Abstract

We describe a modified three-flat method. In a Cartesian coordinate system, a flat can be expressed as the sum of even–odd, odd–even, even–even, and odd–odd functions. The even–odd and the odd–even functions of each flat are obtained first, and then the even–even function is calculated. All three functions are exact. The odd–odd function is difficult to obtain. In theory, this function can be solved by rotating the flat 90°, 45°, 22.5°, etc. The components of the Fourier series of this odd–odd function are derived and extracted from each rotation of the flat. A flat is approximated by the sum of the first three functions and the known components of the odd–odd function. In the experiments, the flats are oriented in six configurations by rotating the flats 180°, 90°, and 45° with respect to one another, and six measurements are performed. The exact profiles along every 45° diameter are obtained, and the profile in the area between two adjacent diameters of these diameters is also obtained with some approximation. The theoretical derivation, experiment results, and error analysis are presented.

© 1993 Optical Society of America

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References

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  1. G. Schulz, “Ein interferenzverfahren zur absolute ebnheitsprufung langs beliebiger zntralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  2. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed., (North-Holland, Amsterdam, 1976), Chap. 4.
    [CrossRef]
  3. J. Schwider, “Ein interferenzverfahren zur absolutprufung von planflachennormalen. II,” Opt. Acta 14, 389–400 (1967).
    [CrossRef]
  4. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  5. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  6. C. Ai, H. Albrecht, J. C. Wyant, “Absolute testing of flats using shearing technique,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper WN1.
  7. J. Grzanna, G. Schulz, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992).
    [CrossRef] [PubMed]
  8. W. Primak, “Optical flatness standard II: reduction of interferograms,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 375–381 (1989).
  9. C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Applications, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng., 1776, 73–83 (1992).
  10. It is straightforward that from Eqs. (13), we have M1 − M3 = A − A90°. Therefore, the contribution of B is completely removed, regardless the resulting profile of B, and hence Aoo,2oddθ can be derived. Similarly, (M1)90° − M3 = (Bx)90° − Bx, and the contribution of A is completely canceled, and Boo,2oddθ is obtained. In our previous paper,9 there were eight measurements, and the derivation of Coo,2oddθ was similar to that given above. With the eight-measurement algorithm, because of the subtraction, the contribution of the surface that is not rotated is removed. The measurement error of a surface does not affect the other two surfaces. However, in this six-measurement algorithm, it is not so simple. From Eqs. (13),(M6)90°−M6+(M1)90°−M3=(Cx)90°−Cx+(B)90°+(Bx)90° −B−Bx.(A1)Again the contribution of A is completely deleted. Because only the odd–odd part is unknown, we only consider this part by subtracting the odd–even, even–odd, and even terms from each term in Eq. (A1). Therefore, all B terms of Eq. (A1) become(Boo)90°+[(Boo)x]90°−Boo−(Boo)x=0.(A2)After the subtraction, Eq. (A1) is the same as Eqs. (22). This substration operation is equivalent to applying an odd–odd operation to Eq. (A1) to filter non-odd–odd terms.
  11. D. Malacala, ed., “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, (New York, 1978), 489–505.
  12. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), 349–393.
    [CrossRef]

1992 (1)

1990 (1)

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

1984 (1)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

1967 (2)

G. Schulz, “Ein interferenzverfahren zur absolute ebnheitsprufung langs beliebiger zntralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

J. Schwider, “Ein interferenzverfahren zur absolutprufung von planflachennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

Ai, C.

C. Ai, H. Albrecht, J. C. Wyant, “Absolute testing of flats using shearing technique,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper WN1.

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Applications, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng., 1776, 73–83 (1992).

Albrecht, H.

C. Ai, H. Albrecht, J. C. Wyant, “Absolute testing of flats using shearing technique,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper WN1.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), 349–393.
[CrossRef]

Fritz, B. S.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

Grzanna, J.

J. Grzanna, G. Schulz, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Primak, W.

W. Primak, “Optical flatness standard II: reduction of interferograms,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 375–381 (1989).

Schulz, G.

J. Grzanna, G. Schulz, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992).
[CrossRef] [PubMed]

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

G. Schulz, “Ein interferenzverfahren zur absolute ebnheitsprufung langs beliebiger zntralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed., (North-Holland, Amsterdam, 1976), Chap. 4.
[CrossRef]

Schwider, J.

J. Schwider, “Ein interferenzverfahren zur absolutprufung von planflachennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed., (North-Holland, Amsterdam, 1976), Chap. 4.
[CrossRef]

Wyant, J. C.

C. Ai, H. Albrecht, J. C. Wyant, “Absolute testing of flats using shearing technique,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper WN1.

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Applications, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng., 1776, 73–83 (1992).

