Abstract

Hermite–Gaussian and Laguerre–Gaussian beams with complex arguments of the type introduced by Siegman [ J. Opt. Soc. Am. 63, 1093 ( 1973)] are shown to arise naturally in correction terms of a perturbation expansion whose leading term is the fundamental paraxial Gaussian beam. Additionally, they can all be expressed as derivatives of the fundamental Gaussian beam and as paraxial limits of multipole complex-source point solutions of the reduced-wave equation.

© 1986 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [Crossref]
  2. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical beam eigenfunctions,”J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [Crossref]
  3. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,”J. Opt. Soc. Am. 67, 699–700 (1977).
    [Crossref]
  4. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [Crossref]
  5. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,”J. Opt. Soc. Am. 69, 575–578 (1979).
    [Crossref]
  6. M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [Crossref]
  7. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [Crossref]
  8. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  9. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [Crossref]
  10. E. Zauderer, Partial Differential Equations of Applied Mathmatics (Wiley-Interscience, New York, 1983).
  11. A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953) Vol. 2, Chap. X.
  12. A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953) Vol. 2, Chap. XI.
  13. B. Van der Pol, “A generalization of Maxwell’s definition of solid harmonics to waves in n dimensions,” Physica (The Hague) 3, 393–397 (1936).
    [Crossref]
  14. A. Erdelyi, “Zur Theorie der Kugelwellen,” Physica (The Hague) 4, 107–120 (1937).
    [Crossref]
  15. E. Zauderer, “Gaussian beams and their generalizations,” Phys. Rev. A 31, 3139–3144 (1985).
    [Crossref] [PubMed]

1985 (2)

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 (1)

1977 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial 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]

1937 (1)

A. Erdelyi, “Zur Theorie der Kugelwellen,” Physica (The Hague) 4, 107–120 (1937).
[Crossref]

1936 (1)

B. Van der Pol, “A generalization of Maxwell’s definition of solid harmonics to waves in n dimensions,” Physica (The Hague) 3, 393–397 (1936).
[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]

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex source point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[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, “Zur Theorie der Kugelwellen,” Physica (The Hague) 4, 107–120 (1937).
[Crossref]

Felsen, L. B.

Fukumitsu, O.

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 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 optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

McKnight, W. B.

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

Pattanayak, D. N.

Shin, S. Y.

Siegman, A. E.

Takenaka, T.

Van der Pol, B.

B. Van der Pol, “A generalization of Maxwell’s definition of solid harmonics to waves in n dimensions,” Physica (The Hague) 3, 393–397 (1936).
[Crossref]

Yokota, M.

Zauderer, E.

E. Zauderer, “Gaussian beams and their generalizations,” Phys. Rev. A 31, 3139–3144 (1985).
[Crossref] [PubMed]

E. Zauderer, Partial Differential Equations of Applied Mathmatics (Wiley-Interscience, New York, 1983).

Electron. Lett. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Phys. Rev. A (4)

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

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

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

E. Zauderer, “Gaussian beams and their generalizations,” Phys. Rev. A 31, 3139–3144 (1985).
[Crossref] [PubMed]

Physica (The Hague) (2)

B. Van der Pol, “A generalization of Maxwell’s definition of solid harmonics to waves in n dimensions,” Physica (The Hague) 3, 393–397 (1936).
[Crossref]

A. Erdelyi, “Zur Theorie der Kugelwellen,” Physica (The Hague) 4, 107–120 (1937).
[Crossref]

Proc. IEEE (1)

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

Other (3)

E. Zauderer, Partial Differential Equations of Applied Mathmatics (Wiley-Interscience, New York, 1983).

A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953) Vol. 2, Chap. X.

A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953) Vol. 2, Chap. XI.

