Abstract

The phase errors caused by spurious reflection in Twyman-Green and Fizeau interferometers are studied. A practical algorithm effectively eliminating the error is presented. Two other algorithms are reviewed, and the results obtained using the three algorithms are compared.

© 1988 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), p. 409.
  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 (1983).
    [CrossRef] [PubMed]
  3. J. C. Wyant, “Review of Phase Shifting Interferometry,” in Technical Digest, OSA Spring Meeting (Optical Society of America, Washington, DC, 1984).
  4. C. Ai, J. C. Wyant, “Effect of Piezoelectric Transducer Nonlinearity on Phase Shift Interferometry,” Appl. Opt. 26, 1112 (1987).
    [CrossRef] [PubMed]
  5. C. Ai, “Phase Measurement Accuracy Limitation in Phase Shifting Interferometry,” Ph.D. Dissertation, U. Arizona, Tucson (1987).
  6. J. C. Wyant, “Use of an ac Heterodyne Lateral Shear Interferometer with Real-time Wavefront Correction Systems,” Appl. Opt. 14, 2622 (1975).
    [CrossRef] [PubMed]
  7. P. Hariharan, “Digital Phase-Stepping Interferometry: Effects of Multiply Reflected Beams,” Appl. Opt. 26, 2506 (1987).
    [CrossRef] [PubMed]

1987 (2)

1983 (1)

1975 (1)

Ai, C.

C. Ai, J. C. Wyant, “Effect of Piezoelectric Transducer Nonlinearity on Phase Shift Interferometry,” Appl. Opt. 26, 1112 (1987).
[CrossRef] [PubMed]

C. Ai, “Phase Measurement Accuracy Limitation in Phase Shifting Interferometry,” Ph.D. Dissertation, U. Arizona, Tucson (1987).

Bruning, J. H.

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.

Burow, R.

Elssner, K.-E.

Grzanna, J.

Hariharan, P.

Merkel, K.

Schwider, J.

Spolaczyk, R.

Wyant, J. C.

Appl. Opt. (4)

Other (3)

J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.

C. Ai, “Phase Measurement Accuracy Limitation in Phase Shifting Interferometry,” Ph.D. Dissertation, U. Arizona, Tucson (1987).

J. C. Wyant, “Review of Phase Shifting Interferometry,” in Technical Digest, OSA Spring Meeting (Optical Society of America, Washington, DC, 1984).

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Figures (6)

Fig. 1
Fig. 1

Phase error due to the extraneous beam. For a given Θ, the error is a function of the phase Φ and has a frequency equal to the spatial frequency of the interference fringe. The solid curve is for Θ = 0°, and the dashed curve is for Θ = 90°

Fig. 2
Fig. 2

Schematic of the relation of the resulting phase error to the test beam and the extraneous beam. The maximum error occurs when the phasor is perpendicular to the sum of and .

Fig. 3
Fig. 3

(a) Three-beam interference of the test, reference, and extraneous beams. (b) Two-beam interference of the reference and extraneous beams. (c) Two-beam interference of the test and extraneous beams. The vertical fringes in (a) are due to the spurious reflection.

Fig. 4
Fig. 4

Resulting phase of three-beam interference obtained using (a) the simple arctangent formula, (b) the new algorithm, and (c) Schwider’s algorithm. It should be noted that the phase error of (a) has the same pattern as the intensity distribution in Fig. 3(c).

Fig. 5
Fig. 5

Cross sections of the resulting phases of Fig. 4. (a), (b), and (c) are the cross sections of Figs. 4(a), (b), and (c), respectively.

Fig. 6
Fig. 6

Schematic of the interference due to scattered light. B and C are the test and reference beams. Their scattered beams, b and c, are from point Q on the imaging lens.

