Abstract

A general description of phase measurement by digital heterodyne techniques is presented in which the heterodyning is explained as a filtering process in the frequency domain. Examples of commonly used algorithms are given. Special emphasis is given to the analysis of systematic errors. Gaussian error propagation is used to derive equations for the random phase errors of common algorithms.

© 1990 Optical Society of America

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References

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  1. J. H. Bruning, D. R. 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]
  2. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).
  3. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  4. C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1981).
  5. K. Creath, “Comparison of phase measuring algorithms,” in Surface Characterization and Testing, K. Creath, ed., Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
    [CrossRef]
  6. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, 1968).
  7. E. Hardtwig, Fehler- und Ausgleichsrechnung (Bibliographisches Institut, Mannheim, 1968).
  8. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  9. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  10. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–351 (1984).
    [CrossRef]
  11. W. R. C. Rowley, J. Hamon, “Quelques mesures de dyssymétrie de profils spectraux,” R. Opt. Theor. Instrum. 42, 519–523 (1963).
  12. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
    [CrossRef]
  13. J. B. Hayes, “Linear methods of computer controlled optical figuring,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1984).
  14. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaszyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  15. P. Hariharan, “Digital phase shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  16. Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef] [PubMed]
  17. K. Freischlad, “Wavefront sensing by heterodyne shearing interferometry,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1986).
  18. C. L. Koliopoulos, “Radial grating lateral shear heterodyne interferometer,” Appl. Opt. 19, 1523–1528 (1980).
    [CrossRef] [PubMed]
  19. C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase-shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
    [CrossRef] [PubMed]
  20. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  21. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  22. Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
    [CrossRef] [PubMed]
  23. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
    [CrossRef] [PubMed]
  24. K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
    [CrossRef]
  25. P. Hariharan, “Digital phase stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]

1987

1985

1984

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–351 (1984).
[CrossRef]

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

1983

1982

1980

1975

1974

1972

1966

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

1963

W. R. C. Rowley, J. Hamon, “Quelques mesures de dyssymétrie de profils spectraux,” R. Opt. Theor. Instrum. 42, 519–523 (1963).

Ai, C.

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

Cheng, Y.-Y.

Creath, K.

K. Creath, “Comparison of phase measuring algorithms,” in Surface Characterization and Testing, K. Creath, ed., Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
[CrossRef]

Elssner, K.-E.

Freischlad, K.

K. Freischlad, “Wavefront sensing by heterodyne shearing interferometry,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1986).

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

Gallagher, J. E.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–351 (1984).
[CrossRef]

Grzanna, J.

Hamon, J.

W. R. C. Rowley, J. Hamon, “Quelques mesures de dyssymétrie de profils spectraux,” R. Opt. Theor. Instrum. 42, 519–523 (1963).

Hardtwig, E.

E. Hardtwig, Fehler- und Ausgleichsrechnung (Bibliographisches Institut, Mannheim, 1968).

Hariharan, P.

Hayes, J. B.

J. B. Hayes, “Linear methods of computer controlled optical figuring,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1984).

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, 1968).

Herriott, D. R.

Ichioka, Y.

Inuiya, M.

Koliopoulos, C. L.

C. L. Koliopoulos, “Radial grating lateral shear heterodyne interferometer,” Appl. Opt. 19, 1523–1528 (1980).
[CrossRef] [PubMed]

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1981).

Merkel, K.

Mertz, L.

Morgan, C. J.

Rosenfeld, D. P.

Rowley, W. R. C.

W. R. C. Rowley, J. Hamon, “Quelques mesures de dyssymétrie de profils spectraux,” R. Opt. Theor. Instrum. 42, 519–523 (1963).

Schwider, J.

Spolaszyk, R.

White, A. D.

Womack, K. H.

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

Wyant, J. C.

Appl. Opt.

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

J. H. Bruning, D. R. 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]

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaszyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

P. Hariharan, “Digital phase shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

Y.-Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

C. L. Koliopoulos, “Radial grating lateral shear heterodyne interferometer,” Appl. Opt. 19, 1523–1528 (1980).
[CrossRef] [PubMed]

C. Ai, J. C. Wyant, “Effect of piezoelectric transducer nonlinearity on phase-shift interferometry,” Appl. Opt. 26, 1112–1116 (1987).
[CrossRef] [PubMed]

Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
[CrossRef] [PubMed]

L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
[CrossRef] [PubMed]

P. Hariharan, “Digital phase stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
[CrossRef] [PubMed]

Metrologia

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
[CrossRef]

Opt. Eng.

