Abstract

The accuracy of phase shifting interferometers is impaired by mechanical drifts and vibrations, intensity variations, nonlinearities of the photoelectric detection device, and, most seriously, by inaccuracies of the reference phase shifter. The phase shifting procedure enables the detection of most of the errors listed above by a special Lissajous display technique described here. Furthermore, it is possible to correct phase shifter inaccuracies by using an iterative process relying solely on the interference pattern itself and the Fourier sums used in phase shifting interferometry.

© 1988 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 (1974).
    [CrossRef] [PubMed]
  2. J. C. Wyant, “Interferometric Optical Metrology: Basic Principles and New Systems,” Laser Focus (May1982), p. 65.
  3. J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Elect. Opt. (Nov.1985), p. 118.
  4. B. S. Fritz, “Absolute Calibration of an Optical Flat,” Opt. Eng. 23, 379 (1984).
    [CrossRef]
  5. J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
    [CrossRef]
  6. 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 (1983).
    [CrossRef] [PubMed]
  7. C. L. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Thesis, U. Arizona (1981).
  8. P. Carre, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13 (1966).
    [CrossRef]
  9. Y.-Y. Cheng, J. C. Wyant, “Phase Shifter Calibration in Phase-Shifting Interferometry,” Appl. Opt. 24, 3049 (1985).
    [CrossRef] [PubMed]
  10. K. A. Stetson, W. R. Brohinsky, “Electrooptic Holography and Its Applications to Hologram Interferometry,” Appl. Opt. 24, 3631 (1985).
    [CrossRef] [PubMed]
  11. J. J. Snyder, “Algorithm for Fast Digital Analysis of Interference Fringes,” Appl. Opt.’ 19, 1223 (1980).
    [CrossRef] [PubMed]

1986 (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

1985 (3)

1984 (1)

B. S. Fritz, “Absolute Calibration of an Optical Flat,” Opt. Eng. 23, 379 (1984).
[CrossRef]

1983 (1)

1982 (1)

J. C. Wyant, “Interferometric Optical Metrology: Basic Principles and New Systems,” Laser Focus (May1982), p. 65.

1980 (1)

1974 (1)

1966 (1)

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

Basila, D.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Bushan, B.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

Carre, P.

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

Cheng, Y.-Y.

Creath, K.

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Elect. Opt. (Nov.1985), p. 118.

Elssner, K.-E.

Fritz, B. S.

B. S. Fritz, “Absolute Calibration of an Optical Flat,” Opt. Eng. 23, 379 (1984).
[CrossRef]

Gallagher, J. E.

Grzanna, J.

Herriott, D. R.

Koliopoulos, C. L.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

C. L. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Thesis, U. Arizona (1981).

Merkel, K.

Rosenfeld, D. P.

Schwider, J.

Snyder, J. J.

Spolaczyk, R.

Stetson, K. A.

White, A. D.

Wyant, J. C.

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Elect. Opt. (Nov.1985), p. 118.

Y.-Y. Cheng, J. C. Wyant, “Phase Shifter Calibration in Phase-Shifting Interferometry,” Appl. Opt. 24, 3049 (1985).
[CrossRef] [PubMed]

J. C. Wyant, “Interferometric Optical Metrology: Basic Principles and New Systems,” Laser Focus (May1982), p. 65.

Appl. Opt. (5)

J. Trib. (1)

J. C. Wyant, C. L. Koliopoulos, B. Bushan, D. Basila, “Development of a 3-D Noncontact Digital Optical Profiler,” J. Trib. 108, 1 (1986).
[CrossRef]

Laser Focus (1)

J. C. Wyant, “Interferometric Optical Metrology: Basic Principles and New Systems,” Laser Focus (May1982), p. 65.

Laser Focus/Elect. Opt. (1)

J. C. Wyant, K. Creath, “Recent Advances in Interferometric Optical Testing,” Laser Focus/Elect. Opt. (Nov.1985), p. 118.

