Abstract

We present a new type of phase-shifting interferometer, which utilizes a polarizing prism to form quadrature phase-shifted fringe patterns onto a single imaging sensor. By changing the direction of linear polarization of the incident light orthogonally, four phase-shifted fringe patterns in quadrature are obtained by taking images twice; thus it is possible to reduce phase errors caused by mechanical vibrations and air turbulence that occur in temporal phase-shifting interferometers. We present the principle of this interferometer with its theoretical analysis, using the Jones matrix, along with experimental results.

© 2008 Optical Society of America

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References

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  1. J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501-598.
  2. K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).
  3. S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
    [CrossRef]
  4. J. W. Baer and W. P. Lotz, “Figure testing of 300 mm Zerodur mirrors at cryogenic temperatures,” Proc. SPIE 4822, 35-40(2002).
    [CrossRef]
  5. R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-365 (1984).
  6. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
    [CrossRef]
  7. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
    [CrossRef]
  8. F. M. Santoyo, D. Kerr, and R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362-4363 (1988).
    [CrossRef]
  9. S. Almazan and D. Malacara, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3525-3531 (2003).
  10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  11. S. Nakadate and M. Isshiki, “Real-time fringe pattern processing and its applications,” Proc. SPIE 2544, 74-86 (1995).
  12. S. Nakadate, “Phase detection of equidistant fringes for highly sensitive optical sensing. II. Experiments,” J. Opt. Soc. Am. A 5, 1265-1269 (1988).
    [CrossRef]

2004 (1)

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

2003 (1)

S. Almazan and D. Malacara, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3525-3531 (2003).

2002 (1)

J. W. Baer and W. P. Lotz, “Figure testing of 300 mm Zerodur mirrors at cryogenic temperatures,” Proc. SPIE 4822, 35-40(2002).
[CrossRef]

2001 (1)

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

2000 (1)

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
[CrossRef]

1999 (1)

S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
[CrossRef]

1995 (1)

S. Nakadate and M. Isshiki, “Real-time fringe pattern processing and its applications,” Proc. SPIE 2544, 74-86 (1995).

1988 (2)

1984 (1)

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-365 (1984).

1982 (1)

Almazan, S.

S. Almazan and D. Malacara, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3525-3531 (2003).

Baer, J. W.

J. W. Baer and W. P. Lotz, “Figure testing of 300 mm Zerodur mirrors at cryogenic temperatures,” Proc. SPIE 4822, 35-40(2002).
[CrossRef]

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
[CrossRef]

Ina, H.

Isshiki, M.

S. Nakadate and M. Isshiki, “Real-time fringe pattern processing and its applications,” Proc. SPIE 2544, 74-86 (1995).

Kerr, D.

Kobatake, H.

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

Kobayashi, S.

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
[CrossRef]

Li, C.

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

Lotz, W. P.

J. W. Baer and W. P. Lotz, “Figure testing of 300 mm Zerodur mirrors at cryogenic temperatures,” Proc. SPIE 4822, 35-40(2002).
[CrossRef]

Maki, T.

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

Malacara, D.

S. Almazan and D. Malacara, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3525-3531 (2003).

Maruyama, S.

S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
[CrossRef]

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Moore, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-365 (1984).

Nakadate, S.

S. Nakadate and M. Isshiki, “Real-time fringe pattern processing and its applications,” Proc. SPIE 2544, 74-86 (1995).

S. Nakadate, “Phase detection of equidistant fringes for highly sensitive optical sensing. II. Experiments,” J. Opt. Soc. Am. A 5, 1265-1269 (1988).
[CrossRef]

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Santoyo, F. M.

Schwider, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
[CrossRef]

Shibata, T.

S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
[CrossRef]

Smythe, R.

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-365 (1984).

Takeda, M.

Tsukamoto, K.

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
[CrossRef]

Tyrer, R.

Wyant, J. C.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

Appl. Opt. (1)

Exp. Therm. Fluid. Sci. (1)

S. Maruyama, T. Shibata, and K. Tsukamoto, “Measurement of diffusion fields of solutions using real-time phase shift interferometer and rapid heat-transfer control system,” Exp. Therm. Fluid. Sci. 19, 34-48 (1999).
[CrossRef]

J. Jpn. Soc. Microgravity Appl. (1)

K. Tsukamoto, C. Li, H. Kobatake, and T. Maki, “In situ observation of crystal growth in microgravity,” J. Jpn. Soc. Microgravity Appl. 18, 190-196 (2001).

