Abstract

In this paper we present a simple and effective method, based on appropriate superpositions of Bessel–Gauss beams, which in the Fresnel regime is able to describe in analytic form the three-dimensional evolution of important waves as Bessel beams, plane waves, Gaussian beams, and Bessel–Gauss beams when truncated by finite apertures. One of the by-products of our mathematical method is that one can get in a few seconds, or minutes, high-precision results, which normally require quite lengthy numerical simulations. The method works in electromagnetism (optics, microwaves) as well as in acoustics.

© 2012 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  2. F. Gori and G. Guattari, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  3. J. J. Wen and M. A. Breazele, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  4. D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
    [CrossRef]
  5. H. E. H. Figueroa, M. Z. Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).
  6. E. Recami and M. Z. Rached, “Localized waves: a review,” in Advances in Imaging and Electron Physics, Vol. 156 (Academic, 2009), pp. 235–353.
    [CrossRef]
  7. E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
    [CrossRef]
  8. M. Z. Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
    [CrossRef]
  9. C. A. Dartora and K. Z. Nobrega, “Study of Gaussian and Bessel beam propagation using a new analytic approach,” Opt. Commun. 285, 510–516 (2012).
    [CrossRef]
  10. M. Z. Rached, E. Recami, and M. Balma, “Proposte di antenne generatrici di fasci non-diffrattivi di microonde (Proposal of apertures generating nondiffracting beams of microwaves),” http://arxiv.org/abs/1108.2027 .

2012

C. A. Dartora and K. Z. Nobrega, “Study of Gaussian and Bessel beam propagation using a new analytic approach,” Opt. Commun. 285, 510–516 (2012).
[CrossRef]

2009

M. Z. Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

2004

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

2003

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

1988

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

1987

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

Balma, M.

M. Z. Rached, E. Recami, and M. Balma, “Proposte di antenne generatrici di fasci non-diffrattivi di microonde (Proposal of apertures generating nondiffracting beams of microwaves),” http://arxiv.org/abs/1108.2027 .

Breazele, M. A.

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

Dartora, C. A.

C. A. Dartora and K. Z. Nobrega, “Study of Gaussian and Bessel beam propagation using a new analytic approach,” Opt. Commun. 285, 510–516 (2012).
[CrossRef]

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

Ding, D.

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

Figueroa, H. E. H.

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gori, F.

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

Guattari, G.

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

Nobrega, K. Z.

C. A. Dartora and K. Z. Nobrega, “Study of Gaussian and Bessel beam propagation using a new analytic approach,” Opt. Commun. 285, 510–516 (2012).
[CrossRef]

Nóbrega, K. Z.

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

Rached, M. Z.

M. Z. Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

M. Z. Rached, E. Recami, and M. Balma, “Proposte di antenne generatrici di fasci non-diffrattivi di microonde (Proposal of apertures generating nondiffracting beams of microwaves),” http://arxiv.org/abs/1108.2027 .

E. Recami and M. Z. Rached, “Localized waves: a review,” in Advances in Imaging and Electron Physics, Vol. 156 (Academic, 2009), pp. 235–353.
[CrossRef]

Recami, E.

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

M. Z. Rached, E. Recami, and M. Balma, “Proposte di antenne generatrici di fasci non-diffrattivi di microonde (Proposal of apertures generating nondiffracting beams of microwaves),” http://arxiv.org/abs/1108.2027 .

E. Recami and M. Z. Rached, “Localized waves: a review,” in Advances in Imaging and Electron Physics, Vol. 156 (Academic, 2009), pp. 235–353.
[CrossRef]

Wen, J. J.

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

Zhang, Y.

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

E. Recami, M. Z. Rached, K. Z. Nóbrega, C. A. Dartora, and H. E. H. Figueroa, “On the localized superluminal solutions to the Maxwell equations,” IEEE J. Sel. Top. Quantum Electron. 9, 59–73 (2003), special issue on “Nontraditional Forms of Light.”
[CrossRef]

J. Acoust. Soc. Am.

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

D. Ding and Y. Zhang, “Notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 116, 1401–1405 (2004).
[CrossRef]

Opt. Commun.

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

C. A. Dartora and K. Z. Nobrega, “Study of Gaussian and Bessel beam propagation using a new analytic approach,” Opt. Commun. 285, 510–516 (2012).
[CrossRef]

Phys. Rev. A

M. Z. Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79, 013816 (2009).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

H. E. H. Figueroa, M. Z. Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).

E. Recami and M. Z. Rached, “Localized waves: a review,” in Advances in Imaging and Electron Physics, Vol. 156 (Academic, 2009), pp. 235–353.
[CrossRef]

M. Z. Rached, E. Recami, and M. Balma, “Proposte di antenne generatrici di fasci non-diffrattivi di microonde (Proposal of apertures generating nondiffracting beams of microwaves),” http://arxiv.org/abs/1108.2027 .

