Abstract

It is shown that the Hermite–Gauss- and Laguerre–Gauss-type solutions of paraxial optics whose corresponding Hermite and Laguerre polynomials have complex arguments are closely related to a hidden symmetry in the parabolic equation. These solutions are generated from the fundamental Gaussian beam solution by applying the powers of the infinitesimal operators of this symmetry group. The Fourier spectrum of these solutions is obtained from the Fourier spectrum of the fundamental Gaussian beam solution in a similar manner. The Gaussian beam solutions containing Hermite polynomials with complex arguments that were derived by Siegman [ J. Opt. Soc. Am. 63, 1093 ( 1973)] represent a limiting case of the more-general solutions considered here. The generalized Gaussian beam solutions show a sharp mode picture with exact zeros in the field distribution only in the focal or waist plane. Unconventional applications of the parabolic approximation in optics including focus wave modes are demonstrated.

© 1989 Optical Society of America

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References

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  1. M. Lax, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
  3. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 10, 1550–1567 (1966).
    [CrossRef]
  4. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975). (Russian translation, Sovyetskoye Radio, Moscow, 1980).
  5. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972. (Russian translation, Mir, Moscow, 1974).
  6. D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
    [CrossRef]
  7. T. Ooya, M. Tateiba, O. Fukumitsu, “Transmission and reflection of a Gaussian beam at normal incidence on a dielectric slab,”J. Opt. Soc. Am. 65, 537–541 (1975).
    [CrossRef]
  8. A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. (Leipzig) 25, 113–135 (1970).
  9. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical beam eigenfunctions,”J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  10. P. A. Bélanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [CrossRef]
  11. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,”J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [CrossRef]
  12. N. Kh. Ibragimov, Gruppy Preobrasovanij v Matematicheskoi Fizike (Groups of Transformations in Mathematical Physics) (Nauka, Moscow, USSR, 1983).
  13. A. O. Barut, R. Raczka, Theory of Group Representations and Applications (PWN-Polish Scientific Publishers, Warszawa, Poland1977). [Russian translation in two volumes (Mir, Moscow, USSR, 1980).]
  14. H. Bateman, A. Erdélyi, Higher Transcendental Functions Vol. 2, (McGraw-Hill, New York, 1953) (Russian translation, Nauka, Moscow, 1974).
  15. A. Wünsche, “New exact vector solutions for quasi-plane, quasi-monochromatic waves in dispersive media,” J. Opt. Soc. Am. A (to be published).
  16. A. Wünsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A (to be published).

1984 (1)

1977 (1)

1975 (2)

1973 (1)

1970 (1)

A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. (Leipzig) 25, 113–135 (1970).

1966 (2)

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 10, 1550–1567 (1966).
[CrossRef]

1961 (1)

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

Ashkin, A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Barut, A. O.

A. O. Barut, R. Raczka, Theory of Group Representations and Applications (PWN-Polish Scientific Publishers, Warszawa, Poland1977). [Russian translation in two volumes (Mir, Moscow, USSR, 1980).]

Bateman, H.

H. Bateman, A. Erdélyi, Higher Transcendental Functions Vol. 2, (McGraw-Hill, New York, 1953) (Russian translation, Nauka, Moscow, 1974).

Bélanger, P. A.

Boyd, G. D.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

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

Erdélyi, A.

H. Bateman, A. Erdélyi, Higher Transcendental Functions Vol. 2, (McGraw-Hill, New York, 1953) (Russian translation, Nauka, Moscow, 1974).

Fukumitsu, O.

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).

Ibragimov, N. Kh.

N. Kh. Ibragimov, Gruppy Preobrasovanij v Matematicheskoi Fizike (Groups of Transformations in Mathematical Physics) (Nauka, Moscow, USSR, 1983).

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Kogelnik, H.

Lax, M.

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

Li, T.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972. (Russian translation, Mir, Moscow, 1974).

Ooya, T.

Pratesi, R.

Raczka, R.

