Abstract

We use the scalar Kirchhoff–Huygens diffraction integral to obtain analytic expressions for both axial and transverse intensity distributions, assuming normal incidence on a circular aperture for four types of incident field: (1) plane wave, (2) Bessel beam, (3) Gaussian beam, and (4) Bessel–Gauss beam. We use the Fresnel approximation to obtain the axial intensity as a function of distance from the aperture. We consider both Fresnel and Fraunhofer diffraction for the case of the transverse intensity distributions. For the axial case, we find that the Bessel–Gauss beam performs worse than the Bessel beam, in terms both of the magnitude of intensity and of its ability to extend a distance from the aperture. In the transverse case, we find that the Bessel–Gauss beam performance in terms of remaining nearly diffraction free over a given distance is highly dependent on the relationship among the aperture radius, the beam waist parameter, and the transverse wave number.

© 1991 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  3. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  4. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 2.
  5. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 5.
  6. G. Barton, Elements of Green’s Functions and Propagation (Clarendon, Oxford, 1989), Chap. 13.
  7. E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and the Rayleigh–Sommerfield theories of diffraction at an aperture,”J. Opt. Soc. Am. 54, 587–594 (1964).
    [Crossref]
  8. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,”J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [Crossref]
  9. 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]
  10. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [Crossref]
  11. R. E. English, N. George, “Diffraction from a circular aperture: on-axis field strength,” Appl. Opt. 26, 2360–2363 (1987).
    [Crossref] [PubMed]
  12. B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).
  13. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), Chap. 8, pp. 438–439.
  14. G. N. Watson, Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1958), Chap. 16.
  15. R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated Gaussian field distribution,”J. Opt. Soc. Am. 61, 31–35 (1971).
    [Crossref]
  16. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [Crossref] [PubMed]
  17. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).
  18. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 259, Eq. (27).
  19. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I.

1988 (1)

1987 (4)

R. E. English, N. George, “Diffraction from a circular aperture: on-axis field strength,” Appl. Opt. 26, 2360–2363 (1987).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

1971 (1)

1966 (1)

1965 (1)

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]

1964 (1)

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Barton, G.

G. Barton, Elements of Green’s Functions and Propagation (Clarendon, Oxford, 1989), Chap. 13.

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), Chap. 8, pp. 438–439.

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

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]

Durnin, J.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

English, R. E.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I.

George, N.

Godfrey, B. B.

B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 5.

Luke, Y. L.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 259, Eq. (27).

Marchand, E. W.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 2.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Schell, R. G.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Tyras, G.

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1958), Chap. 16.

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]

Wolf, E.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

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]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Other (9)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 2.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 5.

G. Barton, Elements of Green’s Functions and Propagation (Clarendon, Oxford, 1989), Chap. 13.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964).

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, New York, 1962), p. 259, Eq. (27).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. I.

B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), Chap. 8, pp. 438–439.

G. N. Watson, Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1958), Chap. 16.

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Figures (17)

Fig. 1
Fig. 1

Circular aperture in cylindrical coordinates.

Fig. 2
Fig. 2

Axial intensity versus distance from aperture: comparison of plane-wave, Bessel beam, and Gaussian beam incidence (α = 1000).

Fig. 3
Fig. 3

Axial intensity versus distance from aperture: comparison of plane-wave, Bessel beam, and Gaussian beam incidence (α = 5000).

Fig. 4
Fig. 4

Axial intensity versus distance from aperture: comparison of plane-wave, Bessel beam, and Gaussian beam incidence (α = 15,000).

Fig. 5
Fig. 5

Axial intensity versus distance from aperture: comparison of plane-wave, Gaussian beam, and Bessel–Gauss beam incidence (α = 1000).

Fig. 6
Fig. 6

Axial intensity versus distance from aperture: comparison of plane-wave, Gaussian beam, and Bessel–Gauss beam incidence (α = 5000).

Fig. 7
Fig. 7

Axial intensity versus distance from aperture: comparison of plane-wave, Gaussian beam, and Bessel–Gauss beam incidence (α = 15,000).

Fig. 8
Fig. 8

Comparison of Bessel beam and Bessel–Gauss beam transverse diffracted intensities (w0a).

Fig. 9
Fig. 9

Comparison of Bessel beam and Bessel–Gauss beam transverse diffracted intensities (w0a).

Fig. 10
Fig. 10

Comparison of Bessel beam and Bessel–Gauss beam transverse diffracted intensities (w0a).

Fig. 11
Fig. 11

Transverse intensity distribution for Bessel–Gauss beam incidence (r0 = 0.5 m).

