Abstract

This paper describes some practical methods to calibrate the phase shifter in phase-shifting interferometry (PSI). The phase shifter used in the experiment is a piezoelectric transducer (PZT) that has a nonlinearity of <1%. Using the quantitative method described in this paper, the repeatability in the measurement of the phase-shifting angle is ~0.046° rms, and the 3σ value is 0.139°. A calibration-insensitive phase calculation algorithm is discussed and compared with other synchronous detection equations (e.g., the three-bucket or the four-bucket method). Experimental results verify the calibration-insensitive mechanism of the self-calibrating algorithm.

© 1985 Optical Society of America

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References

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  1. J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978).
  2. C. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1981).
  3. 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]
  4. J. B. Hayes, “Linear Methods of Computer Controlled Optical Figuring,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1984).
  5. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13 (1966).
    [CrossRef]

1983 (1)

1966 (1)

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

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978).

Burow, R.

Carré, P.

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

Elssner, K. E.

Grzanna, J.

Hayes, J. B.

J. B. Hayes, “Linear Methods of Computer Controlled Optical Figuring,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1984).

Koliopoulos, C.

C. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1981).

Merkel, K.

Schwider, J.

Spolaczyk, R.

Appl. Opt. (1)

Metrologia (1)

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

Other (3)

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978).

C. Koliopoulos, “Interferometric Optical Phase Measurement Techniques,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1981).

J. B. Hayes, “Linear Methods of Computer Controlled Optical Figuring,” Ph.D. Dissertation, Optical Sciences Center, U. Arizona (1984).

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Figures (4)

Fig. 1
Fig. 1

Phase error vs phase for a phase step of 88° rather than 90°.

Fig. 2
Fig. 2

Display of five frames of phase-shifted fringe patterns for PZT calibration: (a) β = 90°, (b) β = 96°.

Fig. 3
Fig. 3

Averaged rms phase error vs different phase-shifting angles (β) for three different phase calculation algorithms.

Fig. 4
Fig. 4

Surface roughness measurement vs different phase-shifting angle (β) for both the four-bucket algorithm and the phase-shifting angle compensated algorithm.

Equations (24)

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I ( x , y ) = I 0 { 1 + γ cos [ ϕ ( x , y ) + β ] } .
A ( x , y ) = 2 . 5 β 1 . 5 β I 0 { 1 + γ cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
B ( x , y ) = 1 . 5 β 0 . 5 β I 0 { 1 + γ cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
D ( x , y ) = + 0 . 5 β + 1 . 5 β I 0 { 1 + γ cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
E ( x , y ) = + 1 . 5 β + 2 . 5 β I 0 { 1 + γ cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) .
A ( x , y ) = I 0 ( 1 + γ { cos ϕ [ sin ( 5 β 2 ) sin ( 3 β 2 ) ] + sin ϕ [ cos ( 3 β 2 ) cos ( 5 β 2 ) ] } ) ,
B ( x , y ) = I 0 ( 1 + γ { cos ϕ [ sin ( 3 β 2 ) sin ( β 2 ) ] + sin ϕ ( β 2 ) cos ( 3 β 2 ) ] } ) ,
D ( x , y ) = I 0 ( 1 + γ { cos ϕ [ sin ( 3 β 2 ) sin ( β 2 ) ] + sin ϕ [ cos ( 3 β 2 ) cos ( β 2 ) ] } ) .
E ( x , y ) = I 0 ( 1 + γ { cos ϕ [ sin ( 5 β 2 ) sin ( 3 β 2 ) ] + sin ϕ [ cos ( 5 β 2 ) cos ( 3 β 2 ) ] } ) .
β = cos 1 [ ( 1 2 ) ( A E B D ) ] .
A ( x , y ) = 2 β β { I 1 + I 2 cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
B ( x , y ) = β 0 { I 1 + I 2 cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
C ( x , y ) = 0 + β { I 1 + I 2 cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) ,
D ( x , y ) = + β + 2 β { I 1 + I 2 cos [ ϕ ( x , y ) + ψ ( t ) ] } d ψ ( t ) .
A ( x , y ) = I 1 + I 2 { cos ϕ [ sin ( 2 β ) sin β ] } + sin ϕ [ cos β cos ( 2 β ) ] } ,
B ( x , y ) = I 1 + I 2 [ cos ϕ sin β + sin ϕ ( 1 cos β ) ] ,
C ( x , y ) = I 1 + I 2 [ cos ϕ sin β + sin ϕ ( cos β 1 ) ] ,
D ( x , y ) = I 1 + I 2 { cos ϕ [ sin ( 2 β ) sin β ] } + sin ϕ [ cos ( 2 β ) cos β ] } .
U = B C , M = T W , V = A D , N = 3 U V , W = A + D , P = U + V , T = B + C , Q = V U .
β = cos 1 ( Q 2 U ) ,
β = tan 1 ( N P Q ) .
ϕ ( x , y ) = tan 1 [ ( Q M ) tan β ] .
ϕ ( x , y ) = tan 1 ( N P M ) ,
ϕ ( x , y ) = tan 1 [ [ ( B C ) + ( A D ) ] [ 3 ( B C ) ( A D ) ] ( B + C ) ( A + D ) ] .

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