Abstract

A detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z=0 is modulated by a Gaussian envelope is presented. We call such a field a Helmholtz–Gauss (HzG) beam. A simple closed-form expression for the paraxial propagation of the HzG beams is written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex scaled version of the transverse shape of the nondiffracting beam. The general expression for the angular spectrum of the HzG beams is also derived. We introduce for the first time closed-form expressions for the Mathieu–Gauss beams in elliptic coordinates and for the parabolic Gauss beams in parabolic coordinates. The properties of the considered beams are studied both analytically and numerically.

© 2005 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. Micely, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
    [CrossRef]
  4. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
    [CrossRef]
  5. S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
    [CrossRef]
  6. M. A. Bandres, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
    [CrossRef] [PubMed]
  7. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  8. Y. Li, H. Lee, E. Wolf, “New generalized Bessel–Gauss beams,” J. Opt. Soc. Am. A 21, 640–646 (2004).
    [CrossRef]
  9. A. P. Kiselev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
    [CrossRef]
  10. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  11. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

2004 (3)

2002 (1)

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

2001 (1)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

2000 (1)

1987 (3)

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. Micely, 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]

Allison, I.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

Bandres, M. A.

Chávez-Cerda, S.

M. A. Bandres, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
[CrossRef] [PubMed]

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Courtial, J.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

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

Eberly, J. H.

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

Gori, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Guattari, G.

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

Gutiérrez-Vega, J. C.

M. A. Bandres, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
[CrossRef] [PubMed]

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Iturbe-Castillo, M. D.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Kiselev, A. P.

A. P. Kiselev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
[CrossRef]

Lee, H.

Li, Y.

MacVicar, I.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

Micely, J. J.

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

New, G. H. C.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

O’Neil, A. T.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

Padgett, M. J.

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

Padovani, C.

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

Rami´rez, G. A.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Rodri´guez-Dagnino, R. M.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Tepichi´n, E.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

Wolf, E.

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

Opt. Commun. (2)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001).
[CrossRef]

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

Opt. Lett. (2)

Opt. Spectrosc. (1)

A. P. Kiselev, “New structures in paraxial Gaussian beams,” Opt. Spectrosc. 96, 479–481 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

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

Quantum Semiclassic. Opt. (1)

S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-Vega, A. T. O’Neil, I. MacVicar, J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” Quantum Semiclassic. Opt. 4, S52–S57 (2002).
[CrossRef]

Other (2)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 2000).

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Figures (7)

Fig. 1
Fig. 1

Normalized axial irradiance distribution of a HzG beam as a function of z for γ=1,3,,15. The vertical dashed lines are located at the maximum distance zmax=zR/γ. For numerical purposes, λ=632.8 nm and θ0=0.05°.

Fig. 2
Fig. 2

(a)–(c) Transverse amplitude distribution of a CG beam at different z planes; (d), (e) propagation of the amplitude and phase profiles along the planes (y, z) and (x, z); (f)–(h) amplitude and phase distribution of the angular spectrum at different z planes.

Fig. 3
Fig. 3

(a)–(c) Transverse amplitude distribution of a first-order BG beam at different z planes, (d) propagation of the amplitude and phase profiles along the plane (x, z) in the range [-1.2zmax, 1.2zmax].

Fig. 4
Fig. 4

(a)–(c) Transverse amplitude distribution of a second-order even MG beam at different z planes, (d) propagation of the amplitude pattern along the planes (y, z) and (x, z) in the range [0, 1.2zmax], (e)–(g) amplitude and phase distributions of the angular spectrum as a function of the normalized coordinates (u/kt, v/kt).

Fig. 5
Fig. 5

(a)–(c) Transverse amplitude and phase distributions of a seventh-order HMG beam at different z planes, (d)–(f) amplitude and phase distributions of the angular spectrum as a function of the normalized coordinates (u/kt, v/kt).

Fig. 6
Fig. 6

(a)–(c) Transverse amplitude distribution of an even PG beam with a=3 at different z planes, (d) propagation of the amplitude pattern along the plane (x, z) in the range [0, 1.2zmax], (e)–(g) amplitude and phase distributions of the angular spectrum.

Fig. 7
Fig. 7

(a)–(c) Transverse amplitude and phase distributions of a TPG beam with a=3 at different z planes, (d)–(f) amplitude and phase distributions of the angular spectrum.