K. H. Womack, “Frequency domain description of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).
[CrossRef]

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–351 (1984).
[CrossRef]

Opt. Lett.

R. Opt. Theor. Instrum.

W. R. C. Rowley, J. Hamon, “Quelques mesures de dyssymétrie de profils spectraux,” R. Opt. Theor. Instrum. 42, 519–523 (1963).

Other

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

C. L. Koliopoulos, “Interferometric optical phase measurement techniques,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1981).

K. Creath, “Comparison of phase measuring algorithms,” in Surface Characterization and Testing, K. Creath, ed., Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
[CrossRef]

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon, Oxford, 1968).

E. Hardtwig, Fehler- und Ausgleichsrechnung (Bibliographisches Institut, Mannheim, 1968).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

K. Freischlad, “Wavefront sensing by heterodyne shearing interferometry,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1986).

J. B. Hayes, “Linear methods of computer controlled optical figuring,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1984).

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Figures (6)

Fig. 1
Fig. 1

Filter functions for common heterodyne procedures.

Fig. 2
Fig. 2

Spectra of filter functions as a function of the normalized frequency νsf: (a) Case 1, (b) case 4, (c) case 6.

Fig. 3
Fig. 3

P–V phase error (δ − 1) as a function of the normalized frequency νsf.

Fig. 4
Fig. 4

Suppression of higher harmonics by the integrating-bucket technique.

Fig. 5
Fig. 5

Influence of a quadratic phase shift: (a) quadratic phase shift, (b) sampling points.

Fig. 6
Fig. 6

Normalized variance of the frequency measurement with additive noise.

Tables (1)

Tables Icon

Table 1 Variance of Phase Error σΦ2 of Common Heterodyne Proceduresa

Equations (97)

