Abstract

Phase errors in a Fizeau phase-shifting interferometer caused by multiple-reflected beams from a retroreflective optics, such as a corner cube and a right-angle prism, are studied. Single- and double-pass configurations are presented, and their measurement results are compared. An attenuator is not needed in a double-pass configuration because light is reflected by the retroreflective optics twice and the reference surface once and hence the intensities match. It is more accurate to test a corner cube or a right-angle prism in a double-pass configuration than in a single-pass configuration. Simulations and experimental results are presented.

© 1993 Optical Society of America

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References

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  1. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  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, 3421–3432 (1983).
    [CrossRef] [PubMed]
  3. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26, 2506–2507 (1987).
    [CrossRef] [PubMed]
  4. C. Ai, J. C. Wyant, “Effect of spurious reflection on phase shift interferometry,” Appl. Opt. 27, 3039–3045 (1988).
    [CrossRef] [PubMed]
  5. G. Bonsch, H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).
  6. R. A. Nicolaus, “Evaluation of Fizeau interferences by phase-shifting interferometry,” Optik 87, 23–26 (1991).
  7. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometry with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  8. In the right-hand side the main interference pattern has a many tilt fringes because of the term D; i.e., the phase of D term varies over the pupil. On the other hand, the phase of term F is equal to a constant over the pupil. Therefore the phase error approximately equals ±sin−1(F/D). For R1 = 4% and R2 = 50%, phase error is ±0.08λ.
  9. P. Hariharan, “Digital phase-stepping interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  10. D. A. Thomas, J. C. Wyant, “Determination of the dihedral angle errors of a corner cube from its Twyman–Green interferogram,” J. Opt. Soc. Am. 67, 467–472 (1977).
    [CrossRef]
  11. C. Ai, K. L. Smith, “Accurate measurement of the dihedral angle of a corner cube,” Appl. Opt. 31, 519–527 (1992).
    [CrossRef] [PubMed]

1992 (1)

1991 (1)

R. A. Nicolaus, “Evaluation of Fizeau interferences by phase-shifting interferometry,” Optik 87, 23–26 (1991).

1989 (1)

G. Bonsch, H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

1988 (1)

1987 (2)

1983 (1)

1977 (1)

1975 (1)

1974 (1)

Ai, C.

Bohme, H.

G. Bonsch, H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

Bonsch, G.

G. Bonsch, H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Elssner, K.-E.

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Merkel, K.

Nicolaus, R. A.

R. A. Nicolaus, “Evaluation of Fizeau interferences by phase-shifting interferometry,” Optik 87, 23–26 (1991).

Rosenfeld, D. P.

Schwider, J.

Smith, K. L.

Spolaczyk, R.

Thomas, D. A.

White, A. D.

Wyant, J. C.

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

Optik (2)

G. Bonsch, H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik 82, 161–164 (1989).

R. A. Nicolaus, “Evaluation of Fizeau interferences by phase-shifting interferometry,” Optik 87, 23–26 (1991).

Other (1)

In the right-hand side the main interference pattern has a many tilt fringes because of the term D; i.e., the phase of D term varies over the pupil. On the other hand, the phase of term F is equal to a constant over the pupil. Therefore the phase error approximately equals ±sin−1(F/D). For R1 = 4% and R2 = 50%, phase error is ±0.08λ.

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Figures (9)

Fig. 1
Fig. 1

Multiple reflection between the reference (top) and the test (bottom) surfaces, r and r′ are the coefficients of the reflection of the two surfaces (r = r1 and r′ = r2).

Fig. 2
Fig. 2

(a) Intensity profile and (b) phase error for the initial phase from 0° to 360°. The solid curve is for R1 = 4% and R2 = 50% and the dashed curve is for R1 = 4% and R2 = 4%. The intensities are derived from Eq. (1), and the dotted curves are the best-fit sinusoidal curves.

Fig. 3
Fig. 3

Schematic of a Fizeau interferometer. R and T represent the reference and the test surfaces, respectively, and the three beams reflected by them are r, t, and s. These notations are also used in Figs. 4 and 5.

Fig. 4
Fig. 4

(a) Lower portion of test beam t reflected by a right-angle prism, (b) Multiple reflections in the single-pass configuration. s is the secondary reflection by the prism, and r is the reference beam.

Fig. 5
Fig. 5

Double-pass configuration. The lower portion of the reference surface is blocked, r is the reference beam, and t is the test beam, which is reflected by the prism twice and by the reference surface once.

Fig. 6
Fig. 6

A right-angle prism tested in a single-pass configuration with an attenuator of T = 70%, where the reference surface is not tilted: (a) interferogram (b) x profile of OPD along the middle of the pupil. The spikes around x = 0 are due to diffraction.

Fig. 7
Fig. 7

Same as Fig. 6 but the reference surface is tilted in horizontal direction such that in the right-hand side the main interference fringes are vertical and the ghost fringes are horizontal, and vice versa in the left-hand side.

Fig. 8
Fig. 8

Isometric contours of the wave-front deformation of a corner cube, tested in a single-pass configuration with an attenuator of T = 20%, (a) before and (b) after the odd aberrations of the resultant OPD's are subtracted. In (a) the contours are asymmetric with respect to the center. (Interval, 0.05λ.)

