Abstract

Diffraction from secondary mirror spiders can significantly affect the image quality of optical telescopes; however, these effects vary drastically with the chosen image-quality criterion. Rigorous analytical calculations of these diffraction effects are often unwieldy, and virtually all commercially available optical design and analysis codes that have a diffraction-analysis capability are based on numerical Fourier-transform algorithms that frequently lack an adequate sampling density to model narrow spiders. The effects of spider diffraction on the Strehl ratio (or peak intensity of the diffraction image), full width at half-maximum of the point-spread function, the fractional encircled energy, and the modulation transfer function are discussed in detail. A simple empirical equation is developed that permits accurate engineering calculations of fractional encircled energy for an arbitrary obscuration ratio and spider configuration. Performance predictions are presented parametrically in an attempt to provide insight into this sometimes subtle phenomenon.

© 1995 Optical Society of America

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References

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  1. A. Couder, “Dealing with spider diffraction,” in Amateur Telescope Making, Advanced (Book Two), A. G. Ingalls, ed. (Scientific American, New York, 1946), pp. 260–262.
  2. R. E. Cox, “Spider diffraction in moderate-size telescopes,” Sky Telesc.166–171 (Sept.1960).
  3. R. C. Ludden, “A 10-inch reflector fashioned in wood,” Sky Telesc.112–114 (Feb.1969).
  4. C. H. Werenskiold, “A note on curved spiders,” Sky Telesc.262–263 (Oct.1969).
  5. R. E. Cox, “Secondary mirrors and spiders,” Telesc. Making 7, 4–7 (Spring1980).
  6. W. A. Rhodes, “An antidiffraction mask for reflectors,” Sky Telesc.289–290 (Apr.1957).
  7. Kenneth Novak & Co., Catalog, Box 69, Ladysmith, Wisc. 54848.
  8. C. H. Werenskiold, “Improved telescope spider design,” J. R. Astron. Soc. Can. 35, 268–273 (1941).
  9. E. Everhart, J. Kantorski, “Diffraction effects produced by obscurations in reflecting telescopes of modest size,” Astron. J. 64, 455–463 (1959).
    [CrossRef]
  10. J. L. Richter, “Spider diffraction: a comparison of curved and straight legs,” Appl. Opt. 23, 1907–1913 (1984).
    [CrossRef] [PubMed]
  11. L. M. Beyer, L. C. Clune, “Intensity and encircled energy for circular pupils obscured by strut supported central obscurations,” Appl. Opt. 27, 5067–5071 (1988).
    [CrossRef] [PubMed]
  12. P. P. Clark, J. W. Howard, E. R. Freniere, “Asymptotic approximation to the encircled energy function for arbitrary aperture shapes,” Appl. Opt. 23, 353–357 (1984).
    [CrossRef] [PubMed]
  13. P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 107, 63–69 (1977).
  14. H. F. A. Tschunko, P. J. Sheehan, “Aperture configuration and imaging performance,” Appl. Opt. 10, 1432–1438 (1971).
    [CrossRef] [PubMed]
  15. A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 61, 113.
  17. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 194, 216.
  18. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, New York, 1950), Chap. 2, p. 21.
  19. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 112.
  20. A. A. Dantzler, “Encircled energy correction method for ray-trace programs,” Appl. Opt. 27, 5001–5002, (1988).
    [CrossRef] [PubMed]
  21. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8, p. 381.
  22. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 10, p. 458.
  23. M. V. Klein, Optics (Wiley, New York, 1970), Chap. 7, p. 298.
  24. H. F. A. Tschunko, “Imaging performance of annular apertures,” Appl. Opt. 13, 1820–1823 (1974).
    [CrossRef] [PubMed]
  25. These photographs were provided by H. F. A. Tschunko, NASA/GSFC.
  26. R. R. Butts, C. B. Hogge, “The modulation transfer function of an annular aperture with supporting struts,” Air Force Weapons Laboratory report AFWL-TR-75-311 (Feb.1976).

1988

1984

1980

R. E. Cox, “Secondary mirrors and spiders,” Telesc. Making 7, 4–7 (Spring1980).

1974

1971

1969

R. C. Ludden, “A 10-inch reflector fashioned in wood,” Sky Telesc.112–114 (Feb.1969).

C. H. Werenskiold, “A note on curved spiders,” Sky Telesc.262–263 (Oct.1969).

1960

R. E. Cox, “Spider diffraction in moderate-size telescopes,” Sky Telesc.166–171 (Sept.1960).

1959

E. Everhart, J. Kantorski, “Diffraction effects produced by obscurations in reflecting telescopes of modest size,” Astron. J. 64, 455–463 (1959).
[CrossRef]

1957

W. A. Rhodes, “An antidiffraction mask for reflectors,” Sky Telesc.289–290 (Apr.1957).

1941

C. H. Werenskiold, “Improved telescope spider design,” J. R. Astron. Soc. Can. 35, 268–273 (1941).

Beyer, L. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8, p. 381.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 112.

