Abstract

Simply by push-pull shear action at their outer edge, large circular telescope mirrors can be laterally supported free from bending distortion, if the upright mirror is kept in balance by a cosine distribution of weak axial forces applied at its outer rim and possibly also at the central hole. The flexure-induced comalike wave-front aberration of a thin 8-m meniscus mirror was reduced to an rms value of 0.5 nm over the full aperture; the largest path difference was reduced to 2.4 nm. A comparable result has also been calculated for a meniscus mirror of different geometry and a material with different elastic constants. Realizing such possibilities in practice demands accurate engineering. A helpful artifice is investigated for the correct application of the tangential supporting forces.

© 1994 Optical Society of America

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References

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  1. G. Schwesinger, “Lateral support of very large telescope mirrors by edge forces only,” J. Mod. Opt. 38, 1507–1516 (1991).
    [CrossRef]
  2. L. Noethe, European Southern Observatory, Garching, Germany (personal communication, 1992).
  3. F. I. Niordson, Shell Theory, Vol. 29 of North-Holland Series in Applied Mathematics and Mechanics (North-Holland, Amsterdam, 1985), p. 105.

1991 (1)

G. Schwesinger, “Lateral support of very large telescope mirrors by edge forces only,” J. Mod. Opt. 38, 1507–1516 (1991).
[CrossRef]

Niordson, F. I.

F. I. Niordson, Shell Theory, Vol. 29 of North-Holland Series in Applied Mathematics and Mechanics (North-Holland, Amsterdam, 1985), p. 105.

Noethe, L.

L. Noethe, European Southern Observatory, Garching, Germany (personal communication, 1992).

Schwesinger, G.

G. Schwesinger, “Lateral support of very large telescope mirrors by edge forces only,” J. Mod. Opt. 38, 1507–1516 (1991).
[CrossRef]

J. Mod. Opt. (1)

G. Schwesinger, “Lateral support of very large telescope mirrors by edge forces only,” J. Mod. Opt. 38, 1507–1516 (1991).
[CrossRef]

Other (2)

L. Noethe, European Southern Observatory, Garching, Germany (personal communication, 1992).

F. I. Niordson, Shell Theory, Vol. 29 of North-Holland Series in Applied Mathematics and Mechanics (North-Holland, Amsterdam, 1985), p. 105.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the mirror meniscus and applied edge forces: θ, azimuth angle counted from the direction of gravity; w, component of the elastic deflection along the outer normal in the plane θ = 0; a, radius of curvature of the middle surface; R, outer radius of the meniscus; R 0, radius of the central hole; z (= ζa), distance of the center of gravity from the outer midedge; N, support force with radial component N r and tangential shearing component N t ; t (=τa), distance between the outer mid-edge and the plane of support forces N; V cos θ, axial force at the outer edge; V 0 cos θ, axial force at the inner edge (central hole).

Fig. 2
Fig. 2

Deflections w of mirror 1 over its normalized radius for optimized shear fractions β assuming that (1) both edges are free of corrective forces (curve A) and (2) the outer edge is bent by a corrective moment of suitable amount equivalent to the parameter value τ (curve B).

Fig. 3
Fig. 3

Deflections w of mirror 1 over its normalized radius for optimized shear fractions β assuming (1) both edges to be free of corrective forces (same as curve A as in Fig. 2), (2) at the inner edge a corrective distribution of axial forces V 0 cos θ, represented by the parameter value ɛ0 (curve C), (3) in addition to the axial forces of case (2) a corrective bending moment at the outer edge, equivalent to the parameter value τ (curve D).

Fig. 4
Fig. 4

Deflections w of mirror 2 over its normalized radius for optimized shear fractions β assuming that (1) both edges are free of corrective forces (curve E) and (2) at the inner edge a corrective distribution of axial forces V 0 cos θ, corresponding to the parameter value ɛ0 (curve F).

Fig. 5
Fig. 5

Extraperipheral application of the tangential supporting force.

Tables (2)

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Table 1 Geometry and Elastic Constants of Two Analyzed Meniscus Mirrors

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Table 2 Result of Reducing rms w by Optimum Choice of Shear Fraction β and Parameters τ and ɛ0 a

Equations (11)

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N r = P r cos θ ,
N t = P t sin θ ,
0 2 π V 0 cos θ × R 0 cos θ × R 0 d θ = π R 0 2 V 0 = ɛ 0 G a .
ɛ 0 - ɛ + τ - ζ = 0.
β = P t P t + P r .
M = N t b = P t b sin θ .
U r = - d M / d s = - ( d M / d θ ) / R = - P t b cos θ / R .
U t = M d θ / d s = M / R = P t b sin θ / R .
P r = P r + P t b / R ,
P t = P t ( 1 - b / R ) .
β = P t / ( P t + P r ) = P t ( 1 - b / R ) / ( P t + P r ) , β = β ( 1 - b / R ) .

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