Abstract

Within the limits of the paraxial approximation used in treating gaussian beams, the ordinary boundary-diffraction-wave theory is also applicable to diffraction problems that involve gaussian incident beams. The total field diffracted by an aperture is thus given by the interference of two component waves: a boundary-diffraction wave and, if allowed by geometrical considerations, the unperturbed wave that would propagate freely to the observation point in the absence of the diffracting aperture. To this end, the gaussian field distribution must be described by properly defined complex amplitude and phase functions. Examples are calculated for gaussian beams with cylindrical symmetry. The general equation for the ray paths associated with the gaussian beams is also derived; it is used to show that the shadow boundary behind the diffracting screen follows a hyperbola.

© 1974 Optical Society of America

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References

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  1. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
    [Crossref]
  2. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
    [Crossref]
  3. A. Rubinowicz, in Progress in Optics, edited by E. Wolf (North–Holland, Amsterdam, and Wiley, New York, 1965), p. 199.
    [Crossref]
  4. G. A. Maggi, Ann. di Mat. IIa,  16, 21 (1888).
    [Crossref]
  5. A. Rubinowicz, Ann. Phys. 53, 257 (1917).
    [Crossref]
  6. T. Young, Philos. Trans. R. Soc. Lond. 20, 26 (1802).
  7. J. W. Y. Lit and R. Tremblay, J. Opt. Soc. Am. 59, 559 (1969).
    [Crossref]
  8. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 59, 79 (1969).
    [Crossref]
  9. A. E. Siegman, An Introduction to Lasers and Masers (McGraw–Hill, New York, 1971), Ch. VIII.
  10. C. S. Williams, Appl. Opt. 12, 872 (1973).
    [Crossref] [PubMed]
  11. The sign convention is chosen here to be consistent with that used in Refs. 1 and 2.
  12. G. A. Deschamps, Electron. Lett. 7, 684 (1971).
    [Crossref]
  13. J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971).
    [Crossref]
  14. A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
    [Crossref]
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. III.
  16. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chs. VII and VIII.
  17. R. G. Schell and G. Tyras, J. Opt. Soc. Am. 61, 31 (1971).
    [Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. III.

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[Crossref]

Keller, J. B.

Lit, J. W. Y.

Maggi, G. A.

G. A. Maggi, Ann. di Mat. IIa,  16, 21 (1888).
[Crossref]

Marchand, E. W.

Miyamoto, K.

Rubinowicz, A.

A. Rubinowicz, Ann. Phys. 53, 257 (1917).
[Crossref]

A. Rubinowicz, in Progress in Optics, edited by E. Wolf (North–Holland, Amsterdam, and Wiley, New York, 1965), p. 199.
[Crossref]

Schell, R. G.

Siegman, A. E.

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).
[Crossref]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw–Hill, New York, 1971), Ch. VIII.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chs. VII and VIII.

Streifer, W.

Tremblay, R.

Tyras, G.

Williams, C. S.

Wolf, E.

Young, T.

T. Young, Philos. Trans. R. Soc. Lond. 20, 26 (1802).

Ann. di Mat. IIa (1)

G. A. Maggi, Ann. di Mat. IIa,  16, 21 (1888).
[Crossref]

Ann. Phys. (1)

A. Rubinowicz, Ann. Phys. 53, 257 (1917).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (1)

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[Crossref]

J. Opt. Soc. Am. (7)

Philos. Trans. R. Soc. Lond. (1)

T. Young, Philos. Trans. R. Soc. Lond. 20, 26 (1802).

Other (5)

A. Rubinowicz, in Progress in Optics, edited by E. Wolf (North–Holland, Amsterdam, and Wiley, New York, 1965), p. 199.
[Crossref]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw–Hill, New York, 1971), Ch. VIII.

The sign convention is chosen here to be consistent with that used in Refs. 1 and 2.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1964), Ch. III.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chs. VII and VIII.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the diffraction problem.

Fig. 2
Fig. 2

Diffraction of gaussian beam by a circular aperture.

Fig. 3
Fig. 3

Light ray path in a medium with cylindrical symmetry.

Fig. 4
Fig. 4

Illustrating the evaluation of Fj.

Fig. 5
Fig. 5

Illustrating the contour integral on Γj around Qj.

Fig. 6
Fig. 6

Defining the shadow boundary for a diffracting aperture S.

Equations (65)

