Abstract

Conventional phase-shifting algorithms based on a least-squares estimate use N samples over an incomplete period of the sampled waveform. We introduce a class of phase-shifting algorithms having N + 1 samples symmetrically disposed over one full period of the sampled waveform. Fourier analysis techniques are used to derive these algorithms and modify them to improve their performance in the presence of phase-shift errors. The algorithms can be used in phase measurement systems having periodic, but not necessarily sinusoidal, waveforms.

© 1992 Optical Society of America

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References

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  1. 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]
  2. K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef] [PubMed]
  3. P. Harihan, B. F. Oreb, N. Brown, “Real-time holographic interferometry: a microcomputer system for the measurement of vector displacements,” Appl. Opt. 22, 876–880 (1983).
    [CrossRef]
  4. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mésures,” Metrologia 2, 13–16 (1966).
    [CrossRef]
  5. J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May, 1982), pp. 65–71.
  6. 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]
  7. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  8. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  9. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  10. R. Bracewell, The Fourier Transform and Its Applications, USA, Electrical and Electronic Engineering Series (McGraw-Hill, New York, 1965).
  11. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  12. 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]

1990 (1)

1987 (1)

1985 (1)

1984 (1)

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

1983 (2)

1982 (2)

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
[CrossRef] [PubMed]

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May, 1982), pp. 65–71.

1974 (1)

1966 (1)

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

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, USA, Electrical and Electronic Engineering Series (McGraw-Hill, New York, 1965).

Brangaccio, D. J.

Brohinsky, W. R.

Brown, N.

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]

Creath, K.

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]

Eiju, T.

Elssner, K. E.

Freischlad, K.

Gallagher, J. E.

Greivenkamp, J. E.

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

Grzanna, J.

Harihan, P.

Hariharan, P.

Herriott, D. R.

Koliopoulos, C. L.

Merkel, K.

Morgan, C. J.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

Spolaszyk, R.

Stetson, K. A.

White, A. D.

Wyant, J. C.

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May, 1982), pp. 65–71.

Appl. Opt. (5)

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

Laser Focus (1)

J. C. Wyant, “Interferometric optical metrology: basic principles and new systems,” Laser Focus (May, 1982), pp. 65–71.

Metrologia (1)

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. (1)

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

Opt. Lett. (1)

Other (2)

R. Bracewell, The Fourier Transform and Its Applications, USA, Electrical and Electronic Engineering Series (McGraw-Hill, New York, 1965).

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 (7)

Fig. 1
Fig. 1

(a) Periodic interference or grating pattern g(t). (b) |G(ν)| the modulus of the Fourier spectrum of g(t).

Fig. 2
Fig. 2

The (4 + 1) sampling function: (a) numerator f1(t), (b) denominator f2(t).

Fig. 3
Fig. 3

The (4 + 1) sampling function, FT: (a) numerator F1(ν), (b) denominator F2(ν).

Fig. 4
Fig. 4

The (6 + 1) sampling function: (a) numerator f1(t), (b) denominator f2(t).

Fig. 5
Fig. 5

The (6 + 1) sampling function, FT: (a) numerator F1(ν), (b) denominator F2(ν).

Fig. 6
Fig. 6

(a) Modifying sampling function c(t). (b) The FT C(ν) of the modifying sampling function.

Fig. 7
Fig. 7

(a) Modified (6 + 1) sampling function f1′(t) with error-correcting properties. (b) The modified (6 + 1) sampling function FT F1′(ν).

Equations (77)

