Abstract

Light beams extending beyond the paraxial region are investigated by the correction scheme of Lax et al. [ Phys. Rev. A 11, 1365 ( 1975)] All higher-order corrections for the Laguerre–Gaussian and Hermite–Gaussian beams are obtained explicity. It is shown that the corrections for both Gaussian beams are described in the same form. When the field distribution of a light beam is specified at some transverse plane, we can deal with its propagation beyond the paraxial approximation by superposing the corrected beams.

© 1985 Optical Society of America

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References

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  1. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
  2. G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
    [CrossRef]
  3. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  4. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  5. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  6. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  7. M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  8. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  9. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  10. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  11. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  12. A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. X. [Note that the coefficient 22n is dropped in the right-hand side of formula (32) at p. 195.]

1983 (1)

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

1981 (1)

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

1979 (2)

1977 (1)

1975 (1)

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

1973 (1)

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961 (2)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave 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]

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]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

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]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Erdelyi, A.

A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. X. [Note that the coefficient 22n is dropped in the right-hand side of formula (32) at p. 195.]

Felsen, L. B.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Goubau, G.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave 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]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[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]

Pattanayak, D. N.

Schwering, F.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

Bell Syst. Tech. J. (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[CrossRef]

J. Opt. Soc. Am. (3)

Phys. Rev. A (4)

M. Lax, W. H. Louisell, 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]

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 wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (1)

A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. X. [Note that the coefficient 22n is dropped in the right-hand side of formula (32) at p. 195.]

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Equations (63)