A. O. Barut, R. Raczka, Theory of Group Representations and Applications (PWN-Polish Scientific Publishers, Warszawa, Poland1977). [Russian translation in two volumes (Mir, Moscow, USSR, 1980).]

Ronchi, L.

Siegman, A. E.

Tateiba, M.

Wünsche, A.

A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. (Leipzig) 25, 113–135 (1970).

A. Wünsche, “New exact vector solutions for quasi-plane, quasi-monochromatic waves in dispersive media,” J. Opt. Soc. Am. A (to be published).

A. Wünsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A (to be published).

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975). (Russian translation, Sovyetskoye Radio, Moscow, 1980).

Ann. Phys. (Leipzig) (1)

A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. (Leipzig) 25, 113–135 (1970).

Appl. Opt. (1)

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).

J. Opt. Soc. Am. (3)

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

Phys. Rev. (1)

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second-harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Phys. Rev. A (1)

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

Other (7)

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975). (Russian translation, Sovyetskoye Radio, Moscow, 1980).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972. (Russian translation, Mir, Moscow, 1974).

N. Kh. Ibragimov, Gruppy Preobrasovanij v Matematicheskoi Fizike (Groups of Transformations in Mathematical Physics) (Nauka, Moscow, USSR, 1983).

A. O. Barut, R. Raczka, Theory of Group Representations and Applications (PWN-Polish Scientific Publishers, Warszawa, Poland1977). [Russian translation in two volumes (Mir, Moscow, USSR, 1980).]

H. Bateman, A. Erdélyi, Higher Transcendental Functions Vol. 2, (McGraw-Hill, New York, 1953) (Russian translation, Nauka, Moscow, 1974).

A. Wünsche, “New exact vector solutions for quasi-plane, quasi-monochromatic waves in dispersive media,” J. Opt. Soc. Am. A (to be published).

A. Wünsche, “Embedding of focus wave modes into a wider class of approximate wave equation solutions,” J. Opt. Soc. Am. A (to be published).

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Equations (92)

