Abstract

Approximate, but accurate, analytical expressions for the far-field divergence angle of a Gaussian beam normally incident on a circular aperture are derived. A first equation is obtained based on the concept of Gaussian transform, in which the Bessel function present in the far-field diffraction integral is approximated by a Gaussian function. Refining this approach yields another simple, practical closed-form formula with such a level of accuracy that we propose that it can be used as an exact reference. All approximations hold for any combination of Gaussian beam width and aperture radius.

© 2000 Optical Society of America

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References

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  1. J. F. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
    [CrossRef]
  2. A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
    [CrossRef]
  3. J. P. Campbell, L. G. DeShazer, “Near fields of truncated-Gaussian apertures,” J. Opt. Soc. Am. 59, 1427–1429 (1969).
    [CrossRef]
  4. A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
    [CrossRef]
  5. G. O. Olaofe, “Diffraction by Gaussian apertures,” J. Opt. Soc. Am. 60, 1654–1657 (1970).
    [CrossRef]
  6. R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated-Gaussian field distribution,” J. Opt. Soc. Am. 61, 31–35 (1971).
    [CrossRef]
  7. P. Belland, J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
    [CrossRef] [PubMed]
  8. K. Tanaka, N. Saga, H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt. 24, 1102–1106 (1985).
    [CrossRef] [PubMed]
  9. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  10. K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
    [CrossRef]
  11. K. Uehara, H. Kikuchi, “Transmission of a Gaussian beam through a circular aperture,” Appl. Opt. 25, 4514–4516 (1986).
    [CrossRef] [PubMed]
  12. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  13. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
    [CrossRef]
  14. D. Ding, X. Liu, “Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 63–95.
  16. J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995), Chap. 3, pp. 63–85.
  17. S. D. Conte, C. DeBoor, Elementary Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1972), pp. 32–33.

1999 (1)

1996 (1)

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

1991 (1)

1986 (1)

1985 (1)

1982 (1)

1972 (2)

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
[CrossRef] [PubMed]

K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

1971 (1)

1970 (1)

1969 (1)

1967 (1)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

1965 (2)

J. F. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Avizonis, P. V.

Belland, P.

Buck, A. L.

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

Campbell, J. P.

Chai, A. S.

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Conte, S. D.

S. D. Conte, C. DeBoor, Elementary Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1972), pp. 32–33.

Crenn, J. P.

DeBoor, C.

S. D. Conte, C. DeBoor, Elementary Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1972), pp. 32–33.

DeShazer, L. G.

Ding, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 63–95.

Holmes, D. A.

Kauffman, J. F.

J. F. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

Kenney, C. S.

Kikuchi, H.

Korka, J. E.

Kukumitsu, O.

K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Lenz, G.

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

Liu, X.

Mizokami, H.

Olaofe, G. O.

Overfelt, P. L.

Saga, N.

Schell, R. G.

Shibukawa, M.

K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Tanaka, K.

K. Tanaka, N. Saga, H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt. 24, 1102–1106 (1985).
[CrossRef] [PubMed]

K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

Tyras, G.

Uehara, K.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995), Chap. 3, pp. 63–85.

Wertz, H. J.

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Antennas Propag. (2)

J. F. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. Tanaka, M. Shibukawa, O. Kukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749–755 (1972).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
[CrossRef]

Proc. IEEE (1)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 63–95.

J. T. Verdeyen, Laser Electronics, 3rd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1995), Chap. 3, pp. 63–85.

S. D. Conte, C. DeBoor, Elementary Numerical Analysis, 2nd ed. (McGraw-Hill, New York, 1972), pp. 32–33.

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Figures (6)

Fig. 1
Fig. 1

Geometry and main characteristics for the diffraction of a Gaussian beam by a circular aperture.

Fig. 2
Fig. 2

Functional dependence, with respect to the radial position in the far field, of the product between the Gaussian and the Bessel functions present in the diffraction integral. Legend and labels are common to all plots and are shown in (d).

Fig. 3
Fig. 3

(a) Divergence angle as a function of the beam width for three different values of aperture radius. (b) Percentage error referenced to the numerical integration for the method used in (a).

Fig. 4
Fig. 4

(a) Divergence angle as a function of the beam width for different methods and R = 20 mm. (b) Percentage error referenced to the numerical integration for the methods used in (a). (c) Divergence angle as a function of the aperture radius for the methods used in (a) and w 0 = 5 mm. (d) Percentage error referenced to the numerical integration for the methods used in (c).

Fig. 5
Fig. 5

(a) Divergence angle as a function of the beam width for three different values of aperture radius. (b) Percentage error referenced to the numerical integration for the method used in (a).

Fig. 6
Fig. 6

Relative error of Eq. (18) referenced to the numerical integration for different decimal places used to estimate the constants K 1 and K 2.

Tables (1)

Tables Icon

Table 1 Variation of the Divergence Angle for a Fixed Beam-Width Aperture Radius Ratio

Equations (24)

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UGρ=U0 exp-ρ2w02,
Ur; z=expjkzexpj kr22zjλz×2π 0 Uρ; 0J02πrρλzρdρ,
Ur; z=2π expjkzexpj kr22zjλz0R U0×exp-ρ2w02J02πrρλzρdρ.
θtan θ=rez.
Ur; z=expjkzexpj kr22zjλz 2π2w02U0 exp-ρ2w02.
θ=λπw0.
Ur; z=expjkzexpj kr22z4π2R2jλzJ1kRrzkRrz,
θ=2.5838λ2πR.
Pρ=exp-ρ2w02J02πrρλz,
exp-ρ2w02J02πrρλzexp-ρ2w02exp-ρ2w02r=exp-ρ2w02r,
intr=0Rexp-ρ2w02exp-ρ2w02rρdρ=0Rexp-ρ2w02rρdρ=w02r21-exp-R2w02r.
C=int0e=w022e1-exp-R2w02.
intre=w02re21-exp-R2w02re=C.
1w02+2πreKλz2=12C.
θ=rez=Kλ2π12C-1w021/2=Kλ2πw0e1-exp-R2w02-11/2.
θ=λπe-1 w0e1-exp-R2w02-11/20.2428λw0e1-exp-R2w02-11/2.
θ=1πee-1λR2.5155 λ2πR.
θ0.2494λw0e1-exp-R2w02-11/2.
θλ4w0e1-exp-R2w02-11/2.
1w02r=1K1w02+1w02r=1K1w02+1K2λz/2πr2.
θ=K2K1λ2πw0e1-exp-R2K12w02-11/2.
θR/w0  1=K2K1λ2πw0e-1,
θR/w0  1=K2λ2πRe.
θ0.2428λw0e1-exp-R1.0271w02-11/2.

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