. 11

Value of the phase error α0r ε(α0r ) versus α0r with f(x, y) = 1: (a) ε(α0r ) = H exp(-Aα 0r) with H = 0.84, A = 0.37; (b) ε(α0r) = ε1 + ε20r/π) with ε1 = 0.38, ε2 = -0.14; (c) ε(α0r) = ε1 + ε20r/π) with ε1 = 0.84, ε2 = -0.9764.

Tables (6)

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Table 1 Relative Comparison of Different Unwrapping Algorithms with Intentionally Noisy Data (Low-Contrast Fringes)

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Table 2 Several Focus and Voltage Conditions Investigated for the LCPDI in Empty-Cavity Measurements

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Table 3 Absorbance (OD) at 543 nm

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Table 4 Peak-to-Valley Residual Phase Errors (2π rad) Due to Linear and Quadratic Spatially Uniform Phase-Shift Errors for the Five- and Six-Frame Algorithms

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Table 5 Comparison of Residual Errors (2π rad) Obtained with Five-and Six-Frame Algorithms with both Experimental and Simulated Interference Images

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Table 6 Comparison of Residual Errors (2π rad) Obtained with Five-and Six-Frame Algorithms and only Linear and Quadratic Phase-Shift-Error Termsa

Equations (12)

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tanϕ=ΔI3-ΔI1ΔI0+ΔI4-2ΔI2×I0obj+I4obj+2I2objI3obj+I1obj,
Ix, y, αr=I0x, y1+γx, ycosαr-ϕx, y  for r=1, 2,  , m,
αr=α0r1+εα0r=α0r1+ε1x, y+ε2x, yα0rπ+ε3x, yα0rπ2++εpx, yα0rπp-1 for r=1, 2,  , m,
α01=-π, α02=-π/2, α03=0,α04=π/2, α05=π.
tanϕ=35I1-6I2-17I3+17I4+6I5-5I6I1-26I2+25I3+25I4-26I5+I6
Δϕ=ϕperturbed-ϕideal.
εα0r=Hexp-Aα0rfx, y,
fx, y=gx, y/max[gx, y],
gx, y=1-exp-|Bx2+Cy2|exp-|Dx2+Ey21/2/F|G|sintan-1Kx/My|,
αr=α0r1+Hexp-Aα0rfx, y+Θ.
ε1x, y=Hfx, y, ε2x, y=-AHπfx, y,
αr=α0r1+εα0r+Θα0r1+ε1x, y+ε2x, yα0rπ+Θ forr = 1, 2,  , m,

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