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( 2 x 2 + 2 y 2 + i 4 ζ ) ψ ( x , y , ζ ) = 0 , ζ 2 k 0 z ,             ζ = k 0 2 z .
ψ 0 , 0 ( x , y , ζ ) = A 0 γ 1 1 / 2 γ 2 1 / 2 ( γ 1 + i ζ ) 1 / 2 ( γ 2 + i ζ ) 1 / 2 × exp ( - x 2 γ 1 + i ζ - y 2 γ 2 + i ζ ) ,
L 2 x 2 + 2 y 2 + i 4 ζ
ψ ( x , y , ζ ) = d x d y d ζ f ( x , y , ζ ) × ψ ( x - x , y - y , ζ - ζ ) = f ˜ ( - i x , - i y , - i ζ ) ψ ( x , y , ζ ) .
f ˜ ( k x , k y , k ζ ) d x d y d ζ f ( x , y , ζ ) × exp [ - i ( k x x + k y y + k ζ ζ ) ] .
D 1 ( α 1 ) - 2 x + ( α 1 - i ζ ) x = ( α 1 - i ζ ) exp ( + x 2 α 1 - i ζ ) x exp ( - x 2 α 1 - i ζ ) , D 2 ( α 2 ) - 2 y + ( α 2 - i ζ ) y = ( α 2 - i ζ ) exp ( + y 2 α 2 - i ζ ) y exp ( - y 2 α 2 - i ζ ) .
x = lim α 1 1 α 1 D 1 ( α 1 ) ,             y = lim α 2 1 α 2 D 2 ( α 2 ) .
[ D 1 ( α 1 ) , L ] = 0 ,             [ D 2 ( α 2 ) , L ] = 0.
[ D 1 ( α 1 ) , D 1 ( β 1 ) ] = - 2 ( α 1 - β 1 ) ,             [ D 1 ( α 1 ) , D 2 ( α 2 ) ] = 0 , [ D 2 ( α 2 ) , D 2 ( β 2 ) ] = - 2 ( α 2 - β 2 ) .
D ( α 0 , α 1 , α 2 , α 3 , α 4 ) = α 0 + α 1 x + α 2 y + α 3 D 1 ( 0 ) + α 4 D 2 ( 0 ) , D 1 ( 0 ) - 2 x - i ζ x ,             D 2 ( 0 ) - 2 y - i ζ y .
[ D 1 ( α 1 ) , x ] = 2 ,             [ D 1 ( α 1 ) , y ] = 0 , [ D 1 ( α 1 ) , ζ ] = i x , [ D 2 ( α 2 ) , x ] = 0 ,             [ D 2 ( α 2 ) , y ] = 2 , [ D 2 ( α 2 ) , ζ ] = i y ,
[ D 1 ( α 1 ) , x ] = α 1 - i ζ ,             [ D 1 ( α 1 ) , y ] = 0 ,             [ D 1 ( α 1 ) , ζ ] = 0 , [ D 2 ( α 2 ) , x ] = 0 ,             [ D 2 ( α 2 ) , y ] = α 2 - i ζ ,             [ D 2 ( α 2 ) , ζ ] = 0.
D 1 m ( α 1 ) = ( α 1 - i ζ ) m exp ( + x 2 α 1 - i ζ ) m x m exp ( - x 2 α 1 - i ζ ) , D 2 n ( α 2 ) = ( α 2 - i ζ ) n exp ( + y 2 α 2 - i ζ ) n y n exp ( - y 2 α 2 - i ζ ) ,
f [ D 1 ( α 1 ) , D 2 ( α 2 ) ] = exp ( + x 2 α 1 - i ζ + y 2 α 2 - i ζ ) × f [ ( α 1 - i ζ ) x , ( α 2 - i ζ ) y ] × exp ( - x 2 α 1 - i ζ - y 2 α 2 - i ζ ) .
( x , y , ζ ) exp [ - μ 1 D 1 ( α 1 ) - μ 2 D 2 ( α 2 ) ] ( x , y , ζ ) × exp [ + μ 1 D 1 ( α 1 ) + μ 2 D 2 ( α 2 ) ] = [ x - μ 1 ( α 1 - i ζ ) , y - μ 2 ( α 2 - i ζ ) , ζ ] .
