Abstract

It is shown that the conventional Hermite–Gaussian wave functions can be generated by assigning complex-source coordinates to multipole field expressions.

© 1985 Optical Society of America

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References

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  1. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  2. J. A. Arnaud, “Degenerate optical cavities. II: Effect of misalignments,” Appl. Opt. 8, 1909–1917 (1969).
    [CrossRef] [PubMed]
  3. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
    [CrossRef]
  4. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  5. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  6. A. L. Cullen, P. K. Yu, “Complex-source-point theory of the elecromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
    [CrossRef]
  7. P. K. Yu, K. M. Luk, “High-order azimuthal modes in the open resonator,” Electron.Lett. 19, 539–541 (1983).
    [CrossRef]
  8. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. A. 63, 1093–1094 (1973).
    [CrossRef]
  9. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  10. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  11. K. M. Luk, P. K. Yu, “Complex-source point theory of Gaussian beam and resonator,” IEE Proc. J. Optoelectron. 2, 105–113 (1985).

1985

K. M. Luk, P. K. Yu, “Complex-source point theory of Gaussian beam and resonator,” IEE Proc. J. Optoelectron. 2, 105–113 (1985).

1983

P. K. Yu, K. M. Luk, “High-order azimuthal modes in the open resonator,” Electron.Lett. 19, 539–541 (1983).
[CrossRef]

1979

A. L. Cullen, P. K. Yu, “Complex-source-point theory of the elecromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
[CrossRef]

1977

1976

1973

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. A. 63, 1093–1094 (1973).
[CrossRef]

1971

J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
[CrossRef]

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

1969

1966

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

Arnaud, J. A.

Cullen, A. L.

A. L. Cullen, P. K. Yu, “Complex-source-point theory of the elecromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
[CrossRef]

Deschamps, G. A.

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

Felsen, L. B.

Kogelnik, H.

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

Li, T.

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

Luk, K. M.

K. M. Luk, P. K. Yu, “Complex-source point theory of Gaussian beam and resonator,” IEE Proc. J. Optoelectron. 2, 105–113 (1985).

P. K. Yu, K. M. Luk, “High-order azimuthal modes in the open resonator,” Electron.Lett. 19, 539–541 (1983).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. A. 63, 1093–1094 (1973).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Yu, P. K.

K. M. Luk, P. K. Yu, “Complex-source point theory of Gaussian beam and resonator,” IEE Proc. J. Optoelectron. 2, 105–113 (1985).

P. K. Yu, K. M. Luk, “High-order azimuthal modes in the open resonator,” Electron.Lett. 19, 539–541 (1983).
[CrossRef]

A. L. Cullen, P. K. Yu, “Complex-source-point theory of the elecromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
[CrossRef]

Appl. Opt.

Electron. Lett.

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

Electron.Lett.

P. K. Yu, K. M. Luk, “High-order azimuthal modes in the open resonator,” Electron.Lett. 19, 539–541 (1983).
[CrossRef]

IEE Proc. J. Optoelectron.

K. M. Luk, P. K. Yu, “Complex-source point theory of Gaussian beam and resonator,” IEE Proc. J. Optoelectron. 2, 105–113 (1985).

J. Opt. Soc. A.

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. A. 63, 1093–1094 (1973).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE

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

Proc. R. Soc. London Ser. A

A. L. Cullen, P. K. Yu, “Complex-source-point theory of the elecromagnetic open resonator,” Proc. R. Soc. London Ser. A 366, 155–171 (1979).
[CrossRef]

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Equations (16)

