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References

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  1. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [CrossRef]
  2. J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Amer. 59, 1427 (1969).
    [CrossRef]
  3. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

1969 (1)

J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Amer. 59, 1427 (1969).
[CrossRef]

1967 (1)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

1966 (1)

Buck, A. L.

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

Campbell, J. P.

J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Amer. 59, 1427 (1969).
[CrossRef]

DeShazer, L. G.

J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Amer. 59, 1427 (1969).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

Kogelnik, H.

Li, T.

Appl. Opt. (1)

J. Opt. Soc. Amer. (1)

J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Amer. 59, 1427 (1969).
[CrossRef]

Proc. IEEE (1)

A. L. Buck, Proc. IEEE 55, 448 (1967).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 59.

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Figures (1)

Fig. 1
Fig. 1

The solid curve (—) represents the solution of Eq. (5). The aperture inside radius is fixed at a and the outside radius is b while w is the spot size which maximizes the on-axis irradiance If in the focal plane z = R. The value of w/b computed for a/b = 0 agrees with the work of Buck.1 From Eq. (5) it can be shown that w / b 2 1 2 as a/b → 1. The dashed curve (---) is a plot of Eq. (7) which optimizes w/b as a function of a/b for maximum power transmission through the aperture.

Equations (7)

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υ ( r ) = [ 2 P / π w 2 ] 1 2 exp [ ( r / w ) 2 i ( π r 2 / λ R ) ] ,
u 0 = 2 π exp ( 2 π i z / λ ) i λ z a b [ υ ( r ) exp [ i ( π r 2 / λ z ) ] r d r .
I 0 = 2 P π w 2 · { exp [ ( a / w ) 2 ] exp [ ( b / w ) 2 ] } 2 + 4 exp [ ( a 2 + b 2 ) / w 2 ] sin 2 [ π ( b 2 a 2 ) ( z R ) / 2 λ z R ] ( λ z / π w 2 ) 2 + [ ( z R ) / R ] 2 .
I f = 2 π w 2 P / λ 2 R 2 · { exp [ ( a / w ) 2 ] exp [ ( b / w ) 2 ] } 2 .
exp [ 1 ( a / b ) 2 ( w / b ) 2 ] = ( w / b ) 2 + 2 ( w / b ) 2 + 2 ( a / b ) 2 .
P t = P { exp [ 2 ( a / w ) 2 ] exp [ 2 ( b / w ) 2 ] } .
w 2 = ( b 2 a 2 ) / ln ( b / a ) .

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