Abstract

The theoretical basis is developed for a technique to fabricate nonaxisymmetric mirrors. Stresses are applied to a mirror blank that would have the effect of elastically deforming a desired surface into a sphere. A sphere is then polished into the blank, and upon release of the applied stress, the spherical surface deforms into the desired one. The method can be applied iteratively, so arbitrary accuracy should be possible. Calculations of the stresses and deformations are carried out in detail for an off-axis section of a paraboloid. For a very general class of surfaces, it is sufficient to only impose appropriate stresses at the edge of the blank plus a uniform pressure on the back.

© 1980 Optical Society of America

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References

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  1. 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.
  2. J. Nelson, Proc. Soc. Photo-Opt. Instrum. Eng. 172, (January1979).
  3. G. Gabor, Proc. Soc. Photo-Opt. Instrum. Eng. 172, (January1979).
  4. T. S. Mast, J. E. Nelson, “Figure Control for a Segmented Telescope Mirror,” Lawrence Berkeley Laboratory Report LBL-8621 (March1979).
  5. J. E. Nelson, G. Gabor, L. K. Hunt, J. Lubliner, T. S. Mast, Appl. Opt. 19, 2341 (1980).
    [CrossRef] [PubMed]
  6. E. Everhart, Appl. Opt. 5, 713 (1966).
    [CrossRef] [PubMed]
  7. G. LeMaitre, Nouv. Rev. Opt. 6, 361 (1974).
    [CrossRef]
  8. A. S. Leonard, in Proceedings 69, Joint Convention of Western Amateur Astronomers and Association of Lunar and Planetary Observers (1969).
  9. L. W. Alvarez, Lawrence Berkeley Laboratory; private communication (1978).
  10. W. A. Eul, W. W. Woods, Optical Telescope Technology, NASA SP-233 (U.S. GPO, Washington, D.C., 1969), p. 207.
  11. S. Timoshenko, Strength of Materials (Van Nostrand, New York, 1955).
  12. A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1927).
  13. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959).

1980 (1)

1979 (2)

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

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

1974 (1)

G. LeMaitre, Nouv. Rev. Opt. 6, 361 (1974).
[CrossRef]

1969 (1)

A. S. Leonard, in Proceedings 69, Joint Convention of Western Amateur Astronomers and Association of Lunar and Planetary Observers (1969).

1966 (1)

Alvarez, L. W.

L. W. Alvarez, Lawrence Berkeley Laboratory; private communication (1978).

Eul, W. A.

W. A. Eul, W. W. Woods, Optical Telescope Technology, NASA SP-233 (U.S. GPO, Washington, D.C., 1969), p. 207.

Everhart, E.

Gabor, G.

Hunt, L. K.

LeMaitre, G.

G. LeMaitre, Nouv. Rev. Opt. 6, 361 (1974).
[CrossRef]

Leonard, A. S.

A. S. Leonard, in Proceedings 69, Joint Convention of Western Amateur Astronomers and Association of Lunar and Planetary Observers (1969).

Love, A. E. H.

A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1927).

Lubliner, J.

Mast, T. S.

J. E. Nelson, G. Gabor, L. K. Hunt, J. Lubliner, T. S. Mast, Appl. Opt. 19, 2341 (1980).
[CrossRef] [PubMed]

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. E. Nelson, G. Gabor, L. K. Hunt, J. Lubliner, T. S. Mast, Appl. Opt. 19, 2341 (1980).
[CrossRef] [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, Strength of Materials (Van Nostrand, New York, 1955).

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).

Woods, W. W.

W. A. Eul, W. W. Woods, Optical Telescope Technology, NASA SP-233 (U.S. GPO, Washington, D.C., 1969), p. 207.

Appl. Opt. (2)

Nouv. Rev. Opt. (1)

G. LeMaitre, Nouv. Rev. Opt. 6, 361 (1974).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

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

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

Proceedings 69, Joint Convention of Western Amateur Astronomers and Association of Lunar and Planetary Observers (1)

A. S. Leonard, in Proceedings 69, Joint Convention of Western Amateur Astronomers and Association of Lunar and Planetary Observers (1969).