Butts, R. R.

R. R. Butts, C. B. Hogge, “The modulation transfer function of an annular aperture with supporting struts,” Air Force Weapons Laboratory report AFWL-TR-75-311 (Feb.1976).

Clark, P. P.

Clune, L. C.

Couder, A.

A. Couder, “Dealing with spider diffraction,” in Amateur Telescope Making, Advanced (Book Two), A. G. Ingalls, ed. (Scientific American, New York, 1946), pp. 260–262.

Cox, R. E.

R. E. Cox, “Secondary mirrors and spiders,” Telesc. Making 7, 4–7 (Spring1980).

R. E. Cox, “Spider diffraction in moderate-size telescopes,” Sky Telesc.166–171 (Sept.1960).

Dantzler, A. A.

Everhart, E.

E. Everhart, J. Kantorski, “Diffraction effects produced by obscurations in reflecting telescopes of modest size,” Astron. J. 64, 455–463 (1959).
[CrossRef]

Freniere, E. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 194, 216.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 61, 113.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 10, p. 458.

Hogge, C. B.

R. R. Butts, C. B. Hogge, “The modulation transfer function of an annular aperture with supporting struts,” Air Force Weapons Laboratory report AFWL-TR-75-311 (Feb.1976).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, New York, 1950), Chap. 2, p. 21.

Howard, J. W.

Kantorski, J.

E. Everhart, J. Kantorski, “Diffraction effects produced by obscurations in reflecting telescopes of modest size,” Astron. J. 64, 455–463 (1959).
[CrossRef]

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970), Chap. 7, p. 298.

Ludden, R. C.

R. C. Ludden, “A 10-inch reflector fashioned in wood,” Sky Telesc.112–114 (Feb.1969).

Meinel, A. B.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.

Meinel, M. P.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.

Peters, P. J.

P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 107, 63–69 (1977).

Rhodes, W. A.

W. A. Rhodes, “An antidiffraction mask for reflectors,” Sky Telesc.289–290 (Apr.1957).

Richter, J. L.

Sheehan, P. J.

Tschunko, H. F. A.

Werenskiold, C. H.

C. H. Werenskiold, “A note on curved spiders,” Sky Telesc.262–263 (Oct.1969).

C. H. Werenskiold, “Improved telescope spider design,” J. R. Astron. Soc. Can. 35, 268–273 (1941).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8, p. 381.

Woolf, N. J.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.

Appl. Opt.

Astron. J.

E. Everhart, J. Kantorski, “Diffraction effects produced by obscurations in reflecting telescopes of modest size,” Astron. J. 64, 455–463 (1959).
[CrossRef]

J. R. Astron. Soc. Can.

C. H. Werenskiold, “Improved telescope spider design,” J. R. Astron. Soc. Can. 35, 268–273 (1941).

Sky Telesc.

W. A. Rhodes, “An antidiffraction mask for reflectors,” Sky Telesc.289–290 (Apr.1957).

R. E. Cox, “Spider diffraction in moderate-size telescopes,” Sky Telesc.166–171 (Sept.1960).

R. C. Ludden, “A 10-inch reflector fashioned in wood,” Sky Telesc.112–114 (Feb.1969).

C. H. Werenskiold, “A note on curved spiders,” Sky Telesc.262–263 (Oct.1969).

Telesc. Making

R. E. Cox, “Secondary mirrors and spiders,” Telesc. Making 7, 4–7 (Spring1980).

Other

Kenneth Novak & Co., Catalog, Box 69, Ladysmith, Wisc. 54848.

A. Couder, “Dealing with spider diffraction,” in Amateur Telescope Making, Advanced (Book Two), A. G. Ingalls, ed. (Scientific American, New York, 1946), pp. 260–262.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, pp. 149–201.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 61, 113.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 194, 216.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, New York, 1950), Chap. 2, p. 21.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 112.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 8, p. 381.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 10, p. 458.

M. V. Klein, Optics (Wiley, New York, 1970), Chap. 7, p. 298.

P. J. Peters, “Aperture shaping—a technique for the control of the spatial distribution of diffracted energy,” in Stray Light Problems in Optical Systems, J. D. Lytle, H. E. Morrow, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 107, 63–69 (1977).

These photographs were provided by H. F. A. Tschunko, NASA/GSFC.

R. R. Butts, C. B. Hogge, “The modulation transfer function of an annular aperture with supporting struts,” Air Force Weapons Laboratory report AFWL-TR-75-311 (Feb.1976).

