Abstract

The optical performance of pairs of toroidal mirrors in grazing incidence has been studied analytically and numerically. Two types of toroidal surface are possible: football and bicycle tire. In grazing incidence and for configurations that compensate up to second-order aberrations, there are significant differences in performance between the two types. For football-type tori the best configuration appears to be Z-shaped with tangential and sagittal foci at the middle point between the mirrors. For bicycle tire-type tori the best configuration is U-shaped with the tangential focus at the middle point and the sagittal at infinity.

© 1983 Optical Society of America

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References

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  1. J. H. Underwood, Space Sci.Instrum. 1, 289 (1975).
  2. R. C. Chase, J. K. Silk, Appl.Opt. 14, 2096 (1975).
    [CrossRef] [PubMed]
  3. R. J. Speer, Space Sci.Instrum. 2, 463 (1976).
  4. A. M. Malvezzi, L. Garifo, G. Tondello, Appl. Opt. 20, 2560 (1981).
    [CrossRef] [PubMed]
  5. D. E. Aspnes, Appl.Opt. 21, 2642 (1982).
    [CrossRef] [PubMed]
  6. H. Haber, J. Opt. Soc. Am. 40, 153 (1950).
    [CrossRef]

1982 (1)

D. E. Aspnes, Appl.Opt. 21, 2642 (1982).
[CrossRef] [PubMed]

1981 (1)

1976 (1)

R. J. Speer, Space Sci.Instrum. 2, 463 (1976).

1975 (2)

J. H. Underwood, Space Sci.Instrum. 1, 289 (1975).

R. C. Chase, J. K. Silk, Appl.Opt. 14, 2096 (1975).
[CrossRef] [PubMed]

1950 (1)

Aspnes, D. E.

D. E. Aspnes, Appl.Opt. 21, 2642 (1982).
[CrossRef] [PubMed]

Chase, R. C.

R. C. Chase, J. K. Silk, Appl.Opt. 14, 2096 (1975).
[CrossRef] [PubMed]

Garifo, L.

Haber, H.

Malvezzi, A. M.

Silk, J. K.

R. C. Chase, J. K. Silk, Appl.Opt. 14, 2096 (1975).
[CrossRef] [PubMed]

Speer, R. J.

R. J. Speer, Space Sci.Instrum. 2, 463 (1976).

Tondello, G.

Underwood, J. H.

J. H. Underwood, Space Sci.Instrum. 1, 289 (1975).

Appl. Opt. (1)

Appl.Opt. (2)

R. C. Chase, J. K. Silk, Appl.Opt. 14, 2096 (1975).
[CrossRef] [PubMed]

D. E. Aspnes, Appl.Opt. 21, 2642 (1982).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Space Sci.Instrum. (2)

J. H. Underwood, Space Sci.Instrum. 1, 289 (1975).

R. J. Speer, Space Sci.Instrum. 2, 463 (1976).

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Figures (3)

Fig. 1
Fig. 1

Configurations with two identical toroidal mirrors M and M1. The distances between the source A and M, between M and the middle point C, between C and M1 and between M1 and the image point B are all equal. The upper part is a view of the tangential plane of the system. The lower part is a view of the sagittal plane. Two orientations are possible in the tangential plane: U-shaped, solid line; or Z-shaped, dotted line. Two configurations are considered in the sagittal plane: one with the sagittal focus in C (s = 0) and one with the sagittal focus at infinity (s = ∞).

Fig. 2
Fig. 2

The two different toroidal surfaces. The upper part is a bicycle tire type of torus. The axis of rotational symmetry of the surface (dotted line) is parallel to the z axis. In the lower part, a football type of torus is drawn. In this case the axis of rotational symmetry is contained in the (x = y) plane. P(u, w, l) is a point on the surface of the toroidal surface.

Fig. 3
Fig. 3

Calculated image plane patterns for the two configurations Z0/0 (left) and U∞/0 (right) in the case of football type of tori (solid lines) and bicycle tire type of tori (dotted lines). A 5-mrad semiaperture fan of rays has been considered. The figures refer to (from top to bottom): a point source A on-axis (x = 0,z = 0 mm), a point source A′ off-axis in the tangential direction (x = 1,z = 0 mm), and a point source off-axis in the sagittal direction (x = 0,z = 1 mm). The origin of the coordinates is at the geometrical conjugate of the on-axis point source A. The length scales are in micrometers and they are all equal except in the middle-right figure, where the scale of the x axis has been increased by a factor of 10. The axes cross at each figure at the predicted position of the geometrical image.

Tables (1)

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Table I Aberrations in the x and z Directions for Toroidal Mirror Pairs in the U∞/0 and Z0/0 Configurations a

Equations (2)

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[ ( R u ) 2 + w 2 ( R ρ ) ] 2 = ρ 2 l 2 .
[ ( ρ u ) 2 + l 2 + ( R ρ ) ] 2 = R 2 w 2 .

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