Abstract

This paper investigates Fresnel diffraction from plane-wave gaussian beams truncated by circular apertures. Kirchhoff’s integral has been evaluated analytically and (analytical) expressions are derived, in terms of infinite series, for the irradiance and total power of the diffracted field. Some graphs are presented for Fraunhofer conditions. The results obtained greatly reduce the labor of numerically computing the irradiance and the power.

© 1970 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Ch. VIII.
  2. J. P. Campbell and L. G. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [Crossref]
  3. G. N. Watson, Theory of Bessel Functions (Cambridge University Press, New York, 1966), p. 132.
  4. Reference 1, p. 396.
  5. Reference 3, p. 136.
  6. Reference 1, p. 398.
  7. Modern Computing Methods, edited by E. T. Goodwin (H. M. Stationery Office, London, 1962), Ch. 13.

1969 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Ch. VIII.

Campbell, J. P.

DeShazer, L. G.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, New York, 1966), p. 132.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Ch. VIII.

J. Opt. Soc. Am. (1)

Other (6)

G. N. Watson, Theory of Bessel Functions (Cambridge University Press, New York, 1966), p. 132.

Reference 1, p. 396.

Reference 3, p. 136.

Reference 1, p. 398.

Modern Computing Methods, edited by E. T. Goodwin (H. M. Stationery Office, London, 1962), Ch. 13.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Ch. VIII.

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Figures (4)

Fig. 1
Fig. 1

The geometry of the problem.

Fig. 2
Fig. 2

Normalized irradiance of the diffraction vs distance from the axis. The inset shows an exploded view of the oscillations of the irradiance for large radial distances. The innermost scale refers to the cases η=0 and η=1, the center scale to the case η=5, and the outermost to the case η=10.

Fig. 3
Fig. 3

The (gaussian) amplitude of the incident field vs normalized radius R=ρ/a.

Fig. 4
Fig. 4

The fraction L(x) of the total power contained within circles of normalized radius x (=kωa) in the Fraunhofer diffraction pattern of a gaussian aperture. As in Fig. 2, the inset shows an exploded view of L(x) for large values of x; three scales are given: one for η=0 and η=1, one for each of the cases η=5 and η=10.

Tables (1)

Tables Icon

Table I Comparison of L(x) with increasing η.

Equations (22)

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U ( ρ , φ , z ) = A 0 S U ( ρ , φ ) e i k s S ( 1 + cos χ ) ρ d ρ d φ ,
U ( ρ , r ) = A r 0 a ρ U ( ρ ) J 0 ( k ρ ω ) exp ( i k ρ 2 / 2 r ) d ρ ,
U ( ρ ) = exp ( - ρ 2 / α 0 2 ) ,
U ( ρ , r ) = A r 0 a ρ J 0 ( k ρ ω ) exp ( - ρ 2 / α 2 ) d ρ ,
T = 0 a exp ( - ρ 2 / α 2 ) ρ J 0 ( k ω ρ ) d ρ .
T = T ( x ) = ( a 2 / x 2 ) 0 x exp ( - μ ζ 2 ) ζ J o ( ζ ) d ζ ,
0 x u n J n - 1 ( u ) d u = x n J n ( x ) ,             n = 1 , 2 , 3 , ,
T ( x ) = a 2 η exp ( - η / 2 ) n = 1 ( η / x ) n J n ( x ) ,
lim x 0 [ J n ( x ) / x n ] = 1 / 2 n n ! ,
T ( 0 ) = a 2 η exp ( - η / 2 ) n = 1 η n 2 n n ! .
U ( ρ , r ) U ( 0 , r ) = 1 exp ( η / 2 ) - 1 n = 1 ( η / x ) n J n ( x ) .
( d / d x ) [ J n ( x ) / x n ] = - J n + 1 ( x ) / x n ,
exp [ 1 2 x ( t - 1 / t ) ] = n = - t n J n ( x )
U ( ρ , r ) U ( 0 , r ) = 1 [ exp ( η / 2 ) - 1 ] × [ exp ( η / 2 - x 2 / 2 η ) - n = 0 ( - x / η ) n J n ( x ) ] .
I = U ( ρ , r ) / U ( 0 , r ) 2 .
I = ( 2 / x ) J 1 ( x ) 2 .
I 0 = A a 2 [ 1 - exp ( - η / 2 ) ] / η 2
L * ( x 0 ) = B 0 x 0 ( η / ζ ) n J n ( ζ ) 2 ζ d ζ ,
J n - j ( x ) J j ( x ) x n - 1 d x = - 1 2 ( n - 1 ) x n - 2 × [ J n - j - 1 ( x ) J j - 1 ( x ) + J n - j ( x ) J j ( x ) ] ,
L ( x ) = 1 - Re η x 2 η 2 [ exp ( Re η ) - 1 ] n = 2 ( η / x ) n × 1 n - 1 j = 1 n - 1 p j ( J n - j - 1 J j - 1 + J n - j J j ) ( x )
L ( x ) 1 - J 0 2 ( x ) - J 1 2 ( x ) ,
L ( x ) = Re η η 2 [ exp ( Re η ) - 1 ] × { η 2 exp ( Re η ) 2 Re ( η ) [ 1 - exp ( - x 2 Re η / η 2 ) ] + 2 Re [ η exp ( - η / 2 ) n = 0 ( - x / n ) n j = 1 ( x / η ) j J n + j ( x ) ] + n = 0 x 2 [ - x / η ¯ j = 0 n p j ( J n - j J j + J n - j + 1 J j + 1 ) ( x ) ] } .