Abstract

It has been known for some time that, if curved legs rather than the usual straight ones are used in the spider that supports the secondary optics in certain telescopes, the visible diffraction effect is reduced. Fraunhofer theory is used to calculate the diffraction effects due to the curved leg spider. Calculated and photographic diffraction patterns are compared for straight and curved leg spiders.

© 1984 Optical Society of America

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References

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  1. A. Couder, in Amateur Telescope Making Advanced, A. G. Ingalls, Ed. (Scientific American, New York, 1949), p. 620.
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  3. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1952).
  4. E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Couder, A.

A. Couder, in Amateur Telescope Making Advanced, A. G. Ingalls, Ed. (Scientific American, New York, 1949), p. 620.

Emde, F.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

Jahnke, E.

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1952).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Other

A. Couder, in Amateur Telescope Making Advanced, A. G. Ingalls, Ed. (Scientific American, New York, 1949), p. 620.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge U.P., London, 1952).

E. Jahnke, F. Emde, Tables of Functions (Dover, New York, 1945).

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Figures (15)

Fig. 1
Fig. 1

Curved leg spiders. In each case the large circle is the telescope aperture and the small one is the aperture obstruction due to the secondary optic. The half-angles of subtense of the curved spider legs are 30°, 60°, and 45°, respectively.

Fig. 2
Fig. 2

Crescent aperture diffraction. The thin, circular arc aperture on the left gives rise to the Fraunhofer diffraction pattern shown at right.

Fig. 3
Fig. 3

Close in diffraction patterns. The solid line depicts the Fraunhofer diffraction brightness in stellar magnitudes for the circular aperture; the broad dashed curve is that for a moderate annulus with = 0.3; the narrow dashed curve is that for the narrow annulus with = 0.99.

Fig. 4
Fig. 4

Off-center rectangular slit geometry. This is the construction for one straight spider leg.

Fig. 5
Fig. 5

Construction for the translation factor. It is straightforward to evaluate Fraunhofer’s integral in the ξ,η coordinate system, but the function is needed in the ξ′,η′ reference frame.

Fig. 6
Fig. 6

Eccentric aperture. The left sketch shows a telescope aperture with an eccentric obstruction. The sketch on the right shows a telescope focused on an off-axis object, the effective aperture is like that at left.

Fig. 7
Fig. 7

Geometric representation of the diffraction intensity due to the eccentric aperture. The diffraction intensity I at field point w,ψ (x = kSAw) is shown.

Fig. 8
Fig. 8

Construction for the crescent shaped aperture. The aperture area has radial interval (1 − δ)RρR and tangential range Φ − σψ ≤ Φ + σ.

Fig. 9
Fig. 9

Diffraction brightness as a function of azimuth angle ψ for the illustrated aperture. The curve is normalized so that B = 0 at x =0.

Fig. 10
Fig. 10

Construction for the curved spider leg translation factor.

Fig. 11
Fig. 11

Definitions of the P(σ,ν) and Q(σ,ν) functions. The functions are asymmetric about ν = 0: P(σ,−ν) = P(σ,ν) and Q(σ,−ν) = Q(σ,ν).

Fig. 12
Fig. 12

Obtuse angle searchlight effect. When the aperture is like that at left, with 90° < σ < 180°, the diffraction pattern is that at right. The notation 2 × light means that the diffraction intensity is twice that of light.

Fig. 13
Fig. 13

Far-field diffraction envelopes. The lines are the envelopes of the rapid spatially varying brightness for the various cases. Plot a represents the circular aperture, b depicts that of the annular aperture with = 0.3. Plot c shows the background brightness midway between the straight leg spider with δ = 0.01 and = 0.3. Plot d is that for the same aperture but along the spider leg. Plot e is the diffraction pattern envelope for the curved leg spider with = 0.3, δ = 0.029 (= 0.02/b), σ = 30°, γ = 133.0°, b = 0.7, c = 0.889, and with ψ =0.

Fig. 14
Fig. 14

Diffraction photographs. The six diffraction images were taken of a distant pinhole light source with the masks shown in Fig. 15 in front of the 305-mm EFL lens. The separation between the unenlarged image centers of (b) and (c) is 3.6 mm.

Fig. 15
Fig. 15

Diffraction masks. The six masks were used in front of the lens to form the images shown in Fig. 14. The largest extent of any actual mask is 50 mm.

Equations (38)