Metrologia (1)

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

Opt. Eng. (1)

B. S. Fritz, “Absolute Calibration of an Optical Flat,” Opt. Eng. 23, 379 (1984).
[CrossRef]

Other (1)

C. L. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Thesis, U. Arizona (1981).

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Figures (8)

Fig. 1
Fig. 1

Schematic of a planeness Fizeau interferometer used in the experiments.

Fig. 2
Fig. 2

Scan line from the CCD array.

Fig. 3
Fig. 3

Lissajous figure for an almost ideal adjustment of all parameters of the interferometer and the PSI process. The number of phase steps is R = 40. The central spot indicates the averages due to Eq. (8).

Fig. 4
Fig. 4

Deformations of the Lissajous figure due to nonlinearities of the photoelectric detector (R = 64).

Fig. 5
Fig. 5

Intensity variations due to mode shifting of an unstabilized He–Ne laser during the warmup time (R = 40).

Fig. 6
Fig. 6

High frequency mechanical vibrations. Note the loss in contrast of the interference pattern. The irregularities of the Lissajous figure are due to stochastic phase shifts in the interferometer (R = 64).

Fig. 7
Fig. 7

(a) Lissajous figure for an uncalibrated phase shifter but with linear slope of the piezovoltage (R = 32). (b) The same as Fig. 6 but after one iteration cycle. Note the closing of the gap in the Lissajous figure. (c) Display of the -curves for the runs documented in (a) lower curve and (b) upper curve. The ordinate values are given in radians and the abscissas values represent the phase step number.

Fig. 8
Fig. 8

Demonstration of the convergence of the phase correcting algorithm underlying Eqs. (28)(30) (R = 40): (a) starting sequence of voltages is rather unsatisfactory, (b) the same after one iteration cycle, (c) the same after five iteration cycles, (d) the same after eight iteration cycles, (e) comparison of -curves for runs (a)–(d), (f) same as (e), but for the driving voltages of the PZT.

Equations (40)