Fig. 9
Fig. 9

Isometric contours of the resultant OPD of a corner cube tested in a double-pass configuration without using an attenuator. (Interval, 0.05λ.)

Tables (2)

Tables Icon

Table 1 Characteristics for Different Reflectances of the Test Sample

Tables Icon

Table 2 Values of C, D, E, and F

Equations (20)

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I r / I i = [ ( r 1 r 2 ) 2 + 4 r 1 r 2 sin 2 ( δ / 2 ) ] / [ ( 1 r 1 r 2 ) 2 + 4 r 1 r 2 sin 2 ( δ / 2 ) ] ,
phase = tan 1 [ ( I 3 I 1 ) / ( I 0 I 2 ) ]
modulation = ac / dc = { 0.5 [ ( I 3 I 1 ) 2 + ( I 0 I 2 ) 2 ] 1 / 2 } / { ( I 0 + I 1 + I 2 + I 3 ) / 4 } ,
r 1 exp i [ ϕ i ( x ) + 2 n ϕ r ( x ) + η ] ,
( 1 + r 1 ) ( r 2 ) ( 1 r 1 ) × exp i [ ϕ i ( x ) + 2 ( n 1 ) ϕ r ( x ) + 2 ϕ t ( x ) ] ,
( 1 + r 1 ) ( r 2 ) ( 1 r 1 ) r 1 r 2 exp i [ ϕ i ( x ) + 2 ( n 1 ) ϕ r ( x ) + 4 ϕ t ( x ) 2 ϕ r ( x ) η ) ] ,
r 1 exp i [ ϕ i ( x ) + 2 ϕ r ( x ) + η ] from the reference surface , ( 1 R 1 ) r 2 exp i [ ϕ i ( x ) + 2 ϕ t ( x ) ] from the first reflection of the test surface , ( 1 R 1 ) r 2 ( r 1 r 2 ) exp i [ ϕ i ( x ) + 4 ϕ t ( x ) 2 ϕ r ( x ) η ] from the secondary reflection .
I = C + D cos [ 2 ϕ t ( x ) 2 ϕ r ( x ) η ] + E cos [ 4 ϕ t ( x ) 4 ϕ r ( x ) 2 η ] + F cos [ 2 ϕ t ( x ) 2 ϕ r ( x ) η ] ,
C = r 1 2 + ( 1 R 1 ) 2 r 2 2 + ( 1 R 1 ) 2 r 2 4 r 1 2 , D = 2 ( 1 R 1 ) ( r 1 r 2 ) , E = 2 ( 1 R 1 ) ( r 1 r 2 ) 2 , F = 2 ( 1 R 1 ) 2 r 2 2 ( r 1 r 2 ) .
ϕ out ( x ) = ϕ in ( x ) + ϕ p ( x ) + ϕ p ( x ) , if x > 0 ,
ϕ out ( x ) = O [ ϕ in ( x ) + ϕ p ( x ) + ϕ p ( x ) ] = ϕ in ( x ) + ϕ p ( x ) + ϕ p ( x )
ϕ t , e ( x ) = ϕ p ( x ) + ϕ p ( x ) .
ϕ in ( x ) = ϕ in ( x ) + ϕ t , e ( x ) 2 ϕ r ( x ) η ,
ϕ out ( x ) = O { ϕ in ( x ) + ϕ t , e ( x ) } = ϕ in ( x ) + 2 ϕ t , e ( x ) 2 ϕ r ( x ) η .
r 1 exp i [ ϕ i ( x ) + 2 ϕ r ( x ) + η ] from the reference surface , ( 1 R 1 ) r 2 exp i [ ϕ i ( x ) + ϕ t , e ( x ) ] from the first reflection of the test surface , ( 1 R 1 ) r 2 2 r 1 exp i [ ϕ i ( x ) + 2 ϕ t , e ( x ) 2 ϕ r ( x ) η ] from the secondary reflection .
I = C + D cos [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r ( x ) η ] + E cos [ 2 ϕ t , e ( x ) 4 ϕ r , e ( x ) 2 η ] + F cos [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r ( x ) η ] ,
I 3 I 1 = 2 D sin [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r , e ( x ) 2 ϕ r , o ( x ) ] + 2 F sin [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r , e ( x ) + 2 ϕ r , o ( x ) ] , I 0 I 2 = 2 D cos [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r , e ( x ) 2 ϕ r , o ( x ) ] + 2 F cos [ 2 ϕ i , o ( x ) + ϕ t , e ( x ) 2 ϕ r , e ( x ) + 2 ϕ r , o ( x ) ] .
r 1 exp i [ ϕ i ( x ) + 2 ϕ r ( x ) + η ] from the reference surface , ( 1 R 1 ) r 2 2 r 1 exp i [ ϕ i ( x ) + 2 ϕ t , e ( x ) 2 ϕ r ( x ) η ) ] from the test surface ,
I = r 1 2 + ( 1 R 1 ) 2 r 2 4 r 1 2 2 ( 1 R 1 ) r 2 2 r 1 2 × cos [ 2 ϕ t , e ( x ) 4 ϕ r , e ( x ) 2 η ] .
tan 1 [ ( I 3 I 1 ) / ( I 0 I 2 ) ] = 2 [ ϕ t , e ( x ) 2 ϕ r , e ( x ) ] .

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