Abstract

Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite–gaussian eigenfunctions ψn (x,z) consisting of a hermite polynomial of real argument Hn [√2x/w(z)] times the complex gaussian function exp [−jkx2/2q(z)], in which q(z) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψˆn(x,z) of the form Hn[√cx]exp [−cx2], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)1/2x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψˆn, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ϕˆn(cx). The new eigenfunctions ψˆn are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψˆn functions are not spherical for n > 1. However, they may still be useful as a basis set for other optical resonator and beam-propagation problems.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
    [Crossref]

1966 (1)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Proc. IEEE (1)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (21)

Equations on this page are rendered with MathJax. Learn more.

u ( x , y , z ) = ψ ( x , y , z ) e j k z .
2 ψ x 2 + 2 ψ y 2 2 k ψ z = 0 ,
ψ n ( x , z ) = ( 1 w ( z ) ) 1 2 H n ( 2 x w ( z ) ) e [ j k / 2 q ( z ) ] x 2 e j ( n + 1 2 ) ϕ ( z ) ,
ψ ˆ n ( x , y ) = A ( q ) H n { ( j k / 2 q ) x } e ( j k / 2 q ) x 2 .
ψ ˆ n ( x , z ) = A ( q ) H n ( c x ) e c x 2 ,
H n ( c x ) 2 c x H n ( c x ) + 2 n H n ( c x ) = 0 ,
q A d A d q = n + 1 2 .
ψ ˆ n ( x , z ) = ( q 0 / q ) ( n + 1 ) / 2 H n ( c x ) e c x 2 ,
ψ n ( x ) = H n ( 2 x w ) e x 2 / w 2 ,
ψ ˆ n ( x ) = H n ( x w ) e x 3 / w 3 .
d 2 ψ n d x 2 + ( 2 n + 1 x 2 ) ψ n = 0 ,
d 2 ψ ˆ n d x 2 + 2 c x d ψ ˆ n d x + 2 ( n + 1 ) c ψ ˆ n = 0 .
L ψ ˆ n = λ n ψ ˆ n ,
L [ d 2 d x 2 + 2 c x d d x ] , λ n = 2 ( n + 1 ) c .
L + [ d 2 d x 2 d d x ( 2 c * x ) ] .
d 2 ϕ ˆ n d x 2 c * x d ϕ ˆ n d x ( 2 c * + μ n ) ϕ ˆ n = 0 .
ϕ ˆ n ( x ) = H n ( c * x ) ,
μ n = 2 ( n + 1 ) c * .
ϕ ˆ n * ( x ) ψ ˆ m ( x ) d x = H n ( c x ) H m ( c x ) e c x 2 d x = K n δ n m .
u ( x ) = n a n ψ ˆ n ( x ) ,
a n = ϕ ˆ n * ( x ) u ( x ) d x .