Abstract

We introduce the generalized Helmholtz–Gauss (gHzG) beam and analyze its propagation through optical systems described by ABCD matrices with real and complex elements. The transverse mathematical structure of the gHzG beam is form invariant under paraxial transformations and reduces to those of ordinary HzG and modified HzG beams as special cases. We derive a closed-form expression for the fractional Fourier transform of gHzG beams.

© 2006 Optical Society of America

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  1. J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A 22, 289 (2005).
    [CrossRef]
  2. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, J. Opt. Soc. Am. A 23, 1994 (2006).
    [CrossRef]
  3. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, Opt. Eng. 45, 068001 (2006).
    [CrossRef]
  4. Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
    [CrossRef]
  5. F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
    [CrossRef]
  6. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000).
    [CrossRef]
  7. C. J. R. Sheppard and T. Wilson, IEE J. Microwaves, Opt. Acoust. 2, 105 (1978).
    [CrossRef]
  8. M. Santarsiero, Opt. Commun. 132, 1 (1996).
    [CrossRef]
  9. A. Belafhal and L. Dalil-Essakali, Opt. Commun. 177, 181 (2000).
    [CrossRef]
  10. L. W. Casperson, D. G. Hall, and A. A. Tovar, J. Opt. Soc. Am. A 14, 3341 (1997).
    [CrossRef]
  11. S. Ruschin, J. Opt. Soc. Am. A 11, 3224 (1994).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science, 1986).
  13. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]

2006 (2)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, Opt. Eng. 45, 068001 (2006).
[CrossRef]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, J. Opt. Soc. Am. A 23, 1994 (2006).
[CrossRef]

2005 (1)

2003 (1)

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

2000 (2)

1997 (1)

1996 (1)

M. Santarsiero, Opt. Commun. 132, 1 (1996).
[CrossRef]

1994 (1)

1993 (1)

1987 (1)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves, Opt. Acoust. 2, 105 (1978).
[CrossRef]

Bandres, M. A.

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, Opt. Eng. 45, 068001 (2006).
[CrossRef]

J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A 22, 289 (2005).
[CrossRef]

Belafhal, A.

A. Belafhal and L. Dalil-Essakali, Opt. Commun. 177, 181 (2000).
[CrossRef]

Bouchal, Z.

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

Casperson, L. W.

Chávez-Cerda, S.

Dalil-Essakali, L.

A. Belafhal and L. Dalil-Essakali, Opt. Commun. 177, 181 (2000).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hall, D. G.

Iturbe-Castillo, M. D.

López-Mariscal, C.

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, Opt. Eng. 45, 068001 (2006).
[CrossRef]

Mendlovic, D.

Ozaktas, H. M.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Ruschin, S.

Santarsiero, M.

M. Santarsiero, Opt. Commun. 132, 1 (1996).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves, Opt. Acoust. 2, 105 (1978).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Tovar, A. A.

Wilson, T.

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves, Opt. Acoust. 2, 105 (1978).
[CrossRef]

Czech. J. Phys. (1)

Z. Bouchal, Czech. J. Phys. 53, 537 (2003).
[CrossRef]

IEE J. Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, IEE J. Microwaves, Opt. Acoust. 2, 105 (1978).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

M. Santarsiero, Opt. Commun. 132, 1 (1996).
[CrossRef]

A. Belafhal and L. Dalil-Essakali, Opt. Commun. 177, 181 (2000).
[CrossRef]

Opt. Eng. (1)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, Opt. Eng. 45, 068001 (2006).
[CrossRef]

Opt. Lett. (1)

Other (1)

A. E. Siegman, Lasers (University Science, 1986).

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Figures (2)

Fig. 1
Fig. 1

Propagation of an ordinary MG 2 ( r , ε = 8 ) beam through a GRIN medium from p = 0 to p = 2.5 . (a), (b) Constant Gaussian width. (c), (d) Variable Gaussian width.

Fig. 2
Fig. 2

Propagation of a MG 4 ( r , ε = 8 ) ( κ 1 = 18394 + i 4751 m 1 , q 1 = 0.331 i 0.102 m ) through two free-space regions separated by a complex thin lens. Transverse field at (a) z = 0 , (b) after the complex lens z = 1 m , (c) at z = 2 m , (d) Evolution along the plane ( z , x ) .

Equations (10)

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U 1 ( r 1 ) = exp ( i k r 1 2 2 q 1 ) W ( r 1 ; κ 1 ) , q 1 < 0 ,
W ( r 1 ; κ 1 ) = π π g ( ϕ ) exp [ i κ 1 r 1 cos ( ϕ θ 1 ) ] d ϕ ,
U 2 ( r 2 ) = k exp ( i k ζ ) i 2 π B U 1 ( r 1 ) exp [ i k 2 B ( A r 1 2 2 r 1 r 2 + D r 2 2 ) ] d 2 r 1 ,
U 2 ( r 2 ) = exp ( κ 1 κ 2 B i 2 k ) GB ( r 2 , q 2 ) W ( r 2 ; κ 2 ) ,
GB ( r 2 , q 2 ) = exp ( i k ζ ) A + B q 1 exp ( i k r 2 2 2 q 2 )
q 2 = A q 1 + B C q 1 + D , κ 2 = κ 1 A + B q 1 .
r gen = q 1 2 κ 1 k q 1 + ( κ 1 q 1 + κ 1 q 1 k q 1 ) ( z z 1 ) ,
[ A B C D ] = [ cos ( p π 2 ) a sin ( p π 2 ) sin ( p π 2 ) a cos ( p π 2 ) ] .
q 2 = a q 1 cos ( p π 2 ) + a sin ( p π 2 ) q 1 sin ( p π 2 ) + a cos ( p π 2 ) ,
κ 2 = κ 1 q 1 q 1 cos ( p π 2 ) + a sin ( p π 2 ) .

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