Abstract

Wave-front or surface errors may be divided into rotationally symmetric and nonrotationally symmetric terms. It is shown that if either the test part or the reference surface in an interferometrie test is rotated to N equally spaced positions about the optical axis and the resulting wave fronts are averaged, then errors in the rotated member with angular orders that are not integer multiples of the number of positions will be removed. Thus if the test piece is rotated to N equally spaced positions and the data rotated back to a common orientation in software, all nonrotationally symmetric errors of the interferometer except those of angular order kNθ are completely removed. It is also shown how this method may be applied in an absolute test, giving both rotationally symmetric and nonsymmetric components of the surface. A general proof is given that assumes only that the surface or wave-front information can be described by some arbitrary set of orthognal polynomials in a radial coordinate r and terms in sin θ and cos θ. A simulation, using Zernike polynomials, is also presented.

© 1996 Optical Society of America

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References

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  1. C. J. Evans, “Compensation for errors introduced by nonzero fringe densities in phase measuring interferometers,” CIRP Ann. Int. Inst. Prod. Eng. Res., 42/1, 577–580 (1993).
  2. See, for example, M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1991), Chap. 1, p. 43.
  3. C. E. learned the method from A. Slomba, then of United Technologies, who claimed that it was well known at Perkin Elmer (now Hughes Danbury Optical Systems) in Danbury, Conn.
  4. G. W. Ritchey, “On the modern reflecting telescope and the making and testing of optical mirrors,” Smithson. Contrib. Knowl. 34, (1904),as quoted by J. Ojeda-Castenada, “Focault, wire and phase modulation techniques,” in Optical Shop Testing, 1st ed., D. Malacara, ed. (Wiley, New York, 1978).
  5. K. L. Shu, “Ray-trace analysis and data reduction methods for the Ritchey–Commontest,” Appl. Opt. 22, 1879–1886 (1983).
  6. F. M. Kuchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.
  7. G. Schulz, J. Schwider, “Precise measurement of planeness,” Appl. Opt. 6, 1077–1084 (1967);“Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (Pergamon, Oxford, 1976), pp. 94–167.
  8. G. Shulz, J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring-error compensation,” Appl. Opt. 31, 3767–3780 (1992).
  9. K.-E. Elssner, A. Vogel, J. Grzanna, G. Schulz, “Establishing a flatness standard,” Appl. Opt. 33, 2437–2446 (1994).
  10. B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  11. C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 73–83 (1992).
  12. C. Ai, J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).
  13. C. Ai, L.-Z. Shao, R. E. Parks, “Abolute testing of flats: using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).
  14. S. J. Mack, Tropel, N. Y. Fairport (personal communication) 1991.
  15. F. M. Kuchel, “Verfahren und Vorrichtung zur interferometrischen Absolutprüfung von Planflächen,” European patentEP 0 441 153 B1 (14August1991).
  16. A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).
  17. L. A. Selberg, “Absolute testing of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 181–184.
  18. K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).
  19. R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 56–63 (1978).
  20. Kuchel, in reviewing a draft of this manuscript, derived an alternative proof based on a Fourier series expansion of any arbitrary azimuthal profile that is concentric with the axis about which the part is rotated.
  21. The notation used here is that proposed by the NAPM IT 11 Standards Committee on Interferometric Testing, in which the subscript refers to the maximum radial order whereas the superscript gives the azimuthal order, with a negative sign indicating the sine rather than cosine) term.

1994

1993

C. Ai, J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).

C. J. Evans, “Compensation for errors introduced by nonzero fringe densities in phase measuring interferometers,” CIRP Ann. Int. Inst. Prod. Eng. Res., 42/1, 577–580 (1993).

1992

1984

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

1983

1973

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

1967

1904

G. W. Ritchey, “On the modern reflecting telescope and the making and testing of optical mirrors,” Smithson. Contrib. Knowl. 34, (1904),as quoted by J. Ojeda-Castenada, “Focault, wire and phase modulation techniques,” in Optical Shop Testing, 1st ed., D. Malacara, ed. (Wiley, New York, 1978).

Ai, C.

C. Ai, J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).

C. Ai, L.-Z. Shao, R. E. Parks, “Abolute testing of flats: using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 73–83 (1992).

Elssner, K.-E.

Evans, C. J.

C. J. Evans, “Compensation for errors introduced by nonzero fringe densities in phase measuring interferometers,” CIRP Ann. Int. Inst. Prod. Eng. Res., 42/1, 577–580 (1993).

Fairport, N. Y.

