Abstract

The characteristic polynomials associated with the algorithms used in digital phase detection are used to investigate the effects of additive noise on phase measurements. First, it is shown that a loss factor η can be associated with any algorithm. This parameter describes the influence of the algorithm on the global signal-to-noise ratio (SNR). Second, the variance of the phase error is shown to depend mainly on the global SNR. The amplitude of a modulation of this variance at twice the signal frequency depends on a single parameter β. The material presented here extends previously published results, and as many as 19 algorithms are analyzed.

© 1997 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 Science, New York, 1990), Vol. 28, pp. 271–359.
  2. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase-shifting,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  4. The inequalities derive from the fact that η is equal to the magnitude of the Hermitian product of the vectors {c0, c1 …, cM–1} and {ζ0, ζ-1,…, ζ-(M-1)} divided by the product of their norms.
  5. If the development is carried on to the second order, one finds that 〈Δφ〉 = (β/2SNR2)sin[2(φ - θ)]. This expression corresponds to the modulation of 〈Δφ〉 mentioned in Ref. 9 and to the curves plotted therein. However, the amplitude of this modulation is much less than the standard deviation of Δφ, even in the worst cases of low SNR and high β. As an example, for algorithm (16) and SNR = 2, the ratio of the amplitude of the modulation of the phase-error average over its standard deviation is β/ 2  SNR = 0.07. So, the modulation of the phase-error average is usually hidden in the phase noise.
  6. C. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
    [CrossRef]
  7. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  8. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  9. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  10. C. L. Koliopoulos, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).
  11. J. B. Hayes, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).
  12. W. R. C. Rowley, J. Hamon, “Quelques mesures de dissymétrie de profils spectraux,” Rev. Opt. Théor. Instrum. 42, 519–523 (1963).
  13. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  14. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real time wave-front correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  15. J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
    [CrossRef] [PubMed]
  16. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  17. 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]
  18. B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
    [CrossRef]
  19. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  20. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  21. Y. Surrel, “Phase-stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]

1996 (1)

1995 (5)

J. Schmit, K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34, 3610–3619 (1995).
[CrossRef] [PubMed]

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[CrossRef]

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

1993 (2)

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

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

1992 (1)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

1991 (1)

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

1990 (2)

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

C. Brophy, “Effect of intensity error correlation on the computed phase of phase-shifting interferometry,” J. Opt. Soc. Am. A 7, 537–541 (1990).
[CrossRef]

1987 (1)

1975 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

1963 (1)

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

Brophy, C.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

de Groot, P.

Eiju, T.

Falkenstörfer, O.

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

Farrant, D. I.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

Frankena, H. J.

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Freischlad, K.

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

Hamon, J.

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

Hariharan, P.

Hayes, J. B.

J. B. Hayes, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).

Hibino, K.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

Koliopoulos, C. L.

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

C. L. Koliopoulos, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

Oreb, B. F.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

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]

Rathjen, C.

Rowley, W. R. C.

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

Schmit, J.

Schreiber, H.

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

Schwider, J.

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

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

Smorenburg, C.

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Streibl, N.

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

Surrel, Y.

van Wingerden, J.

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Wyant, J. C.

Zhao, B.

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Zöller, A.

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

Appl. Opt. (1)

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase-shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

Appl. Opt. (6)

J. Opt. Soc. Am. A (3)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting for nonsinusoidal wave forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
[CrossRef]

K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
[CrossRef]

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

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng. (2)

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

B. Zhao, Y. Surrel, “Phase shifting: six-sample self-calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34, 2821–2822 (1995).
[CrossRef]

Rev. Opt. Théor. Instrum. (1)

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

Other (5)

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

The inequalities derive from the fact that η is equal to the magnitude of the Hermitian product of the vectors {c0, c1 …, cM–1} and {ζ0, ζ-1,…, ζ-(M-1)} divided by the product of their norms.