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( 2 + k 2 ) Ψ ( r , z ) = 0 ,
Ψ ( r , z ) = ψ ( r , z ) e jkz .
( 2 2 j k z ) ψ = 0 .
[ t 2 + 4 σ 2 σ 4 f 2 σ 2 σ ( σ 2 σ ) ] ψ = 0 ,
f = 1 / k w 0 , σ = [ 1 j ( z / z 0 ) ] 1 , z 0 = k w 0 2 / 2 .
t 2 = 1 ρ ρ ( ρ ρ ) + 1 ρ 2 2 θ 2 ,
t 2 = 2 ξ 2 + 2 η 2 ,
ψ = s = 0 f 2 s ψ ( 2 s ) .
P ψ ( 0 ) = 0 ,
P ψ ( 2 s + 2 ) = ψ ( 2 s ) , s = 0 , 1 ,
P = t 2 + 4 σ 2 σ ,
= 4 σ 2 σ ( σ 2 σ ) .
ψ ( 0 ) ( r , 0 ) = ψ ( r , 0 ) ,
ψ ( 2 s ) ( r , 0 ) = 0 , s = 1 , 2 , .
U ( r , 0 ) = m , n a m n ψ m n ( 0 ) ( r , 0 ) .
a m n = U ( r , 0 ) ϕ m n * ( r , 0 ) d S ,
ψ m n ( 0 ) ϕ s t * d S = δ m s δ n t ,
U ( r , z ) = m , n a m n s f 2 s ψ m n ( 2 s ) ( r , z ) e jkz .
ψ m n ( 0 ) = ( 1 ) n n ! σ m + n + 1 ρ m L n m ( σ ρ 2 ) e σ ρ 2 e j m θ , m = 0 , ± 1 , , n = 0 , 1 ,
L n m ( υ ) = 1 n ! υ m e υ d n d υ n ( e υ υ m + n ) .
ϕ m n = ( 1 ) n π ( m + n ) ! w 0 2 σ * n ρ m L n m ( σ * ρ 2 ) e j m θ .
ψ m n par ( 2 ) = σ 1 ψ m n + 2 ( 0 ) .
ψ m n ( 2 ) = ( 1 σ 1 ) ψ m n + 2 ( 0 ) .
ψ m n ( 4 ) = 2 ζ ψ m n + 3 ( 0 ) + 1 2 ζ 2 ψ m n + 4 ( 0 ) ,
ψ m n ( 6 ) = 5 ζ ψ m n + 4 ( 0 ) 2 ζ 2 ψ m n + 5 ( 0 ) + 1 6 ζ 3 ψ m n + 6 ( 0 ) ,
ζ = 1 σ 1 .
ψ m n ( 2 s ) = p = 1 s c p ( 2 s ) ζ p ψ m n + s + p ( 0 ) , s = 1 , 2 , ,
( ζ p ψ m n ( 0 ) ) = P [ 1 p + 1 ζ p + 1 ψ m n + 2 ( 0 ) 2 ( 1 δ 0 p ) ζ p ψ m n + 1 ( 0 ) + ( 1 δ 1 p ) p ζ p 1 ψ m n ( 0 ) ]
ψ m n ( 2 s ) = p = 1 s c p ( 2 s ) ( ζ p ψ m n + s + p ( 0 ) ) = P { p = 1 s c p ( 2 s ) [ 1 p + 1 ζ p + 1 ψ m n + s + p + 2 ( 0 ) 2 ( 1 δ 0 p ) ζ p ψ m n + s + p + 1 ( 0 ) + ( 1 δ 1 p ) p ζ p 1 ψ m n + s + p ( 0 ) ] } .
ψ m n ( 2 s + 2 ) = p = 1 s + 1 [ 1 p c p 1 ( 2 s ) 2 c p ( 2 s ) + ( p + 1 ) c p + 1 ( 2 s ) ] ζ p ψ m n + s + p , ( 0 )
c 0 ( 2 s ) = c s + 1 ( 2 s ) = c s + 2 ( 2 s ) = 0 .
c p ( 2 s + 2 ) = 1 p c p 1 ( 2 s ) 2 c p ( 2 s ) ( p + 1 ) c p + 1 ( 2 s ) .
c 1 ( 2 ) = 1 .
c p ( 2 s ) = ( 1 ) s + p ( 2 s ) ! s ( p 1 ) ! ( s p ) ! ( s + p ) ! , s = 1 , 2 , , p = 1 , 2 , , s .
ψ ˆ m n ( 0 ) = 2 ( m + n ) σ ( m + n / 2 ) + 1 H m ( σ ξ ) H n ( σ η ) × exp [ σ ( ξ 2 + η 2 ) ] , m , n = 0 , 1 , ,
H m ( υ ) = ( 1 ) m e υ 2 d m d υ m e υ 2 .
ϕ ˆ m n = 1 π m ! n ! w 0 2 σ * ( m + n / 2 ) H m ( σ * ξ ) H n ( σ * η ) .
( 1 ) n 2 2 n n ! L n ( σ ρ 2 ) = q = 0 n ( n q ) H 2 n 2 q ( σ ξ ) H 2 q ( σ η )
ψ ˆ 00 ( 2 s ) = p = 1 s c p ( 2 s ) ζ p q = 0 s + p ( s + p q ) ψ ˆ 2 s + 2 p 2 q 2 q ( 0 ) .
G m n ( 2 s ) = ( 2 ) ( m + n ) m + n ξ m η n ψ ˆ 00 ( 2 s ) .
( 2 ) ( m + n ) m + n ξ m η n ψ ˆ 00 ( 0 ) = ψ ˆ m n ( 0 ) ,
ψ ˆ m n ( 2 s ) = p = 1 s c p ( 2 s ) ζ p κ ( m , n , s + p ) ,
κ ( m , n , t ) = q = 0 t ( t q ) ψ ˆ m + 2 t 2 q n + 2 q . ( 0 )
ψ ˆ m ( 0 ) = 2 m σ ( m + 1 / 2 ) H m ( σ ξ ) e σ ξ 2 .
ψ ˆ m ( 2 s ) = p = 1 s c p ( 2 s ) ζ p ψ ˆ m + 2 s + 2 p . ( 0 )
τ = σ ρ 2 ,
ψ m n ( 0 ) = λ m n ψ m n ( 0 ) ,
= τ 2 τ 2 + ( 1 + τ ) τ + 1 4 τ 2 θ 2 ,
λ m n = [ ( m / 2 ) + n + 1 ] .
+ = τ * 2 τ * 2 + ( 1 τ * ) τ * + 1 4 τ * 2 θ 2 1 .
μ m n = λ m n ,
P = 4 σ τ ( τ τ ) + σ τ 2 θ 2 + 4 ( σ τ τ + σ 2 σ ) ,
= 4 ( σ τ τ + σ 2 σ ) 2 .
( σ τ τ + σ 2 σ ) ψ m n ( 0 ) = ( 1 ) n + 1 n ! σ ( m / 2 ) + n + 2 × e j m θ ( τ d 2 d τ 2 + d d τ m 2 4 τ ) Λ m n ( τ ) ,
Λ m n ( τ ) = τ m / 2 n m ( τ ) e τ .
υ d d υ n m ( υ ) = ( n + 1 ) n + 1 m ( υ ) ( m + n + 1 υ ) n m ( υ ) ,
( σ τ τ + σ 2 σ ) ψ m n ( 0 ) = ψ m n + 1 ( 0 ) .
ψ m n ( 0 ) = 4 ψ m n + 2 ( 0 ) .
ψ m n par ( 2 ) = σ ( m / 2 ) + n + 2 W ( τ ) e j m θ .
[ τ d 2 d τ 2 + ( 1 + τ ) d d τ + ( m 2 + n + 2 m 2 4 τ ) ] × W ( τ ) = ( 1 ) n + 2 ( n + 2 ) ! Λ m n + 2 ( τ ) .
W ( τ ) = ( 1 ) n + 2 ( n + 2 ) ! Λ m n + 2 ( τ ) .
( ζ p ψ m n ( 0 ) = 4 ζ p ψ m n + 2 ( 0 ) 8 p ζ p 1 ψ m n + 1 ( 0 ) + 4 p ( p 1 ) ζ p 2 ψ m n ( 0 ) .
P ( ζ p ψ m n ( 0 ) ) = 4 p ζ p 1 ψ m n ( 0 ) .

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