Abstract

A Gaussian beam weakly diffracted by a circular aperture can be approximated in the far field by another Gaussian beam with slightly different characteristics. Equations giving the intensity, the divergence, and the radius of the modified beam are derived in simple practical form for experimentalists. These approximated formulas show that, even in the case of negligible power losses through the aperture, the diffracted beam characteristics may appreciably differ from those of the incident beam. In a first approximation, diffraction effects may be ignored only if the ratio a/r0 of the aperture radius a to the 1/e intensity beam radius r0 in the aperture plane is larger than 3.

© 1982 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  3. P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
    [CrossRef]
  4. P. Belland, Thèse de Doctorat d’Etat, Université de Paris VI, CNRS AO 1215 (Nov.1975); also, Report EUR-CEA-FC-806 (Jan.1976).
  5. J. J. Degnan, Appl. Phys. 11, 1 (1976).
    [CrossRef]
  6. J. F. Kauffman, IEEE Trans. Antennas Propag. AP-13, 473 (1965).
    [CrossRef]
  7. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [CrossRef]
  8. J. P. Campbell, L. G. DeShazer, J. Opt. Soc. Am. 59, 1427 (1969).
    [CrossRef]
  9. G. O. Olaofe, J. Opt. Soc. Am. 60, 1654 (1970).
    [CrossRef]
  10. R. G. Schell, G. Tyras, J. Opt. Soc. Am. 61, 31 (1971).
    [CrossRef]
  11. K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
    [CrossRef]
  12. B. J. Klein, J. J. Degnan, Appl. Opt. 13, 2134 (1974).
    [CrossRef] [PubMed]
  13. T. Takenaka, M. Kakeya, O. Fukumitsu, J. Opt. Soc. Am. 70, 1323 (1980).
    [CrossRef]
  14. L. D. Dickson, Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  15. D. Véron, Submillimeter Interferometry of High Density Plasmas, in Infrared and Millimeter Waves, Vol. 2 (Academic, New York, 1979), Chap. 2.
  16. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 391.

1980 (1)

1976 (1)

J. J. Degnan, Appl. Phys. 11, 1 (1976).
[CrossRef]

1975 (1)

P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
[CrossRef]

1974 (1)

1972 (1)

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

1971 (1)

1970 (2)

1969 (1)

1967 (1)

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

1966 (1)

1965 (1)

J. F. Kauffman, IEEE Trans. Antennas Propag. AP-13, 473 (1965).
[CrossRef]

Belland, P.

P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
[CrossRef]

P. Belland, Thèse de Doctorat d’Etat, Université de Paris VI, CNRS AO 1215 (Nov.1975); also, Report EUR-CEA-FC-806 (Jan.1976).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Buck, A. L.

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

Campbell, J. P.

Degnan, J. J.

DeShazer, L. G.

Dickson, L. D.

Fukumitsu, O.

T. Takenaka, M. Kakeya, O. Fukumitsu, J. Opt. Soc. Am. 70, 1323 (1980).
[CrossRef]

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Kakeya, M.

Kauffman, J. F.

J. F. Kauffman, IEEE Trans. Antennas Propag. AP-13, 473 (1965).
[CrossRef]

Klein, B. J.

Kogelnik, H.

Li, T.

Olaofe, G. O.

Schell, R. G.

Shibukawa, M.

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Takenaka, T.

Tanaka, K.

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Tyras, G.

Véron, D.

P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
[CrossRef]

D. Véron, Submillimeter Interferometry of High Density Plasmas, in Infrared and Millimeter Waves, Vol. 2 (Academic, New York, 1979), Chap. 2.

Whitbourn, L. B.

P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Appl. Opt. (3)

Appl. Phys. (1)

J. J. Degnan, Appl. Phys. 11, 1 (1976).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. F. Kauffman, IEEE Trans. Antennas Propag. AP-13, 473 (1965).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Tanaka, M. Shibukawa, O. Fukumitsu, IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. D (1)

P. Belland, D. Véron, L. B. Whitbourn, J. Phys. D 8, 2113 (1975).
[CrossRef]

Proc. IEEE (1)

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

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

P. Belland, Thèse de Doctorat d’Etat, Université de Paris VI, CNRS AO 1215 (Nov.1975); also, Report EUR-CEA-FC-806 (Jan.1976).

D. Véron, Submillimeter Interferometry of High Density Plasmas, in Infrared and Millimeter Waves, Vol. 2 (Academic, New York, 1979), Chap. 2.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 391.

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Figures (6)

Fig. 1
Fig. 1

Main characteristics of a propagating Gaussian beam.