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U K ( P ) = S V ( P , Q ) · d a .
V ( P , Q ) = Q × W ( P , Q ) .
U K ( P ) = U B ( P ) + j F j ( P ) .
U B ( P ) = Γ W ( P , Q ) · d 1 .
W ( P , Q ) = e i k s 4 π s ŝ × Q ( i k + ŝ · Q ) U ( Q ) ,
F j ( P ) = lim σ j 0 Γ j W ( P , Q j ) · d 1 .
U K ( P ) = U B ( P ) + U G ( P ) ,
U ( r , z ) = w 0 w ( z ) exp { - r 2 w 2 ( z ) } × exp { i k [ z + r 2 2 R ( z ) - β ( z ) k ] } .
R ( z ) = z [ 1 + ( b / 2 z ) 2 ] ,
w 2 ( z ) = w 0 2 [ 1 + ( 2 z / b ) 2 ] ,
tan β ( z ) = 2 z / b ,
b = k w 0 2 .
1 q ( z ) = 1 R ( z ) + 2 i k w 2 ( z ) ,
d q ( z ) / d z = 1 ,
U ( r , z ) = q ( 0 ) q ( z ) exp { i k [ z + r 2 2 q ( z ) ] } .
U spher ( r , z ) 1 z exp { i k [ z + r 2 2 z ] } .
U ( r ) = A ( r ) exp { i k ϕ ( r ) } ,
A ( r , z ) = q ( 0 ) / q ( z )
ϕ ( r , z ) = z + r 2 / 2 q ( z ) .
Q U ( Q ) = ( i k Q ϕ + Q A A ) U ( Q ) .
ϕ = { r q ( z ) } r ˆ + { 1 - r 2 2 q 2 ( z ) } z ˆ ,
A A = { - 1 q ( z ) } z ˆ .
ϕ 1 ,
r Q b ;
| A A | = 2 w 0 b w ( z ) k ,
λ w 0 .
( ϕ ) 2 = n 2
Q U ( Q ) i k ( Q ϕ ) U ( Q ) .
W ( P , Q ) = U ( Q ) e i k s 4 π s ŝ × p 1 + ŝ · p ,
p Q ϕ .
p = { a q ( z 1 ) } r ˆ + { 1 - a 2 2 q 2 ( z 1 ) } z ˆ .
ŝ = a r ˆ - Z z ˆ s ,
s 2 = a 2 + Z 2 .
U B ( 0 , z ) = - U ( Q ) · a 2 2 · e i k s s · ( 1 + Z q ( z 1 ) - a 2 2 q 2 ( z 1 ) ) / ( s - Z + a 2 q ( z 1 ) [ 1 + Z 2 q ( z 1 ) ] ) .
U B ( 0 , z ) - U ( Q ) e i k s · ( 1 + Z q ( z 1 ) - a 2 2 q 2 ( z 1 ) ) / ( 1 + Z q ( z 1 ) ) 2 .
U B ( 0 , z ) - U ( Q ) exp { i k [ Z + a 2 / 2 Z ] } 1 + Z / q ( z 1 ) .
U B ( 0 , z ) = - ( q ( 0 ) q ( z ) exp { i k z } ) × exp { i k a 2 2 [ 1 Z + 1 q ( z 1 ) ] } .
U K ( P ) = U B ( P ) + U 0 ( P ) .
F 1 ( P ) = - lim a 0 U B ( 0 , z ) = q ( 0 ) q ( z ) exp { i k z } = U 0 ( 0 , z ) .
d r / d s = ϕ .
r = r r ˆ + z z ˆ .
d s = [ ( d r ) 2 + ( d z ) 2 ] 1 2 .
{ d r d s } r ˆ + { d z d s } z ˆ = { r q ( z ) } r ˆ + { 1 - r 2 2 q 2 ( z ) } z ˆ .
d r / d z = r / q ( z ) ,
r ( z ) = r 0 q ( z ) / q ( 0 ) ,
ŝ · p = - 1.
F j = U ( Q j ) Λ j exp { i k s j } ,
Λ j = 1 4 π s j lim σ j 0 Γ j ŝ × p · d 1 1 + ŝ · p .
ξ r ( z 1 + Z ) , = r ( z 1 ) q ( z 1 + Z ) / q ( z 1 ) .
ŝ × p 1 + ŝ · p sin δ 1 - cos δ k = cot δ 2 k ,
k = ( ŝ × p ) / ŝ × p .
Λ j = 1 2 π s j lim σ j 0 0 2 π σ j δ d θ .
δ β - ψ , = σ j s j - { d r d z | r - d r d z | r + σ j } , lim σ j 0 δ σ j = 1 s j + 1 q ( z 1 ) .
Λ j = 1 1 + s j / q ( z 1 ) .
s j Z [ 1 + r 2 / 2 q 2 ( z 1 ) ] .
F j = q ( 0 ) q ( z 1 + Z ) exp { i k [ z 1 + Z + r 2 2 q ( z 1 ) ( 1 + Z q ( z 1 ) ) ] } .
F 1 = q ( 0 ) q ( z 1 + Z ) exp { i k [ z 1 + Z + ξ 2 2 q ( z 1 + Z ) ] } ,
U K ( P ) = U B ( P ) + U 0 ( P ) .
U K ( P ) = U B ( P ) .
ξ max ( z , θ ) = r ( z , θ ) , = ξ 0 ( θ ) [ 1 + X 2 ] 1 2 ,
ξ 0 ( θ ) ξ max ( 0 , θ ) .
ξ max ( z , θ ) = η ( θ ) [ 1 + x 2 1 + X 2 ] 1 2 ,
x = 2 ( z 1 + Z ) / b , Z = 2 z 1 / b .
ξ max ( z , θ ) = η ( θ ) w ( z ) / w 0 .
ξ max ( z , θ ) η ( θ ) [ 1 + Z / z 1 ] .