Abstract

A mirror has been fabricated with the shape of an off-axis section of a paraboloid by grinding and polishing a sphere into a prestressed blank. The applied stresses were then removed allowing the mirror to spring into the desired paraboloidal shape. The 36-cm diam off-axis section deviated 9.9-μm rms from the polished sphere. The final surface deviated 0.03-μm rms from the desired off-axis section.

© 1980 Optical Society of America

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References

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  1. J. Lubliner, J. E. Nelson, “Stressed Mirror Polishing: A Technique for Producing Non-Axisymmetric Mirrors,” Lawrence Berkeley Laboratory Report LBL-9967;Appl. Opt. 19, 2332 (1980).
    [PubMed]
  2. J. Nelson, “The Proposed University of California Ten-Meter Telescope,” in Proceedings, Conference on Optical Telescopes of the Future, December 1977 (Geneva 23: ESO c/o CERN1978), p. 133.
  3. J. Nelson, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).
  4. G. Gabor, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).
  5. T. S. Mast, J. E. Nelson, “Figure Control for a Segmented Telescope Mirror,” Lawrence Berkeley Laboratory Report LBL-8621 (March1979).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.
  7. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).

1979

J. Nelson, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

G. Gabor, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.

Gabor, G.

G. Gabor, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

Lubliner, J.

J. Lubliner, J. E. Nelson, “Stressed Mirror Polishing: A Technique for Producing Non-Axisymmetric Mirrors,” Lawrence Berkeley Laboratory Report LBL-9967;Appl. Opt. 19, 2332 (1980).
[PubMed]

Mast, T. S.

T. S. Mast, J. E. Nelson, “Figure Control for a Segmented Telescope Mirror,” Lawrence Berkeley Laboratory Report LBL-8621 (March1979).

Nelson, J.

J. Nelson, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

J. Nelson, “The Proposed University of California Ten-Meter Telescope,” in Proceedings, Conference on Optical Telescopes of the Future, December 1977 (Geneva 23: ESO c/o CERN1978), p. 133.

Nelson, J. E.

J. Lubliner, J. E. Nelson, “Stressed Mirror Polishing: A Technique for Producing Non-Axisymmetric Mirrors,” Lawrence Berkeley Laboratory Report LBL-9967;Appl. Opt. 19, 2332 (1980).
[PubMed]

T. S. Mast, J. E. Nelson, “Figure Control for a Segmented Telescope Mirror,” Lawrence Berkeley Laboratory Report LBL-8621 (March1979).

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.

Proc. Soc. Photo-Opt. Instrum. Eng.

J. Nelson, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

G. Gabor, Proc. Soc. Photo-Opt. Instrum. Eng. 172 (January1979).

Other

T. S. Mast, J. E. Nelson, “Figure Control for a Segmented Telescope Mirror,” Lawrence Berkeley Laboratory Report LBL-8621 (March1979).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), p. 466.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).

J. Lubliner, J. E. Nelson, “Stressed Mirror Polishing: A Technique for Producing Non-Axisymmetric Mirrors,” Lawrence Berkeley Laboratory Report LBL-9967;Appl. Opt. 19, 2332 (1980).
[PubMed]

J. Nelson, “The Proposed University of California Ten-Meter Telescope,” in Proceedings, Conference on Optical Telescopes of the Future, December 1977 (Geneva 23: ESO c/o CERN1978), p. 133.

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Figures (8)

Fig. 1
Fig. 1

Diagram defining global (X,Y,Z) and local coordinates (x,y,z = ρ,θ,z) of mirror segment on paraboloid.

Fig. 2
Fig. 2

Zernike coefficients describing surface of off-axis section of paraboloid as function of off-axis distance R. Paraboloid has radius of curvature k = 368.80 cm, and segment radius is a = 17.94 cm. Desired surface for fabricated mirror (R = 35.00 cm) is dominated by astigmatism and coma.

Fig. 3
Fig. 3

Schematic showing mirror blank, Invar block, radial arm, and lever and weight system used to apply forces F1 and F2. Twenty-four such assemblies produced shear forces and moments to stress the mirror.

Fig. 4
Fig. 4

Schematic diagram showing geometry of paraboloidal null test. Focus of Twyman-Green interferometer is located at vertex of global paraboloid. Mirror section under test is fixed 35.00 cm from vertex.

Fig. 5
Fig. 5

Interferogram from spherical null test of spherical mirror under stress before polishing. Applied forces and couples and coefficients describing surface are listed in Tables I and II. Surface is antiparabola dominated by 19.4 μm of astigmatism (saddle) and 6.6 μm of coma (internal extremum). Fringes are half wave, 632.8 nm/2. Contour plot of surface is generated from coefficients from fourth-order fit.

Fig. 6
Fig. 6

Interferogram from paraboloidal null test after first polish. Fringes (quarterwave = 632.8 nm/4) show difference between desired and achieved paraboloidal sections. This error surface is dominated by focus and astigmatism, and its rms amplitude is 0.55 μm. Contour plot of surface is generated from coefficients from fourth-order fit.