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D ( p , q ) = A exp [ - i k ( p ξ + q η ) ] d ξ d η .
I D 2 ,
B 2.5 log D 2 ( magnitude ) .
D = 2 π S A 2 k S A w [ J 1 ( k S A w ) - J 1 ( k S A w ) ] .
V ( Φ ) = 2 S A 2 sin [ x δ sin ( ψ - Φ ) ] x 2 sin ( ψ - Φ ) cos ( ψ - Φ ) × { sin [ x cos ( ψ - Φ ) ] - sin [ x cos ( ψ - Φ ) ] } + i cos [ x cos ( ψ - Φ ) ] - i cos [ x cos ( ψ - Φ ) ] } .
V ( Φ ) = 2 S A 2 δ x { sin x - sin x + i cos ( ψ - Φ ) [ cos x - cos x ] }
V ( Φ ) = 2 S A 2 sin x δ x ( 1 - )
D = 2 π S A 2 x [ J 1 ( x ) - J 1 ( x ) ] - V ( 0 ) - V ( 90 ° ) - V ( 180 ° ) - V ( 270 ° ) .
D ( p , q ) = exp [ i k ( p ξ 0 + q η 0 ) ] D ( p , q ) .
ξ 0 = ξ - ξ ,             η 0 = η - η ,
D ( w , ψ ) = exp [ i k C w cos ( ψ - Γ ) ] D ( w , ψ ) ,
w = p 2 + q 2 ,             ψ = tan - 1 q / p ,
C = ξ 0 2 + η 0 2 ,             Γ = tan - 1 η 0 / ξ 0 .
D = 1 x [ J 1 ( x ) - J 1 ( x ) exp ( i c x cos ψ ) ]
I = 1 x 2 [ J 1 2 ( x ) + 2 J 1 2 ( x ) - 2 J 1 ( x ) J 1 ( x ) cos ( c x cos ψ ) ] .
D = Φ - σ Φ + σ ( 1 - δ ) R R exp [ - i k ρ w cos ( ϕ - ψ ) ] ρ d ρ d ϕ .
exp ( ± i z cos ν ) = J 0 ( z ) + 2 n = 1 ( ± i ) n J n ( z ) cos n ν .
D = 2 R 2 { σ [ H 0 ( 0 ) - H 0 ( δ ) ] + n = 1 ( - i ) n n [ H n ( 0 ) - H n ( δ ) ] × [ sin n ( ψ + σ - Φ ) - sin n ( ψ - σ - Φ ) ] } ,
D = 2 R 2 { σ [ H 0 ( 0 ) - H 0 ( δ ) ] + 2 n = 1 ( - i ) n n [ H n ( 0 ) - H n ( δ ) ] × [ sin n σ cos n ( ψ - Φ ) ] } .
H n ( δ ) = 1 y 2 0 ( 1 - δ ) y y J n ( y ) d y ,
H n ( δ ) = 1 y 2 { ( 1 - δ ) y J n + 1 [ ( 1 - δ ) y ] + 2 n m = 0 J n + 2 m + 2 [ ( 1 - δ ) y ] } ,
U ( Φ ) = 2 b 2 S A 2 exp [ i c x cos ( ψ - Φ - γ ) ] { σ [ H 0 ( 0 ) - H 0 ( δ ) ] + 2 n = 1 ( - i ) n n [ H n ( 0 ) - H n ( δ ) ] sin n σ cos n ( ψ - Φ ) } .
b = 1 2 ( 1 - ) csc σ ,
c 2 = 1 4 ( 1 + ) 2 + b 2 cos 2 σ ,
γ = tan - 1 1 + - 2 b cos σ .
J n + 1 ( y ) cos ( y - 3 π 4 - n π 2 ) 1 2 π y ,             for y n .
( - i ) n H n ( δ ) = F ( δ ) cos 2 n π 2 - i G ( δ ) sin 2 n π 2 ,
F ( δ ) = 1 - δ cos [ ( 1 - δ ) y - 3 π 4 ] y 1 2 π y ,
G ( δ ) = 1 - δ sin [ ( 1 - δ ) y - 3 π 4 ] y 1 2 π y , for y 100.
D = 2 R 2 ( [ F ( 0 ) - F ( δ ) ] { σ + n = 1 1 2 n [ sin 2 n ( ψ + σ - Φ ) - sin 2 n ( ψ - σ - Φ ) ] } - i [ G ( 0 ) - G ( δ ) ] n = 1 1 2 n - 1 [ sin ( 2 n - 1 ) ( ψ + σ - Φ ) - sin ( 2 n - 1 ) ( ψ - σ - Φ ) ] ) .
n = 1 1 2 n sin 2 n ν = { - π 4 - 1 2 ν , for - π < ν < 0 , π 4 - 1 2 ν , for 0 < ν < π
n = 1 1 2 n - 1 sin ( 2 n - 1 ) ν = { - π 4 , for - π < ν < 0 , π 4 , for 0 < ν < π .
P ( σ , ν ) = σ + n = 1 1 2 n [ sin 2 n ( ν + σ ) - sin 2 n ( ν - σ ) ] ,
Q ( σ , ν ) = n = 1 1 2 n - 1 [ sin ( 2 n - 1 ) ( ν + σ ) - sin ( 2 n - 1 ) ( ν - σ ) ,
D = 2 R 2 { [ F ( 0 ) - F ( δ ) ] P ( σ , ψ - Φ ) - i [ G ( 0 ) - G ( δ ) ] Q ( σ , ψ - Φ ) } .
U ( Φ ) = 2 b 2 S A 2 exp [ i c x cos ( ψ - Φ - γ ) ] { [ F ( 0 ) - F ( δ ) ] P ( σ , ψ - Φ ) - i [ G ( 0 ) - G ( δ ) ] Q ( σ , ψ - Φ ) } ,             for y 100.
x = 2 π Y λ E S A .
Y = x f λ π 0.00016 x f ( mm ) ,

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