Abstract

A relatively simple transform from an arbitrary solution of the paraxial wave equation to the corresponding exact solution of the Helmholtz wave equation is derived in the condition that the evanescent waves are ignored and is used to study the corrections to the paraxial approximation of an arbitrary free-propagation beam. Specifically, the general lowest-order correction field is given in a very simple form and is proved to be exactly consistent with the perturbation method developed by Lax et al. [Phys. Rev. A 11, 1365 (1975)]. Some special examples, such as the lowest-order correction to the paraxial approximation of a fundamental Gaussian beam whose waist plane has a parallel shift from the z=0 plane, are presented.

© 1998 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  3. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  4. M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  5. G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  6. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  7. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [CrossRef] [PubMed]
  8. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  9. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  10. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [CrossRef]
  11. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989).
    [CrossRef]
  12. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  13. C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
    [CrossRef]
  14. M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
    [CrossRef]

1996

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

1992

1990

1989

1985

1983

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1979

1977

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Ambrosini, D.

Bagini, V.

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Borghi, R.

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Fukumitsu, O.

Gori, F.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Nemoto, S.

Pacileo, A. M.

Pattanayak, D. N.

Pratesi, R.

Ronchi, L.

Saghafi, S.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Siegman, A. E.

Spagnolo, G. S.

Takenaka, T.

Wünsche, A.

Yokota, M.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Phys. Rev. A

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

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Equations (31)

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2ϕ+k2ϕ=0.
ψ(fx, fy, z)=ψ(fx, fy, 0)exp(ikΔz)(forλ2f2<1)ψ(fx, fy, 0)exp(-kδz)(forλ2f21)
ψ(fx, fy, z)=-ϕ(x, y, z) ×exp[-i2π(fxx+fyy)]dxdy.
i2k ϕ(0)z+2x2+2y2ϕ(0)=0.
ψ(0)(fx, fy, z)=ψ(0)(fx, fy, 0)exp(-iπλf2z)
ψ(fx, fy, z)=exp(ikz) ×exp-ikzn=2 (2n-3)!!(2n)!!(λ2f2)n ×ψ(0)(fx, fy, z),
Δ=1-12λ2f2-n=2 (2n-3)!!(2n)!!(λ2f2)n.
F-1[fx2+fy2]=-14π22x2+2y2,
ϕ(x, y, z)=exp(ikz)n=2Tnϕ(0)(x, y, z)
Tn=m=0 (-1)mnm!k2mn-(2n-3)!!(2n)!!ikzm ×2x2+2y2mn.
Tn(pn)=m=0pn (-1)mnm!k2mn-(2n-3)!!(2n)!!ikzm ×2x2+2y2mn,
Aq(p2, p3,pq)=n=2qTn(pn)
ϕq(p2, p3,pq, x, y, z)
=exp(ikz)Aq(p2, p3,pq)ϕ(0)(x, y, z).
tn(pn)=m=0pn 1m!-(2n-3)!!(2n)!!ikzλ2nf2nm,
aq(p2, p3,pq)=n=2qtn(pn),
ψq(p2, p3,pq, fx, fy, z)
=exp(ikz)aq(p2, p3,pq)ψ(0)(fx, fy, z)
ηq(p2, p3,pq, fx, fy, z)
=1-exp[ikz(1-λ2f2/2)]aq(p2, p3,pq)exp(ikzΔ).
Qq+1=kz(2q-1)!!(λ2fmax2)q+1(2q+2)!!1,
Rn(pn+1)=1(pn+1)!(2n-3)!!(2n)!!kzλ2nfmax2npn+11(for2nq)
ϕ(2)=-iz8k32x2+2y22ϕ(0),
ψ(fx, fy, z)exp(ikz)(1-ikzλ4f4/8)ψ(0)(fx, fy, z)
ϕ(2)=iz2k2ϕ(0)z2.
i2k z+2x2+2y2ϕ(2)=-2ϕ(0)z2.
ϕG(0)=1[1+i(z-z0)/l]exp-x2+y22w02[1+i(z-z0)/l],
ϕG(2)=1kw02 -iz/l[1+i(z-z0)/l]2 ×L2x2+y22w02[1+i(z-z0)/l]ϕG(0),
L2(u)=12u2-2u+1.
ϕmn(0)(0, 0, z)=12w0Hm(0)Hn(0) 11+z2/l2 ×exp[-i(m+n+1)arctan(z/l)],
ϕmn(2)(0, 0, z)=-iz2kl21+z2l2-2(m+n+1)2+1-2z2l2-i4(m+n+1)zl ×ϕmn(0)(0, 0, z).

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