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Equations (43)

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2 u + k 2 u = 0
u ( x , y , 0 ) = exp [ - k ( x 2 + y 2 ) / 2 a ]
u = w exp ( i k z )
2 w + 2 i k w z = 0
x = ξ / k ,             y = η / k
v ( ξ , η , z ) = w ( ξ / k , η / k , z )
v ξ ξ + v η η + 2 i v z + 1 k v z z = 0 ,
v ( ξ , η , 0 ) = exp [ - ( ξ 2 + η 2 ) / 2 a ] .
v = V + r = 1 v ( r ) / k r ,
L V V ξ ξ + V η η + 2 i V z = 0 ,
L v ( r ) = v z z ( r - 1 ) ;             r = 1 , 2 , 3 , ,
V ( ξ , η , 0 ) = exp [ - ( ξ 2 + η 2 ) / 2 a ] ,
v ( r ) ( ξ , η , 0 ) = 0 ;             r = 1 , 2 , 3 , .
V ( ξ , η , z ) = 2 a σ exp [ - σ p 2 ] ,
H n ( x ) = ( - 1 ) n exp ( x 2 ) d n d x n [ exp ( - x 2 ) ] ;             n = 0 , 1 , ,
d d x H n ( x ) - 2 x H n ( x ) = - H n + 1 ( x ) .
L n m ( x ) = 1 n ! x - m exp ( x ) d n d x n [ x n + m exp ( - x ) ] ; n , m = 0 , 1 , ,
x d d x L n m ( x ) + ( n + m + 1 - x ) L n m ( x ) = ( n + 1 ) L n + 1 m ( x ) ,
d d x L n m ( x ) - L n m ( x ) = - L n m + 1 ( x )
v ( n , m ) = 2 a ( - 1 ) n + m σ ( n + m ) / 2 + 1 H n ( σ ξ ) H m ( σ η ) exp ( - σ ρ 2 )
v ^ ( n , m ) = 2 a ( - 2 ) n + m i n n ! σ n + m + 1 ρ m L n m ( σ ρ 2 ) exp ( i m ϕ - σ ρ 2 ) .
v ( n , m ) ξ = 2 a ( - 1 ) n + m σ ( n + m ) / 2 + 1 H m ( σ η ) × [ σ H n ( σ ξ ) exp ( - σ ρ 2 ) - 2 σ ξ H n ( σ ξ ) exp ( - σ ρ 2 ) ] = 2 a ( - 1 ) n + m + 1 σ ( n + 1 + m ) / 2 + 1 H n + 1 ( σ ξ ) H m ( σ η ) exp ( - σ ρ 2 ) = v ( n + 1 , m )
v ( n , m ) η = v ( n , m + 1 ) .
v ( n , m ) = n ξ n m η m v ( 0 , 0 ) = n ξ n m η m V ;
v ^ ( n , m ) z = 2 a ( - 2 ) n + m i n n ! exp ( i m ϕ ) × [ ( - 2 i ) ( n + m + l ) σ n + m + 2 ρ m l n m ( σ ρ 2 ) exp ( - σ ρ 2 ) + ( - 2 i ) σ n + m + 3 ρ m + 2 L n m ( σ ρ 2 ) exp ( - σ ρ 2 ) - ( - 2 i ) σ n + m + 3 ρ m + 2 L n m ( σ ρ 2 ) exp ( - σ ρ 2 ) ] = 2 a ( - 2 ) n + m + 1 i n + 1 ( n + 1 ) ! σ n + m + 2 ρ m L n + 1 m ( σ ρ 2 ) × exp ( i m ϕ - σ ρ 2 ) = v ^ ( n + 1 , m ) .
v ^ ( n , 0 ) = n z n v ^ ( 0 , 0 ) = n z n V ,
