Abstract

In interferometric surface and wavefront metrology, three-flat tests are the archetypes of measurement procedures to separate errors in the interferometer reference wavefront from errors due to the test part surface, so-called absolute tests. What is believed to be a new class of solutions of the three-flat problem for circular flats is described in terms of functions that are symmetric or antisymmetric with respect to reflections at a single line passing through the center of the flat surfaces. The new solutions are simpler and easier to calculate than the known solutions based on twofold mirror symmetry or rotation symmetry. Strategies for effective azimuthal averaging and a method for determining the averaging error are also discussed.

© 2006 Optical Society of America

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References

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  1. M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
    [CrossRef]
  2. J. Schwider, "Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen II," Opt. Acta 14, 389-400 (1967).
    [CrossRef]
  3. C. J. Evans and R. N. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  4. C. Ai and J. C. Wyant, "Absolute testing of flats by using even and odd functions," Appl. Opt. 32, 4698-4705 (1993).
    [CrossRef] [PubMed]
  5. R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
    [CrossRef]
  6. P. Clapham and G. Dew, "Surface-coated reference flats for testing fully aluminized surfaces by means of a Fizeau interferometer," J. Sci. Instrum. 44, 899-902 (1967).
    [CrossRef]
  7. R. E. Parks, "Removal of test optics errors," in Advances in Optical Metrology I, N. Balasubramanian and J. C. Wyant, eds., Proc. SPIE 153, 56-63 (1978).
  8. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).
  9. K. R. Freischlad, "Absolute interferometric testing based on reconstruction of rotational shear," Appl. Opt. 40, 1637-1648 (2001).
    [CrossRef]
  10. R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
    [CrossRef]

2001

1998

1997

R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
[CrossRef]

1996

1993

1984

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

1978

R. E. Parks, "Removal of test optics errors," in Advances in Optical Metrology I, N. Balasubramanian and J. C. Wyant, eds., Proc. SPIE 153, 56-63 (1978).

1967

J. Schwider, "Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen II," Opt. Acta 14, 389-400 (1967).
[CrossRef]

P. Clapham and G. Dew, "Surface-coated reference flats for testing fully aluminized surfaces by means of a Fizeau interferometer," J. Sci. Instrum. 44, 899-902 (1967).
[CrossRef]

Ai, C.

Bourgeois, R. P.

R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
[CrossRef]

Clapham, P.

P. Clapham and G. Dew, "Surface-coated reference flats for testing fully aluminized surfaces by means of a Fizeau interferometer," J. Sci. Instrum. 44, 899-902 (1967).
[CrossRef]

Dew, G.

P. Clapham and G. Dew, "Surface-coated reference flats for testing fully aluminized surfaces by means of a Fizeau interferometer," J. Sci. Instrum. 44, 899-902 (1967).
[CrossRef]

Evans, C. J.

Freischlad, K. R.

Fritz, B. S.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

Kestner, R. N.

Küchel, M. F.

M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Magner, J.

R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
[CrossRef]

Parks, R. E.

R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998).
[CrossRef]

R. E. Parks, "Removal of test optics errors," in Advances in Optical Metrology I, N. Balasubramanian and J. C. Wyant, eds., Proc. SPIE 153, 56-63 (1978).

Schwider, J.

J. Schwider, "Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen II," Opt. Acta 14, 389-400 (1967).
[CrossRef]

Shao, L.

Stahl, H. P.

R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
[CrossRef]

Wyant, J. C.

Appl. Opt.

J. Sci. Instrum.

P. Clapham and G. Dew, "Surface-coated reference flats for testing fully aluminized surfaces by means of a Fizeau interferometer," J. Sci. Instrum. 44, 899-902 (1967).
[CrossRef]

Opt. Acta

J. Schwider, "Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen II," Opt. Acta 14, 389-400 (1967).
[CrossRef]

Opt. Eng.

B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).

Optik

M. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Proc. SPIE

R. E. Parks, "Removal of test optics errors," in Advances in Optical Metrology I, N. Balasubramanian and J. C. Wyant, eds., Proc. SPIE 153, 56-63 (1978).

R. P. Bourgeois, J. Magner, and H. P. Stahl, "Results of the calibration of interferometer transmission flats for the LIGO Pathfinder optics," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 86-94 (1997).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(Color online) Reference surface S R and test surface S T in a Fizeau interferometer and coordinate system of the flat surfaces that are being compared. The coordinate system of the interferometer is indicated with bold arrows.

Fig. 2
Fig. 2

(Color online) Measurement sequence (BA, CA, CB) for the comparison of three flats A, B, and C.

