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

References

  • View by:
  • |

  1. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).
  3. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  4. M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005).
    [CrossRef]
  5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000).
    [CrossRef]
  6. M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P.Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
    [CrossRef]
  8. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

J. Opt. Soc. Am.

S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

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

J. Phys. A: Math Gen.

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000).
[CrossRef]

J. Phys. A: Math. Gen

M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005).
[CrossRef]

Opt. Commun.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P.Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Opt. Lett.

T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

Other

M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).

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 (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 ) .

Metrics