Equations (71)

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U0(rt)=exp(-r2/w02)W(rt; kt),
2x2+2y2+kt2W(rt; kt)=0
W(rt; kt)=-ππA(φ)exp[ikt(x cos φ+y sin φ)]dφ,
U(r)=exp-i kt22k zμGB(r)Wxμ, yμ; kt,
GB(r)=exp(ikz)μexp-r2μw02
μ=μ(z)=1+iz/zR,
2x¯2+2y¯2+kt2W(x¯, y¯; kt)=0
W(x¯, y¯; kt)=-ππA(φ)exp[ikt(x¯ cos φ+y¯ sin φ)]dφ.
U(u, v; z)=12π  U(x, y, z)exp(-ixu-iyv)dxdy,
U(u, v; z)=D(z)exp-w02µ4ρ2Ww022iu, w022iv; kt,
D(z)=w022exp(-14kt2w02)exp(ikz)
U(r)=exp[i(k-kt2/2k)z]W(rt; kt),
γ=θ0θG=12ktw0
zmax=w0sin θ0w0kkt=zRγ.
I(z¯)=11+z¯2exp-2γ2z¯21+z¯2,
I(zmax)=γ21+γ2exp-2γ21+γ2.
exp-w02µ4ρ2expktw022ρ
exp(γ2)exp-14w02(ρ-kt)2,
W(rt; kt)=cos(kty)
CG(r)=exp-ikt22k zμGB(r)cosktyμ.
CG(u, v; z)=D(z)exp-μw024ρ2cosh[2γ2(v/kt)].
W(rt; kt)=Jm(ktr)exp(imϕ),
BGm(r)=exp-ikt22kzμGB(r)Jmktrμexp(imϕ),
BGm(u, v; z)=(-i)mD(z)exp-μw024ρ2Im(2γ2ρ/kt)×exp(imϕ),
We(rt; kt)=Jem(ξ, q)cem(η, q),
Wo(rt; kt)=Jom(ξ, q)sem(η, q),
MGme(r)=exp-ikt22k zμGB(r)Jem(ξ¯, q)cem(η¯, q),
x=f0(1+iz/zR)cosh ξ¯ cos η¯,
y=f0(1+iz/zR)sinh ξ¯ sin η¯,
MGme(u, v; z)=D(z)exp-μw024ρ2Jem(ξˆ, q)cem(ηˆ, q),
u=2iw02f0 cosh ξˆ cos ηˆ,
v=2iw02f0 sinh ξˆ sin ηˆ.
HMGm±(r)=MGme(r)±iMGmo(r),
We(ξ, η; kt)=|Γ1|2π2Pe(2ktξ; a)Pe(2ktη; -a),
Wo(ξ, η; kt)=|Γ3|2π2Po(2ktξ; a)Po(2ktη; -a),
PGe(r; a)=exp-ikt22k zμGB(r)|Γ1|2π2×Pe(2kt/μξ; a)Pe(2kt/μη; -a).
PGe(u, v; z)=D(z)exp-μw024ρ2|Γ1|2π2×Pe(-iktw02ξ˜; a)Pe(-iktw02η˜; -a),
TPG±(r; a)=PGe(r; a)±iPGo(r; a),
u0(x, y)=exp(-r2/w02)exp[i(kxx+kyy)].
u(r)=exp(ikz)Ψ(r)
2x2+2y2+2ikzΨ(r)=0.
Ψ(r)=exp[iP(z)]expikr22q(z)expikxxμ(z)+kyyμ(z),
dqdz-1=0,
qdμdz-μ=0,
i2kμ2-kt2q-2kqμ2dPdz=0.
q=q0+z,
μ=μ0q,
P=i ln q+kt22kμ02q+P0,
q0=-izR,μ0=izR,P0=-i ln(-izR)+ikt2zR2k,
q(z)=z-izR,
μ(z)=iq/zR=1+iz/zR,
P(z)=i ln[(1+z2/zR2)1/2]-arctanzzR-kt2zR2kzRq-i.
Ψ(r)=expii ln[(1+z2/zR2)1/2]-arctanzzR-kt2zR2kzRq+i×expikr2q(z)expikxxμ+kyyμ.
1μ=11+iz/zR=zRiq=w0w(z)exp[-i arctan(z/zR)],
Ψ(r)=1μexp-iκzμexp-r2μw02expikxxμ+kyyμ.
u(r)=exp-iκzμexp(ikz)μ×exp-r2μw02expikxxμ+kyyμ.
U(r)=-ππA(φ)u(r)dφ,
U(r)=exp-iκzμexp(ikz)μexp-r2μw02 -ππA(φ)×expikxxμ+kyyμdφ.
U(r)=exp-iκzμexp(ikz)μexp-r2μw02Wxμ, yμ; kt,
U(u, v; z)=12π --U(r)exp(-ixu-iyv)dxdy,
U(u, v; z)=-ππA(φ)F{u(r)}dφ,
F{u(r)}=f(z) --exp-r2μw02+ikxxμ+ikyyμ-ixu-iyvdxdy,
F{u(r)}=D(z)exp-w02μ4ρ2expw022(kxu+kyv),
D(z)=w022exp-14kt2w02exp(ikz)
-ππdφA(φ)exp[i(w02kxu+iw02kyv)/2i]
U(u, v; z)=D(z)exp-w02μ4ρ2Ww022iu, w022i v; kt,
U(r)=exp-r02w02w0w(z)exp-r2-r02w2(z)+ikz+ik(r2-r02)2R(z)-iΘ(z)Wxμ, yμ; kt,
r012ktw02=ktkzR.
GB(r)=w0w(z)exp-r2w2(z)+ikz+ikr22R(z)-iΘ(z)
U(r)=C(z)GB(r)Wxμ, yμ; kt,
C(z)=expr02w2(z)-r02w02exp-ikr022R(z).

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