Appl. Opt. (1)

Opt. Acta (2)

G. Schulz, “Ein interferenzverfahren zur absolute ebnheitsprufung langs beliebiger zntralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

J. Schwider, “Ein interferenzverfahren zur absolutprufung von planflachennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

Opt. Commun. (1)

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Opt. Eng. (1)

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

Other (7)

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed., (North-Holland, Amsterdam, 1976), Chap. 4.
[CrossRef]

C. Ai, H. Albrecht, J. C. Wyant, “Absolute testing of flats using shearing technique,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper WN1.

W. Primak, “Optical flatness standard II: reduction of interferograms,” in Optical Testing and Metrology II, C. P. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.954, 375–381 (1989).

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Applications, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng., 1776, 73–83 (1992).

It is straightforward that from Eqs. (13), we have M1 − M3 = A − A90°. Therefore, the contribution of B is completely removed, regardless the resulting profile of B, and hence Aoo,2oddθ can be derived. Similarly, (M1)90° − M3 = (Bx)90° − Bx, and the contribution of A is completely canceled, and Boo,2oddθ is obtained. In our previous paper,9 there were eight measurements, and the derivation of Coo,2oddθ was similar to that given above. With the eight-measurement algorithm, because of the subtraction, the contribution of the surface that is not rotated is removed. The measurement error of a surface does not affect the other two surfaces. However, in this six-measurement algorithm, it is not so simple. From Eqs. (13),(M6)90°−M6+(M1)90°−M3=(Cx)90°−Cx+(B)90°+(Bx)90° −B−Bx.(A1)Again the contribution of A is completely deleted. Because only the odd–odd part is unknown, we only consider this part by subtracting the odd–even, even–odd, and even terms from each term in Eq. (A1). Therefore, all B terms of Eq. (A1) become(Boo)90°+[(Boo)x]90°−Boo−(Boo)x=0.(A2)After the subtraction, Eq. (A1) is the same as Eqs. (22). This substration operation is equivalent to applying an odd–odd operation to Eq. (A1) to filter non-odd–odd terms.

D. Malacala, ed., “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, (New York, 1978), 489–505.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics XXVI, E. Wolf, ed. (North-Holland, Amsterdam, 1988), 349–393.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Three flats, A(x, y), B(x, y), and C(x, y), of a front view and a rear view. The coordinate systems indicate the orientations of the flats.

Fig. 2
Fig. 2

Six configurations and the corresponding measurements. In each configuration, the flat below is flipped in x and is a flat of a rear view.

Fig. 3
Fig. 3

Simulation results: (a) flat C has a Gaussian shape bump of 1λ (pv); (b) Reconstructed flat C′, where the ripple = 0.11λ (pv); (c) flat C″, obtained from the first 36 Zernike polynomials of flat C.

Fig. 4
Fig. 4

Fizeau interferometer. Flat B is flipped in the horizontal direction (into paper). A and B, two flats; AR, antireflection coating; PZT, piezoelectric transducer.

Fig. 5
Fig. 5

(a)–(f), Optical path difference maps (in waves) of the six measurements, where Mi’s correspond to those in Fig. 2.

Fig. 6
Fig. 6

(a)–(c) Surface profiles (in waves) of flats A, B, and C derived from six measurements in Fig. 5.

Equations (29)

