Abstract

This paper presents an analytical method that allows for unambiguous separation of misalignment from the interferometric measurement of cylindrical optics with rectangular apertures. This method not only removes the misalignment-induced aberration from the measured wavefront data, but also yields the amount of misalignment in the test setup. We verified this method during testing of a convex cylindrical optic.

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References

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  1. M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt.9(3), 533–537 (1970).
    [CrossRef] [PubMed]
  2. G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE.429, 187–193 (1983).
  3. E. W. D. Young and G. C. Dente, “The effects of rigid body motion in interferometric tests of large-aperture, off-axis, aspheric optics,” Southwest Conference on Optics. Albuquerque, NM, SPIE. 540: 59–68 (1983).
  4. F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
    [CrossRef]
  5. P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
    [CrossRef]
  6. Mathworld. “Least-squarea fitting,” Retrieved Jan 3, 2012, from http://mathworld.wolfram.com/LeastSquaresFitting.html .
  7. B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005).
    [CrossRef]

2011 (1)

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

2010 (1)

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
[CrossRef]

2005 (1)

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005).
[CrossRef]

1983 (1)

G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE.429, 187–193 (1983).

1970 (1)

Dente, G. C.

G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE.429, 187–193 (1983).

Geary, J.

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
[CrossRef]

Liu, F.

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
[CrossRef]

Reardon, P.

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
[CrossRef]

Reardon, P. J.

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005).
[CrossRef]

Rimmer, M.

Robinson, B. M.

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

B. M. Robinson and P. J. Reardon, “First-order perturbations of reflective surfaces and their effects in interferometric testing of mirrors,” J. Mod. Opt.52(18), 2625–2636 (2005).
[CrossRef]

Opt. Eng. (2)

F. Liu, B. M. Robinson, P. Reardon, and J. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng.50(4), 043609 (2011).
[CrossRef]

P. Reardon, F. Liu, and J. Geary, “Schmidt-like corrector plate for cylindrical optics,” Opt. Eng.49(5), 053002 (2010).
[CrossRef]

Precision Surface Metrology. J. C. Wyant, SPIE. (1)

G. C. Dente, “Separating misalignment from misfigure in interferograms on off-axis aspheres,” Precision Surface Metrology. J. C. Wyant, SPIE.429, 187–193 (1983).

Other (2)

E. W. D. Young and G. C. Dente, “The effects of rigid body motion in interferometric tests of large-aperture, off-axis, aspheric optics,” Southwest Conference on Optics. Albuquerque, NM, SPIE. 540: 59–68 (1983).

Mathworld. “Least-squarea fitting,” Retrieved Jan 3, 2012, from http://mathworld.wolfram.com/LeastSquaresFitting.html .

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Figures (6)

Fig. 1
Fig. 1

(a) The test layout of a convex cylindrical mirror in the center-of-curvature configuration. (b). The line plot of the test layout illustrates the assigned coordinate system, where the surface under test sits at the origin of the coordinate.

Fig. 2
Fig. 2

Sample interferogram (column No.1) and OPD maps (column No.2) simulated in ray tracing software (CODE V®) under different amount of movements.

Fig. 3
Fig. 3

The artificial interferogram generated from the 2-D Chebyshev surface contour map. Each of them was assigned a P-V of 4λ.

Fig. 4
Fig. 4

Sample interferogram (column No.1) and OPD maps (second column) under different movements. They are simulated by a linear model in Matlab® and yield the same results as those simulated in ray tracing software (CODE V®) in Fig. 2.

Fig. 5
Fig. 5

Interferogram (a) and OPD maps (b) from misalignments using the retrieved results of the reduced matrix.

Fig. 6
Fig. 6

Interferometric measurements done by four metrologists independently, before separation of the misalignment aberrations (left column) and after separation (right column).

Tables (1)

Tables Icon

Table 1 P-V and RMS Error Before and After Separating Misalignment Errors

Equations (11)

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S= x,y (W A i C i ) 2 dxdy ,
δW=2[t n ^ +ϕ(r× n ^ )].
δW(x,y)= A i C i (x,y)
A i = pupil C i (x,y)δW(x,y) 1 1 x 2 1 y 2 dxdy . =( C i ,δW).
δW(x,y)=2(δ r x n x +δ r y n y +δ r z n z ) =2( ε x + ϕ y z ϕ z y) n x +2( ε y + ϕ z x ϕ x z) n y +2( ε z + ϕ x y ϕ y x) n z
A i =( C i ,δW)=2( C i ,δ r x n x )+2( C i ,δ r y n y )+2( C i ,δ r z n z ) =2 ε x ( C i , n x )+2 ε y ( C i , n y )+2 ε z ( C i , n z ) +2 ϕ x ( C i ,y n z z n y )+2 ϕ y ( C i ,z n x x n z )+2 ϕ z ( C i ,x n y y n x ) =2 ε x ( C i , n x )+2 ε y ( C i , n y )+2 ε z ( C i , n z ) +2 ϕ x ( C i , l x )+2 ϕ y ( C i , l y )+2 ϕ z ( C i , l z )
A=.
δ= [ ε x ε y ε z ϕ x ϕ y ϕ z ] T .
X=[ ( C 1 , n x ) ( C 1 , n y ) ( C 1 , n z ) ( C 1 , l x ) ( C 1 , l y ) ( C 1 , l z ) ( C 2 , n x ) ( C 2 , n y ) ( C 2 , n z ) ( C 2 , l x ) ( C 2 , l y ) ( C 2 , l z ) ( C 3 , n x ) ( C 3 , n y ) ( C 3 , n z ) ( C 3 , l x ) ( C 3 , l y ) ( C 3 , l z ) ( C 4 , n x ) ( C 4 , n y ) ( C 4 , n z ) ( C 4 , l x ) ( C 4 , l y ) ( C 4 , l z ) ( C 5 , n x ) ( C 5 , n y ) ( C 5 , n z ) ( C 5 , l x ) ( C 5 , l y ) ( C 5 , l z ) ( C 6 , n x ) ( C 6 , n y ) ( C 6 , n z ) ( C 6 , l x ) ( C 6 , l y ) ( C 6 , l z ) ].
δ= ( X T X ) 1 X T A= X + A.
δ'= [ ε y ε z ϕ y ϕ z ] T = [ 0.0011mm 0.1941mm 0.005° 0.01° ] T

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