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( N ) = 1 4 π N ! u 1 u 2 u N 1 ( x ̂ P x u 1 u 2 u N 1 + ŷ P y u 1 u 2 u N 1 + P z u 1 u 2 u N 1 ) × ( u 1 ) ( u 2 ) ( u N 1 ) 1 r exp [ j ( k r ω t ) ] ,
r = ( x 2 + y 2 + z 2 ) 1 / 2 , u i { x , y , z } , ( ´ i { 1 , 2 , 3 , , N 1 } ) ,
P x x m 2 g z 2 g y n 2 h z 2 h = 4 π ( m + n + 1 ) ! × ( 1 ) m g m ! g ! ( m 2 g ) ! ( 2 k z 0 ) m 2 g ( 1 ) n h n ! h ! ( n 2 h ) ! ( 2 k z 0 ) n 2 h .
( m + n + 1 ) = x ̂ g = 0 [ m / 2 ] ( 1 ) m g m ! g ! ( m 2 g ) ! ( 2 k z 0 x ) m 2 g ( z ) 2 g × n = 0 [ n / 2 ] ( 1 ) n h n ! h ! ( n 2 h ) ! ( 2 k z 0 y ) n 2 h ( z ) 2 h ψ = x ̂ z m H m ( k z 0 x / z ) n H n ( k z 0 y / z ) ψ ,
ψ = 1 r exp ( jkr ) [ exp ( j ω t ) is assumed ] ;
r = [ x 2 + y 2 + ( z + j z 0 ) 2 ] 1 / 2
( x ) α ( y ) β ( z ) γ ψ g = 0 [ α / 2 ] α ! ( j ) α g 2 g g ! ( α 2 g ) ! ( k r ) α g x α 2 g × h = 0 [ β / 2 ] β ! ( j ) β h 2 h h ! ( β 2 h ) ! ( k r ) β h y β 2 h × i = 0 [ γ / 2 ] γ ! ( j ) γ i 2 i i ! ( γ 2 i ) ! ( k r ) γ i ( z + j z 0 ) γ 2 i ψ .
r = ( z + j z 0 ) + x 2 + y 2 2 ( z + j z 0 ) ( x 2 + y 2 ) 2 8 ( z + j z 0 ) 3 + .
( x ) α ( y ) β ( z ) γ ψ g = 0 [ α / 2 ] α ! ( j ) α g 2 g g ! ( α 2 g ) ! ( k z + j z 0 ) α g x α 2 g × h = 0 [ β / 2 ] β ! ( j ) β h 2 h h ! ( β 2 h ) ! ( k z + j z 0 ) β h y β 2 h × ( j k ) γ exp ( k z 0 ) z + j z 0 exp [ jkz j k x 2 + y 2 2 ( z + j z 0 ) ] .
( m + n + 1 ) = x ̂ A j z 0 z + j z 0 exp [ jkz j k x 2 + y 2 2 ( z + j z 0 ) ] × g = 0 [ m / 2 ] ( 1 ) g g ! ( 2 z 0 / k ) m 2 g × c = 0 [ m / 2 ] g ( j ) m c x m 2 g 2 c 2 c c ! ( m 2 g 2 c ) ! ( k z + j z 0 ) m 2 g c × h = 0 [ n / 2 ] ( 1 ) h h ! ( 2 z 0 / k ) n 2 h × d = 0 [ n / 2 ] h ( j ) n d y n 2 h 2 d 2 d d ! ( n 2 h 2 d ) ! ( k z + j z 0 ) n 2 h d ,
w 0 2 = 2 z 0 / k , w ( z ) 2 / w 0 2 = 1 + z 2 / z 0 2 , R ( z ) = z + z 0 2 / z , Φ ( z ) = arctan ( z / z 0 ) .
( m + n + 1 ) = x ̂ w 0 w exp ( ρ 2 w 2 ) exp [ jkz + j Φ j k ρ 2 2 R ] × g = 0 [ m / 2 ] c = 0 [ m / 2 ] g ( 1 ) m c m ! g ! c ! ( m 2 g 2 c ) ! × ( 2 2 x w ) m 2 g 2 c ( 2 w 0 w ) c exp [ j ( m 2 g c ) Φ ] × h = 0 [ n / 2 ] d = 0 [ n / 2 ] h ( 1 ) n d n ! h ! d ! ( n 2 h 2 d ) ! × ( 2 2 y w ) n 2 h 2 d ( 2 w 0 w ) d exp [ j ( n 2 h d ) Φ ] ,
( m + n + 1 ) = x ̂ w 0 w exp ( ρ 2 w 2 ) exp [ jkz + j Φ j k ρ 2 2 R ] × g = 0 [ m / 2 ] c = g [ m / 2 ] ( 1 ) m c + g m ! g ! ( c g ) ! ( m 2 c ) ! × ( 2 2 x 2 ) m 2 c ( 2 w 0 w ) c g exp [ j ( m g c ) Φ ] × h = 0 [ n / 2 ] d = h [ n / 2 ] ( 1 ) n d h n ! h ! ( d h ) ! ( n 2 d ) ! × ( 2 2 y w ) m 2 c d ( 2 w 0 w ) d h exp [ j ( n h d ) Φ ] .
( m + n + 1 ) = x ̂ w 0 w exp ( ρ 2 w 2 ) exp [ jkz + j Φ j k ρ 2 2 R ] × c = 0 [ m / 2 ] ( 1 ) m m ! c ! ( m 2 c ) ! ( 2 2 x w ) m 2 c exp [ j ( m 2 c ) Φ ] × g = 0 c ( c g ) ( 2 w 0 w ) c g exp [ j ( c g ) Φ ] × d = 0 [ n / 2 ] ( 1 ) n n ! d ! ( n 2 d ) ! ( 2 2 y w ) n 2 d exp [ j ( n 2 d ) Φ ] × h = 0 d ( d h ) ( 2 w 0 w ) d h exp [ j ( d h ) Φ ] .
g = 0 c ( c g ) ( 2 w 0 w ) c g exp [ j ( c g ) Φ ] = [ 1 2 w 0 w exp ( j Φ ) ] c = [ exp ( j 2 Φ ) ] c = ( 1 ) c exp ( j 2 c Φ ) ,
Π ( m + n + 1 ) = x ̂ w 0 w exp ( ρ 2 w 2 ) exp [ jkz + j Φ j k ρ 2 2 R ] × c = 0 [ m / 2 ] ( 1 ) m + c m ! c ! ( m 2 c ) ! ( 2 2 x w ) m 2 c exp ( j m Φ ) × d = 0 [ n / 2 ] ( 1 ) n + d n ! d ! ( n 2 d ) ! ( 2 2 y w ) n 2 d exp ( j n Φ ) = H m ( 2 x w ) H n ( 2 y w ) w 0 w exp ( ρ 2 w 2 ) × exp [ jkz + j ( m + n + 1 ) Φ j k ρ 2 2 R ] .

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