Abstract

A new and very general beam solution of the paraxial wave equation in Cartesian coordinates is presented. We call such a field a Cartesian beam. The complex amplitude of the Cartesian beams is described by either the parabolic cylinder functions or the confluent hypergeometric functions, and the beams are characterized by three parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Applying the general expression of the Cartesian beams, we also derive two new and meaningful beam structures that, to our knowledge, have not yet been reported in the literature. Special cases of the Cartesian beams are the standard, elegant, and generalized Hermite–Gauss beams, the cosine-Gauss beams, the Lorentz beams, and the fractional order beams.

© 2007 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, 1986).
  2. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Lett. 29, 144 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. M. A. Bandres and J. C. Gutiérrez-Vega, Proc. SPIE 6290, 6290-0S (2006).
  7. O. El Gawhary and S. Severini, J. Opt. A, Pure Appl. Opt. 8, 409 (2006).
    [CrossRef]
  8. J. C. Gutiérrez-Vega, Opt. Lett. 32, 1521 (2007).
    [CrossRef] [PubMed]
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2007 (1)

2006 (2)

M. A. Bandres and J. C. Gutiérrez-Vega, Proc. SPIE 6290, 6290-0S (2006).

O. El Gawhary and S. Severini, J. Opt. A, Pure Appl. Opt. 8, 409 (2006).
[CrossRef]

2005 (1)

2004 (1)

1989 (1)

1975 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 499 (1975).
[CrossRef]

J. Math. Phys. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, Jr., J. Math. Phys. 16, 499 (1975).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

O. El Gawhary and S. Severini, J. Opt. A, Pure Appl. Opt. 8, 409 (2006).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Proc. SPIE (1)

M. A. Bandres and J. C. Gutiérrez-Vega, Proc. SPIE 6290, 6290-0S (2006).

Other (2)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

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

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Tables (1)

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Table 1 Special Cases of U α e , o ( x ; q 0 , q ̃ 0 )

Equations (24)

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( x 2 + 2 i k z ) U ( x , z ) = 0 ,
GB ( x , q ) = ( 1 + z q 0 ) 1 2 exp [ i k x 2 2 q ( z ) ] ,
[ u 2 i u u α i 2 ] F = 0 ,
z ( χ 2 q 2 ) = 1 k q 2 ,
z Z Z = ( i α 1 2 ) 2 k χ 2 ,
χ ( z ) = [ k ( 1 q ̃ 1 q ) ] 1 2 ,
U α e , o ( x ; q , q ̃ ) = ( q ̃ q ̃ 0 q q 0 ) i α 2 P α e , o ( x χ ( z ) ) × [ GB ( x , q ) GB ( x , q ̃ ) ] 1 2 .
U α e , o ( x ; q , q ̃ ) = ( q ̃ q 0 q q ̃ 0 ) i α 2 1 4 GB ( x , q )
× ( x χ ) 0 , 1 F 1 1 ( 2 1 4 i α 2 , 2 1 2 ; i x 2 2 χ 2 ) .
Ψ 0 e , o ( x 0 ; q 0 , q ̃ 0 ) = exp ( i k x 2 2 q 0 ) exp ( i S 0 x 0 ) ( x 0 + δ 0 χ 0 ) 0 , 1
× F 1 1 ( 2 1 4 i α 2 , 2 1 2 ; i ( x 0 + δ 0 ) 2 2 χ 0 2 ) ,
Ψ 1 e , o ( x 1 ; q 1 , q ̃ 1 ) = k i 2 π B Ψ 0 e , o ( x 0 ; q 0 , q ̃ 0 )
× exp [ i k ( A x 0 2 2 x 0 x 1 + D x 1 2 ) 2 B ] d x 0 .
Ψ 1 e , o ( x 1 ; q 1 , q ̃ 1 ) = ( A + B q ̃ 0 A + B q 0 ) i α 2 1 4 exp ( B S 0 S 1 i 2 k )
× ( x 1 + δ 1 χ 1 ) 0 , 1 F 1 1 ( 2 1 4 i α 2 , 2 1 2 ; i ( x 1 + δ 1 ) 2 2 χ 1 2 )
× GB ( x 1 , q 1 ) exp ( i S 1 x 1 ) ,
GB ( x 1 , q 1 ) = ( A + B q 0 ) 1 2 exp ( i k x 1 2 2 q 1 )
q 1 = A q 0 + B C q 0 + D , q ̃ 1 = A q ̃ 0 + B C q ̃ 0 + D ,
δ 1 = δ 0 ( A + B q 0 ) B S 0 k ,
S 1 = S 0 ( A + B q 0 ) .
χ 1 2 = χ 0 2 ( A + B q 0 ) ( A + B q ̃ 0 ) ,
χ 1 2 = ( A + B q 0 ) 2 [ χ 0 2 + B k ( A + B q 0 ) ] .
Φ α e , o ( x , z ) = exp ( i k x 2 4 z ) z 1 4 + i α 2 P α e , o ( x z k ) .
Θ α e , o ( x , 0 ) = P α e , o ( x W 0 ) exp ( x 2 w 0 2 ) .

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