Fig. 2
Fig. 2

Changes in the characteristics of a Gaussian beam passing through a circular aperture, with the beam-waist in the aperture plane. Beam contours at 1/e intensity: - - - initial, without diffraction; — real, with diffraction; … fictitious, used to calculate r 0 , θ m , and I 0 (see text).

Fig. 3
Fig. 3

Comparison between the three formulas, giving the radius ratio r 0 / r 0 (or the divergence ratio θ m / θ m ) as a function of a/r0, when the beam waist is in the aperture plane: 1, first method, Eq. (26); 2, second method, Eq. (35); 3, simplified form, Eq. (37).

Fig. 4
Fig. 4

Comparison between the three formulas, giving the intensity ratio I 0 / I 0 as a function of a/r0, when the beam waist is in the aperture plane: 1, first method, Eq. (27); 2, second method, Eq. (36); 3, simplified form, Eq. (39).

Fig. 5
Fig. 5

General curves giving the ratio r m / r m (or θ m / θ m ) as a function of x = a/r0, when the beam waist is at a distance l from the aperture, with p = l λ / r m 2 as a parameter: (a) p = 0, 1, 5, 7, and 10; (b) p = 20 and 30.

Fig. 6
Fig. 6

General curves giving the ratio I m / I m as a function of x = a/r0, when the beam is at a distance l from the aperture, with p = l λ / r m 2 as a parameter: (a) p = 0, 1, 5, 7, and 10; (b) p = 20 and 30.

Equations (63)