Other (7)

L. W. Alvarez, Lawrence Berkeley Laboratory; private communication (1978).

W. A. Eul, W. W. Woods, Optical Telescope Technology, NASA SP-233 (U.S. GPO, Washington, D.C., 1969), p. 207.

S. Timoshenko, Strength of Materials (Van Nostrand, New York, 1955).

A. E. H. Love, Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1927).

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

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

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 (4)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Coefficients describing deflections needed to transform a sphere into parabola as defined by Eq. (7). Paraboloid with k = 40 m and segments with a = 0.7 m is assumed, and coefficients as function of off-axis distance are shown. Best fitting sphere is assumed. Root-mean-square deflection is also shown.

Fig. 3
Fig. 3

Diagram showing application of shear force V and couple M at the edge of the plate and a uniform pressure q on the back of the plate. These are types of external forces needed to deflect a sphere into an off-axis paraboloid.

Fig. 4
Fig. 4

Maximum stress induced in mirror segment during bending is shown as function of off-axis distance. Lowest stress sphere is used, and paraboloidal segment with k = 40 m, a = 0.70 m, and h = 0.10 m. Material (CerVit) has E = 9 × 105 kg/cm2 and ν = 0.25.

Equations (59)

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m = 0 n = 0 , m ( α mn cos n θ + β mn sin θ ) ρ m · m n even
n ~ ( 0 d ) n d .
Z = ( X 2 + Y 2 ) / 2 k .
X = R + cx sz , Y = y , Z = ( R 2 / 2 k ) + cz + sx .
R 2 2 k + cz + sx = 1 2 k [ ( R + cx sz ) 2 + y 2 ] .
c s 2 z 2 2 ( k + c 2 sx ) z + c ( c 2 x 2 + y 2 ) = 0 ,
z = 1 c s 2 [ k + c 2 sx ( k 2 + 2 c 2 skx c 2 s 2 y 2 ) 1 / 2 ] .
z = c 2 k ( c 2 x 2 + y 2 ) c 3 s 2 k 2 x ( c 2 x 2 + y 2 ) + c 3 s 2 8 k 3 ( c 2 x 2 + y 2 ) ( 5 c 2 x 2 + y 2 ) + O ( 2 r 5 / k 4 ) ,
z = z 0 [ l 2 y 2 ( x x 0 ) 2 ] 1 / 2 ,
z = z 0 ( l 2 x 0 2 ) 1 / 2 ( x 0 l + x 0 3 2 l 3 ) x + r 2 2 l + x 0 2 4 l 3 ( 3 x 2 + y 2 ) x 0 x r 2 2 l 3 + r 2 8 l 3 + O ( x 0 r 5 l 5 , r 6 l 5 ) .
z = r 2 2 l + r 4 8 l 3 + O ( r 6 / l 5 ) .
w = 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 + neglected terms .
