Abstract

Computer simulations predict the expected rms measurement error in a phase-shifting interferometer in the presence of mechanical vibrations. The simulations involve a numerical resolution of a nonlinear mathematical model and are performed over a range of vibrational frequencies and amplitudes for three different phase-shift algorithms. Experimental research with an interference microscope and comparison with analytical solutions verify the numerical model.

© 1996 Optical Society of America

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References

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  1. K. Kinnstätter, Q. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
  2. L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).
  3. J. van Wingerden, H. H. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
  4. K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Creath, ed., Proc. Soc. Photo-Opt. Instrum. Eng.680, 19–28 (1986).
  5. P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995); “Errata,” J. Opt. Soc. Am. A 12, 2212 (1995).
  6. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.
  7. J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).
  8. J. Schwider, “Phase shifting interferometry: reference phase error reduction,” Appl. Opt. 28, 3889–3892 (1989).
  9. J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), postdeadline paper PD-4.
  10. P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

1995 (1)

1991 (1)

1989 (1)

1988 (1)

1984 (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Bushan, B.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

Creath, K.

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), postdeadline paper PD-4.

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

de Groot, P.

P. de Groot, “Vibration in phase shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995); “Errata,” J. Opt. Soc. Am. A 12, 2212 (1995).

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

Frankena, H. H.

George, O. E.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

Kinnstätter, K.

Koliopoulos, C. L.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

Lohmann, Q. W.

Schmit, J.

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), postdeadline paper PD-4.

Schwider, J.

Selberg, L. A.

L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).

Smorenburg, C.

Streibl, N.

van Wingerden, J.

Wyant, J. C.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

Appl. Opt. (3)

ASLE Trans. (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, O. E. George, “An optical profilometer for surface characterization of magnetic media,” ASLE Trans. 27, 101–105 (1984).

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

Other (5)

J. Schmit, K. Creath, “Some new error-compensating algorithms for phase-shifting interferometry,” in Optical Fabrication and Testing, Vol. 13 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), postdeadline paper PD-4.

P. de Groot, “Long-wavelength laser diode interferometer for surface flatness measurement,” in Optical Measurements and Sensors for the Process Industries, C. Gorecki, R. W. Preater, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2248, 136–140 (1994).

L. A. Selberg, “Interferometer accuracy and precision,” in Optical Fabrication and Testing, D. R. Campbell, C. W. Johnson, M. Lorenzen, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1400, 24–32 (1990).

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

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992), Chap. 14.

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Figures (6)

Fig. 1
Fig. 1

Expected rms measurement error averaged over all phase angles for the three-bucket algorithm. The vibrational frequencies are normalized to 1/4 of the camera frame rate. The lower curve in each graph is the analytical prediction based on the linear approximation of Ref. 5.

Fig. 2
Fig. 2

Expected measurement error averaged over all phase angles for the Schmit–Creath five-bucket algorithm.

Fig. 3
Fig. 3

Expected measurement error averaged over all phase angles for the seven-bucket algorithm.

Fig. 4
Fig. 4

Measurement error as a function of vibrational frequency for the three-, five-, and seven-bucket algorithms and a vibrational amplitude of 1/2 fringe. Although there are significant differences between the algorithms below a normalized frequency of one, all fail catastrophically above this frequency.

Fig. 5
Fig. 5

Experimental system for verifying the numerical simulation. The waveform generator and PZT provide controlled mechanical vibrations.

Fig. 6
Fig. 6

Comparison of experimental and numerical simulations of vibrational errors in the seven-bucket algorithm.