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F ( x , y ) = F ee + F oo + F oe + F eo ,
F ee ( x , y ) = F ( x , y ) + F ( x , y ) + F ( x , y ) + F ( x , y ) 4 , F oo ( x , y ) = F ( x , y ) F ( x , y ) F ( x , y ) + F ( x , y ) 4 , F eo ( x , y ) = F ( x , y ) + F ( x , y ) F ( x , y ) F ( x , y ) 4 , F oe ( x , y ) = F ( x , y ) F ( x , y ) + F ( x , y ) F ( x , y ) 4 .
[ F ( x , y ) ] θ = F ( x cos θ y sin θ , x sin θ + y cos θ ) .
[ F ( x , y ) ] 180 ° = F ee + F oo F oe F eo , [ F ( x , y ) ] x = F ee F oo F oe + F eo .
F oo ( x , y ) = m = 1 f 2 m sin ( 2 m θ ) ,
F oo = F oo , 2 θ = F oo , 2 odd θ + F oo , 2 even θ ,
F oo , 2 even θ = m = even f 2 m sin ( 2 m θ ) = m = 1 f 4 m sin ( 4 m θ ) = F oo , 4 θ , F oo , 2 odd θ = m = odd f 2 m sin ( 2 m θ ) .
F oo , 4 θ = F oo , 4 odd θ + F oo , 4 even θ ,
F oo , 4 odd θ = m = odd f 4 m sin ( 4 m θ ) , F oo , 4 even θ = m = even f 4 m sin ( 4 m θ ) .
F oo , 2 θ = F oo , 2 odd θ + F oo , 4 odd θ + F oo , 8 odd θ + F oo , 16 odd θ + ,
[ F oo , 2 θ ] 90 ° = F oo , 2 odd θ + F oo , 2 even θ ,
[ F oo , 4 θ ] 45 ° = F oo , 4 odd θ + F oo , 4 even θ ,
M 1 = A + B x , M 2 = A 180 ° + B x , M 3 = A 90 ° + B x , M 4 = A 45 ° + B x , M 5 = A + C x , M 6 = B + C x .
M 1 = A ee + A oo + A oe + A eo + B ee B oo B oe + B eo , M 2 = A ee + A oo A oe A eo + B ee B oo B oe + B eo , M 5 = A ee + A oo + A oe + A eo + C ee C oo C oe + C eo .
A oe + A eo = ( M 1 M 2 ) / 2 , B oe + B eo = { [ M 1 ( M 1 ) 180 ° ] / 2 ( A oe + A eo ) } x , C oe + C eo = { [ M 5 ( M 5 ) 180 ° ] / 2 ( A oe + A eo ) } x .
m 1 = [ M 1 + ( M 1 ) 180 ° ] / 2 = A ee + A oo + B ee B oo , m 5 = [ M 5 + ( M 5 ) 180 ° ] / 2 = A ee + A oo + C ee C oo , m 6 = [ M 6 + ( M 6 ) 180 ° ] / 2 = B ee + B oo + C ee C oo .
A ee = [ m 1 + m 5 m 6 + ( m 1 + m 5 m 6 ) x ] / 4 , B ee = [ m 1 + ( m 1 ) x 2 A ee ] / 2 , C ee = [ m 5 + ( m 5 ) x 2 A ee ] / 2 .
A oo = A oo , 2 θ = A oo , 2 odd θ + A oo , 2 even θ , B oo = B oo , 2 θ = B oo , 2 odd θ + B oo , 2 even θ , C oo = C oo , 2 θ = C oo , 2 odd θ + C oo , 2 even θ ,
( A oo , 2 θ ) 90 ° = A oo , 2 odd θ + A oo , 2 even θ , ( B oo , 2 θ ) 90 ° = B oo , 2 odd θ + B oo , 2 even θ , ( C oo , 2 θ ) 90 ° = C oo , 2 odd θ + C oo , 2 even θ .
m 1 = M 1 ( A oe + A eo + A ee ) ( B oe + B eo + B ee ) x , m 3 = M 3 ( A oe + A eo + A ee ) 90 ° ( B oe + B eo + B ee ) x , m 6 = M 6 ( B oe + B eo + B ee ) ( C oe + C eo + C ee ) x .
m 1 = A oo B oo , m 3 = ( A oo ) 90 ° B oo , m 6 = B oo C oo ,
A oo , 2 odd θ = ( m 1 m 3 ) / 2 , B oo , 2 odd θ = [ ( m 1 ) 90 ° m 3 ] / 2 , C oo , 2 odd θ = [ ( m 6 ) 90 ° m 6 + ( m 1 ) 90 ° m 3 ] / 2 .
m 1 = M 1 ( A oe + A eo + A ee + A oo , 2 odd θ ) ( B oe + B eo + B ee + B oo , 2 odd θ ) x , m 4 = M 4 ( A oe + A eo + A ee + A oo , 2 odd θ ) 45 ° ( B oe + B eo + B ee + B oo , 2 odd θ ) x , m 6 = M 6 ( B oe + B eo + B ee + B oo , 2 odd θ ) ( C oe + C eo + C ee + C oo , 2 odd θ ) x .
m 1 = A oo , 4 θ B oo , 4 θ , m 4 = ( A oo , 4 θ ) 45 ° B oo , 4 θ , m 6 = B oo , 4 θ C oo , 4 θ .
A oo , 4 odd θ = ( m 1 m 4 ) / 2 , B oo , 4 odd θ = [ ( m 1 ) 45 ° m 4 ] / 2 , C oo , 4 odd θ = [ ( m 6 ) 45 ° m 6 + ( m 1 ) 45 ° m 4 ] / 2 .
A = A ee + A oe + A eo + A oo , 2 odd θ + A oo , 4 odd θ, B = B ee + B oe + B eo + B oo , 2 odd θ + B oo , 4 odd θ, C = C ee + C oe + C eo + C oo , 2 odd θ + C oo , 4 odd θ,
M 1 = A + B x = A ( y ) + B ( y ) , M 2 = A 180 ° + B x = A ( y ) + B ( y ) , M 5 = A + C x = A ( y ) + C ( y ) , M 6 = B + C x = B ( y ) + C ( y ) .
(M6)90°M6+(M1)90°M3=(Cx)90°Cx+(B)90°+(Bx)90°BBx.
(Boo)90°+[(Boo)x]90°Boo(Boo)x=0.

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