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
Related Articles
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)

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, “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).

2005 (1)

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]

2000 (1)

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

1993 (1)

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]

1971 (1)

1970 (1)

Abramowitz, M.

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

Alieva, T.

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]

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

Allen, L.

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]

Arnaud, J. A.

Bastiaans, M. J.

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]

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

Beijersbergen, M.W.

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]

Collins, S. A.

Luneburg, R. K.

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

Stegun, I. A.

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

van der Veen, H. E. L. O.

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]

Woerdman, J. P.

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]

Wünsche, A.

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

J. Opt. Soc. Am. (2)

J. Phys. A: Math Gen. (1)

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

J. Phys. A: Math. Gen (1)

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. (1)

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]

Other (3)

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).

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

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