S. J. Mack, Tropel, N. Y. Fairport (personal communication) 1991.

Freishlad, K.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

Fritz, B. S.

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

Grzanna, J.

Jensen, A. E.

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

Kaiser, W.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

Kuchel, F. M.

F. M. Kuchel, “Verfahren und Vorrichtung zur interferometrischen Absolutprüfung von Planflächen,” European patentEP 0 441 153 B1 (14August1991).

F. M. Kuchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.

Kuchel, M.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

Mack, S. J.

S. J. Mack, Tropel, N. Y. Fairport (personal communication) 1991.

Mantravadi, M. V.

See, for example, M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1991), Chap. 1, p. 43.

Parks, R. E.

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 56–63 (1978).

C. Ai, L.-Z. Shao, R. E. Parks, “Abolute testing of flats: using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

Ritchey, G. W.

G. W. Ritchey, “On the modern reflecting telescope and the making and testing of optical mirrors,” Smithson. Contrib. Knowl. 34, (1904),as quoted by J. Ojeda-Castenada, “Focault, wire and phase modulation techniques,” in Optical Shop Testing, 1st ed., D. Malacara, ed. (Wiley, New York, 1978).

Schulz, G.

Schuster, K. H.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

Schwider, J.

Selberg, L. A.

L. A. Selberg, “Absolute testing of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 181–184.

Shao, L.-Z.

C. Ai, L.-Z. Shao, R. E. Parks, “Abolute testing of flats: using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

Shu, K. L.

Shulz, G.

Slomba, A.

C. E. learned the method from A. Slomba, then of United Technologies, who claimed that it was well known at Perkin Elmer (now Hughes Danbury Optical Systems) in Danbury, Conn.

Tropel,

S. J. Mack, Tropel, N. Y. Fairport (personal communication) 1991.

Vogel, A.

Wegmann, U.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

Wyant, J. C.

C. Ai, J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4705 (1993).

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 73–83 (1992).

Appl. Opt.

CIRP Ann. Int. Inst. Prod. Eng. Res.

C. J. Evans, “Compensation for errors introduced by nonzero fringe densities in phase measuring interferometers,” CIRP Ann. Int. Inst. Prod. Eng. Res., 42/1, 577–580 (1993).

J. Opt. Soc. Am.

A. E. Jensen, “Absolute calibration method for laser Twyman–Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313 (1973).

Opt. Eng.

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

Smithson. Contrib. Knowl.

G. W. Ritchey, “On the modern reflecting telescope and the making and testing of optical mirrors,” Smithson. Contrib. Knowl. 34, (1904),as quoted by J. Ojeda-Castenada, “Focault, wire and phase modulation techniques,” in Optical Shop Testing, 1st ed., D. Malacara, ed. (Wiley, New York, 1978).

Other

C. Ai, J. C. Wyant, “Absolute testing of flats decomposed to even and odd functions,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 73–83 (1992).

See, for example, M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1991), Chap. 1, p. 43.

C. E. learned the method from A. Slomba, then of United Technologies, who claimed that it was well known at Perkin Elmer (now Hughes Danbury Optical Systems) in Danbury, Conn.

F. M. Kuchel, “Absolute measurement of flat mirrors in the Ritchey–Common test,” in Optical Fabrication and Testing, OSA 1986 Technical Digest Series (Optical Society of America, Washington, D.C., 1986), pp. 114–119.

L. A. Selberg, “Absolute testing of spherical surfaces,” in Optical Fabrication and Testing, Vol. 13 of OSA 1994 Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 181–184.

K. Freishlad, M. Kuchel, K. H. Schuster, U. Wegmann, W. Kaiser, “Real time wavefront measurement with lambda/10fringe spacing for the optical shop,” in Optical Testing and Metrology III: Recent Advances in Industrial Optical Inspection, C. Grover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1332, 18–24 (1990).

R. E. Parks, “Removal of test optics errors,” in Advances in Optical Metrology I, N. Balasubramanian, J. C. Wyant, eds., Proc. Soc. Photo-Opt. Instrum. Eng.153, 56–63 (1978).

Kuchel, in reviewing a draft of this manuscript, derived an alternative proof based on a Fourier series expansion of any arbitrary azimuthal profile that is concentric with the axis about which the part is rotated.

The notation used here is that proposed by the NAPM IT 11 Standards Committee on Interferometric Testing, in which the subscript refers to the maximum radial order whereas the superscript gives the azimuthal order, with a negative sign indicating the sine rather than cosine) term.

