Abstract

An approximate analysis is derived for the propagation of Bessel, Bessel–Gauss, and Gaussian beams with a finite aperture. This treatment is based on the fact that the circ function can be expanded into an approximate sum of complex Gaussian functions, so that these three beams are typically expressed as a combination of a set of infinite-aperture Bessel–Gauss beams. Correspondingly, the evaluation of the diffracted field distribution of the beams is reduced to the summation of Bessel–Gauss functions. From analytical results, the present approach provides a good description of the diffracted beams in the region far (greater than a factor of the Fresnel distance) from the aperture. A possible extension of this method to other apertured beams is also discussed.

© 1999 Optical Society of America

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References

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  1. Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. 43, 125–132 (1897).
    [CrossRef]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [CrossRef]
  5. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  6. D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
    [CrossRef]
  7. J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990).
    [CrossRef] [PubMed]
  8. J. Y. Lu, J. F. Greenleaf, “Pulse-echo imaging using a nondiffracting beam transducer,” Ultrasound Med. Biol. 17, 265–281 (1991).
    [CrossRef] [PubMed]
  9. 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, 732–945 (1991).
    [CrossRef]
  10. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  11. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  12. D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).
  13. E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
    [CrossRef]
  14. In most cases, it is not desired that the aperture be placed at the waist of the Gaussian or Bessel–Gauss beam. However, in certain circumstances, that is, when a laser beam is focused through a finite-aperture lens or a pinhole, the waist size of the beam is on the order of the aperture size. See, for example, Ref. 16 and references therein; see also R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated Gaussian field distribution,” J. Opt. Soc. Am. 61, 31–35 (1971).
    [CrossRef]
  15. E. W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966), Chap. 6.
  16. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996).
    [CrossRef]
  17. I. S. Gradsthteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

1996 (1)

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

1993 (1)

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

1991 (3)

1990 (1)

J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990).
[CrossRef] [PubMed]

1989 (2)

A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[CrossRef] [PubMed]

D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
[CrossRef]

1988 (1)

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987 (3)

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]

1980 (1)

E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

1971 (1)

1897 (1)

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. 43, 125–132 (1897).
[CrossRef]

Breazeale, M. A.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Cavanagh, E.

E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

Cheney, E. W.

E. W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966), Chap. 6.

Cook, B. D.

E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

Ding, D.

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Du, G.

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Durnin, J.

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

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

Eberly, J. H.

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

Friberg, A. T.

Gori, F.

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

Gradsthteyn, I. S.

I. S. Gradsthteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Greenleaf, J. F.

J. Y. Lu, J. F. Greenleaf, “Pulse-echo imaging using a nondiffracting beam transducer,” Ultrasound Med. Biol. 17, 265–281 (1991).
[CrossRef] [PubMed]

J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990).
[CrossRef] [PubMed]

Guattari, G.

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

Herman, R. M.

Hsu, D. K.

D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
[CrossRef]

Kenney, C. S.

Lenz, G.

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

Lin, J.

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Lu, J. Y.

J. Y. Lu, J. F. Greenleaf, “Pulse-echo imaging using a nondiffracting beam transducer,” Ultrasound Med. Biol. 17, 265–281 (1991).
[CrossRef] [PubMed]

J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990).
[CrossRef] [PubMed]

Margetan, F. J.

D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
[CrossRef]

Miceli, J. J.

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

Overfelt, P. L.

Padovani, C.

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

Rayleigh, Lord

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. 43, 125–132 (1897).
[CrossRef]

Ryzhik, I. M.

I. S. Gradsthteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Schell, R. G.

Shui, Y.

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Thompson, D. O.

D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
[CrossRef]

Turunen, J.

Tyras, G.

Vasara, A.

Wen, J. J.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Wiggins, T. A.

Zhang, D.

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Acta Acust. (China) (1)

D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).

Appl. Phys. Lett. (1)

D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (2)

E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980).
[CrossRef]

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

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

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

Philos. Mag. (1)

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. 43, 125–132 (1897).
[CrossRef]

Phys. Rev. Lett. (1)

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

Ultrasound Med. Biol. (1)

J. Y. Lu, J. F. Greenleaf, “Pulse-echo imaging using a nondiffracting beam transducer,” Ultrasound Med. Biol. 17, 265–281 (1991).
[CrossRef] [PubMed]

Other (2)

E. W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966), Chap. 6.

I. S. Gradsthteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

(a) Real and imaginary parts of the Gaussian expansion [Eq. (10)] for the circ function, (b) magnitude and phase of the Gaussian expansion.

Fig. 2
Fig. 2

Comparison of the relative intensity distributions at the axis for the plane wave diffracted from a circular aperture, evaluated by use of the Gaussian expansion technique and the exact analytical expression. For the greater distance, the results from these two methods (not shown here) have almost no difference.

Fig. 3
Fig. 3

Relative intensity distributions for plane-wave, Bessel, Bessel–Gauss, and Gaussian beams on the beam axis. (a) α=2 and B=25/36, (b) α=30 and B=100, (c) α=30 and B=16.