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g ( t ) = n = 0 a n cos ( 2 π n ν g t + Φ n ) ,
p ( t ) = g f 1 = - g ( τ ) f 1 ( τ + t ) d τ ,
q ( t ) = g f 2 = - g ( τ ) f 2 ( τ + t ) d τ .
p ( 0 ) = n = 0 a n 2 [ exp ( i Φ n ) F 1 * ( n ν g ) + exp ( - i Φ n ) F 1 * ( - n ν g ) ] ,
q ( 0 ) = n = 0 a n 2 [ exp ( i Φ n ) F 2 * ( n ν g ) + exp ( - i Φ n ) F 2 * ( - n ν g ) ] ,
r = p ( 0 ) q ( 0 ) .
a n F 1 * ( n ν g ) = - i A δ ( n , m ) ,
a n F 2 * ( n ν g ) = A δ ( n , m ) ,
δ ( n , m ) = 1 for m = n , δ ( n , m ) = 0 for m n .
F 1 ( m ν g ) = i F 2 ( m ν g ) .
r = tan ( Φ m ) .
f 1 ( t ) = - f 1 ( - t ) ,             i . e . , f 1 ( t ) is real and odd ,
f 2 ( t ) = f 2 ( - t ) ,             i . e . , f 2 ( t ) is real and even .
f 1 ( t ) = n = 0 N α n δ ( t - t n ) ,
t n = n T f N - T f 2
f 2 ( t ) = n = 0 N β n δ ( t - t n ) ,
α n = - α N - n             for n = 0 - M ,
n = 0 N α n = 0 .
F 1 ( ν ) = - 2 i n = 0 M α n sin ( 2 π ν t n ) .
β N - n = β n             for n = 0 - M .
F 2 ( ν ) = β M exp ( - 2 π i ν t M ) + 2 n = 0 M - 1 β n cos ( 2 π ν t n ) ,
F 2 ( ν ) = 2 n = 0 M β n cos ( 2 π ν t n ) .
F 2 ( 0 ) = 0.
β M + 2 n = 0 M - 1 β n = 0 ,
2 n = 0 M β n = 0 .
α n = - sin ( 2 π t n T f ) = sin ( 2 π n N )             for n = 0 - N ,
β n = cos ( 2 π t n T f ) = - cos ( 2 π n N )             for n = 1 - ( N - 1 ) ,
F 1 ( ν ) = - 2 i n = 1 M sin ( 2 π n N ) sin [ 2 π ν ( n T f N - T f 2 ) ] .
F 1 ( m ν f ) = i N / 2 if m = k N + 1 F 1 ( m ν f ) = - i N / 2 if m = k N - 1 F 1 ( m ν f ) = 0 otherwise }
F 1 ( m ν f ) = i N / 2 if m = 2 k N + 1 = - i N / 2 if m = 2 k N - 1 = i N / 2 if m = ( 2 k + 1 ) N - 1 = - i N / 2 if m = ( 2 k + 1 ) N + 1 = 0 otherwise } .
F 2 ( ν ) = 1 - cos ( π ν T f ) - 2 n = 1 M - 1 cos ( 2 π n N ) cos [ 2 π ν ( n T f N - T f 2 ) ]
F 2 ( ν ) = - cos ( π ν T f ) - 2 n = 1 M cos ( 2 π n N ) cos [ 2 π ν ( n T f N - T f 2 ) ]
F 2 ( m ν f ) = N / 2 if m = k N ± 1 F 2 ( m ν f ) = 0 otherwise } ,
F 2 ( m ν f ) = N / 2 if m = 2 k N ± 1 = - N / 2 if m = ( 2 k + 1 ) N ± 1 = 0 otherwise } .
α 0 = 0 ,             α 1 = 1 ,             α 2 = 0 ,             α 3 = - 1 ,             α 4 = 0.
β 0 = - ½ ,             β 1 = 0 ,             β 2 = 1 ,             β 3 = 0 ,             β 4 = - ½ .
p ( 0 ) = g ( - T f 4 ) - g ( T f 4 ) .
q ( 0 ) = - g ( - T f 2 ) 2 + g ( 0 ) - g ( T f 2 ) 2 .
I 1 = g ( - T f 2 ) ,             I 2 = g ( - T f 4 ) ,             I 3 = g ( 0 ) , I 4 = g ( T f 4 ) ,             I 5 = g ( T f 2 ) ,