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s ( t ) = I [ 1 + V cos ( 2 π ν s t + Φ ) ] ,
s 0 = I [ 1 + V cos ( Φ ) ] for 2 π ν s t = 0 ,
s 1 = I [ 1 V sin ( Φ ) ] for 2 π ν s t = π / 2 ,
s 2 = I [ 1 V cos ( Φ ) ] for 2 π ν s t = π ,
s 3 = I [ 1 + V sin ( Φ ) ] for 2 π ν s t = 3 π / 2 .
Φ = arctan s 1 + s 3 s 0 s 2 .
Φ = arctan s ( T s 4 ) + s ( 3 T s 4 ) s ( 0 ) s ( T s 2 ) ,
c i = s ( t ) f i ( t ) d t , i = 1 , 2 .
c i = 2 Re [ 0 S ( ν ) F i * ( ν ) d ν ] , i = 1 , 2 ,
r = 2 Re [ 0 S ( ν ) F 1 * ( ν ) d ν ] 2 Re [ 0 S ( ν ) F 2 * ( ν ) d ν ] .
S ( ν ) = n = s n δ ( ν n ν s ) ,
s n = | s n | e i Φ n ,
r = c 1 c 2 = s 0 F 1 ( 0 ) + 2 Re [ n = 1 s n F 1 * ( n ν s ) ] s 0 F 2 ( 0 ) + 2 Re [ n = 1 s n F 2 * ( n ν s ) ] .
s n F 1 * ( n ν s ) = i A δ ( n , m ) ,
s n F 2 * ( n ν s ) = A δ ( n , m ) ,
F 1 ( m ν s ) = i F 2 ( m ν s ) .
r = 2 | s m | | F 2 ( m ν s ) | sin ( Φ m Ψ ) 2 | s m | | F 2 ( m ν s ) | cos ( Φ m Ψ ) ,
F 2 ( ν ) = | F 2 ( ν ) | e i Ψ ( ν ) ,
Φ m = arctan ( r ) + Ψ ( m ν s ) .
c i = n = 1 N s ( t n ) f i ( t n ) , i = 1 , 2 .
d ( t ) = n = δ ( t n Δ ) ,
F 1 ( 0 ) = F 2 ( 0 ) = 0 ,
tan Φ = s ( 0 ) + s ( T s 4 ) s ( T s 2 ) s ( 3 T s 4 ) s ( 0 ) + s ( T s 4 ) + s ( T s 2 ) s ( 3 T s 4 ) ;
f 1 ( t ) = δ ( t ) + δ ( t T f 4 ) δ ( t T f 2 ) δ ( t 3 T f 4 ) ,
f 2 ( t ) = δ ( t ) + δ ( t T f 4 ) + δ ( t T f 2 ) δ ( t 3 T f 4 ) ,
F 1 ( ν ) = 4 i cos ( π 4 ν ν f ) sin ( π 2 ν ν f ) exp ( 3 π 4 i ν ν f ) ,
F 2 ( ν ) = 4 sin ( π 4 ν ν f ) sin ( π 2 ν ν f ) exp ( 3 π 4 i ν ν f ) .
r = tan ( Φ + 3 π 4 ν s ν f ) tan ( π 4 ν s ν f ) ,
r = tan ( Φ + 3 π 4 ) .
tan Φ = s ( T s 4 ) + s ( 3 T s 4 ) s ( 0 ) s ( T s 2 ) .
F 1 ( ν ) = 2 i sin ( π 2 ν ν f ) exp ( i π ν ν f ) ,
F 2 ( ν ) = 2 i sin ( π 2 ν ν f ) exp ( i π 2 ν ν f ) ,
r = sin ( Φ + π ) cos ( Φ + π 2 ) ,
= ν s ν f ν f .
r = tan ( Φ ) .
tan Φ = s ( 0 ) s ( T s 4 ) s ( T s 4 ) s ( T s 2 ) .
F 1 ( ν ) = 2 i sin ( π 4 ν ν f ) exp ( i π 4 ν ν f ) ,
F 2 ( ν ) = 2 i sin ( π 4 ν ν f ) exp ( i 3 π 4 ν ν f ) ,
r = sin ( Φ + π 4 ν s ν f ) cos ( Φ + π 4 ν s ν f + π 2 ) ,
r = tan ( Φ + π 4 ) .
tan Φ = s ( 0 ) 2 s ( T s 4 ) + s ( T s 2 ) s ( 0 ) s ( T s 2 ) .
F 1 ( ν ) = 4 sin 2 ( π 4 ν ν f ) exp ( i π 2 ν ν f ) ,
F 2 ( ν ) = 4 i sin ( π 4 ν ν f ) cos ( π 4 ν ν f ) exp ( i π 2 ν ν f ) ,
r = tan ( π 4 ν s ν f ) tan ( Φ + π 2 ) ,
r = tan ( Φ ) .
tan Φ = 3 2 [ s ( T s 3 ) + s ( 2 T s 3 ) ] s ( 0 ) 1 2 [ s ( T s 3 ) + s ( 2 T s 3 ) ] .
F 1 ( ν ) = i 3 sin ( π 3 ν ν f ) exp ( i π ν ν f ) ,
F 2 ( ν ) = 1 cos ( π 3 ν ν f ) exp ( i π ν ν f ) .