J. Opt. Soc. Am. (1)

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

Opt. Eng. (3)

R. Smythe and R. Moore, “Instantaneous phase measuring interferometry,” Opt. Eng. 23, 361-365 (1984).

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phase-shifted interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960-966 (2000).
[CrossRef]

S. Almazan and D. Malacara, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3525-3531 (2003).

Proc. SPIE (3)

S. Nakadate and M. Isshiki, “Real-time fringe pattern processing and its applications,” Proc. SPIE 2544, 74-86 (1995).

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304-314 (2004).
[CrossRef]

J. W. Baer and W. P. Lotz, “Figure testing of 300 mm Zerodur mirrors at cryogenic temperatures,” Proc. SPIE 4822, 35-40(2002).
[CrossRef]

Other (1)

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501-598.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of a phase-shifting interferometer with orthogonal linear polarizations.

Fig. 2
Fig. 2

Quadrature phase-shifting interferomgrams. (a) Phase shifted fringes with incident linearly polarized light at an azimuth of 45°. (b) Phase-shifted fringes with incident linearly polarized light at an azimuth of 135°.

Fig. 3
Fig. 3

Equidistant and straight fringes used to calculate the phase difference between two interferograms separated by two orthogonal polarizations.

Fig. 4
Fig. 4

Phase distribution calculated with fringe patterns shown in Fig. 2.

Equations (28)

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E = ( E x ( t ) E y ( t ) ) = exp ( i ω t ) ( E 0 x exp ( i φ x ) E 0 y exp ( i φ y ) ) = ( E 0 x E 0 y exp ( i δ ) ) ,
E 45 ° = 1 2 ( 1 1 ) ,
E 135 ° = 1 2 ( 1 1 ) .
E 45 ° , λ / 4 = 1 2 ( 1 1 ) .
E 45 ° , λ / 4 = [ T λ / 4 ] 45 ° E 45 ° , λ / 4 = 1 2 ( 1 i i 1 ) ,
[ T λ / 4 ] 45 ° = ( 1 i i 1 ) .
E 45 ° , λ / 4 = 1 2 ( 1 i 1 i ) .
E 45 ° , λ / 4 = [ T λ / 4 ] 135 ° E 45 ° , λ / 4 = 1 2 ( i i ) ,
[ T λ / 4 ] 135 ° = ( 1 i i 1 ) .
E 45 ° , λ / 8 = [ T λ / 8 ] 0 ° E 45 ° = 1 2 ( exp ( i π / 8 ) exp ( i π / 8 ) ) ,
[ T λ / 8 ] 0 ° = ( exp ( i π / 8 ) 0 0 exp ( i π / 8 ) ) .
E 45 ° , λ / 8 = [ T λ / 8 ] 0 ° E 45 ° , λ / 8 = 1 2 ( exp ( i π / 4 ) exp ( i π / 4 ) ) .
E 45 ° , λ / 8 = 1 2 ( exp ( i π / 4 ) exp ( i π / 4 ) ) .
1 2 ( i i ) + 1 2 ( exp [ i ( π / 4 + φ ) ] exp [ i ( π / 4 φ ) ] ) = 1 2 ( i + exp [ i ( π / 4 + φ ) ] i + exp [ i ( π / 4 φ ) ] ) ,
I 1 = | 1 2 { i + exp [ i ( π / 4 + φ ) ] } | 2 = 1 sin θ ,
I 2 = | 1 2 { i + exp [ i ( π / 4 φ ) ] } | 2 = 1 + cos θ .
E 135 ° , λ / 4 = 1 2 ( 1 1 ) .
E 135 ° , λ / 4 = [ T λ / 4 ] 45 ° E 135 ° , λ / 4 = 1 2 ( 1 i 1 i ) ,
E 135 ° , λ / 4 = 1 2 ( 1 i 1 + i ) .
E 135 ° , λ / 4 = [ T λ / 4 ] 135 ° E 135 ° , λ / 4 = 1 2 ( i i ) .
E 135 ° , λ / 8 = [ T λ / 8 ] 0 ° E 135 ° = 1 2 ( exp ( i π / 8 ) exp ( i π / 8 ) ) .
E 135 ° , λ / 8 = 1 2 ( exp ( i π / 8 ) exp ( i π / 8 ) ) .
E 135 ° , λ / 8 = [ T λ / 8 ] 0 ° E 135 ° , λ / 8 = 1 2 ( exp ( i π / 4 ) exp ( i π / 4 ) ) ,
E 135 ° , λ / 8 = 1 2 ( exp ( i π / 4 ) exp ( i π / 4 ) ) .
1 2 ( i i ) + 1 2 ( exp [ i ( π / 4 + φ ) ] exp [ i ( π / 4 φ ) ] ) = 1 2 ( i exp [ i ( π / 4 + φ ) ] i + exp [ i ( π / 4 φ ) ] ) .
I 3 = | 1 2 { i exp [ i ( π / 4 + φ ) ] } | 2 = 1 + sin θ ,
I 4 = | 1 2 { i + exp [ i ( π / 4 φ ) ] } | 2 = 1 cos θ .
θ = tan 1 ( I 3 I 1 I 2 I 4 ) .

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