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Figures (22)

Fig. 1
Fig. 1

Diffraction effects of secondary mirror spiders on telescope image quality.

Fig. 2
Fig. 2

Relationship among the complex pupil function, the PSF, and the OTF. Frequently used image-quality criteria associated with each function are indicated.

Fig. 3
Fig. 3

Parametric plot of the ratio of the peak irradiance in the diffraction-limited PSF produced by an annular aperture of obscuration ratio ɛ and four spiders of width δD divided by that produced by an annular aperture without spiders.

Fig. 4
Fig. 4

Diffraction-limited irradiance distribution in the focal plane of a telescope depending on the dimensions of the pupil function in the exit pupil, the focal length of the telescope, the wavelength, and the incident field strength.

Fig. 5
Fig. 5

Diffraction-limited PSF for an annular aperture with a narrow opaque strut consisting of two parts: an image core and a diffraction flare perpendicular to the strut.

Fig. 6
Fig. 6

Encircled energy caused by the diffraction-limited annular apertures. This figure can be used as a set of characteristic curves from which to obtain values of EE annulus(r), which are necessary when the empirical equation is applied to various aperture configurations.

Fig. 7
Fig. 7

Fractional encircled energy resulting from diffraction-limited narrow rectangular apertures. This figure can be used as a set of characteristic curves from which to obtain values of EE rect(r), which are necessary when the empirical equation is applied to various aperture configurations.

Fig. 8
Fig. 8

(a) Annular aperture with a 0.7 obscuration ratio and four spiders whose width is 3% of the aperture diameter. (b) Corresponding PSF clearly showing complex interference effects. (c) Three-dimensional isometric plot of I n (x, y) on a log scale when determined by the first step of our hybrid approach to making rigorous calculations.

Fig. 9
Fig. 9

Comparison of predictions from an empirical equation (continuous curve) with rigorous calculations (bold dots) indicating a less than 0.5% error.

Fig. 10
Fig. 10

Aperture configuration with small obscuration and three narrow spiders producing a relatively mild scalloping of the first few rings of the PSF.

Fig. 11
Fig. 11

Parametric plot of fractional encircled energy versus circle radius for different spider widths.

Fig. 12
Fig. 12

Variations in the nature of the diffraction flares with spider width.

Fig. 13
Fig. 13

Corresponding fractional encircled energy curves providing insight into the image-degradation effects of secondary mirror spiders of varying widths.

Fig. 14
Fig. 14

Curved spider producing two searchlight beams emanating in opposite directions from the image core. This is intuitive if one approximates the curve as a set of straight segments.

Fig. 15
Fig. 15

Fractional encircled energy curves compared for a variety of spider configurations. The spider width is held constant.

Fig. 16
Fig. 16

Comparison of an asymptotic approximation, an empirical formula, and a rigorous calculation for a few discrete data points. The inset shows a comparison of the asymptotic approximation and an exact calculation over a wider range of circle radii.

Fig. 17
Fig. 17

HST aperture and diffraction-limited performance including the effects of central obscuration, spiders, and pads.

Fig. 18
Fig. 18

MTF for an annular aperture with obscuration ratio ɛ.

Fig. 19
Fig. 19

Analytical solution for the MTF profile in the x direction (θ = 0).

Fig. 20
Fig. 20

Effect of telescope secondary mirror spiders on MTF. Note the improvement at high spatial frequencies.

Fig. 21
Fig. 21

Sensitivity of the MTF to the central obscuration ratio.

Fig. 22
Fig. 22

Sensitivity of the MTF to the spider width.

Tables (5)

Tables Icon

Table 1 Fractional Encircled Energy from Diffraction-Limited Annular Apertures

Tables Icon

Table 2 Fractional Encircled Energy from Narrow Rectangular Apertures

Tables Icon

Table 3 Validation of Empirical Equation

Tables Icon

Table 4 Encircled Energy for Different Wavelengths and Spider Widths

Tables Icon

Table 5 HST Image Degradation

Equations (34)

