Abstract

The effect of intensity noise on the computed phase of phase-shifting interferometry is investigated. Using a simple Taylor expansion, one can easily show the sensitivity of the error in computed phase to frame-to-frame intensity noise correlation. The sensitivity of various algorithms to statistically independent intensity noise and intensity quantization noise is considered. Intensity-error cross correlations are found to increase the error in computed phase for the case of intensity quantization error.

© 1990 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. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.
  2. 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]
  3. 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]
  4. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed. Proc. Soc. Photo-Opt. Instrum. Eng.680, 19 (1986).
    [CrossRef]
  5. K. Kinnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef] [PubMed]
  6. C. Ai, J. C. Wyant, “Effect of spurious reflection of phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988).
    [CrossRef] [PubMed]
  7. K. A. Stetson, W. R. Brohinsky, “Electrooptic holography and its application to holographic interferometry,” Appl. Opt. 24, 3631–3637 (1985).
    [CrossRef] [PubMed]
  8. J. H. Bruning, D. R. Herriot, 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]
  9. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  10. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  11. P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  12. J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

1988 (2)

1987 (1)

1985 (1)

1984 (1)

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

1983 (1)

1982 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Ai, C.

Bhushan, B.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.

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

Eiju, T.

Elssner, K. E.

Gallagher, J. E.

George, O. E.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

Grievenkamp, J. E.

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

Grzanna, J.

Hariharan, P.

Herriot, D. R.

Kinnstaetter, K.

Koliopoulos, C. L.

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

Lohmann, A. W.

Merkel, K.

Morgan, C. J.

Oreb, B. F.

Rosenfeld, D. P.

Schwider, J.

Spolaczyk, R.

Stetson, K. A.

Streibl, N.

White, A. D.

Wyant, J. C.

C. Ai, J. C. Wyant, “Effect of spurious reflection of phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988).
[CrossRef] [PubMed]

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

Appl. Opt. (6)

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (3)

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 351–398.

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

J. C. Wyant, C. L. Koliopoulos, B. Bhushan, O. E. George, presented at the 38th Annual Meeting of the American Society of Lubrication Engineers (1983).

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Figures (2)

Fig. 1
Fig. 1

Illustration of intensity quantization error for a 90° algorithm. The circles represent actual intensities; the horizontal hash marks on the vertical axis represent possible intensity measurements. ΔIn and ΔIn+4 are always the same, ΔIn and ΔIn+2 are always opposite in sign, and ΔIn and ΔIn+1 are approximately uncorrelated.

Fig. 2
Fig. 2

Diagram used in the construction of a 13-frame, 75° algorithm. Each vector is rotated 75° from its preceding vector. Note that if the In of Eq. (12) are viewed as vectors, the numerator of Eq. (12) has a resultant that lies on the vertical axis of the diagram; the denominator has a resultant that lies on the horizontal axis.

Tables (2)

Tables Icon

Table 1 Proportionality Constant between the Average Phase Variance and the Relative Intensity Variance for the Case of Uncorrelated Intensity Noise

Tables Icon

Table 2 Average Phase Variance Due to Intensity Quantization Error Versus Q, the Number of Integer Levels Representing the Modulated Intensitya

Equations (30)

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I n = I 0 + M I 0 cos ( ϕ + δ n ) .
ϕ = ϕ + ( ϕ / I n ) Δ I n + O ( Δ I 2 ) ,
( Δ ϕ ) 2 n m ( ϕ / I n ) ( ϕ / I m ) Δ I n Δ I m .
V ϕ n m ( ϕ / I n ) ( ϕ / I m ) Δ I n Δ I m ,
ϕ = tan 1 ( A n I n / B n I n )
V ϕ [ 1 ( C M I 0 ) 2 ] n m [ A n cos ( ϕ ) B n sin ( ϕ ) ] × [ A m cos ( ϕ ) B m sin ( ϕ ) ] Δ I n Δ I m ,
V ϕ [ V I ( C M I 0 ) 2 ] [ A n 2 cos 2 ( ϕ ) + B n 2 sin 2 ( ϕ ) 2 A n B n sin ( ϕ ) cos ( ϕ ) ] ,
V ϕ ¯ [ V I 2 ( C M I 0 ) 2 ] ( A n 2 + B n 2 ) .
k = 1 / ( 2 C 2 ) ( A n 2 + B n 2 ) .
V ϕ ¯ [ 2 ( C Q ) 2 ] n m ( A n A m + B n B m ) Δ I n Δ I m .
Δ I n Δ I n + 4 p 1 / 12 p = 0 , ± 1 , ± 2 , ± 3 , Δ I n Δ I n + 2 p 1 / 12 p = ± 1 , ± 3 , ± 5 , Δ I n Δ n + p 0 p = ± 1 , ± 3 , ± 5 .
V ϕ ¯ ( 1 3 Q 2 )
ϕ = tan 1 { K [ ( I 1 I 11 ) ( I 2 I 10 ) ( I 3 I 13 ) + ( I 4 I 12 ) ( 4 I 7 I 5 I 6 I 8 I 9 ] [ ( I 1 I 13 ) + ( I 2 I 12 ) ( I 3 I 11 ) ( I 4 I 10 ) + ( I 5 I 9 ) + ( I 6 I 8 ) } ,
K = 4 2 sin ( 15 ° ) + 2 sin ( 30 ° ) + 2 sin ( 45 ° ) + 2 sin ( 60 ° ) + 2 sin ( 75 ° ) 2 + 2 cos ( 15 ° ) + 2 cos ( 30 ° ) + 2 cos ( 45 ° ) + 2 cos ( 60 ° ) + 2 cos ( 75 ° ) = 0.899095 .
V ϕ ¯ [ 1 6 ( C Q ) 2 ] [ n ( A n 2 + B n 2 ) 2 ( A 1 A 13 + B 1 B 13 ) ] .
V ϕ ¯ ( 1 12.87 Q 2 ) ,
Δ I n Δ I m = Δ I 2 ( ± r ) | n m | ( r < 1 ) .
V ϕ ¯ [ Δ I 2 2 ( M I 0 ) 2 ] ( 1 r 2 ) ,
k = V ϕ ¯ / [ V I / ( M I 0 ) 2 ]
Arg = I 3 I 2 I 1 I 2
Arg = I 4 I 2 I 1 I 3
Arg = 2 ( I 4 I 2 ) 2 I 3 I 2 I 5
Arg = I n I n 2 + I 2 I 1 I 3 + I n 1
Arg = 3 ( I 3 I 2 ) 2 I 1 I 2 I 3
1 / ( Q 2 V ϕ ¯ )
ϕ = tan 1 [ ( I 4 I 2 ) / ( I 1 I 3 ) ] .
ϕ e = tan 1 [ ( q 4 q 2 ) / ( q 1 q 3 ) ] ,
( q 1 , q 2 , q 3 , q 4 ) = { ( 2 , 0 , 2 , 0 ) ω 1 = [ 0.000 0.253 ] ( 2 , 1 , 2 , 1 ) ω 2 = [ 0.253 0.723 ] ( 1 , 1 , 1 , 1 ) ω 3 = [ 0.723 π / 4 ] ,
V k = ω k 2 / 12 + [ ϕ e ( k ) ϕ ¯ k ] 2 ,
V ϕ ¯ = ( k | ω k | V k ) / ( π / 4 ) ,

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