ψ ( x , y , ζ ) exp [ - μ 1 D 1 ( α 1 ) - μ 2 D 2 ( α 2 ) ] ψ ( x , y , ζ ) = exp { + μ 1 [ 2 x - μ 1 ( α 1 - i ζ ) ] + μ 2 [ 2 y - μ 2 ( α 2 - i ζ ) ] } × ψ [ x - μ 1 ( α 1 - i ζ ) , y - μ 2 ( α 2 - i ζ ) , ζ ] ,
ψ m , n ( x , y , ζ ) ( - i ) m + n ( α 1 + γ 1 ) m / 2 ( α 2 + γ 2 ) n / 2 × D 1 m ( α 1 ) D 2 n ( α 2 ) ψ 0 , 0 ( x , y , ζ ) ,
m x m exp ( - x 2 p 2 ) = ( - 1 ) m p m H m ( x p ) exp ( - x 2 p 2 ) ,
ψ m , n ( x , y , ζ ) = A 0 γ 1 1 / 2 γ 2 1 / 2 ( γ 1 + i ζ ) 1 / 2 ( γ 2 + i ζ ) 1 / 2 × exp ( - x 2 γ 1 + i ζ - y 2 γ 2 + i ζ ) × i m ( α 1 - i ζ γ 1 + i ζ ) m / 2 H m { [ α 1 + γ 1 ( α 1 - i ζ ) ( γ 1 + i ζ ) ] 1 / 2 x } × i n ( α 2 - i ζ γ 2 + i ζ ) n / 2 H n { [ α 2 + γ 2 ( α 2 - i ζ ) ( γ 2 + i ζ ) ] 1 / 2 y } .
ψ m , n ( x , y , ζ ) lim α 1 lim α 2 α 1 - m / 2 α 2 - n / 2 ψ m , n ( x , y , ζ ) = ( - i ) m + n m + n x m y n ψ 0 , 0 ( x , y , ζ )
ψ m , n ( x , y , ζ ) = A 0 γ 1 1 / 2 γ 2 1 / 2 ( γ 1 + i ζ ) ( m + 1 ) / 2 ( γ 2 + i ζ ) ( n + 1 ) / 2 × exp ( - x 2 γ 1 + i ζ - y 2 γ 2 + i ζ ) × i m H m [ x ( γ 1 + i ζ ) 1 / 2 ] i n H n [ y ( γ 2 + i ζ ) 1 / 2 ] .
L n ( 0 ) ( x 2 + y 2 ) = ( - 1 ) n 2 2 n n ! k = 0 n ( n k ) H 2 k ( x ) H 2 ( n - k ) ( y ) ,
L n ( ν ) ( u ) = 1 n ! e u u - ν n u n ( e - u u n + ν ) = 1 n ! u - ν ( u - 1 ) n u n + ν .
exp ( - z z * ) L n ( 0 ) ( z z * ) = 1 n ! n ( z z * ) n [ exp ( - z z * ) ( z z * ) n ] = ( - 1 ) n n ! 2 n z n z * n exp ( - z z * ) ,
exp ( + λ z z * ) m + n z m z * n exp ( - λ z z * ) = ( - λ ) n m ! z n - m L m n - m ( λ z z * ) = ( - λ ) m n ! z * m - n L n m - n ( λ z z * ) .
α 1 = α 2 α 0 ,             γ 1 = γ 2 γ 0 w 0 2 .
r ( x , y ) ,             ( x , y ) , D ( α 0 ) [ D 1 ( α 0 ) , D 2 ( α 0 ) ] ,
D ( α 0 ) = - 2 r + ( α 0 - i ζ ) = ( α 0 - i ζ ) exp ( + r 2 α 0 - i ζ ) × exp ( - r 2 α 0 - i ζ )
D 2 ( α 0 ) = ( α 0 - i ζ ) 2 exp ( + r 2 α 0 - i ζ ) × 2 exp ( - r 2 α 0 - i ζ ) .
ψ 2 n ( 0 ) ( r , ζ ) ( - 1 ) n 2 2 n n ! ( α 0 + γ 0 ) n [ D 2 ( α 0 ) ] n ψ 0 ( 0 ) ( r , ζ ) = 1 2 2 n n ! k = 0 n ( n k ) ψ 2 k , 2 ( n - k ) ( r , ζ ) ,
