Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems

Martin J. Bastiaans and Tatiana Alieva

Open Access Open Access

Abstract

Based on the common Hermite-Gaussian modes, a general class of orthonormal sets of Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. It is shown that these modes remain in their class of orthonormal Hermite-Gaussian-type modes, when they propagate through first-order optical systems. A propagation law for the generating function is formulated.

©2005 Optical Society of America

Full Article  |  PDF Article
More Like This
First-order optical systems with unimodular eigenvalues

Martin J. Bastiaans and Tatiana Alieva
J. Opt. Soc. Am. A 23(8) 1875-1883 (2006)

Generalized mode propagation in first-order optical systems with loss or gain

Moshe Nazarathy, Amos Hardy, and Joseph Shamirt
J. Opt. Soc. Am. 72(10) 1409-1420 (1982)

Mode mapping in paraxial lossless optics

Tatiana Alieva and Martin J. Bastiaans
Opt. Lett. 30(12) 1461-1463 (2005)

View More...

Cited By

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

Alert me when this article is cited.


Equations (31)

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

𝓗 n , m ( r ; w x , w y ) = 𝓗 n ( x ; w x ) 𝓗 m ( y ; w y )
𝓗 n ( x ; w ) = 2 1 4 ( 2 n n ! w ) − 1 2 H n ( 2 π x ⁄ w ) exp ( − π x 2 ⁄ w 2 ) ,
∫ 𝓗 n ( x ; w ) 𝓗 l ( x ; w ) d x = δ n l
exp ( − s 2 + 2 s z ) = ∑ n = 0 ∞ H n ( z ) s k n ! ,
2 1 2 ( w x w y ) − 1 ⁄ 2 exp [ − ( s x 2 + s y 2 ) + 2 2 π ( s x x ⁄ w x + s y y ⁄ w y ) − π ( x 2 ⁄ w x 2 + y 2 ⁄ w y 2 ) ]
= ∑ n = 0 ∞ ∑ m = 0 ∞ 𝓗 n , m ( r ; w x , w y ) ( 2 n + m n ! m ! ) 1 2 s x n s y m .
2 1 2 ( det K ) 1 2 exp ( − s t M s + 2 2 π s t K r − π r t L r )
= ∑ n = 0 ∞ ∑ m = 0 ∞ 𝓗 n , m ( r ; K , L ) ( 2 n + m n ! m ! ) 1 2 s x n s y m ,
K = ( w x 0 0 w y ) − 1 = W − 1 , L = W − 2 , M = I .
( r o q o ) = ( A B C D ) ( r i q i ) .
A B t = B A t , C D t = D C t , A D t − B C t = I ,
A t C = C t A , B t D = D t B , A t D − C t B = I .
f o ( r o ) = exp ( i ϕ ) det i B ∬ f i ( r i ) exp [ i π ( r i t B − 1 A r i − 2 r i t B − 1 r o + r o t D B − 1 r o ) ] d r i ,
∬ f i , 1 ( r ) f i , 2 * ( r ) d r = ∬ f o , 1 ( r ) f o , 2 * ( r ) d r ,
K o = K i ( A + B i L i ) − 1 ,
i L o = ( C + D i L i ) ( A + B i L i ) − 1 ,
M o = M i − 2 i K i ( A + B i L i ) − 1 B K i t .
( I i L o ) K o − 1 = ( A B C D ) ( I i L i ) K i − 1 .
∫ ∫ 𝓗 n , m ( r ; K , L ) 𝓗 l , k * ( r ; K , L ) d r = δ nl δ mk ,
M − K [ ( L + L * ) ⁄ 2 ] − 1 K t = 0 ,
K [ ( L + L * ) ⁄ 2 ] − 1 K * t = I ,
M − 1 = M * = K * K − 1 = ( K * K − 1 ) t ,
( L + L * ) ⁄ 2 = K t K * = ( K t K * ) t ,
a t d + b t c = d t a + c t b and a t c − b t d = c t a − d t b ,
a t d − b t c + d t a − c t b = 2 I and a t c + b t d = c t a + d t b ,
W − 1 ( a + i b ) = ( cos γ x ⁄ exp ( i γ x ) 0 0 cos γ y ⁄ exp ( i γ y ) ) W ( d − i c ) = ( exp ( i γ 1 ) 0 0 exp ( i γ 2 ) ) ;
w − 1 ( a + i b ) = cos γ exp ( i γ ) w ( d − i c ) = 1 2 ( exp ( i γ 1 ) − i exp ( i γ 2 ) − i exp ( i γ 1 ) exp ( i γ 2 ) ) ;
K i , o = ( a i , o + i b i , o ) − 1 ,
L i , o = ( d i , o − i c i , o ) ( a i , o + i b i , o ) − 1 ,
M i , o = ( a i , o + i b i , o ) − 1 ( a i , o − i b i , o ) ,
( a o b o c o d o ) = ( A B C D ) ( a i b i c i d i ) .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use