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Figures (13)

Fig. 1.
Fig. 1.

Field given by Eq. (15), representing a Bessel beam at z=0, with kρ=4.07·104m1 and truncated by a finite circular aperture with radius R=3.5mm. The coefficients An are given by Eq. (19), with q=0, L=3R2, qR=6/L, and N=23.

Fig. 2.
Fig. 2.

Intensity of a Bessel beam truncated by a finite aperture, as given by solution (11).

Fig. 3.
Fig. 3.

Orthogonal projection of the intensity shown in Fig. 2.

Fig. 4.
Fig. 4.

Oscillations of the field intensity on the z axis, when adopting N=500 in Eq. (11).

Fig. 5.
Fig. 5.

Field given by Eq. (15), when representing a Gaussian beam truncated at z=0. The dotted line depicts the ideal Gaussian curve, when truncation is absent.

Fig. 6.
Fig. 6.

Square magnitude of the field emanated by a finite aperture, in the case of a Gaussian beam, according to the solution (11).

Fig. 7.
Fig. 7.

Orthogonal projection corresponding to Fig. 6.

Fig. 8.
Fig. 8.

Field in Eq. (15), in the case now of a Bessel–Gauss beam truncated at z=0. Here we adopted the values L=10R2, qR=q, and N=30; see the text for details.

Fig. 9.
Fig. 9.

Square magnitude of the field emanated by a finite aperture in the case of a truncated Bessel–Gauss beam, represented by solution (11).

Fig. 10.
Fig. 10.

Orthogonal projection for the case in Fig. 9.

Fig. 11.
Fig. 11.

Field at z=0, with kρ=0, as given by Eq. (15) in the new case of a plane wave truncated at z=0. We have here adopted the values L=6R2, qR=8/L, and N=150. The coefficients An being given by relations (19), with q=0.

Fig. 12.
Fig. 12.

Square magnitude of the resulting field, emanated by the finite aperture, as given by solution (11) in the present case of a truncated plane wave.

Fig. 13.
Fig. 13.

Orthogonal projection corresponding to Fig. 12.

Equations (23)

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Ψ(ρ,z)=ikzexp[i(kz+kρ22z)]0Ψ(ρ,0)exp(ikρ22z)J0(kρρz)ρdρ,
Ψ(ρ,0)=ΨG(ρ,0)=Aexp(qρ2),
ΨG(ρ,z)=ikA2zQexp[ik(z+ρ22z)]exp[k2ρ24Qz2],
Q=qik/2z.
Ψ(ρ,0)=ΨBG(ρ,0)=AJ0(kρρ)exp(qρ2),
ΨBG(ρ,z)=ikA2zQexp[ik(z+ρ22z)]J0(ikkρρ2zQ)exp[14Q(kρ2+k2ρ2z2)],
ΨB(ρ,z)=AJ0(kρρ)exp(ikzz),
Ψ(ρ,z)=ik2zexp[ik(z+ρ22z)]n=1NAnQnexp[k2ρ24Qnz2],
Qn=qnik2z.
V(ρ)=n=1NAnexp(qnρ2).
Ψ(ρ,z)=ik2zexp[ik(z+ρ22z)]n=NNAnQnJ0(ikkρρ2zQn)exp[14Qn(kρ2+k2ρ2z2)],
V(ρ)=J0(kρρ)n=NNAneqnρ2.
qn=qR+iqIn,
qIn=2πLn,
V(ρ)=J0(kρρ)exp(qRρ2)n=Anexp(i2πnLρ2),
exp(qRρ2)n=Anexp(i2πnLρ2),
G(r)=n=Anexp(i2πnr/L)for|r|L/2,
G(r)={exp(qRr)exp(qr)for|r|R20forR2<|r|<L/2,
An=1LR2R2exp(qRr)exp(qr)exp(i2πnLr)dr=1L(qRq)i2πn{exp[(qRqi2πLn)R2]exp[(qRqi2πLn)R2]}.
r=ρ2,
n=Anexp(i2πnρ2/L)={exp(qRρ2)exp(qρ2)for|ρ|R0forR<|ρ|<L/2,
exp(qRρ2)n=Anexp(i2πnρ2/L)={exp(qρ2)for0ρR0forR<ρL/2exp(qRρ2)f(ρ)0forρ>L/2,
exp(qRρ2)n=Anexp(i2πnρ2/L)exp(qρ2)circ(ρ/R),

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