Abstract

The total energy integral functions for apodization T1 = (1 − r2)n and T2 = (2rr2)n are compared for unobstructed apertures. The energy values increase linearily for T2 and with the (2n + 3) power for T1. For obstructed apertures, these values for T1 increase with higher powers at small image radii and assume linear characteristics only at larger radii. This transition occurs as the obstruction approaches zero. Partial energy functions for a region centered some distance from the central region show the same general trend as the image irradiance functions.

© 1983 Optical Society of America

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References

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  1. H. F. A. Tschunko, Appl. Opt. 13, 1820 (1974).
    [CrossRef] [PubMed]
  2. H. F. A. Tschunko, Appl. Opt. 17, 1075 (1978).
    [CrossRef] [PubMed]
  3. H. F. A. Tschunko, Appl. Opt. 18, 3770 (1979).
    [CrossRef] [PubMed]
  4. H. F. A. Tschunko, Appl. Opt. 22, 133 (1983).
    [CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Total energy integral functions over the image radius x for zero central obstruction and for transmissions T1 and T2 with apodization exponents n.

Fig. 2
Fig. 2

Image radii x for levels of total energy E over apodization exponent n or central irradiance Io for zero central obstruction and apodization T1.

Fig. 3
Fig. 3

Area with radius R around center C a distance L from radiation source S for partial energy integral functions E.

Fig. 4
Fig. 4

Total energy integral functions E over image radius x for central obstruction ratio a = 0.5 and a = 0.8 for n = 0 to n = 1.0 and apodization T1.

Fig. 5
Fig. 5

Total energy integral functions E over image radius x for apodization T1, n = 2 and obstruction ratios a = 0 to 1.

Fig. 6
Fig. 6

Partial energy integral functions E for central obstruction a = 0 and apodization T1, n = 0 for D = 1 to D = 100 optical units

Fig. 7
Fig. 7

Partial energy integral functions E for central obstruction a = 0.5 and apodization T1, n = 0.5 for D = 1 to D = 100 optical units.

Fig. 8
Fig. 8

Asymptotic envelopes of the maxima of the partial energy integrals for central obstruction a = 0 and apodization T1, n = 0 and n = 1, D = 1, 2, 4, 10, 20, 33, 50, and 100 optical units.

Equations (13)

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I 0 = [ r = a r = 1 T ( r ) 2 r d r ] 2 .
I ( x ) = ( 1 / I 0 ) [ a 1 T ( r ) J 0 ( x · r ) 2 π r d r ] 2 ,
F = a 1 [ T ( r ) ] 2 2 r d r .
E ( x ) = ( I 0 / F ) 0.5 0 x I ( x ) · x · d x .
I 0 = 1 / ( n + 1 ) 2 ,
F = 1 / ( 2 n + 1 ) ,
E = ( n + 1 ) 2 / ( 2 n + 1 ) = F / I 0 .
E = L R L + R I ( x ) · x · · d x ,
= arc cos [ ( L 2 + R 2 x 2 ) / ( L · D ) ] .
I ( x ) 2.5 ( 1 a ) 2 · x 3 ,
E ( x ) 1 0.6 ( 1 a ) 1 · x 1 ,
E ( L ) 0.15 · 10 n · D 5 / 3 · L ( 2 n 3 ) ,
E ( L ) 0.2 · D 5 / 3 · L 3 .

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