ψ 0 ( 0 ) ( r , ζ ) = A 0 γ 0 γ 0 + i ζ exp ( - r 2 γ 0 + i ζ ) γ 0 , 0 ( r , ζ ) .
ψ 2 n ( 0 ) ( r , ζ ) = A 0 γ 0 γ 0 + i ζ exp ( - r 2 γ 0 + i ζ ) × ( α 0 - i ζ γ 0 + i ζ ) n L n ( 0 ) [ ( α 0 + γ 0 ) r 2 ( α 0 - i ζ ) ( γ 0 + i ζ ) ] .
ψ 2 n ( 0 ) ( r , ζ ) lim α 0 α - n γ 2 n ( 0 ) ( r , ζ ) = ( - 1 ) n 2 2 n n ! ( 2 ) n ψ 0 ( 0 ) ( r , ζ ) ,
ψ 2 n ( 0 ) ( r , ζ ) = A 0 γ 0 ( γ 0 + i ζ ) n + 1 × exp ( - r 2 γ 0 + i ζ ) L n ( 0 ) ( r 2 γ 0 + i ζ ) .
z ± x ± i y ,             z ± 1 2 ( x i y ) .
D ± ( α 0 ) 1 2 [ D 1 ( α 0 ) i D 2 ( α 0 ) ] = - z + ( α 0 - i ζ ) z ± ,
D ± ( α 0 ) = ( α 0 - i ζ ) exp ( + z + z - α 0 - i ζ ) z ± exp ( - z + z - α 0 - i ζ ) .
ψ m + n ( n - m ) ( z + , z - , ζ ) ( - i ) m + n ( m ! n ! ) 1 / 2 ( α 0 + γ 0 ) ( m + n ) / 2 × D + m ( α 0 ) D - n ( α 0 ) ψ 0 ( 0 ) ( z + , z - , ζ ) ,
ψ m + n ( n - m ) ( z + , z - , ζ ) = A 0 γ 0 γ 0 + i ζ exp ( - z + z - γ 0 + i ζ ) × i n - m ( m ! n ! ) 1 / 2 ( α 0 + γ 0 ) ( n - m ) / 2 ( α 0 - i ζ ) m ( γ 0 + i ζ ) n × z + n - m L m ( n - m ) [ ( α 0 + γ 0 ) z + z - ( α 0 - i ζ ) ( γ 0 + i ζ ) ] = A 0 γ 0 γ 0 + i ζ exp ( - z + z - γ 0 + i ζ ) × i m - n ( n ! m ! ) 1 / 2 ( α 0 + γ 0 ) ( m - n ) / 2 ( α 0 - i ζ ) n ( γ 0 + i ζ ) m × z - m - n L n ( m - n ) [ ( α 0 + γ 0 ) z + z - ( α 0 - i ζ ) ( γ 0 + i ζ ) ] .
ψ m + n ( n - m ) ( z + , z - , ζ ) lim α 0 α 0 - ( m + n ) / 2 ψ m + n ( n - m ) ( z + , z - , ζ ) = ( - i ) m + n ( m ! n ! ) 1 / 2 m + n z + m z - n ψ 0 ( 0 ) ( z + , z - , ζ ) ,
ψ m + n ( n - m ) ( z + , z - , ζ ) = A 0 γ 0 ( γ 0 + i ζ ) n + 1 exp ( - z + z - γ 0 + i ζ ) × ( m ! n ! ) 1 / 2 ( i z + ) n - m L m ( n - m ) ( z + z - γ 0 + i ζ ) = A 0 γ 0 ( γ 0 + i ζ ) m + 1 exp ( - z + z - γ 0 + i ζ ) × ( n ! m ! ) 1 / 2 ( i z - ) m - n L n ( m - n ) ( z + z - γ 0 + i ζ ) .
ψ ˜ ( k x , k y , k ζ ) d x d y d ζ × exp [ - i ( k x x + k y y + k ζ ζ ) ] ψ ( x , y , ζ ) .
f ( k x , k y , k ζ ) ψ ˜ ( k x , k y , k ζ ) ( k x 2 + k y 2 + 4 k ζ ) ψ ˜ ( k x , k y , k ζ ) = 0.
ψ ˜ ( k x , k y , k ζ ) = 2 π χ ( k x , k y ) δ [ k ζ + ¼ ( k x 2 + k y 2 ) ] ,
ψ ˜ ( k x , k y , ζ ) d x d y exp [ - i ( k x x + k y y ) ] ψ ( x , y , ζ ) .