Fig. 12
Fig. 12

Transverse intensity distribution for Bessel–Gauss beam incidence (r0 = 0.75 m).

Fig. 13
Fig. 13

Transverse intensity distribution for Bessel–Gauss beam incidence (r0 = 1.0 m).

Fig. 14
Fig. 14

Transverse intensity distribution for Bessel–Gauss beam incidence (r0 = 1.2 m).

Fig. 15
Fig. 15

Fraunhofer diffraction pattern of a Bessel beam (η = 2.05).

Fig. 16
Fig. 16

Fraunhofer diffraction pattern of a Bessel beam (η = 4.05).

Fig. 17
Fig. 17

Fraction of total energy contained within circles of prescribed radii in the Fraunhofer diffraction pattern of a Bessel beam (η = 0, 505, 10.05).

Tables (1)

Tables Icon

Table 1 Zeros of Fraunhofer Transverse Intensity for Bessel Beam Incidence as a Function of Beam Sharpness

Equations (54)

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ψ ( ρ , z ) = i k 0 2 π A ψ inc ( ρ , 0 ) exp ( - i k 0 R ) R d A ,
R = [ ρ 2 + ρ 2 + z 2 - 2 ρ ρ cos ( ϕ - ϕ ) ] 1 / 2 .
R = ( ρ 2 + z 2 ) 1 / 2 ,
R = z [ 1 + 1 2 ( ρ 2 ) 2 = 1 8 ( ρ z ) 4 + ] ,
R z + ρ 2 2 z ,
ψ ( 0 , z ) = i k 0 2 π exp ( - i k 0 z ) z × 0 2 π 0 z ψ inc ( ρ , 0 ) exp ( - i k 0 ρ 2 / 2 z ) ρ d ρ d ϕ
ψ inc ( ρ , 0 ) = ψ 0 exp ( - ρ 2 / w 0 2 ) J 0 ( α ρ )
ψ inc PW ( ρ , z ) = ψ 0 exp ( - i k 0 z )
ψ inc B ( ρ , z ) = ψ 0 J 0 ( α ρ ) exp ( - i β z ) ,
ψ ( 0 , z ) = ψ 0 i k 0 exp ( - i k 0 z ) z 0 a J 0 ( α ρ ) exp ( - q ρ 2 ) ρ d ρ ,
q = 1 w 0 2 + i k 0 2 z .
ψ ( 0 , z ) = i k 0 exp ( i k 0 z ) 2 q z [ exp ( - q a 2 ) n = 1 ( 2 q a α ) n J n ( α a ) ] ,             | 2 q a α | < 1 ,
ψ ( 0 , z ) = i k 0 exp ( - i k 0 z ) 2 q z × [ exp ( - α 2 / 4 q ) - exp ( - q a 2 ) n = 0 ( - α 2 q a ) n J n ( α a ) ] , | 2 q a α | > 1.
I BG I 0 = ( k 0 2 q z ) 2 exp [ - 2 ( a / w 0 ) 2 ] ( S 1 2 + S 2 2 ) ,             | 2 q a α | < 1 ,
S 1 = n = 1 ( 2 a α ) n q n cos ( n σ ) J n ( α a ) ,
S 2 = n = 1 ( 2 a α ) n q n sin ( n σ ) J n ( α a ) ,
σ = tan - 1 ( L z ) .
I BG I 0 = ( k 0 2 q z ) 2 { exp [ - α 2 / ( 2 w 0 2 q 2 ) ] - 2 S 3 exp ( - r 1 / w 0 2 ) × cos ( k 0 r 1 2 z ) - 2 S 4 exp ( - r 1 / w 0 2 ) sin ( k 0 r 1 2 z ) + exp [ - 2 ( a 2 / w 0 2 ) ] × ( S 3 2 + S 4 2 ) } ,             | 2 q a α | > 1 ,
S 3 = n = 0 ( - 1 ) n ( α 2 a ) n 1 q n cos ( n σ ) J n ( α a ) ,
S 4 = n = 0 ( - 1 ) n ( α 2 a ) n 1 q n sin ( n σ ) J n ( α a ) ,
r 1 = α 2 4 q 2 + a 2 .
lim w 0 q = i k 0 2 z ,
