Abstract

We have evaluated the principal optical characteristics of paraboloid-hyperboloid x-ray telescopes by a ray-tracing procedure; we find that our results for resolution, focal plane curvature, and finite source distance effects may be approximated in terms of the design parameters by simple empirical formulas.

© 1972 Optical Society of America

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References

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  1. R. Giacconi, B. Rossi, J. Geophy. Res. 65, 773 (1960).
    [CrossRef]
  2. H. Wolter, Ann. Phys. 10, 94 (1952).
    [CrossRef]
  3. R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
    [CrossRef]
  4. J. D. Mangus, J. H. Underwood, Appl. Opt. 8, 95 (1969).
    [CrossRef] [PubMed]
  5. This term is easily derived for short cone optics and must be present for any two surface mirror with nonzero first surface slope at the intersection of the two surfaces. The focal plane coordinates of an off-axis ray in the x-z plane, which strikes near the intersection are given by xi=-Z0 tanθ(cos2 2α/cos24α) (1+cos2ϕ tan22α)1-tanθ cosϕ tan4αyi=-Z0 tanθ(cos22α/cos24α) (sin2ϕ tan22α)1-tanθ cosϕ tan4α, where ϕ is the azimuthal angle at which the ray strikes the surface. This result does not depend on ξ.
  6. H. Wolter, Ann. Phys. 10, 286 (1952).
    [CrossRef]

1969 (2)

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

J. D. Mangus, J. H. Underwood, Appl. Opt. 8, 95 (1969).
[CrossRef] [PubMed]

1960 (1)

R. Giacconi, B. Rossi, J. Geophy. Res. 65, 773 (1960).
[CrossRef]

1952 (2)

H. Wolter, Ann. Phys. 10, 94 (1952).
[CrossRef]

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

Giacconi, R.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

R. Giacconi, B. Rossi, J. Geophy. Res. 65, 773 (1960).
[CrossRef]

Mangus, J. D.

Reidy, W. P.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Rossi, B.

R. Giacconi, B. Rossi, J. Geophy. Res. 65, 773 (1960).
[CrossRef]

Underwood, J. H.

Vaiana, G. S.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

VanSpeybroeck, L. P.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Wolter, H.

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

H. Wolter, Ann. Phys. 10, 94 (1952).
[CrossRef]

Zehnpfennig, T. F.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Ann. Phys. (2)

H. Wolter, Ann. Phys. 10, 94 (1952).
[CrossRef]

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

Appl. Opt. (1)

J. Geophy. Res. (1)

R. Giacconi, B. Rossi, J. Geophy. Res. 65, 773 (1960).
[CrossRef]

Space Sci. Rev. (1)

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Other (1)

This term is easily derived for short cone optics and must be present for any two surface mirror with nonzero first surface slope at the intersection of the two surfaces. The focal plane coordinates of an off-axis ray in the x-z plane, which strikes near the intersection are given by xi=-Z0 tanθ(cos2 2α/cos24α) (1+cos2ϕ tan22α)1-tanθ cosϕ tan4αyi=-Z0 tanθ(cos22α/cos24α) (sin2ϕ tan22α)1-tanθ cosϕ tan4α, where ϕ is the azimuthal angle at which the ray strikes the surface. This result does not depend on ξ.

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Figures (10)

Fig. 1
Fig. 1

A representation of the type of paraboloid-hyperboloid telescopes discussed in this paper. The back hyperboloid focus is confocal with the paraboloid focus. The front focus of the hyperboloid is also the focus of the telescope.

Fig. 2
Fig. 2

The rms blue circle radius is given as a function of the incident angle for one value of (A/Z02) and several values of α. The solid lines were calculated using Eq. (4) while the points are the results of the Monte Carlo ray-tracing procedure.

Fig. 3
Fig. 3

The rms blue circle radius is given as a function of α for several values of (A/Z02). θ is fixed at 5′. The curves were calculated using Eq. (4).

Fig. 4
Fig. 4

The rms blur circle radius is given as a function of α for two values of (A/Z02). θ is fixed at 10′. The curves were calculated using Eq. (4).

Fig. 5
Fig. 5

The optimum value of α is given as a function of θ for several values of (A/Z02). The curves were calculated using Eq. (5).

Fig. 6
Fig. 6

The effective area of a typical telescope as a function of x-ray wavelength, λ, for two values of θ. The mirror surface is made of a nickel alloy, kanigen.

Fig. 7
Fig. 7

A plot of the vignetting factor, V(α,θ), as a function of θ for several values of α.

Fig. 8
Fig. 8

The rms blur circle radius is given as a function of the finite source distance divided by the focal length. The results are shown for several values of (Lp/Z0) and α.

Fig. 9
Fig. 9

The rms blur circle radius for a flat focal plane at θ = 10′ is given as a function of effective area at θ = 0 for specific telescopes. The effect of Lp and the number of surfaces are shown. Nine points have been calculated, and the lines have been sketched in to interpolate approximately.

Fig. 10
Fig. 10

The telescope effective area is given as a function of the mirror wall thickness for a three-surface telescope and a five-surface telescope.

Equations (16)

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r p 2 = P 2 + 2 P Z + [ 4 e 2 P d / ( e 2 - 1 ) ]             ( paraboloid ) , r h 2 = e 2 ( d + Z ) 2 - Z 2             ( hyperboloid ) .
L h L p = P e d + ( e - 1 ) L p ξ 1 + ( ξ L p / Z 0 ) .
A effective = A ( 1 / N ) U 0 , σ 2 = σ x 2 + σ y 2 , σ x 2 = ( U x x / U 0 ) - ( U x / U 0 ) 2 ,
U 0 = i i p i h , U x = i x i i p i h , U x x = i x i 2 i p i h .
σ D = ( ξ + 1 ) 10 tan 2 θ tan α ( L p Z 0 ) + 4 tan θ tan 2 α ,
( / 2 ) α ( / 2 )             0 θ 3 0 0.035 ( L p / Z 0 ) 0.176 ( 1 / 4 ) ξ 4.
A = ( 2 π r 0 ) ( L p tan α ) ,
σ D ( 1 40 π ) tan 2 θ tan 3 α ( A Z 0 2 ) + 4 tan θ tan 2 α .
tan α + = [ ( 3 / 320 π ) tan θ ( A / Z 0 2 ) ] 1 5 .
δ = 0.055 ( ξ + 1 ) ( r 2 L p / Z 0 2 ) ( 1 / tan α ) 2 .
A effective = ( A ) [ V ( α , θ ) ] [ R ( α , λ ) ] ,
V ( α , θ ) 1 - ( 2 / 3 ) ( θ / α )             ( θ < α ) .
( 1 / p ) + ( 1 / q ) = 1 / z 0 ,
σ D = 4 ( L p / Z 0 ) tan α ( Z 0 / p ) 2
A ( P ) [ 1 - ( 5 Z 0 / 2 p ) ] / [ 1 + ( 5 Z 0 / 2 p ) ] .
xi=-Z0tanθ(cos22α/cos24α)(1+cos2ϕtan22α)1-tanθcosϕtan4αyi=-Z0tanθ(cos22α/cos24α)(sin2ϕtan22α)1-tanθcosϕtan4α,

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