If the development is carried on to the second order, one finds that 〈Δφ〉 = (β/2SNR2)sin[2(φ - θ)]. This expression corresponds to the modulation of 〈Δφ〉 mentioned in Ref. 9 and to the curves plotted therein. However, the amplitude of this modulation is much less than the standard deviation of Δφ, even in the worst cases of low SNR and high β. As an example, for algorithm (16) and SNR = 2, the ratio of the amplitude of the modulation of the phase-error average over its standard deviation is β/ 2  SNR = 0.07. So, the modulation of the phase-error average is usually hidden in the phase noise.

C. L. Koliopoulos, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1981).

J. B. Hayes, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).

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Tables (1)

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Table 1 Examples of Algorithms that can be Found in the Literature with Their Characteristic Diagrama

Equations (42)

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δ * = δ 1 + ε .
I k = A 1 + γ   frng φ + k δ * ,
φ * = arctan k = 0 M - 1 b k I k k = 0 M - 1 a k I k .
S φ = k = 0 M - 1 c k I k ,
S φ = m = - γ m   exp im φ .
P x = k = 0 M - 1 c k x k .
P N x = ζ x - 1 x - ζ 2 x - ζ 3 x - ζ N - 1
= ζ x N - 1 x - ζ = 1 - ζ - N x N 1 - ζ - 1 x
= 1 + ζ - 1 x + ζ - 2 x 2 + + ζ - N - 1 x N - 1 ,
Δ I k Δ I m = σ 2 δ km ,
S = S exp i φ ,
S * = S * exp i φ * ,
S * = S + Δ S ,
snr 2 = Ĩ * 2 σ 2 Ĩ 2 σ 2 = A 2 γ 2 2 σ 2 ,
r 2 = k = 0 M - 1 c k 2 = k = 0 M - 1 a k 2 + b k 2 .
S = A γ 2 P ζ .
SNR 2 = S * 2 σ 2 r 2 S 2 σ 2 r 2 = A 2 γ 2 4 σ 2 r 2 P ζ 2 = snr 2 P ζ 2 2 r 2 ,
SNR = η M 2 1 / 2 snr ,
η = P ζ r M   0 < η 1
S * = S exp i φ 1 + exp - i φ Δ S S
= S * exp i φ + Δ φ
S * exp i φ 1 + i Δ φ .
Δ S S = J + iK ,
J = k = 0 M - 1   a k Δ I k S ,
K = k = 0 M - 1   b k Δ I k S ,
S * = S exp i φ 1 + cos   φ - i   sin   φ J + iK S exp i φ 1 + i K   cos   φ - J   sin   φ .
Δ φ K   cos   φ - J   sin   φ .
Δ φ 0 .
Δ φ 2 = K 2 cos 2   φ + J 2 sin 2   φ - 2 JK sin   φ   cos   φ = 1 2 J 2 + K 2 - J 2 - K 2 cos 2 φ - 2 JK sin 2 φ .
β = k = 0 M - 1   c k 2 r 2 ,
θ = 1 2 arg k = 0 M - 1   c k 2 ,
k = 0 M - 1   c k 2 = k = 0 M - 1 a k 2 - b k 2 + 2 ia k b k = r 2 β   exp 2 i θ .
J 2 + K 2 = 1 SNR 2 ,
J 2 - K 2 = β   cos 2 θ SNR 2 ,
JK = β   sin 2 θ 2 SNR 2 .
Δ φ 2 = 1 2 SNR 2 1 - β   cos 2 φ - θ .
P x x - a = k = 0 M   d k x k = c 0 + c 1 x + + c M - 1 x M - 1 x - a = - ac 0 + c 0 - ac 1 x + c 1 - ac 2 x 2 + + c M - 2 - ac M - 1 x M - 1 + c M - 1 x M .
k = 0 M d k 2 = a 2 c 0 2 + k = 0 M - 2 c k 2 + a 2 c k + 1 2 - 2 ac k c k + 1 + c M - 1 2
= 1 + a 2 k = 0 M - 1   c k 2 - 2 a k = 0 M - 2   c k c k + 1 .
k = 0 M - 1   c k 2 = N sin 2 2 π / N ,
k = 0 M - 2   c k c k + 1 = N   cos 2 π / N sin 2 2 π / N ,
a 2 - 2 a   cos 2 π / N + 1 = 0 ,

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