Abstract

Phase-shifting interferometry suffers from two main sources of error: phase-shift miscalibration and detector nonlinearity. Algorithms that calculate the phase of a measured wave front require a high degree of tolerance for these error sources. An extended method for deriving such error-compensating algorithms patterned on the sequential application of the averaging technique is proposed here. Two classes of algorithms were derived. One class is based on the popular three-frame technique, and the other class is based on the 4-frame technique. The derivation of algorithms in these classes was calculated for algorithms with up to six frames. The new 5-frame algorithm and two new 6-frame algorithms have smaller phase errors caused by phase-shifter miscalibration than any of the common 3-, 4- or 5-frame algorithms. An analysis of the errors resulting from algorithms in both classes is provided by computer simulation and by an investigation of the spectra of sampling functions.

© 1995 Optical Society of America

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References

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  1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1990), Vol. 28, Chap. 4, pp. 271–359.
    [CrossRef]
  2. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
    [CrossRef]
  3. J. H. Bruning, D. H. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703, (1974).
    [CrossRef] [PubMed]
  4. J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus65–71 (1982).
  5. J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
    [CrossRef]
  6. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt Eng. 23, 350–352 (1984).
  7. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–373 (1982).
    [CrossRef] [PubMed]
  8. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  9. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  10. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  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]
  12. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  13. J. C. Wyant, K. N. Prettyjohns, “Optical profiler using improved phase shifting interferometry,” U.S. patent4, 639, 139 (27January1987).
  14. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, 1993), Chap. 4, pp. 94–140.
  15. K. Creath, P. Hariharan, “Phase shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–26 (1994).
    [CrossRef] [PubMed]
  16. K. Freischald, C. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  17. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
  18. M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
    [CrossRef]

1994 (1)

1993 (2)

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
[CrossRef] [PubMed]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

1992 (1)

1991 (1)

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

1990 (1)

1987 (1)

1984 (3)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt Eng. 23, 350–352 (1984).

1983 (1)

1982 (2)

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus65–71 (1982).

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–373 (1982).
[CrossRef] [PubMed]

1974 (1)

Bhushan, B.

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Creath, K.

K. Creath, P. Hariharan, “Phase shifting errors in interferometric tests with high-numerical-aperture reference surfaces,” Appl. Opt. 33, 24–26 (1994).
[CrossRef] [PubMed]

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
[CrossRef]

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, 1993), Chap. 4, pp. 94–140.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Freischald, K.

Gallagher, J. E.

George, O. E.

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt Eng. 23, 350–352 (1984).

Grzanna, J.

Hariharan, P.

Herriott, D. H.

Koliopoulos, C.

Koliopoulos, C. K.

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

Kujawinska, M.

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Larkin, K. G.

Merkel, K.

Morgan, C. J.

Oreb, B. F.

Prettyjohns, K. N.

J. C. Wyant, K. N. Prettyjohns, “Optical profiler using improved phase shifting interferometry,” U.S. patent4, 639, 139 (27January1987).

Rosenfeld, D. P.

Schreiber, H.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Schwider, J.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

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]

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1990), Vol. 28, Chap. 4, pp. 271–359.
[CrossRef]

Spolaczyk, R.

Streibl, N.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Surrel, Y.

White, A. D.

Wojciak, J.

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Wyant, J. C.

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus65–71 (1982).

J. C. Wyant, K. N. Prettyjohns, “Optical profiler using improved phase shifting interferometry,” U.S. patent4, 639, 139 (27January1987).

Zoller, A.

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

Appl. Opt. (5)

ASLE Trans. (1)

J. C. Wyant, C. K. Koliopoulos, B. Bhushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–113 (1984).
[CrossRef]

J. Opt. Soc. Am. A (2)

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic system and principles,” Laser Focus65–71 (1982).

Opt Eng. (1)

J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt Eng. 23, 350–352 (1984).

Opt. Eng. (2)

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Opt. Lasers Eng. (1)

M. Kujawinska, J. Wojciak, “High accuracy Fourier transform fringe pattern analysis,” Opt. Lasers Eng. 14, 325–339 (1991).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. C. Wyant, K. N. Prettyjohns, “Optical profiler using improved phase shifting interferometry,” U.S. patent4, 639, 139 (27January1987).

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Technique, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, 1993), Chap. 4, pp. 94–140.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1990), Vol. 28, Chap. 4, pp. 271–359.
[CrossRef]

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, Chap. 5, pp. 349–383.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Phase error for three intensity fringes caused by 20% phase-shift miscalibration.

Fig. 2
Fig. 2

P-V phase error versus percent of phase-shift miscalibration.

Fig. 3
Fig. 3

P-V phase error versus percent of second-order detector nonlinearity error.

Fig. 4
Fig. 4

Phase error for three intensity fringes caused by 10% second-order detector nonlinearity error.

Fig. 5
Fig. 5

Sampling functions.

Fig. 6
Fig. 6

Frequency spectra of sampling functions.