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Strehl ratio S = ( A annulus - A spiders A annulus ) 2 ,
A annulus = π D 2 ( 1 - ɛ 2 ) / 4 ,
A spiders = N D 2 δ ( 1 - ɛ ) / 2 ,
S = [ 1 - 2 N δ π ( 1 + ɛ ) ] 2 .
U ( x 2 , y 2 ) = exp ( i k f ) i λ f exp [ i π ( x 2 2 + y 2 2 ) λ f ] × F { U 1 ( x 1 , y 1 ) } ξ = x 2 / λ f , η = y 2 / λ f .
U 1 ( x 1 , y 1 ) = B ( x 1 , y 1 ) T 1 ( x 1 / D , y 1 / D ) exp [ i k W ( x 1 , y 1 ) ] ,
U 1 ( x 1 , y 1 ) = B T 1 ( x 1 / D , y 1 / D ) ,
I ( x 2 , y 2 ) = U ( x 2 , y 2 ) 2 = B 2 λ 2 f 2 | F [ T 1 ( x 1 D , y 1 D ) ] | ξ = x 2 / λ f , η = y 2 / λ f | 2 .
I ( 0 , 0 ) = B 2 λ 2 f 2 A apeture 2 .
I n ( x , y ) = I ( x 2 , y 2 ) I ( 0 , 0 ) = 1 A aperture 2 | F { T 1 ( x 1 d , y 1 D ) } | ξ = x 2 / λ f , η = y 2 / λ f | 2 ,
E E ( r 2 ) = ϕ = 0 2 π r = 0 r I ( x 2 , y 2 ) r 2 d r 2 d ϕ x 2 = 0 y 2 = 0 I ( x 2 , y 2 ) d x 2 d y 2 .
x 2 = 0 y 2 = 0 I ( x 2 , y 2 ) d x 2 d y 2 = B 2 λ 2 f 2 x 2 = 0 y 2 = 0 | F { T 1 ( x 1 D , y 1 D ) } | ξ = x 2 / λ f , η = y 2 / λ f | 2 × λ 2 f 2 d ξ d η ,
x 2 = 0 y 2 = 0 I ( x 2 , y 2 ) d x 2 d y 2 = B 2 x 1 = 0 y 1 = 0 | T 1 ( x 1 D , y 1 D ) | 2 × d x 1 d y 1 .
| T 1 ( x 1 D , y 1 D ) | 2 = T 1 ( x 1 D , y 1 D ) .
x 2 = 0 y 2 = 0 I ( x 2 , y 2 ) d x 2 d y 2 = B 2 A aperture .
I ( x 2 , y 2 ) = B 2 A aperture 2 λ 2 f 2 I n ( x , y )
E E ( r ) = A aperture D 2 ϕ = 0 2 π r = 0 r I n ( x , y ) r d r d ϕ .
E E ( r ) = ϕ = 0 2 π r = 0 r Part A r d r d ϕ + ϕ = 0 2 π r = 0 r Part B r d r d ϕ x 2 = 0 y 2 = 0 ( Part A + Part B ) d x d y .
E E ( r ) = ( A annulus - 2 A spiders ) E E annulus ( r ) + A spiders E E rect ( r ) A annulus - A spiders ,
I n ( x , y ) = 1 ( 1 - ɛ 2 ) 2 [ 2 J 1 ( π r ) π r - ɛ 2 2 J 1 ( ɛ π r ) ɛ π r ] ,
I n ( x , y ) = sinc 2 ( δ x ) sinc 2 ( β y ) ,
E E rect ( r ) = β δ ϕ = 0 2 π r = 0 r sinc 2 ( δ x ) sinc 2 ( β y ) r d r d ϕ ,
E E rect ( r ) β δ x = - r r sinc 2 ( δ x ) sinc 2 ( β y ) d x .
E E HST ( r ) = [ ( A ann - 2 A spid - 2 A pad ) E E ann ( r ) + A spid E E rect ( r ) + A pad E E circ ( r ) ] / ( A aperture ) ,
A pad = M π γ 2 D 2 / 4 ,             M = number of pads
E E dust ( r ) = [ ( A ann - 2 A spid - 2 A pad - 2 A dust ) E E ann ( r ) + A spid E E rect ( r ) + A p a d E E circ ( r ) + A dust E E dust ( r ) ] / ( A aperture ) ,
A aperture = A ann - A spid - A pad - A dust ,             Case 1 ,
A aperture = A ann - A spid - A pad ,             Case 2.
MTF = - f ( α , β ) f ( α - x , β - y ) d α d β - f ( α , β ) 2 d α d β .
MTF = R f f A T ,             R f f = - f ( α , β ) f ( α - x , β - y ) d α d β .
MTF = ( D 2 / 2 ) { cos - 1 ( r / D ) - ( r / D ) [ 1 - ( r / D ) 2 ] 1 / 2 } π D 2 / 4 .
MTF = R f f π ( D 1 2 - D 2 2 ) / 4 ,
R f f = α = 0 2 ɛ f ( α ) f ( α - x ) d α + α = 2 ɛ 1 - ɛ f ( α ) f ( α - x ) d α + α = 1 - ɛ 1 + ɛ f ( α ) f ( α - x ) d α + α = 1 + ɛ 2 f ( α ) f ( α - x ) d α ,
A ann / A T = π D 2 ( 1 - ɛ 2 ) / 4 π D 2 ( 1 - ɛ 2 ) / 4 - 2 D 2 δ ( 1 - ɛ ) = 1.108.

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