Fig. 4
Fig. 4

Relative intensity distributions in the radial direction for Bessel–Gauss and Bessel beams. (a) α=30 and B=3.14E-9, z=0.500 m. The two curves are indistinguishable. (b) α=30 and B=25, z=0.500 m. (c) α=30 and B=3.306, z=0.75 m. (d) α=30 and B=3.306, z=1.20 m.

Fig. 5
Fig. 5

Fraunhofer diffraction pattern of a Bessel beam. The solid and dotted curves correspond to Figs. 15 and 16, respectively, of Ref. 9. The symbols in this figure conform the definitions in the text. Here η=20.

Fig. 6
Fig. 6

Comparison of the Airy intensity patterns for the plane wave diffracted from a circular aperture, evaluated by use of the Gaussian expansion technique and the exact analytical expression.

Tables (1)

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Table 1 Expansion and Gaussian Coefficients, Ak and Bka

Equations (35)

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q¯(ξ, η)=2iη0 expiξ2+ξ2ηJ02ξξηq¯(ξ)ξdξ,
q¯GB(ξ; α, B)=J0(αξ)exp(-Bξ2).
0Jv(αt)Jv(βt)exp(-γ2t2)tdt
=12γ-2 exp-14γ-2(α2+β2)Iv12αβγ-2
Iv(iz)=ivJv(z),
q¯GB(ξ, η; α, B)=exp-Bξ21+iBη1+iBηJ0αξ1+iBη×exp-i4α2η1+iBη,
B=(a/w0)2
α=αa.
q¯B(ξ, η; α, 0)=J0(αξ)exp[-(i/4)α2η]
H(ξ)=10ξ<10ξ>1
H(ξ)=k=1NAk exp(-Bkξ2).
q¯(ξ)=f1(ξ)0ξ<10ξ>1,
q¯(ξ)=f(ξ)·H(ξ),ξ[0, ),
q¯(ξ)=J0(αξ)0ξ<10ξ>1
q¯(ξ)=J0(αξ)·H(ξ),ξ[0, ).
q¯(ξ, η)=2iη01 expiξ2+ξ2ηJ02ξξηf1(ξ)ξdξ=2iη0 expiξ2+ξ2η×J02ξξηf(ξ)H(ξ)ξdξ=2iη0 expiξ2+ξ2η×J02ξξηf(ξ)k=1NAk exp(-Bkξ2)ξdξ.
q¯(ξ, η)=2iη01 expiξ2+ξ2ηJ02ξξηJ0(αξ)×exp(-Bξ2)ξdξ=2iη0 expiξ2+ξ2ηJ02ξξηJ0(αξ)×exp(-Bξ2)H(ξ)ξdξ=2iη0 expiξ2+ξ2ηJ02ξξηJ0(αξ)×k=1NAk exp(-Bkξ2)ξdξ.
q¯(ξ, η)=k=1NAkexp-Bkξ21+iBkη1+iBkηJ0αξ1+iBkη×exp-i4α2η1+iBkη=k=1NAkq¯GB(ξ, η; α, Bk),
0F2(u)du=1201k2(ξ)/ξdξ=1201[k(ξ)/ξ]2dξ<-.
hn(ξ)=a1ξp1+a2ξp2++anξpn,
ξhn(ξ)
=exp(-u/2)[a1 exp(-p1u)+a2 exp(-p2u)++an exp(-pnu)]=a1 exp[-(1/2+p1)u]+a2 exp[-(1/2+p2)u]++an exp[-(1/2+pn)u].
ξhn(ξ)=a1 exp(-B1x2)+a2 exp(-B2x2)++an exp(-Bnx2),
f(x)=k=1NAk exp(-Bkx2),
Q=0f0(x)-k=1NAk exp(-Bkx2)2dx.
k=1NaikAk=bi,(i=1, 2,, N),
aik=0 exp[-(Bi+Bk)x2]dx=(π)1/2/2(Bi+Bk)-1/2,
bi=0f0(x) exp(-Bix2)dx.
Eexp(-x2/w02)Lp2x2w02,0x<a0,x>a,
q¯(ξ)=exp(-Bξ)Lp(2Bξ2),0ξ<10,ξ>1,
q¯(w)=201 exp(-Bξ2)Lp(2Bξ2)J0(kawξ)ξdξ,
q¯(w)=20 exp(-Bξ2)Lp(2Bξ2)×J0(kawξ)H(ξ)ξdξ=2k=1NAk0 exp(-Bkξ2)Lp(2Bξ2)×J0(kawξ)ξdξ.
0x exp-12αx2Ln12βx2J0(xy)dx
=(α-β)nαn+1exp12αy2Lnβy22α(β-α),
q¯(w)=k=1NAk(B-Bk)p2(Bk)p+1exp-(kaw)24Bk×LpB(kaw)22Bk(B-Bk).

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