Fig. 3
Fig. 3

(Color online) Function W(x, y) and its decomposition into even (symmetric) and odd (antisymmetric) components. The coordinates are pixel numbers. The origin (x, y) = (0, 0) is at the center of the images.

Fig. 4
Fig. 4

(Color online) Measurement sequence comparing three flats, which can be solved for the flat surfaces.

Fig. 5
Fig. 5

(Color online) Simulation of a three-flat test based on simple mirror symmetry using Eq. (25) with six-position averaging. Top row, simulated flat wavefronts; middle row, flat solutions of the flat test; bottom row, difference between true wavefronts and the test solution. The coordinates are pixel numbers. The origin (x, y) = (0, 0) is at the center of the images.

Fig. 6
Fig. 6

(Color online) Rotationally invariant component of W A , shown in the top left corner of Fig. 5, calculated with Eq. (35) (left) and with Eq. (33) (middle). The difference is shown on the right.

Equations (44)

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W ( x , y ) = W R ( x , y ) + W T ( x , y ) ,
[ W 1 ( x , y ) W 2 ( x , y ) W 3 ( x , y ) ] = [ 1 0 1 0 1 0 0 1 0 1 0 1 ] [ W A ( x , y ) W B ( x , y ) W B ( x , y ) W C ( x , y ) ] ,
W ( x , y ) = W I ( x , y ) + W V ( x , y ) .
[ W 1 I ( x , y ) W 2 I ( x , y ) W 3 I ( x , y ) ] = [ 1 1 0 1 0 1 0 1 1 ] [ W A I ( x , y ) W B I ( x , y ) W C I ( x , y ) ] .
[ W A I ( x , y ) W B I ( x , y ) W C I ( x , y ) ] = 1 2 [ 1 1 1 1 1 1 1 1 1 ] [ W 1 I ( x , y ) W 2 I ( x , y ) W 3 I ( x , y ) ] .
W R ( r ) = def [ W ( r , ϕ ) ] R = def 1 2 π 0 2 π W ( r , ϕ ) d ϕ .
[ [ W ( r , ϕ ) ] R ] R = [ W ( r , ϕ ) ] R = W R ( r ) ,
W ( r , ϕ ) = W R ( r ) + Ω ( r , ϕ ) .
[ Ω ( r , ϕ ) ] R = [ W ( r , ϕ ) W R ( r ) ] R = W R ( r ) W R ( r ) = 0.
W R ( r ) = 1 2 π 0 2 π W ( r , ϕ ) d ϕ = lim N ( 1 2 π k = 0 N 1 W ( r , ϕ k ) Δ ϕ k ) .
W R ( r ) 1 N k = 0 N 1 W ( r , k Δ ϕ ) 1 N k = 0 N 1 W ( r , ϕ k Δ ϕ ) .
W x ( x , y ) = def [ W ( x , y ) ] x = def W ( x , y ) .
W e ( x , y ) = def 1 2 [ W ( x , y ) + W x ( x , y ) ] ,
W o ( x , y ) = def 1 2 [ W ( x , y ) W x ( x , y ) ] ,
[ W e ] x = W e ,
[ W o ] x = W o .
W = W e + W o .
W e = 1 2 ( W + W x ) = W R + 1 2 ( Ω + Ω x ) ,
W o = 1 2 ( W W x ) = 1 2 ( Ω Ω x ) .
[ W e ] R = W R ,
[ W o ] R = 0.
W 2 ( x , y ) = 1 N k = 0 N 1 { W B ( x , y ) + [ W A ( x , y ) ] k Δ ϕ } = W B ( x , y ) + 1 N k = 0 N 1 [ W A ( x , y ) ] k Δ ϕ W A R ( x , y ) .
[ W 1 ( x , y ) W 2 ( x , y ) W 3 ( x , y ) W 4 ( x , y ) ] = [ 1 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 ] [ W A ( x , y ) W B ( x , y ) W B ( x , y ) W C ( x , y ) [ W A ( x , y ) ] R ] .
[ W A ( x , y ) W B ( x , y ) W B ( x , y ) W C ( x , y ) [ W A ( x , y ) ] R ] = [ W A e ( x , y ) W B e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A R ] + [ W A o ( x , y ) W B o ( x , y ) W B o ( x , y ) W C o ( x , y ) 0 ] .
[ W 1 o ( x , y ) W 2 o ( x , y ) W 3 o ( x , y ) W 4 o ( x , y ) ] = [ 1 0 1 0 0 0 1 0 1 0 0 1 0 1 0 1 ] [ W A o ( x , y ) W B o ( x , y ) W B o ( x , y ) W C o ( x , y ) ] .