[ - ( k x 2 + k y 2 ) + i 4 ζ ] ψ ˜ ( k x , k y , ζ ) = 0.
ψ ˜ ( k x , k y , ζ ) = χ ( k x , k y ) exp [ - i ¼ ( k x 2 + k y 2 ) ζ ] .
L ˜ - ( k x 2 + k y 2 ) + i 4 ζ .
D ˜ 1 ( α 1 ) - i 2 k x + i ( α 1 - i ζ ) k x = - i 2 exp [ + 1 4 ( α 1 - i ζ ) k x 2 ] k x × exp [ - 1 4 ( α 1 - i ζ ) k x 2 ] , D ˜ 2 ( α 2 ) - i 2 k y + i ( α 2 - i ζ ) k y = - i 2 exp [ + 1 4 ( α 2 - i ζ ) k y 2 ] k y × exp [ - 1 4 ( α 2 - i ζ ) k y 2 ] .
i k x = lim α 1 1 α 1 D ˜ 1 ( α 1 ) ,             i k y = lim α 2 1 α 2 D ˜ 2 ( α 2 ) .
[ D ˜ 1 ( α 1 ) , L ˜ ] = 0 ,             [ D ˜ 2 ( α 2 ) , L ˜ ] = 0.
[ D ˜ 1 ( α 1 ) , D ˜ 1 ( β 1 ) ] = - 2 ( α 1 - β 1 ) ,             [ D ˜ 1 ( α 1 ) , D ˜ 2 ( α 2 ) ] = 0 , [ D ˜ 2 ( α 2 ) , D ˜ 2 ( β 2 ) ] = - 2 ( α 2 - β 2 ) .
[ D ˜ 1 ( α 1 ) , i k x ] = 2 ,             [ D ˜ 1 ( α 1 ) , i k y ] = 0 , [ D ˜ 1 ( α 1 ) , ζ ] = - k x , [ D ˜ 2 ( α 2 ) , i k x ] = 0 ,             [ D ˜ 2 ( α 2 ) , i k y ] = 2 , [ D ˜ 2 ( α 2 ) , ζ ] = - k y ,
[ D ˜ 1 ( α 1 ) , i k x ] = α 1 - i ζ ,             [ D ˜ 1 ( α 1 ) , i k y ] = 0 , [ D ˜ 1 ( α 1 ) , ζ ] = 0 , [ D ˜ 2 ( α 2 ) , i k x ] = 0 ,             [ D ˜ 2 ( α 2 ) , i k y ] = α 2 - i ζ , [ D ˜ 2 ( α 2 ) , ζ ] = 0 ,
D ˜ 1 m ( α 1 ) = ( - i 2 ) m exp [ + 1 4 ( α 1 - i ζ ) k x 2 ] m k x m × exp [ - 1 4 ( α 1 - i ζ ) k x 2 ] , D ˜ 2 n ( α 2 ) = ( - i 2 ) n exp [ + 1 4 ( α 2 - i ζ ) k x 2 ] n k y n × exp [ - 1 4 ( α 2 - i ζ ) k y 2 ] ,
f [ D ˜ 1 ( α 1 ) , D ˜ 2 ( α 2 ) = exp { + 1 4 [ ( α 1 - i ζ ) k x 2 + ( α 2 - i ζ ) k y 2 ] } × f ( - i 2 k x , - i 2 k y ) × exp { - 1 4 [ ( α 1 - i ζ ) k x 2 + ( α 2 - i ζ ) k y 2 ] } .
( k x , k y , ζ ) exp [ - μ 1 D ˜ 1 ( α 1 ) - μ 2 D ˜ 2 ( α 2 ) ] ( k x , k y , ζ ) × exp [ + μ 1 D ˜ 1 ( α 1 ) + μ 2 D ˜ 2 ( α 2 ) ] = ( k x + i 2 μ 1 , k y + i 2 μ 2 , ζ ) .
ψ ˜ ( k x , k y , ζ ) exp [ - μ 1 D ˜ 1 ( α 1 ) - μ 2 D ˜ 2 ( α 2 ) ] ψ ˜ ( k x , k y , ζ ) = exp [ - i μ 1 ( α 1 - i ζ ) ( k x + i μ 1 ) - i μ 2 ( α 2 - i ζ ) ( k y + i μ 2 ) ] × ψ ˜ ( k x + i 2 μ 1 , k y + i 2 μ 2 , ζ ) ,
ψ ˜ ( k x , k y , ζ ) = χ ( k x , k y ) exp [ - i ¼ ( k x 2 + k y 2 ) ζ ] ,