I PW I 0 = 2 - 2 cos ( k 0 a 2 2 z ) ,
z m = a 2 2 m λ 0 ,             m = 1 , 2 , ,
z l = a 2 l λ 0 ,             l = 1 , 2 , .
z 1 = a 2 λ 0 ,
I B I 0 = S 1 2 + S 2 2 ,             | k 0 a α z | < 1 ,
lim w 0 S 1 = n = 1 ( k 0 a α z ) n cos n π 2 J n ( α a )
lim w 0 S 2 = n = 1 ( k 0 a α z ) n sin n π 2 J n ( α a ) .
I B I 0 = 1 - 2 S 3 cos ( k 0 r 1 2 z ) - 2 S 4 sin ( k 0 r 1 2 z ) + S 3 2 + S 4 2 ,             | k 0 a α z | > 1 ,
lim w 0 S 3 = n = 0 ( - 1 ) n ( α z k 0 a ) n cos n π 2 J n ( α a ) ,
lim w 0 S 4 = n = 0 ( - 1 ) n ( α z k 0 a ) n sin n π 2 J n ( α a ) .
z max = k 0 a α .
z max = a [ ( 2 π α λ 0 ) 2 - 1 ] 1 / 2 = β a α .
I G I 0 = ( k 0 2 q z ) 2 { 1 - 2 exp [ - ( a / w 0 ) 2 ] cos ( k 0 a 2 2 z ) + exp [ - 2 ( a / ω 0 ) 2 ] } .
| 2 q a α | = 1
z max = | k 0 a / [ α 2 - ( 2 a w 0 2 ) 2 ] 1 / 2 | .
I ( ρ , z ) I 0 = ( k 0 a 2 r ) 2 n = 0 ( - 1 ) n ( q a 2 ) n n ! k = 0 B k ,
B k = ( - 1 ) k ( α a 2 ) 2 k F 2 1 [ - k , - k ; 1 ; ( λ α ) 2 ] ( k ! ) 2 ( n + k + 1 ) ,             | γ α | < 1
B k = ( - 1 ) k ( γ a 2 ) 2 k F 2 1 [ - k , - k ; 1 ; ( α γ ) 2 ] ( k ! ) 2 2 ( n + k + 1 ) ,             | α γ | < 1 ,
γ = k 0 ρ r ,
r max = k 0 ρ α ,
I ( ρ , z ) I 0 = 4 [ t J 0 ( η ) J 1 ( t ) - η J 1 ( η ) J 0 ( t ) t 2 - η 2 ] 2 ,             t 2 η 2 ,
R r - ρ ρ cos ( ϕ - ϕ ) r + ρ 2 2 r ,
ψ ( ρ , z ) = i k 0 2 π exp ( - i k 0 r ) r 0 2 π 0 a J 0 ( α ρ ) exp ( - ρ 2 / w 0 2 ) × exp { - i k 0 [ ρ 2 2 r - ρ ρ cos ( ϕ - ϕ ) r ] } ρ d ρ d ϕ
ψ ( ρ , z ) = i k 0 2 π exp ( - i k 0 r ) r 0 2 π 0 a J 0 ( α ρ ) exp ( - q ρ 2 ) × exp [ i k 0 ρ ρ r cos ( ϕ - ϕ ) ] ρ d ρ d ϕ ,
q = 1 w 0 2 + i k 0 2 r .
2 π J 0 ( k 0 ρ ρ r ) ,
ψ ( ρ , z ) = i k 0 exp ( - i k 0 r ) r 0 a J 0 ( α ρ ) J 0 ( γ ρ ) exp ( - q ρ 2 ) ρ d ρ ,
M = a 2 0 1 J 0 ( α a x ) J 0 ( γ a x ) exp ( - q a 2 x 2 ) x d x .
M = a 2 n = 0 ( - 1 ) n ( q a 2 ) n n ! 0 1 J 0 ( α a x ) J 0 ( γ a x ) x 2 n + 1 d x .
0 1 J 0 ( α a x ) J 0 ( γ a x ) x 2 n + 1 d x = k = 0 ( - 1 ) k ( α a 2 ) 2 k F 2 1 [ - k , - k ; 1 ; ( γ α ) 2 ] ( k ! ) 2 2 ( n + k + 1 ) ,             | γ α | < 1 ,
0 1 J 0 ( α a x ) J 0 ( γ a x ) x 2 n + 1 d x = k = 0 ( - 1 ) k ( γ a 2 ) 2 k F 2 1 [ - k , - k , 1 ; ( α γ ) 2 ] ( k ! ) 2 2 ( n + k + 1 ) , | α γ | < 1.
ψ ( ρ , z ) = i k 0 exp ( - i k 0 r ) a 2 r n = 0 ( - 1 ) n ( q a 2 ) n n ! ( k = 0 B k ) .

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