Fig. 7
Fig. 7

Interferogram from paraboloidal null test after the second polish. Fringes (quarterwave = 632.8 nm/4) show the difference between the desired and achieved paraboloidal sections. Error surface is dominated by astigmatism, and rms amplitude is 0.16 μm. Contour plot of surface is generated from coefficients from fourth-order fit.

Fig. 8
Fig. 8

Interferogram from paraboloidal null test with mirror in best fit position. Fringes (quarterwave = 632.8 nm/4) show difference between desired and achieved surfaces. The rms error is 0.03 μm. Contour plot of surface is generated from coefficients from fourth-order fit.

Tables (2)

Tables Icon

Table I Expansion Coefficients Describing Mirror Surface

Tables Icon

Table II Forces and Couples Applied to Stress Mirror

Equations (41)

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z ( ρ , θ ) = α 20 ρ 2 + α 22 ρ 2 cos 2 θ + α 31 ρ 3 cos θ + α 33 ρ 3 cos 3 θ + α 40 ρ 4 + α 42 ρ 4 cos 2 θ ,
α 20 = a 2 2 k ( 1 2 + 9 8 4 5 4 6 + . . . ) ( focus ) , α 22 = a 2 R 2 4 k 3 ( 1 3 2 2 + 15 8 4 + . . . ) ( astigmatism ) , α 31 = a 3 R 2 k 3 ( 1 11 4 2 + 21 4 4 + . . . ) ( cosma ) , α 33 = a 3 R 3 8 k 5 ( 1 3 2 + 6 4 + . . . ) , α 40 = 3 a 4 R 2 8 k 5 ( 1 4 2 + 41 4 4 + . . . ) ( spherical aberration ) , α 42 = a 4 R 2 4 k 5 ( 1 5 2 + 115 8 4 + . . . ) .
z ( ρ , θ ) = a 2 2 l ρ 2 + a 4 8 l 3 ρ 4 + a 6 16 l 5 ρ 6 + . . . .
z ( ρ , θ ) = n = 0 m = n n C n m Z n m ( ρ , θ ) n m even .
σ = ( n = 0 m = n n w n m C n m ) 1 / 2 , where w n m = 1 + δ m , 0 2 ( n + 1 ) ,
δ R = R 8 C 22 δ C 22 + 3 C 31 δ C 31 16 C 22 2 + 3 C 31 2 δ ϕ = 8 C 22 δ C 2 2 + 3 C 31 δ C 3 1 16 C 22 2 + 3 C 31 2 } ,
C 00 = α 00 + α 20 2 + α 40 3 + . . .
C 1 1 = α 1 1 + 2 3 α 3 1 + . . .
1 4
[ k = 368.80 cm R = 35.00 cm ϕ = 0 . 0 ]
[ k = 368.68 cm R = 35.79 cm ϕ = 0.0043 ]
[ k = 368.83 cm R = 35.07 cm ϕ = 0.0060 ]
C 11 = α 11 + 2 3 α 31 + . . .
1 4
C 2 2 = α 2 2 + 3 4 α 4 2 + . . .
1 6
C 20 = α 20 2 + α 40 2 + . . .
1 3
C 22 = α 22 + 3 4 α 42 + . . .
1 6
1 8
C 3 1 = α 3 1 3 + . . .
1 8
C 31 = α 31 3 + . . .
1 8
1 8
1 10
C 4 2 = α 4 2 4 + . . .
1 10
C 40 = α 40 6 + . . .
1 5
C 42 = α 42 4 + . . .
1 10
1 10
D 4 ω = q .
ω = q 00 32 D 3 + ν 1 + ν ρ 2 + q 00 64 D ρ 4 ,
q ( ρ , θ ) = n = 0 m = n n q n m Z n m ( ρ , θ ) n m even .
ω ( ρ , θ ) = n = 0 m = n n C n m Z n m ( ρ , θ ) n m even .
C 20 = 7 + 2 ν 480 ( 1 + ν ) q 20 D , C 22 = 303 + 18 ν ν 2 3840 ( 1 ν ) ( 3 + ν ) q 22 D , C 31 = 41 + 7 ν 5760 ( 3 + ν ) q 31 D , C 40 = 1 576 q 20 D , C 42 = 1 2304 15 + ν 3 + ν q 22 D , C 51 = 1 1608 q 31 D , C 60 = 1 5760 q 20 D , C 62 = 1 5760 q 22 D , C 71 = 1 13440 q 31 D .
i j = q i j q 00 , r i j = C i j C 40 0 ,
r 20 = 4.80 20 , r 51 = 0.23 31 , r 22 = 12.61 22 , r 60 = 0.067 20 , r 31 = 0.88 31 , r 62 = 0.067 22 , r 40 = 0.67 20 , r 71 = 0.029 31 . r 42 = 0.78 22 ,

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