Equations (34)

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g ( θ , t ) = Q { 1 + V cos [ θ + ϕ ( t ) ] } ,
ϕ ( t ) = 2 π ν 0 t ,
τ = β 2 π ν 0 ,
g ¯ j = 1 τ τ / 2 + τ / 2 g ( θ , t j + t ' ) d t ' ,
t j = j τ .
φ = tan 1 ( T ) + const ,
T = j = 0 J 1 s j g ¯ j / j = 0 J 1 c j g ¯ j
T = g ¯ 3 g ¯ 2 g ¯ 1 g ¯ 2 .
g ( θ , t ) = Q { 1 + V cos [ ϕ ( t ) + θ n ( t ) ] } .
n ( t ) = A cos ( 2 π ν t + α ) .
E = 1 2 π ( π π π π { Δ φ ( θ , α ) ave [ Δ φ ( α ) ] } 2 d θ d α ) 1 / 2 ,
ave [ Δ φ ( α ) ] = 1 2 π π π Δ φ ( θ , α ) d θ .
g ( θ , α , t ) = Q { 1 + V cos [ ϕ ( t ) + θ A cos ( 2 π ν t + α ) ] } .
g ¯ j ( θ , α ) = n = 0 N 1 g ( θ , α , t j + t n ) ,
t n = ( n N 1 2 ) τ N .
Δ φ ( θ , α ) = φ ( θ , α ) θ ,
φ = tan 1 [ T ( θ , α ) ] + const ,
Τ ( θ , α ) = j = 0 J 1 s j g ¯ j ( θ , α ) / j = 0 J 1 c j g ¯ j ( θ , α ) .
E = ( 1 K 1 M k = 0 K 1 m = 0 M 1 { Δ φ ( α k , θ m ) ave [ Δ φ ( α k ) ] } 2 ) 1 / 2 ,
ave [ Δ φ ( α k ) ] = 1 M m = 0 M 1 Δ φ ( α k , θ m ) .
T = ( g ¯ 5 g ¯ 1 ) + 4 ( g ¯ 2 g ¯ 4 ) ( g ¯ 1 + g ¯ 5 ) + 2 ( g ¯ 2 + g ¯ 4 ) 6 g ¯ 3 .
T = ( g ¯ 0 g ¯ 6 ) 7 ( g ¯ 2 g ¯ 4 ) 4 ( g ¯ 1 + g ¯ 5 ) 8 g ¯ 3 .
E ( ν ) = ½ A | P 1 ( ν ) + P 2 ( ν ) | ,
P 1 ( ν ) = ¼ { F C * ( ν + 1 ) F C * ( ν 1 ) + i [ F S * ( ν + 1 ) F S * ( ν 1 ) ] } ,
P 2 ( ν ) = ¼ { F S * ( ν + 1 ) F S * ( ν 1 ) i [ F C * ( ν + 1 ) F C * ( ν 1 ) ] } ,
E ( ν ) = A 2 8 | i [ F S * ( ν + 1 ) + F C * ( ν 1 ) ] [ F S * ( ν 1 ) + F C * ( ν + 1 ) ] | .
θ = tan 1 ( j = 0 J 1 s j g ¯ j / j = 0 J 1 c j g ¯ j )
F S ( ν ˆ ) = H S ( ν ˆ ) B ( ν ˆ ) , F C ( ν ˆ ) = H C ( ν ˆ ) B ( ν ˆ ) ,
H S ( ν ˆ ) = 1 q j s j exp ( i ϕ j ν ˆ ) , H C ( ν ˆ ) = 1 q j c j exp ( i ϕ j ν ˆ ) ,
q = j s j sin ( ϕ j ) = j c j cos ( ϕ j ) ,
B ν = sin ( ν β / 2 ) ν sin ( β / 2 ) .
E 3 ( ν ) = A 2 2 sin ( ν π 4 ) × [ ( B ν + 1 2 + B ν 1 2 ) + ( B ν + 1 2 B ν 1 2 ) sin ( ν π 2 ) ] 1 / 2 ,
E 5 ( ν ) = A 2 64 | ( B ν + 1 i B ν 1 ) × [ 8 cos ( ν π 2 ) 2 cos ( ν π ) 6 ] × ( B ν + 1 + i B ν 1 ) [ 4 sin ( ν π 2 ) + 2 sin ( ν π ) ] | ,
E 7 ( ν ) = A [ 1 2 ( B ν + 1 2 + B ν 1 2 ) ] 1 / 2 | 1 2 cos ( ν π 2 ) sin 2 ( ν π 4 ) + 1 32 [ cos ( 3 ν π 2 ) cos ( ν π 2 ) ] | .

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