χ ( k x , k y ) = exp [ - i μ 1 α 1 ( k x + i μ 1 ) - i μ 2 α 2 ( k y + i μ 2 ) ] × χ ( k x + i 2 μ 1 , k y + i 2 μ 2 ) .
ψ ˜ 0 , 0 ( k x , k y , ζ ) = A 0 π γ 1 1 / 2 γ 2 1 / 2 × exp [ - ¼ ( γ 1 + i ζ ) k x 2 - ¼ ( γ 2 + i ζ ) k y 2 ] χ 0 , 0 ( k x , k y ) exp [ - i ¼ ( k x 2 + k y 2 ) ζ ] .
ψ ˜ m , n ( k x , k y , ζ ) = ( - i ) m + n ( α 1 + γ 1 ) m / 2 ( α 2 + γ 2 ) n / 2 × D ˜ 1 m ( α 1 ) D ˜ 2 n ( α 2 ) ψ ˜ 0 , 0 ( k x , k y , ζ )
χ m , n ( k x , k y ) = A 0 π γ 1 1 / 2 γ 2 1 / 2 exp [ - ¼ ( γ 1 k x 2 + γ 2 k y 2 ) ] × H m [ ½ ( α 1 + γ 1 ) 1 / 2 k x ] H n [ ½ ( α 2 + γ 2 ) 1 / 2 k y ] .
ψ ˜ m , n ( k x , k y , ζ ) lim α 1 lim α 2 α 1 - m / 2 α 2 - n / 2 ψ ˜ m , n ( k x , k y , ζ ) = k x m k y n ψ ˜ 0 , 0 ( k x , k y , ζ )
ψ ˜ 0 ( 0 ) ( k , ζ ) = A 0 π γ 0 exp [ - ¼ ( γ 0 + i ζ ) k 2 ] χ 0 ( 0 ) ( k ) exp ( - i ¼ k 2 ζ ) .
ψ ˜ 2 n ( 0 ) ( k , ζ ) ( - 1 ) n 2 2 n n ! ( α 0 + γ 0 ) n [ D ˜ 2 ( α 0 ) ] n ψ ˜ 0 ( 0 ) ( k , ζ ) = 1 2 2 n n ! k = 0 n ( n h ) ψ ˜ 2 k , 2 ( n - k ) ( k , ζ )
χ 2 n ( 0 ) ( k ) = A 0 π γ 0 exp ( - ¼ γ 0 k 2 ) ( - 1 ) n × L n ( 0 ) [ ¼ ( α 0 + γ 0 ) k 2 ] ,
ψ ˜ 2 n ( 0 ) ( k , ζ ) lim α 1 α 0 - n ψ ˜ 2 n ( 0 ) ( k , ζ ) = 1 2 2 n n ! ( k 2 ) n ψ ˜ 0 ( 0 ) ( k , ζ ) ,
k ± k x ± i k y ,             k ± 1 2 ( k x i k y ) .
D ˜ ± ( α 0 ) ½ [ D ˜ 1 ( α 0 ) i D ˜ 2 ( α 0 ) ] = - i 2 k ± + i ½ ( α 0 - i ζ ) k ,
D ˜ ± ( α 0 ) = - i 2 exp [ + 1 4 ( α 0 - i ζ ) k + k - ] k ± × exp [ - 1 4 ( α 0 - i ζ ) k + k - ] .
ψ ˜ m + n ( n - m ) ( k + , k - , ζ ) = ( - i ) m + n ( m ! n ! ) 1 / 2 ( α 0 + γ 0 ) ( m + n ) / 2 × D ˜ + m ( α 0 ) D ˜ - n ( α 0 ) ψ ˜ 0 ( 0 ) ( k + , k - , ζ ) ,
χ m + n ( n - m ) ( k + , k - ) = A 0 π γ 0 exp ( - 1 4 γ 0 k + k - ) ( - 1 ) m × ( m ! n ! ) 1 / 2 [ 1 4 ( α 0 + γ 0 ) ] ( n - m ) / 2 k + n - m × L m ( n - m ) [ 1 4 ( α 0 + γ 0 ) k + k - ] = A 0 π γ 0 exp ( - 1 4 γ 0 k + k - ) ( - 1 ) n × ( n ! m ! ) 1 / 2 [ 1 4 ( α 0 + γ 0 ) ] ( m - n ) / 2 k - m - n × L n ( m - n ) [ 1 4 ( α 0 + γ 0 ) k + k - ] .
ψ ˜ m + n ( n - m ) ( k + , k - , ζ ) lim α 1 α 0 - ( m + n ) / 2 ψ ˜ m + n ( n - m ) ( k + , k - , ζ ) = 1 2 m + n ( m ! n ! ) 1 / 2 k - m k + n ψ ˜ 0 ( 0 ) ( k + , k - , ζ ) ,