C. Ai, L.-Z. Shao, R. E. Parks, “Abolute testing of flats: using even and odd functions,” in Optical Fabrication and Testing, Vol. 24 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

S. J. Mack, Tropel, N. Y. Fairport (personal communication) 1991.

F. M. Kuchel, “Verfahren und Vorrichtung zur interferometrischen Absolutprüfung von Planflächen,” European patentEP 0 441 153 B1 (14August1991).

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Figures (3)

Fig. 1
Fig. 1

(a) Coma (left) rotated 180° (right). The average of the two wave fronts is obviously zero. (b) Rotating astigmatism 180° and averaging achieve nothing.

Fig. 2
Fig. 2

Synthetic wave front generated from one Zernike term with a coefficient of 0.1 (arbitrary units) for each angular order up to 4θ. [Peak to valley (P.V.) is 0.64; synthetic fringes spaced at 0.03 intervals.]

Fig. 3
Fig. 3

Resulting wave front obtained after averaging the wave front in Fig. 2 at three equally spaced angular positions (P.V. is 0.19; synthetic fringes spaced at 0.03 intervals).

Tables (2)

Tables Icon

Table 2 Fractional Uncertainty in the Coefficient of Zernike Terms

Equations (28)

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W = T + P ,
W 0 + W 180 2 = T ,
W 0 + W 180 2 = T + P ,
A ( r ,  θ ) = k , l R l k ( r ) ( a l k  cos  k θ + a l k  sin kθ ) ,
A ( r ,  θ + ϕ ) = k , l R l k ( r ) [ a l k  cos  k ( θ + ϕ ) + a l k  sin  k ( θ + ϕ ) ] ,
A ( r , θ + ϕ ) = k , l R l k ( r ) [ cos  k θ ( a l k  cos  k ϕ + a l k sin  k ϕ ) + sin  k θ ( a l k cos  k ϕ a l k sin  k ϕ ) ]
A ( r ,  θ +  ϕ ) = k , l R l k ( r ) ( a l ϕ , k cos   k θ + a l ϕ , k sin   k θ ) ,
a l ϕ , k = a l k cos   k ϕ + a l k sin   k ϕ , a l ϕ , k = a l k cos   k ϕ a l k sin   k ϕ .
i = 0 5 W l k = 6 T l k + i = 0 5 P l k cos ( i k 60 ° ) + i = 0 5 P l k sin ( i k 60 ° ) = 6 T l k + P l k i = 0 5 cos ( i k 60 ° ) + P l k i = 0 5 sin ( i k 60 ° ) ,
i = 0 5 W l k = 6 T l k + P l k i = 0 5 cos ( i k 60 ° ) P l k i = 0 5 sin ( i k 60 ° ) .
i = 0 5 W l 0 = 6 T l 0 + 6 P l 0 ,
i = 0 N 1 cos ( i k 2 π N ) = 0 , k c N , i = 0 N 1 cos ( i k 2 π N ) = N , k = c N ,
i = 0 N 1 sin ( i k 2 π N ) = 0
1 + x + x 2 + x N 1 = 1 x N 1 x ,
cos  θ = exp ( j θ ) + exp ( j θ ) 2 , sin  θ = exp ( j θ ) exp ( j θ ) 2 j ,
i = 0 N 1 cos ( i k 2 π N ) = 1 2 { i = 0 N 1 [ exp ( j 2 π k / N ) ] i + i = 0 N 1 [ exp ( j 2 π k / N ) ] i } .
i = 0 N 1 [ exp ( j 2 π k / N ) ] i = 1 [ exp ( j 2 π k / N ) ] N 1 [ exp ( j 2 π k / N ) ] = 1 exp ( j 2 π k ) 1 [ exp ( j 2 π k / N ) ] .
exp ( j α ) = cos  α + j   sin  α
exp ( j 2 π k ) = 1
d ( a n m ) = m a n m .
u n m = 1 N u ( θ ) 2 π ( m ) a n m P n m .
M 0 = T s + T a + P s + P a , M 1 = T s + T a + P s + ρ 1 ( P a ) , M 2 = T s + T a + P s + ρ 2 ( P a ) ,
M N + 1 = T s + T a + P s ,
M N + 2 = T s + P s + P a .
M 1 M N + 1 = P a M 1 M N + 2 = T a ,
W 1 ( x , y ) = A ( x , y ) + B ( x , y ) , W 2 ( x , y ) = A ( x , y ) + C ( x , y ) , W 3 ( x , y ) = B ( x , y ) + C ( x , y ) .
A ( x , y ) = A ( x , y ) = A ( x , y ) = A ( x , y )
C = W 3 W 1 + W 2 2 ,

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