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I ( ρ , z ) = I ( 0 , z ) exp ( - ρ 2 / r 2 ) = I exp ( - ρ 2 / r 2 )
θ m tan θ m = 1 / k r m ,
r 2 ( z ) = r 2 = r m 2 [ 1 + ( z / k r m 2 ) 2 ] ,
R ( z ) = R = z [ 1 + ( k r m 2 / z ) 2 ] .
P = 0 + 0 2 π I exp ( - ρ 2 / r 2 ) ρ d ρ d ϕ = π r 2 I = π r m 2 I m .
z k r m 2
I ( k r m 2 / z ) 2 I m .
P 0 = π r 0 2 I 0 = P .
P 0 = 0 a 0 2 π I 0 exp ( - ρ 2 / r 0 2 ) ρ d ρ d ϕ = π r 0 2 I 0 [ 1 - exp ( - a 2 / r 0 2 ) ] ,
Δ P / P = exp ( - a 2 / r 0 2 )
I ( ρ , z ) = ( k r 0 2 / z ) 2 I 0 [ exp ( - ρ 2 / 2 r 2 ) - E ( ρ , z ) ] 2
r 0 < a / 2
z a 2 / λ
E ( ρ , z ) = [ exp ( - a 2 / 2 r 0 2 ) ] n = 0 n = m ( - k r 0 2 ρ / a z ) n J n ( k a ρ / z ) .
I ( ρ , z ) = ( k r 0 2 / z ) 2 I 0 exp ( - ρ 2 / r 2 ) .
r 0 4 I 0 [ 1 - E ( 0 , z ) ] 2 = r 0 4 I 0 ,
r 0 4 I 0 [ 1 / e - E ( r 0 , z ) ] 2 = r 0 4 I 0 exp ( - r 2 / r 2 ) .
r / z = 1 / k r 0
r / r = r 0 / r 0 .
k ρ / z = ρ 0 / r 0 2 ,
E ( ρ , z ) = F ( ρ 0 ) = [ exp ( - a 2 / 2 r 0 2 ) ] n = 0 ( - ρ 0 / a ) n J n ( a ρ 0 / r 0 2 ) .
r 0 4 I 0 [ 1 - F ( 0 ) ] 2 = r 0 4 I 0 ,
r 0 4 I 0 [ 1 / e - F ( r 0 ) ] 2 = r 0 4 I 0 exp ( - r 0 2 / r 0 2 ) .
exp ( r 0 2 / 2 r 0 2 ) = e [ 1 - F ( 0 ) ] / [ 1 - e F ( r 0 ) ] .
exp ( r 0 2 / 2 r 0 2 ) e r 0 / r 0 .
r 0 / r 0 = r m / r m = [ 1 - F ( 0 ) ] / [ 1 - e F ( r 0 ) ] .
I 0 / I 0 = I m / I m = [ 1 - e F ( r 0 ) ] 4 / [ 1 - F ( 0 ) ] 2
F ( 0 ) = exp ( - a 2 / 2 r 0 2 )
F ( r 0 ) [ exp ( - a 2 / 2 r 0 2 ) ] n = 0 3 ( - r 0 / a ) n J n ( a / r 0 ) .
I = ( k r 0 2 / z ) 2 I 0 [ 1 - exp ( - a 2 / 2 r 0 2 ) ] 2 ,
I / I = [ 1 - exp ( - a 2 / 2 r 0 2 ) ] 2 .
P 0 = π r 0 2 I 0 = π r 0 2 I 0 [ 1 - exp ( - a 2 / r 0 2 ) ] ,
π r 2 I = π r 2 I [ 1 - exp ( - a 2 / r 0 2 ) ] .
r / r = [ 1 - exp ( - a 2 / 2 r 0 2 ) ] / [ 1 - exp ( - a 2 / r 0 2 ) ] 1 / 2 .
r 0 / r 0 = r m / r m = [ 1 - exp ( - a 2 / 2 r 0 2 ) ] / [ 1 - exp ( - a 2 / r 0 2 ) ] 1 / 2 .
I 0 / I 0 = I m / I m = [ 1 - exp ( - a 2 / r 0 2 ) ] 2 / [ 1 - exp ( - a 2 / 2 r 0 2 ) ] 2 .
r 0 / r 0 = r m / r m 1 - exp ( - a 2 / 2 r 0 2 ) = 1 - Δ P / P
θ m / θ m [ 1 - exp ( - a 2 / 2 r 0 2 ) ] - 1 = [ 1 - Δ P / P ] - 1 .
I 0 / I 0 = I m / I m [ 1 - 2 exp ( - a 2 / 2 r 0 2 ) ] - 1 = [ 1 - 2 Δ P / P ] - 1 .
r 0 2 = r m 2 [ 1 + ( l / k r m 2 ) 2 ] ,
R 0 = l [ 1 + ( k r m 2 / l ) 2 ] ,
I 0 = I m r m 2 / r 0 2 = I m / [ 1 + ( l + k r m 2 ) 2 ] ,
z k r m 2 ,             z a 2 / λ ,             and z l ,
I ( 0 , z ) = I = ( k r m 2 / z ) 2 I m { 1 - 2 [ cos ( k a 2 / 2 R 0 ) ] × [ exp ( - a 2 / 2 r 0 2 ) ] + exp ( - a 2 / r 0 2 ) } .
α = k a 2 / 2 R 0 ,
u = exp ( - a 2 / r 0 2 ) = Δ P / P 1.
I = ( k r m 2 / z ) 2 I m ( 1 - 2 u cos α + u ) = ( k r m 2 / z ) 2 I m
π r 0 2 I 0 ( 1 - u ) = π r m 2 I m .
r m 2 / r m 2 = ( 1 - 2 u cos α + u ) / ( 1 - u ) ,
I m / I m = ( r m / r m ) 2 ( 1 - u ) = ( 1 - u ) 2 / ( 1 - 2 u cos α + u ) .
r m / r m 1 - u cos α = 1 - Δ P / P · cos ( k a 2 / 2 R 0 ) ,
I m / I m [ 1 - 2 u cos α ] - 1 = [ 1 - 2 Δ P / P · cos ( k a 2 / 2 R 0 ) ] - 1 .
θ m / θ m ( 1 - u cos α ) - 1 = [ 1 - Δ P / P · cos ( k a 2 / 2 R 0 ) ] - 1 .
k a 2 / 2 R 0 = ( 2 n + 1 ) π / 2 ,
a n = [ ( 2 n + 1 ) λ R 0 / 2 ] 1 / 2 .
k a 2 / 2 R 0 = ( 2 n + 1 ) π ,
a n = [ ( 2 n + 1 ) λ R 0 ] 1 / 2 = a n 2 .
r m / r m 1 - ( exp { - a 2 2 r m 2 [ ( l / k r m 2 ) 2 ] } ) × cos { k a 2 2 l [ 1 + ( k r m 2 / l ) 2 ] } ,
I m / I m [ 1 - 2 ( exp { - a 2 2 r m 2 [ 1 + ( l / k r m 2 ) 2 ] } ) × cos { k a 2 2 l [ 1 + ( k r m 2 / l ) 2 ] } ] - 1 .
r 0 2 / R 0 = l / k 2 r m 2 ,
k a 2 / 2 R 0 = l a 2 / 2 k r 0 2 r m 2 = ( a 2 / r 0 2 ) ( l λ / 4 π r m 2 ) .
r m / r m = θ m / θ m 1 - [ exp ( - x 2 / 2 ) ] · cos ( p x 2 / 4 π ) ,
I m / I m { 1 - 2 [ exp ( - x 2 / 2 ) ] · cos ( p x 2 / 4 π ) } - 1 .

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