r = p ( 0 ) q ( 0 ) = 2 ( I 2 - I 4 ) 2 I 3 - I 1 - I 5 .
p ( 0 ) = 2 a 1 sin ( Φ 1 ) .
q ( 0 ) = 2 a 1 cos ( Φ 1 ) .
r = p ( 0 ) / q ( 0 ) = tan ( Φ 1 ) ,
tan ( Φ 1 ) = 2 ( I 2 - I 4 ) 2 I 3 - I 1 - I 5 .
α 0 = 0 ,             α 1 = 3 2 ,             α 2 = 3 2 ,             α 3 = 0 , α 4 = - 3 2 ,             α 5 = - 3 2 ,             α 6 = 0.
β 0 = - ½ ,             β 1 = - ½ ,             β 2 = ½ ,             β 3 = 1 , β 4 = ½ ,             β 5 = - ½ ,             β 6 = - ½ .
tan ( Φ 1 ) = 3 ( I 2 + I 3 - I 5 - I 6 ) ( - I 1 - I 2 + I 3 - 2 I 4 - I 5 - I 6 - I 7 ) ,
I 1 = g ( - T g 2 ) ,             I 2 = g ( - T g 3 ) , I 3 = g ( - T g 6 ) ,             I 4 = g ( 0 ) ,             I 5 = g ( T g 6 ) , I 6 = g ( T g 3 ) ,             I 7 = g ( T g 2 ) .
F 1 ( ν ) = 3 i [ sin ( π ν 3 ν f ) + sin ( 2 π ν 3 ν f ) ] .
F 2 ( ν ) = 1 - cos ( π ν ν f ) + cos ( π ν 3 ν f ) - cos ( 2 π ν 3 ν f ) .
r = tan ( Φ 1 ) = Re [ F 1 ( ν g ) ] Re [ F 2 ( ν g ) ] tan ( Φ 1 ) ,
F 1 ( ν g ) F 1 ( ν f ) + Δ ν d F 1 ( ν f ) d ν .
F 2 ( ν g ) F 2 ( ν f ) + Δ ν d F 2 ( ν f ) d ν ,
d F 1 ( ν f ) d ν = i d F 2 ( ν f ) d ν .
d F 1 ( ν f ) d ν = - π i 2 3 ν f .
d F 2 ( ν f ) d ν = π 2 3 ν f .
d F 1 ( ν f ) d ν - i d F 2 ( ν f ) d ν = i π 3 ν f ,
Δ Φ 1 ( e / 2 3 ) sin ( 2 Φ 1 )
C ( ν ) = 2 i c 0 sin ( π ν ν f ) ,
c ( t ) = c 0 { δ ( t - T f 2 ) - δ ( t + T f 2 ) } .
d C ( ν ) d ν = 2 π i c 0 ν f cos ( π ν ν f ) ;
d C ( ν f ) d ν = - i c 0 2 π ν f .
f 1 ( t ) = f 1 ( t ) + c ( t )
F 1 ( ν ) = F 1 ( ν ) + C ( ν ) ,
d F 1 ( ν f ) d ν = d F 1 ( ν ) d ν + d C ( ν ) d ν .
d F 1 ( ν f ) d ν = 2 i π T f N n = 1 M ( 2 n - N ) sin ( 2 π n N ) cos ( 2 π n N ) .
d F 2 ( ν f ) d ν = - 2 π T f N n = 1 M - 1 ( 2 n - N ) cos ( 2 π n N ) sin ( 2 π n N ) ,
d F 2 ( ν f ) d ν = - 2 π T f N n = 1 M ( 2 n - N ) cos ( 2 π n N ) sin ( 2 π n N ) .
d F 1 ( ν f ) d ν = i d F 2 ( ν f ) d ν .
d C ( ν f ) d ν = i d F 2 ( ν f ) d ν - d F 1 ( ν f ) d ν .
c 0 = 1 N n = 1 M - 1 ( N - 2 n ) sin ( 4 π n N ) .
c 0 = 1 N n = 1 M ( N - 2 n ) sin ( 4 π n N ) .
c 0 = 1 2 3 .
tan ( Φ 1 ) = 3 ( I 2 + I 3 - I 5 - I 6 ) + ( I 7 - I 1 ) / 3 ( - I 1 - I 2 + I 3 + 2 I 4 + I 5 - I 6 - I 7 ) .
tan ( Φ 1 ) = n = 1 N - 1 sin ( 2 π n N ) I n + 1 - ( I 1 + I N + 1 ) 2 - n = 1 N - 1 cos ( 2 π n N ) I n + 1 .
I n + 1 = g ( t n )             for n = 0 - N ,
tan ( Φ 1 ) = c 0 ( I N + 1 - I 1 ) + n = 1 N - 1 sin ( 2 π n N ) I n + 1 - ( I 1 + I N + 1 ) 2 - n = 1 N - 1 cos ( 2 π n N ) I n + 1 ,

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