Equations (28)

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A t · exp ( i ϕ t ) from the test arm without extraneous beam , A r · exp ( i ϕ r ) from reference arm without extraneous beam , E · exp ( i η ) from extraneous beam ,
I = A t 2 + A r 2 + E 2 + 2 A t · E · cos ( ϕ t - η ) + 2 A t · A r · cos ( ϕ t - ϕ r ) + 2 A r · E · cos ( η - ϕ r ) .
I n = constant + 2 A t · A r · cos ( ϕ t - ϕ r - n · 90 ° ) + 2 A r · E · cos ( η - ϕ r - n · 90 ° ) ,
ϕ t - ϕ r = tan - 1 I 3 - I 1 I 0 - I 2
Φ = tan - 1 A t · sin ( Φ ) + E · sin ( Θ ) A t · cos ( Φ ) + E · cos ( Θ ) ,
Φ - Φ = tan - 1 [ E · sin ( Θ - Φ ) A t + E · cos ( Θ - Φ ) ] ,
A t · cos ( Θ - Φ ) - E = A t · cos ( η - ϕ t ) - E = 0.
Φ ( x ) = ϕ t 0 + 2 π · f t · x - ϕ r ,
Θ ( x ) = η 0 ± 2 π · f e · x - ϕ r ,
Φ = tan - 1 ( I 1 - I 3 ) - ( I 1 - I 3 ) ( I 0 - I 2 ) - ( I 0 - I 2 ) .
P 4 A t · A r · exp i ( ϕ t - ϕ r ) ,
E = 4 A r · E · exp i ( η - ϕ r ) .
I 0 = A t 2 + A r 2 + E 2 + 2 A t · E · cos ( η - ϕ t ) + Re [ 0.5 · ( P + E ) ] ,
I 1 - I 3 = Im ( P + E ) ,
I 0 - I 2 = Re ( P + E ) ,
( I 1 - I 3 ) = Im [ - P + E ] ,
( I 1 - I 3 ) - ( I 1 - I 3 ) = Im [ ( P + E ) - ( - P + E ) ] .
( I 0 - I 2 ) - ( I 0 - I 2 ) = Re [ ( P + E ) - ( - P + E ) ] .
( I 0 - I 2 ) - ( I 0 - I 2 ) = Re [ ( P + E ) - ( E ) ] ,
( I 1 - I 3 ) - ( I 1 - I 3 ) = Im [ ( P + E ) - ( E ) ] .
R r · exp ( i ϕ r ) from reference arm , T r t · exp ( i ϕ t ) from test arm without extraneous beam , E T · R * · T · { 1 + R * · T · [ 1 + R * · T · ( 1 + · ) ] } from extraneous beam ,
E T · R * · T + T · R * · T · R * · T ,
I = R 2 + T 2 + E 2 + R * · T + R · T * + ( R * · T · R * · T + R * · T · R * · T · R * · T + c . c . ) + ( T * · T · R * · T + T * · T · R * · T · R * · T · R * · T + c . c . ) .
I = constant + ( R * · T + c . c . ) + ( T * · T · R * · T + c . c . ) + ( R * · T · R * · T + T * · T · R * · T · R * · T + c . c . ) .
B · exp [ i ϕ t ( x ) ] from test arm without extraneous beam , C · exp [ i ϕ r ( x ) ] from reference arm without extraneous beam , b · exp [ i η t ( x ) ] from extraneous beam due to test beam , c · exp [ i η r ( x ) ] from extraneous beam due to reference beam ,
I = [ B · exp ( i ϕ t ) - b · exp ( i η t ) + C · exp ( ϕ r ) - c · exp ( i η r ) ] · [ c . c . ] = B 2 + C 2 + b 2 + c 2 - 2 B · b · cos ( ϕ t - η t ) - 2 C · c · cos ( ϕ r - η r ) + 2 b · c · cos ( η t - η r ) + 2 B · C · cos ( ϕ t - ϕ r ) - 2 B · c · cos ( ϕ t - η r ) - 2 b · C · cos ( η t - ϕ r ) .
I = 2 B · C · cos ( ϕ t - ϕ r ) - 2 B · c · cos ( ϕ t - η r ) - 2 b · C · cos ( η t - ϕ r ) + constant .
I 0 - I 2 = 4 B · C · cos ( ϕ t - ϕ r ) - 4 B · c · cos ( ϕ t + Δ - ϕ r ) - 4 b · C · cos ( η t - ϕ r ) .

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