Tables (1)

Tables Icon

Table 1 Algorithms in Classes A and B

Equations (45)

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tan φ = m = 1 M I m sin [ 2 π ( m 1 ) / M ] m = 1 M I m cos [ 2 π ( m 1 ) / M ] ,
tan φ = I 2 I 4 I 1 I 3 ,
tan φ = N 1 + N 2 D 1 + D 2 ,
tan φ M = N D ,
tan φ M + 1 = N 1 + N 2 D 1 + D 2 = N D .
tan φ M + 2 = N + N D + D = N 1 + 2 N 2 + N 3 D 1 + 2 D 2 + D 3 ,
I 2 I 4 I 1 I 3 = N 1 D 1 ,
I 2 I 4 I 5 I 3 = N 2 D 2 ,
tan φ = 2 ( I 2 I 4 ) I 1 + I 5 2 I 3 .
I 1 + 2 I 2 I 3 I 1 I 3 = N 1 D 1 ,
I 2 I 4 I 2 2 I 3 + I 4 = N 2 D 2 .
tan φ = I 1 + 3 I 2 I 3 I 4 I 1 3 I 3 + I 2 + I 4 .
tan ( φ + π 4 ) = 2 ( I 2 I 3 ) I 1 I 2 I 3 + I 4 .
I 1 + 3 I 2 I 3 I 4 I 1 3 I 3 + I 2 + I 4 = N 1 D 1 ,
I 2 + I 3 3 I 4 + I 5 I 2 3 I 3 + I 4 + I 5 = N 2 D 2 .
tan φ = I 1 + 4 I 2 4 I 4 + I 5 I 1 + 2 I 2 6 I 3 + 2 I 4 + I 5 .
tan ( φ + π 4 ) = 3 I 2 3 I 3 I 4 + I 5 I 1 I 2 3 I 3 + 3 I 4 .
α = π 2 ( 1 + δ ) ,
I m ( x , y ) = I ( x , y ) { 1 + γ ( x , y ) cos [ m π 2 ( 1 + δ ) ] } .
I ( x , y ) = I ( x , y ) + δ I 2 ( x , y ) ,
I ( t ) = I [ 1 + γ cos ( 2 π ν t + φ ) ] ,
f N ( t ) = sin ( 2 π ν f t ) , f D ( t ) = cos ( 2 π ν f t ) .
f N ( t ) = m = 1 M α m sin ( 2 π ν f t ) δ ( t t m ) , f D ( t ) = m = 1 M β m cos ( 2 π ν f t ) δ ( t t m ) ,
f N ( t ) = δ ( t π 4 ) δ ( t 3 π 4 ) + δ ( t 5 π 4 ) δ ( t 7 π 4 ) , f D ( t ) = δ ( t π 4 ) δ ( t 3 π 4 ) δ ( t 5 π 4 ) + δ ( t 7 π 4 ) .
f N ( t ) = δ ( t π 2 ) + δ ( t 3 π 2 ) , f D ( t ) = δ ( t ) / 2 δ ( t π ) + δ ( t 2 π ) / 2 .
f N ( t ) = δ ( t 3 π 8 ) + δ ( t 5 π 8 ) , f D ( t ) = 1 2 [ δ ( t π 4 ) δ ( t 3 π 4 ) δ ( t 5 π 4 ) + δ ( t 7 π 4 ) ] .
f N ( t ) = 1 6 [ δ ( t ) 4 δ ( t π 2 ) + 4 δ ( t 3 π 2 ) δ ( t 2 π ) ] , f D ( t ) = 1 6 [ δ ( t ) + 2 δ ( t π 2 ) 6 δ ( t π ) + 2 δ ( t 3 π 2 ) + δ ( t 2 π ) ] .
F N ( ν ) = 2 i sin ( π 4 ν ν f ) 2 i sin ( 3 π 4 ν ν f ) exp ( i 3 π 4 ν ν f ) , F D ( ν ) = 2 cos ( π 4 ν ν f ) + 2 cos ( 3 π 4 ν ν f ) exp ( i 3 π 4 ν ν f ) .
F N ( ν ) = 2 i sin ( π 2 ν ν f ) exp ( i π ν ν f ) , F D ( ν ) = [ 1 + cos ( π ν ν f ) ] exp ( i π ν ν f ) .
F N ( ν ) = 2 i sin ( π 4 ν ν f ) exp ( i 3 π 4 ν ν f ) , F D ( ν ) = [ cos ( 3 π 4 ν ν f ) + cos ( π 4 ν ν f ) ] exp ( i 3 π 4 ν ν f ) .
F N ( ν ) = 1 3 i [ 4 sin ( π 2 ν ν f ) sin ( π ν ν f ) ] exp ( i π ν ν f ) , F D ( ν ) = 1 3 [ 6 + 4 cos ( π 2 ν ν f ) + 2 cos ( π ν ν f ) ] exp ( i π ν ν f ) .
I 1 + 2 I 2 I 3 I 1 I 3
I 2 I 3 I 2 + I 1
I 2 I 4 I 1 I 3
I 1 + I 2 I 3 I 4 I 1 I 2 I 3 + I 4
I 1 + 3 I 2 I 3 I 4 I 1 3 I 3 I 2 + I 4
2 ( I 2 I 3 ) I 1 I 2 I 3 + I 4
2 I 2 2 I 4 I 1 2 I 3 + I 5
I 1 + 2 I 2 2 I 3 2 I 4 + I 5 I 1 2 I 2 2 I 3 + 2 I 4 + I 5
I 1 + 4 I 2 4 I 4 + I 5 I 1 + 2 I 2 6 I 3 + 2 I 4 + I 5
3 I 2 3 I 3 I 4 + I 5 I 1 I 2 3 I 3 + 3 I 4
3 I 2 4 I 4 + I 6 I 1 4 I 3 + 3 I 5
I 1 + 3 I 2 4 I 3 4 I 4 + 3 I 5 + I 6 I 1 3 I 2 4 I 3 + 4 I 4 + 3 I 5 I 6
I 1 + 5 I 2 + 2 I 3 10 I 4 + 3 I 5 + I 6 I 1 + 3 I 2 10 I 3 + 2 I 4 + 5 I 5 I 6
4 ( I 2 I 3 I 4 + I 5 ) I 1 I 2 6 I 3 + 6 I 4 + I 5 I 6

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