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I ( x , y ) = I 0 ( x , y ) { 1 + V ( x , y ) cos [ Φ ( x , y ) - φ ] } ,
φ r = ( r - 1 ) 2 π / R , with r = 1 , 2 , , R .
I r = I 0 + I 0 V cos ( Φ - φ r ) = I 0 + I 0 V cos Φ cos φ r + I 0 V sin Φ sin φ r .
2 r = 1 R I r cos φ r = R I 0 V cos Φ , 2 r = 1 R I r sin φ r = R I 0 V sin Φ , r = 1 R I r = R I 0 .
tan Φ = ( r = 1 R I r sin φ r ) / ( r = 1 R I r cos φ r ) .
ψ r = φ r + ɛ r .
I 1 r = I 10 [ 1 + V 1 cos ( Φ 1 - ψ r ) ] , I 2 r = I 20 [ 1 + V 2 cos ( Φ 2 - ψ r ) ] ,             ( r = 1 , 2 , R ) ,
1 / R r = 1 R U 1 r ,             1 / R r = 1 R U 2 r ,
I r = I 0 { 1 + V cos [ Φ - ψ r + v ( t ) ] } .
v ( t ) = 2 π / λ n = 0 a n sin ( n Ω t + α n ) ,
U r / U 0 = ( 1 / τ 0 τ d t + V / τ { cos ( Φ - ψ r ) 0 τ cos [ k a sin ( ω t - α ) ] d t - sin ( Φ - ψ r ) 0 τ sin [ k a sin ( ω t - α ) ] d t } ) .
0 τ cos [ k a sin ( ω t - α ) ] d t = 0 τ J 0 ( k a ) d t + 2 m = 1 0 τ J 2 m ( k a ) cos 2 m ( ω t - α ) d t , 0 τ sin [ k a sin ( ω t - α ) ] d t = 2 m = 1 J 2 m + 1 ( k a ) sin ( 2 m + 1 ) ( ω t - α ) d t .
U r = U 0 [ 1 + V J 0 ( k a ) cos ( Φ - φ ) ] .
U 1 r / U 10 = 1 + V 1 [ cos ( Φ 1 - ψ r ) J 0 ( a k ) - 2 sin ( Φ 1 - ψ r ) J 1 ( a k ) 0 τ sin ( ω t - α ) d t ] U 2 r / U 20 = 1 + V 1 [ cos ( Φ 2 - ψ r ) J 0 ( a k ) - 2 sin ( Φ 2 - ψ r ) J 1 ( a k ) 0 τ sin ( ω t - α ) d t ] ,
U = U 0 { 1 + V J 0 2 + 4 J 1 2 sin 2 ( ω t - α ) cos ( Φ - ψ + β ) } , tan β = 2 J 1 ( a k ) sin ( ω t - α ) / J 0 ( a k ) .
J 0 ( a k ) + 2 J 1 2 ( a k ) sin 2 ( ω t - α ) / J 0 ( a k ) .
σ = 2 J 1 2 / ( J 0 2 + J 1 2 ) .
σ ( a k ) 2 / 2.
tan β max = 2 J 1 / J 0 a k .
U r = a I r .
U 1 r = U 01 [ 1 + V 1 cos ( Φ 1 - ψ r ) ] , U 2 r = U 02 [ 1 + V 2 cos ( Φ 2 - ψ r ) ] ,
arctan ( r = 1 R U r sin φ r / r = 1 R U r cos φ r ) = Φ ( mod π )
U 01 = 1 / R r = 1 R U 1 r , and U 02 = 1 / R r = 1 R U 2 r .
A r = V 1 cos ( Φ 1 - ψ r ) ; B r = V 2 cos ( Φ 2 - ψ r ) .
A r = V 1 cos ( Φ 1 - ψ r ) ;             B r = - V 2 sin ( Φ 1 + χ - ψ r ) .
V = ( 2 / r = 1 R U r ) ( r = 1 R U r sin φ r ) 2 + ( r = 1 R U r cos φ r ) 2 .
B r V 1 / A r V 2 = - sin ( Φ 1 + χ - ψ r ) / cos ( Φ 1 - ψ r ) ,
ψ r ( mod π ) = Φ 1 + arctan [ tan χ + ( 1 / cos χ ) B r V 1 / A r V 2 ] .
ψ r - φ r = ɛ r ,
q = 1.2 , i . e . , ψ ( 1 ) = ψ ( 0 ) - ɛ / 1.2.
v r ( 1 ) = v r ( 0 ) - v r ( ɛ ) .
Δ Φ = arctan ( r = 1 R ɛ r - C cos 2 Φ - S sin 2 Φ ) / ( R - C sin 2 Φ + S cos 2 Φ ) ,
C = r = 1 R ɛ r cos 2 φ r and S = r = 1 R ɛ r sin 2 φ r .
ɛ r = 1 / M m = 1 M ɛ r m .
U = a I + b I 2 + c I 3 + d I 4 +
I 1 = p + q cos Φ ,             I 3 = p - q cos Φ , I 2 = p + q sin Φ ,             I 4 = p - q sin Φ .
tan Φ ^ = a I 2 + b I 2 2 + c I 2 3 + d I 2 4 + ) - ( a I 4 + b I 4 2 + c I 4 3 + d I 4 4 + ) ( a I 1 + b I 1 2 + c I 1 3 + d I 1 4 + ) - ( a I 3 + b I 3 2 + c I 3 3 + d I 3 4 + ) .
tan Φ ^ = ( k 1 sin Φ - k 3 sin 3 Φ ) / ( k 1 cos Φ + k 3 cos 3 Φ ) ,
k 1 = a 2 q + b 4 p q + c ( 6 p 2 q + 3 / 2 q 3 ) + d ( 8 p 3 q + 6 p q 3 ) , k 3 = c q 3 / 2 + d 2 p q 3 .
r = 1 R U r , r = 1 R U r sin φ r , r = 1 R U r cos φ r , r = 1 R U r sin 2 φ r , r = 1 R U r cos 2 φ r , etc .

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