α 20 = a 2 2 k ( k l 1 + 2 + 9 8 4 + 5 4 6 + . . . ) focus , α 22 = a 2 4 k 2 ( 1 3 2 2 + 15 8 4 + . . . . . ) astigmatism , α 31 = ( a 3 / 2 k 2 ) ( 1 11 4 2 + 21 4 4 + . . . . ) coma , α 33 = ( a 3 / 8 k 2 ) 3 ( 1 3 2 + 6 4 + . . . . . ) α 40 = ( a 4 / 8 k 3 ) [ ( k l ) 3 3 2 ( 1 4 2 + . . . ) ] spherical aberration , α 42 = ( a 4 / 4 k 3 ) 2 ( 1 5 2 + . . . . ) .
l = l 0 ( 1 + a 2 4 l 0 2 ) ,
q = q 0 + q 1 x + q 2 y .
w ( ρ , θ ) = n = 0 [ ( α nn ρ n + α n + 2 , n ρ n + 2 ) cos n θ + ( β nn ρ n + β n + 2 , n ρ n + 2 ) sin θ ] + α 40 ρ 4 + α 51 ρ 5 cos θ + β 51 ρ 5 sin θ .
α 40 = q 0 a 4 / 64 D , α 51 = q 1 a 5 / 192 D , β 51 = q 2 a 5 / 192 D .
M ( θ ) = M 0 + n = 1 ( M n cos n θ + M ¯ n sin n θ ) , V ( θ ) = V 0 + n = 1 ( V n cos n θ + V ̅ n sin n θ ) .
V 0 = q 0 a / 2 M 1 + a V 1 = q 1 a 3 / 4 M ̅ 1 + a V ̅ 1 = q 2 a 3 / 4 .
α 20 = α 20 + 2 ( h a ) 2 α 40 α 31 = α 31 + 3 ν 1 ν ( h a ) 2 α 51 α 33 = α 33 1 1 ν ( h a ) 2 α 51 β 31 = β 31 + 3 ν 1 ν ( h a ) 2 β 51 β 33 = β 33 + 1 1 ν ( h a ) 2 β 51
α nn u = α nn + 2 5 ( 2 ν 1 ν ) ( h a ) 2 ( n + 1 ) α n + 2 , n , β nn u = β nn + 2 5 ( 2 ν 1 ν ) ( h a ) 2 ( n + 1 ) β n + 2 , n ,
α n + 2 , n u = α n + 2 , n , β n + 2 , n u = β n + 2 , n .
w u = α 20 u + n = 1 [ ( α nn u ρ n + α n + 2 , n u ρ n + 2 ) cos n θ + ( β nn u ρ n + β n + 2 , n u ρ n + 2 ) sin n θ ]
M n u = D a 2 { ( 1 ν ) n ( n 1 ) α nn u + ( n + 1 ) [ n + 2 ν ( n 2 ) ] α n + 2 , n u } V n u = D a 3 { ( 1 ν ) n 2 ( n 1 ) α nn u + n ( n + 1 ) ( n 4 ν n ) α n + 2 , n u } .
M 0 = M 0 u + q 0 a 2 [ 3 + ν 16 + 3 ν 80 ( h a ) 2 ] V 0 = q 0 a / 2 M 1 = M 1 u + q 1 a 3 [ 5 + ν 48 + 9 + ν 160 ( h a ) 2 ] V 1 = V 1 u q 1 a 2 [ 17 + ν 48 + 9 + ν 160 ( h a ) 2 ] M ̅ 1 = M ̅ 1 u + q 2 a 3 [ 5 + ν 48 + 9 + ν 160 ( h a ) 2 ] V ̅ 1 = V ̅ 1 u q 2 a 3 [ 17 + ν 48 + 9 + ν 160 ( h a ) 2 ] M 3 = M 3 u + q 1 a h 2 8 V 3 = V 3 u + 3 8 q 1 h 2 .
w B = w ( 1 ) + w ( 2 ) + . . . , M B = M ( 1 ) + M ( 2 ) + . . . , V B = V ( 1 ) + V ( 2 ) + . . . ,
w ( x , y ) = ( k l ) ( x 2 + y 2 ) 2 k 2 + 1 8 k 3 ( r 4 4 R 3 x R 4 ) ,
σ max ( loc ) = Eh 2 ( 1 ν ) k max | k l k + ( r k ) 2 ± 1 ν 2 ( 1 + ν ) ( r k ) 2 | .
p = 2 ( 1 ν ) σ all k Eh .
| k l k + ( r k ) 2 ± 1 ν 2 ( 1 + ν ) ( r k ) 2 | p
l min l l max
l min = k [ 1 + 3 + ν 2 ( 1 + ν ) ( R + a k ) 2 p ] , = k ( 1 + p ) if R a , l max = k [ 1 + p + 1 + 3 ν 2 ( 1 + ν ) ( R a k ) 2 ] if R a .
f ( R ) = 3 + ν 4 ( 1 + ν ) ( R + a k ) 2 if R a = 1 4 ( 1 + ν ) [ ( 3 + ν ) ( R + a k ) 2 ( 1 + 3 ν ) ( R a k ) 2 ] if R a .
