Abstract

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is 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. Special cases of the CiB are the standard, elegant, and generalized Laguerre–Gauss beams; Bessel–Gauss beams; hypergeometric beams; hypergeometric–Gaussian beams; fractional-order elegant Laguerre–Gauss beams; quadratic Bessel–Gauss beams; and optical vortex beams.

© 2008 Optical Society of America

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References

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2008 (1)

2007 (3)

2006 (1)

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

2005 (1)

2004 (1)

M. V. Berry, J. Opt. A, Pure Appl. Opt. 6, 259 (2004).
[CrossRef]

2001 (2)

1999 (1)

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

1989 (1)

1987 (1)

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

1975 (1)

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

J. Math. Phys. (1)

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

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

M. V. Berry, J. Opt. A, Pure Appl. Opt. 6, 259 (2004).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

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

C. F. R. Caron and R. M. Potvliege, Opt. Commun. 164, 83 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

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

Prog. Opt. (1)

M. S. Soskin and M. Vasnetsov, Prog. Opt. 42, 219 (2001).
[CrossRef]

Other (2)

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

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

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Tables (1)

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Table 1 Special Cases of CiB γ m ( r ; q 0 , q ̃ 0 )

Equations (17)

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( x 2 + y 2 + 2 i k z ) U ( r , z ) = 0 ,
GB ( r , q ) = ( 1 + z q 0 ) 1 exp [ i k r 2 2 q ( z ) ] ,
[ ¯ 2 i ρ ¯ γ i ] F = 0 ,
z ( χ 2 q 2 ) = 1 k q 2 ,
z Z Z = ( i γ 1 ) 2 k χ 2 ,
1 χ 2 ( z ) = k ( 1 q ̃ 1 q ) ,
[ ϱ ϱ 1 4 + i γ 2 ϱ + 1 4 ( m 2 ) 2 ϱ 2 ] G ( ϱ ) = 0 .
CiB γ m ( r ; q , q ̃ ) = ( q ̃ q ̃ 0 q q 0 ) i γ 2 ( i r 2 2 χ 2 ) 1 2 M i γ 2 , m 2 ( i r 2 2 χ 2 ) × [ GB ( r , q ) GB ( r , q ̃ ) ] 1 2 exp ( i m θ ) .
CiB γ m ( r ; q , q ̃ ) = ( q ̃ q ̃ 0 q q 0 ) i γ 2 1 2 ( i r 2 2 χ 2 ) m 2 × F 1 1 ( 1 + m i γ 2 , 1 + m , i r 2 2 χ 2 ) GB ( r , q ) exp ( i m θ ) .
Ψ 0 ( r 0 ; q 0 , q ̃ 0 ) = ( i r 0 2 2 χ 0 2 ) m 2 exp ( i m θ ) exp ( i k r 0 2 2 q 0 ) × F 1 1 ( 1 + m i γ 2 , 1 + m , i r 0 2 2 χ 0 2 ) .
Ψ 1 ( r 1 ; q 1 , q ̃ 1 ) = ( i ) m + 1 k B 0 d r 0 Ψ 0 ( r 0 ; q 0 , q ̃ 0 ) × r 0 J m ( k r 0 r 1 B ) exp [ i k 2 B ( A r 0 2 + D r 1 2 ) ] ,
Ψ 1 ( r 1 ; q 1 , q ̃ 1 ) = ( A + B q ̃ 0 A + B q 0 ) i γ 2 1 2 ( i r 1 2 2 χ 1 2 ) m 2 × F 1 1 ( m + 1 i γ 2 , m + 1 ; i r 1 2 2 χ 1 2 ) × GB ( r 1 , q 1 ) exp ( i m θ ) ,
GB ( r 1 , q 1 ) = ( A + B q 0 ) 1 exp ( i k r 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 .
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 ) ] } .
WGB γ m ( r ) = [ ( i r 2 2 W 0 2 ) 1 2 M i γ 2 , m 2 ( i r 2 2 W 0 2 ) ] × exp ( r 2 w 0 2 ) exp ( i m θ ) ,

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