Abstract

The slowly varying envelope approximation is applied to the radiation problems of the Helmholtz equation with a planar single-layer and dipolar sources. The analyses of such problems provide procedures to recover solutions of the Helmholtz equation based on the evaluation of solutions of the parabolic wave equation at a given plane. Furthermore, the conditions that must be fulfilled to apply each procedure are also discussed. The relations to previous work are given as well.

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References

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  1. A. E. Siegman, Lasers, 1st ed. (University Science Books, 1986).
  2. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
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    [CrossRef]
  4. G. P. Agrawal and M. Lax, “Free space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. Q. Cao and X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  14. S. R. Seshadri, “Nonparaxial corrections for the fundamental Gaussian beam,” J. Opt. Soc. Am. A 19, 2134–2141 (2002).
    [CrossRef]
  15. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A 24, 3297–3302 (2007).
    [CrossRef]

2007

2002

2001

1998

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Q. Cao and X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

1992

1990

1985

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

1983

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

1981

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

1979

1975

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

Agrawal, G. P.

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

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

Bélanger, Pierre-A.

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

Borghi, R.

Cao, Q.

Couture, M.

M. Couture and Pierre-A. Bélanger, “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]

Deng, X.

Fukumitsu, O.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

Lax, M.

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

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

Moret, M. A.

Nemoto, N.

Pattanayak, D. N.

Porras, M. A.

Ruiz, B.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Santarsiero, M.

Seshadri, S. R.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers, 1st ed. (University Science Books, 1986).

Takenaka, T.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

Tuovinen, J.

J. Tuovinen, “Accuracy of a Gaussian beam,” IEEE Trans. Antennas Propag. 40, 391–398 (1992).
[CrossRef]

Vaveliuk, P.

Wünsche, A.

Yokota, M.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

Zebende, G. F.

Appl. Opt.

IEEE Trans. Antennas Propag.

J. Tuovinen, “Accuracy of a Gaussian beam,” IEEE Trans. Antennas Propag. 40, 391–398 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. A

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

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

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

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

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

Other

A. E. Siegman, Lasers, 1st ed. (University Science Books, 1986).

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Equations (43)

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2 ψ + k 2 ψ = f ( r t ) δ ( z ) ,
ψ ( r t , z ) = ψ ( r t , z ) .
ψ ( r ) = ψ + ( r ) u ( z ) + ψ ( r ) u ( z ) ,
ψ + ( r t , z ) = ψ ( r t , z ) .
2 ψ ± + k 2 ψ ± = 0 ,
lim z 0 + ψ + ( r t , z ) = lim z 0 ψ ( r t , z ) = ψ ( r ) | z = 0 .
[ ψ + z ψ z | z = 0 = f ( r t ) .
f ( r t ) = 2 ψ + z | z = 0 = 2 ψ - z | z = 0 .
ψ ± ( r ) = U ± ( r ) e ± i k z .
2 U ± ± 2 i k U ± z = 0.
ψ ( r ) = U + ( r ) exp ( i k z ) u ( z ) + U ( r ) exp ( i k z ) u ( z ) ,
ψ ( r ) = [ U + ( r ) u ( z ) + U ( r ) u ( z ) ] p ( z ) ,
p ( z ) = exp ( i k z ) u ( z ) + exp ( i k z ) u ( z ) .
U ( r ) = U + ( r ) u ( z ) + U ( r ) u ( z ) ,
ψ ( r ) = U ( r ) p ( z ) .
p ( z ) 2 U + 2 i k q ( z ) U z + 2 i k δ ( z ) U = f ( r t ) δ ( z ) ,
q ( z ) = 1 i k d p d z = exp ( i k z ) u ( z ) exp ( i k z ) u ( z ) .
U ( r ) | z = 0 = ψ ( r ) | z = 0 .
t 2 U p ± ± 2 i k U p ± z = 0 ,
2 i k U z = 2 i k [ u ( z ) U + z + u ( z ) U z ] ,
2 U z 2 = u ( z ) 2 U + z 2 + u ( z ) 2 U z 2 + [ U + z U - z ] δ ( z ) .
p ( z ) t 2 U p + 2 i k q ( z ) U p z + 2 i k U p δ ( z ) = f ( r t ) δ ( z ) .
U p ( r ) = U p + ( r ) u ( z ) + U p ( r ) u ( z ) ,
2 i k U p ( r ) | z = 0 = f ( r t ) .
t 2 U p ± ± 2 i k U p ± z = 0 , z 0 U p ± | z = 0 = i 2 k f ( r t ) } .
U p ( r ) | z = 0 U ( r ) | z = 0 .
2 ψ + k 2 ψ = g ( r t ) δ ( z ) .
ψ ( r t , z ) = ψ ( r t , z ) .
ψ ( r ) = ψ + ( r ) u ( z ) ψ ( r ) u ( z ) .
lim z = 0 + ψ ( r t , z ) = lim z = 0 ψ ( r t , z ) .
g ( r t ) = [ ψ + ( r ) + ψ ( r ) | z = 0 .
ψ + ( r ) | z = 0 = ψ ( r ) | z = 0 = 1 2 g ( r t ) .
ψ ( r ) = U + ( r ) exp ( i k z ) u ( z ) U ( r ) exp ( i k z ) u ( z ) ,
ψ ( r ) = [ U + ( r ) u ( z ) + U ( r ) u ( z ) ] q ( z ) ,
ψ ( r ) = U ( r ) q ( z ) .
U ( r ) | z = 0 = 1 2 g ( r t ) .
q ( z ) 2 U + 2 i k p ( z ) U z 2 U ( r ) δ ( z ) = g ( r t ) δ ( z ) .
U ( r ) δ ( z ) = U ( r ) | z = 0 δ ( z ) ,
q ( z ) 2 U + 2 i k p ( z ) U z = 0.
2 U ± ± 2 i k U ± z = 0 , z 0 U ± ( r ) | z = 0 = 1 2 g ( r t ) } .
q ( z ) t 2 U p + 2 i k p ( z ) U p z 2 U p δ ( z ) = g ( r t ) δ ( z ) .
U p ( r ) | z = 0 = 1 2 g ( r t ) .
t 2 U p ± ± 2 i k U p ± z = 0 , z 0 U p ± ( r ) | z = 0 = 1 2 g ( r t ) } ,

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