l = 1 2 ( l max + l min ) ,
σ max = Eh 2 ( 1 ν ) k f ( R ) .
M α β = D [ ( 1 ν ) w , α β + ν δ α β 2 w ] ,
M α β = h / 2 h / 2 x 3 σ α β d x 3 .
σ α β = 12 M α β x 3 / h 3 ,
M max = max | M 11 + M 22 2 ± [ ( M 11 M 22 2 ) 2 + M 12 2 ] 1 / 2 | ;
Q α = M α β , β = D 2 w , α .
q = Q α , α = D 4 w .
M rr = D [ 2 w r 2 + ν ( 1 r w r + 1 r 2 2 w θ 2 ) ] ,
M θ θ = D ( 1 r w r + 1 r 2 2 w θ 2 + ν 2 w r 2 ) ,
M r θ = D ( 1 ν ) ( 1 r 2 w r θ 1 r 2 w θ ) ,
Q r = D r ( 2 w r 2 + 1 r w r + 1 r 2 2 w θ 2 ) ,
Q θ = D 1 r θ ( 2 w r 2 + 1 r w r + 1 r 2 2 w θ 2 ) .
V = ( Q r 1 r M r θ θ ) r = a .
M ( θ ) = M 0 + n = 1 ( M n cos n θ + M ̅ n sin n θ ) , V ( θ ) = V 0 + n = 1 ( V n cos n θ + V ̅ n sin n θ ) , q ( r , θ ) = q 0 + q 1 r cos θ + q 2 r sin θ ,
{ M 0 = D a 2 [ ( 2 + ν ) α 20 + 4 ( 3 + ν ) α 40 ] , V 0 = D a 3 ( 32 α 40 ) , M 1 = D a 2 [ 2 ( 3 + ν ) α 31 + 4 ( 5 + ν ) α 51 ] , V 1 = D a 3 [ 2 ( 3 + ν ) α 31 + 4 ( 17 + ν ) α 51 ] , M ̅ 1 = D a 2 [ 2 ( 3 + ν ) β 31 + 4 ( 5 + ν ) β 51 ] , V ̅ 1 = D a 3 [ 2 ( 3 + ν ) β 31 + 4 ( 17 + ν ) β 51 ] n > 1 ; M n = D a 2 { ( 1 ν ) n ( n 1 ) α nn + ( n + 1 ) [ n + 2 ν ( n 2 ) ] α n + 2 , n } , M ̅ n = D a 2 { ( 1 ν ) n ( n 1 ) β nn + ( n + 1 ) [ n + 2 ν ( n 2 ) ] β n + 2 , n } , V n = D a 3 [ ( 1 ν ) n 2 ( n 1 ) α nn + n ( n + 1 ) ( n 4 ν n ) α n + 2 , n ] , V ̅ n = D a 3 [ ( 1 ν ) n 2 ( n 1 ) β nn + n ( n + 1 ) ( n 4 ν n ) β n + 2 , n ] , q 0 = 64 D α 40 / a 4 , q 1 = 192 D α 51 / a 5 , q 2 = 192 D β 51 / a 5 .
u 3 = 1 E [ ( 1 + ν ) χ 1 + ( h 2 4 1 2 ν x 3 2 ) Θ 1 ] .
w 0 = 1 E [ ( 1 + ν ) χ 1 + h 2 4 Θ 1 ] , w 1 = 1 E [ ( 1 + ν ) χ 1 + ( 1 ν 2 ) Θ 1 ] = w 0 ν h 2 8 E Θ 1 .
Θ 1 = E 1 ν 2 w 0 = E 1 ν 2 w 1 .
w 0 = w 1 ν h 2 8 ( 1 ν ) 2 w 1 .
σ α β = E 1 ν 2 { [ ν 2 w 0 δ α β + ( 1 ν ) w 0 , α β ] x 3 + [ h 2 4 x 3 1 6 ( 2 ν ) x 3 3 ] 2 w 0 , α β } .
M α β = D [ ( 1 ν ) w 0 , α β + ν 2 w 0 δ α β + 8 + ν 40 h 2 2 w 0 , α β ] .
w = w 0 + 1 40 8 + ν 1 ν h 2 2 w 0 ,
w = w 1 + 1 10 2 ν 1 ν h 2 2 w 1 .
w 1 = q 0 64 D [ r 4 2 h 2 r 2 + 3 + ν 6 ( 1 ν ) h 4 ] , V = q 0 a / 2 , M = 3 + ν 16 q 0 a 2 + 3 ν 80 q 0 h 2 .
w 1 = q 1 192 D { r 5 cos θ h 2 r 3 1 ν [ ( 3 ν ) cos θ cos 3 θ ] 3 + ν 2 ( 1 + ν ) h 4 r cos θ } , V = q 1 ( 17 + ν 48 a 2 cos θ + 9 ν 160 h 2 cos θ 3 32 h 2 cos 3 θ ) , M = q 1 a ( 5 + ν 48 a 2 cos θ + 9 ν 160 h 2 cos θ + 3 32 h 2 cos 3 θ ) .

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