[ W A o ( x , y ) W B o ( x , y ) W B o ( x , y ) W C o ( x , y ) ] = [ 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 0 ] [ W 1 o ( x , y ) W 2 o ( x , y ) W 3 o ( x , y ) W 4 o ( x , y ) ] .
[ W 1 e ( x , y ) W 2 e ( x , y ) W 3 e ( x , y ) W 4 e ( x , y ) ] = [ 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 ] [ W A e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A R ] ,
[ W A e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A R ] = 1 2 [ 1 0 1 1 1 0 1 1 1 0 1 1 1 2 1 1 ] [ W 1 e ( x , y ) W 2 e ( x , y ) W 3 e ( x , y ) W 4 e ( x , y ) ] .
[ W A W B W C ] = 1 2 [ 1 1 1 2 2 0 0 1 1 1 2 2 2 2 1 1 1 2 2 2 0 ] [ W 1 e W 3 e W 4 e W 1 o W 2 o W 3 o W 4 o ] .
[ W 1 ( x , y ) W 2 ( x , y ) W 3 ( x , y ) W 4 ( x , y ) W 5 ( x , y ) W 6 ( x , y ) ] = [ 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 ] × [ W A ( x , y ) W B ( x , y ) W C ( x , y ) W A ( x , y ) W B ( x , y ) W C ( x , y ) [ W A ( x , y ) ] R [ W B ( x , y ) ] R [ W C ( x , y ) ] R ] .
   [ W A ( x , y ) W B ( x , y ) W C ( x , y ) W A ( x , y ) W B ( x , y ) W C ( x , y ) [ W A ( x , y ) ] R [ W B ( x , y ) ] R [ W C ( x , y ) ] R ] = [ W A e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A R W B R W C R ] + [ W A o ( x , y ) W B o ( x , y ) W C o ( x , y ) W A o ( x , y ) W B o ( x , y ) W C o ( x , y ) 0 0 0 ] .
[ W A e ( x , y ) W B e ( x , y ) W C e ( x , y ) W A R W B R W C R ] = 1 2 [ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 1 0 1 0 1 0 1 2 1 0 1 0 1 0 1 2 ] × [ W 1     e ( x , y ) W 2     e ( x , y ) W 3     e ( x , y ) W 4     e ( x , y ) W 5     e ( x , y ) W 6     e ( x , y ) ] ,
[ W A o ( x , y ) W B o ( x , y ) W C o ( x , y ) W A o ( x , y ) W B o ( x , y ) W C o ( x , y ) ] = [ 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 ] [ W 1 o ( x , y ) W 2 o ( x , y ) W 3 o ( x , y ) W 4 o ( x , y ) W 5 o ( x , y ) W 6 o ( x , y ) ] .
[ W A W B W C ] = 1 2 [ 1 1 1 2 2 0 0 0 0 1 1 1 0 0 2 2 0 0 1 1 1 0 0 0 0 2 2 ] × [ W 1     e W 3     e W 5     e W 1     o W 2     o W 3     o W 4     o W 5     o W 6     o ] .
[ W A o ( x , y ) W B o ( x , y ) W C o ( x , y ) ] = [ 0 0 1 1 0 0 0 1 0 ] [ W 2 o ( x , y ) W 4 o ( x , y ) W 6 o ( x , y ) ] .
[ W A W B W C ] = 1 2 [ 1 1 1 0 0 2 1 1 1 2 0 0 1 1 1 0 2 0 ] [ W 1 e W 3 e W 5 e W 2 o W 4 o W 6 o ] .
W A     R = W 2     e 1 2 ( W 1     e W 3     e + W 4     e ) .
    [ W A R W B R W C R ] = 1 2 [ 1 2 1 0 1 0 1 0 1 2 1 0 1 0 1 0 1 2 ] [ W 1 e W 2 e W 3 e W 4 e W 5 e W 6 e ] .
[ W A R W B R W C R ] = 1 2 [ 1 1 1 1 1 1 1 1 1 ] [ W 1     R W 3     R W 4     R ] .
W α = W B ( x , y ) + W A ( x , y ) , W β = W B ( x , y ) + [ W A ( x , y ) ] Δ ϕ ,
W β W α = ( W A ) Δ ϕ W A ,
W κ = W κ 1 + ( W β W α ) ( κ 2 ) Δ ϕ , 3 κ N .
Ω A ( r , ϕ Δ ϕ ) Ω A ( r , ϕ ) = W β W α .
     W 2 ( x , y ) = W B ( x , y ) + W A ( x , y ) Ω A ( x , y )

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