( 2 x 2 + 2 y 2 + 2 z 2 + 0 c 2 2 t 2 ) φ ( x , y , z , t ) = 0 ,
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] φ 0 ( x , y , z , t ) + c . c . ,             c 2 k 0 2 = ω 0 2 0 .
[ 2 x 2 + 2 y 2 + i 2 k 0 ( z + k 0 ω 0 t ) ] φ 0 ( x , y , z , t ) = 0.
φ ˜ ( k x , k y , k z , ω ) d x d y d z d t exp [ - i ( k x x + k y y + k z z - ω t ) ] φ ( x , y , z , t ) , φ ˜ 0 ( k x , k y , k z , ω ) d x d y d z d t exp [ - i ( k x x + k y y + k z z - ω t ) ] φ 0 ( x , y , z , t ) .
φ ˜ ( k x , k y , k z , ω ) = φ ˜ 0 ( k x , k y , k z - k 0 , ω - ω 0 ) + c . i . ,
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] ψ ( x , y , 2 k 0 z ) + c . c .
φ ˜ ( k x , k y , k z , ω ) = ( 2 π ) 2 χ ( k x , k y ) δ [ k z - k 0 + 1 2 k 0 ( k x 2 + k y 2 ) ] × δ ( ω - ω 0 ) + c . i . ,
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] ψ ( x , y , 2 ω 0 k 0 2 t ) + c . c . ,
φ ˜ ( k x , k y , k z , ω ) = ( 2 π ) 2 χ ( k x , k y ) δ ( k z - k 0 ) × δ [ ω - ω 0 - ω 0 2 k 0 2 ( k x 2 + k y 2 ) ] + c . i . ,
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] f ( k 0 z - ω 0 t ) × ψ { x , y , 1 k 0 2 [ ( 1 + μ ) k 0 z + ( 1 - μ ) ω 0 t ] } + c . c . ,
φ ˜ ( k x , k y , k z , ω ) = 2 π χ ( k x , k y ) 1 k 0 ω 0 × f ˜ ( 1 - μ 2 k z - k 0 k 0 + 1 + μ 2 ω - ω 0 ω 0 ) × δ ( k z k 0 - ω ω 0 + k x 2 + k y 2 2 k 0 2 ) + c . i . ,
f ˜ ( p ) d q exp ( - i p q ) f ( q ) .
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] f ( k 0 z - ω 0 t ) + c . c . ,
φ ˜ ( k x , k y , k z , ω ) = ( 2 π ) 3 δ ( k x ) δ ( k y ) 1 k 0 ω 0 × f ˜ ( ω - ω 0 ω 0 ) δ ( k z k 0 - ω ω 0 ) + c . i . ,
v 0 = ω 0 k 0 ,
φ ( x , y , z , t ) = exp [ i ( k 0 z - ω 0 t ) ] × ψ { x , y , 1 k 0 2 [ ( 1 + μ ) k 0 z + ( 1 - μ ) ω 0 ] } + c . c . ,
φ ˜ ( k x , k y , k z , ω ) = ( 2 π ) 2 χ ( k x , k y ) 1 k 0 ω 0 × δ ( 1 - μ 2 k z - k 0 k 0 + 1 + μ 2 ω - ω 0 ω 0 ) × δ ( k z k 0 - ω ω 0 + k x 2 + k y 2 2 k 0 2 ) + c . i .
u 0 = - 1 - μ 1 + μ ω 0 k 0 = - 1 - μ 1 + μ v 0 .

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