r = 3 sin ( π 3 ν s ν f ) sin ( Φ + π ) cos ( Φ ) + cos ( π 3 ν s ν f ) cos ( Φ + π ) ,
r = tan ( Φ ) .
tan Φ = 2 s ( T s 4 ) + 2 s ( 3 T s 4 ) s ( 0 ) 2 s ( T s 2 ) + s ( T s ) .
F 1 ( ν ) = 4 i sin ( π 2 ν ν f ) exp ( i π ν ν f ) ,
F 2 ( ν ) = 2 [ 1 cos ( π ν ν f ) ] exp ( i π ν ν f ) ,
r = 2 sin ( π 2 ν s ν f ) 1 cos ( π ν s ν f ) tan ( Φ π ν s ν f ) .
r = tan ( Φ ) .
r = δ tan ( Φ ) ,
Δ Φ = arctan ( r ) Φ ,
Δ Φ δ 1 2 sin ( 2 Φ ) .
f 1 ( t ) = δ ( t ) + δ ( t Δ ) δ ( t 2 Δ ) δ ( t 3 Δ ) ,
f 2 ( t ) = δ ( t ) δ ( t Δ ) + δ ( t 2 Δ ) δ ( t 3 Δ ) ,
Δ = 1 4 ν f = T f 4 .
F 1 ( ν ) = 4 i cos ( π 4 ν ν f ) sin ( π 2 ν ν f ) exp ( 3 π i 4 ν ν f ) ,
F 2 ( ν ) = 4 i sin ( π 4 ν ν f ) cos ( π 2 ν ν f ) exp ( 3 π i 4 ν ν f ) .
r = 2 1 tan 2 ( π 4 ν s ν f ) ,
s n = { Γ n ( g 0 g n * + g 0 * g n ) exp [ 2 π i W ( x , y ) n d / λ ] n odd Γ n m = , m 0 , n g m g m n * exp ( 2 π i { W ( x m d , y ) W [ x ( m n ) d , y ] } / λ ) n even ,
g n = 1 / 2 n = 2 n π sin ( n π / 2 ) .
Φ = 2 π λ W ( x , y ) d .
r sin ( Φ ) δ sin ( 3 Φ ) cos ( Φ ) + δ cos ( 3 Φ ) ,
Δ Φ δ sin ( 4 Φ ) .
f 1 ( t ) = δ ( t + 3 π 4 9 α 2 ) + δ ( t + π 4 α 2 ) δ ( t π 4 α 2 ) δ ( t 3 π 4 9 α 2 ) ,
f 1 ( t ) = δ ( t + 3 π 4 9 α 2 ) + δ ( t + π 4 α 2 ) + δ ( t π 4 α 2 ) δ ( t 3 π 4 9 α 2 ) ,
F 1 ( ν ) = 2 i [ sin ( π 4 ν ν f ) exp ( 2 π i α 2 ν ) + sin ( 3 π 4 ν ν f ) exp ( 2 π i 9 α 2 ν ) ] ,
F 2 ( ν ) = 2 [ cos ( π 4 ν ν f ) exp ( 2 π i α 2 ν ) cos ( 3 π 4 ν ν f ) exp ( 2 π i 9 α 2 ν ) ] .
y = n = 1 N a n x n ,
σ y 2 = n = 1 N a n 2 σ n 2 ,
y = f ( x 1 , , x N ) ,
σ y 2 n = 1 N [ f ( x 1 , , x N ) x n ] 2 σ n 2 ,
σ Φ 2 1 ( 2 | S ( ν s ) | | F 2 ( ν s ) | ) 4 n = 1 N [ f 1 ( t n ) c 2 f 2 ( t n ) c 1 ] 2 σ n 2 ,
σ η 2 σ I 2 64 I 2 V 2 5 12 sin 2 ( η ) + 8 sin 4 ( η ) cos 2 ( η ) sin 4 ( η ) 1 sin 2 ( Φ ) ,
1 ν s ν f 1.75
π 4 Φ 3 π 4 .
σ η 2 σ I 2 8 I 2 V 2 1 sin 2 ( Φ ) .
σ Φ 2 = σ I 2 I 2 V 2 sin 4 ( 2 η ) × [ cos 2 ( η ) + 5 12 sin 2 ( η ) + 8 sin 4 ( η ) 4 sin 2 ( η ) cos 2 ( Φ ) ] .
1 ν s ν f 1.5 .
σ Φ 2 = σ I 2 2 I 2 V 2 [ 1 + cos 2 ( Φ ) ] ;
1 2 σ I 2 ( I V ) 2
1 2 I ( I V ) 2
1 2 σ I 2 ( I V ) 2
1 2 I ( I V ) 2
σ I 2 ( I V ) 2 [ 1 + ½ cos ( 2 Φ ) ]
1 ( I V ) 2 { 1 + 1 2 cos ( 2 Φ ) V 2 [ sin ( Φ ) + sin ( 3 Φ ) ] }
σ I 2 ( I V ) 2 [ 1 + ½ cos ( 2 Φ ) ]
1 ( I V ) 2 { 1 + 1 2 cos ( 2 Φ ) V 2 [ sin ( Φ ) + sin ( 3 Φ ) ] }
2 3 σ I 2 ( I V ) 2
1 ( I V ) 2 [ 1 V 2 cos ( 3 Φ ) ]
1 16 σ I 2 ( I V ) 2 [ 7 + cos ( 2 Φ ) ]
I 32 ( I V ) 2 [ 14 + cos ( Φ ) + 2 cos ( 2 Φ ) cos ( 3 Φ ) ]

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