( ξ + i η ) v ^ ( n , m ) = exp ( i ϕ ) [ ρ + i ρ ϕ ] v ^ ( n , m ) = 2 a ( - 2 ) n + m i n n ! σ n + m + 1 { m p m - 1 L n m ( σ ρ 2 ) × exp [ i ( m + 1 ) ϕ - σ ρ 2 ] + 2 σ ρ m + 1 L n m ( σ ρ 2 ) × exp [ i ( m + 1 ) ϕ - σ ρ 2 ] - 2 σ ρ m + 1 L n m ( σ ρ 2 ) × exp [ i ( m + 1 ) ϕ - σ ρ 2 ] - m ρ m - 1 L n m ( σ ρ 2 ) × exp [ i ( m + 1 ) ϕ - σ ρ 2 } = 2 2 a ( - 2 ) n + m i n n ! σ n + m + 1 ρ m + 1 × exp [ i ( m + 1 ) ϕ - σ ρ 2 ] { L n m ( σ ρ 2 ) - L n m ( σ ρ 2 ) } = 2 a ( - 2 ) n + m + 1 i n n ! σ n + m + 2 ρ m + 1 L n m + 1 ( σ ρ 2 ) × exp [ i ( m + 1 ) ϕ - σ ρ 2 ] = v ^ ( n , m + 1 ) ,
v ^ ( 0 , m ) = ( ξ + i η ) m v ^ ( 0 , 0 ) = ( ξ + i η ) m V ,
v ^ ( n , m ) = n z n ( ξ + i η ) m V
n z n V = ( - 2 i ) - n ( 2 ξ 2 + 2 η 2 ) n V .
v ^ ( n , m ) = n z n ( ξ + i η ) m V = ( - 2 i ) - n ( 2 ξ 2 + 2 η 2 ) n ( ξ + i η ) m V = ( - 2 i ) - n r = 0 n s = 0 m ( n r ) ( m s ) i m - s 2 r + s ξ 2 r + s 2 n + m - 2 r - s η 2 n + m - 2 r - s V = ( - 2 i ) - n r = 0 n s = 0 m ( n r ) ( m s ) i m - s v ( 2 r + s , 2 n + m - 2 r - s ) ,
( - 1 ) n + m 2 2 n + m n ! ( x + i y ) m L n m ( x 2 + y 2 ) = r = 0 n s + 0 m ( n r ) ( m s ) ( - i ) m + s H 2 r + s ( x ) H 2 n + m - 2 r - s ( y ) .
2 v ( 1 ) ξ 2 + 2 v ( 1 ) η 2 + 2 i v ( 1 ) z = - 2 V z 2 = - v ^ ( 2 , 0 ) .
V ( ξ , η , 0 ) = v ^ ( n , m ) z = 0 ,
L v ( 2 ) = - v z z ( 1 ) = i v ^ ( n + 3 , m ) - ½ i z v ^ ( n + 4 , m ) .
v ( 2 ) = c ( 2 ) z v ^ ( n + 3 , m ) + d ( 2 ) z 2 v ^ ( n + 4 , m ) ,
V ( ξ , η , 0 ) = v ( n , m ) z = 0 ,
v ( 1 ) = e ( 1 ) z v ( n + 4 , m ) + f ( 1 ) z v ( n + 2 , m + 2 ) + g ( 1 ) z v ( n , m + 4 ) ,
u = ( - i a ) [ ( z - i a ) 2 + ρ 2 / k ] - 1 / 2 exp { i k [ ( z - i a ) 2 + ρ 2 / k ] 1 / 2 } ,
lim k u exp [ - i k ( z - i a ) ] = 2 a σ exp [ - σ ρ 2 ] = V
lim k n ξ n m η m { u exp [ - i k ( z - i a ) ] } = n ξ n m η m V = v ( n , m ) ,
n z n ( ξ + i η ) m { u exp [ - i k ( z - i a ) ] } = exp [ - i k ( z - i a ) ] r = 0 n ( - i k ) r n - r z n - r ( ξ + i η ) m u
lim k n z n ( ξ + i η ) m { u exp [ - i k ( z - i a ) ] } = n z n ( n ξ + i η ) m V = v ^ ( n , m ) .

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