Abstract

A very general beam solution of the paraxial wave equation in elliptic cylindrical coordinates is presented. We call such a field an elliptic beam (EB). The complex amplitude of the EB is described by either the generalized Ince functions or the Whittaker-Hill functions and is characterized by four 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 EB are the standard, elegant, and generalized Ince-Gauss beams, Mathieu-Gauss beams, among others.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008

2007

2006

2005

2004

1970

K. M. Urwin and F. M. Arscott, "Theory of the Whittaker-Hill equation," Proc. R. Soc. Edinb. [Biol],  69, 28-44 (1970).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

Arscott, F. M.

K. M. Urwin and F. M. Arscott, "Theory of the Whittaker-Hill equation," Proc. R. Soc. Edinb. [Biol],  69, 28-44 (1970).

Bandres, M. A.

Bandrés, M. A.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Bentley, J. B.

Chu, S.

Davis, J. A.

Dennis, M. R.

Guizar-Sicairos, M.

Gutierrez-Vega, J. C.

Gutiérrez-Vega, J. C.

Kamikariya, K.

López-Mariscal, C.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Ohtomo, T.

Otsuka, K.

Schwarz, U. T.

Torre, A.

A. Torre, "A note on the general solution of the paraxial wave equation a Lie algebra view," J. Opt. A: Pure Appl. Opt. 10, 55006 (2008).
[CrossRef]

Urwin, K. M.

K. M. Urwin and F. M. Arscott, "Theory of the Whittaker-Hill equation," Proc. R. Soc. Edinb. [Biol],  69, 28-44 (1970).

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

A. Torre, "A note on the general solution of the paraxial wave equation a Lie algebra view," J. Opt. A: Pure Appl. Opt. 10, 55006 (2008).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A: Pure Appl. Opt. 6, S157- S161 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

C. López-Mariscal, M. A. Bandrés, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. R. Soc. Edinb. [Biol]

K. M. Urwin and F. M. Arscott, "Theory of the Whittaker-Hill equation," Proc. R. Soc. Edinb. [Biol],  69, 28-44 (1970).

Proc. SPIE

M. A. Bandres and J. C. Gutiérrez-Vega, "Generalized Ince Gaussian beams," Proc. SPIE 6290, 6290-0S (2006).

Other

M. A. Bandres and M. Guizar-Sicairos, "The paraxial group," doc. ID 101079 (posted 16 October 2008, in press).

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

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Figures (1)

Fig. 1.
Fig. 1.

Intensity and phase patterns at the planes z=0 and z=1.5 zR and along the plane (x,0≤z≤1.5 z R ) for several special cases of the EBs.

Tables (1)

Tables Icon

Table 1. Special cases of the EB [Eq. (7)] at z=0.

Equations (17)

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( x 2 + y 2 + 2 i k z ) U ( r , z ) = 0 ,
GB ( r , q ) = [ q 0 q ( z ) ] exp [ i k r 2 2 q ( z ) ] ,
[ ̅ 2 ρ · ̅ + i γ 1 ] F = 0 ,
z ( χ 2 q 2 ) = i k q 2 ,
z Z Z = ( γ + i ) 2 k χ 2 ,
1 χ 2 ( z ) = ik [ 1 q ˜ ( z ) 1 q ( z ) ] ,
u = x χ ( z ) = 2 ε cosh ξ cos η , v = y χ ( z ) = 2 ε sinh ξ sin η ,
η η N + ε sin ( 2 η ) η N + [ μ ε ( i γ 1 ) cos ( 2 η ) ] N = 0 ,
ξξ E ε sinh ( 2 ) ξ E [ μ ε ( i γ 1 ) cosh ( 2 ξ ) ] E = 0 ,
E B γ σ , m ( r ; ε , q , q ˜ ) = ( q ˜ q 0 q q ˜ 0 ) ( i γ 1 ) 2 C γ σ , m ( i ξ , ε ) C γ σ , m ( η , ε ) GB ( r , q ) .
EB γ σ , m ( r ; ε , q , q ˜ ) = ( q ˜ q 0 q q ˜ 0 ) i γ 2 h c γ σ , m ( i ξ , ε ) h c γ σ , m ( η , ε ) GB ( r , q ) GB ( r , q ˜ ) ,
[ d ηη +μ( ε 2 /8)iγεcos2η+( ε 2 /8)cos4η]hc γ σ,m (η,ε)=0.
Ψ 0 ( r 0 ; ε , q 0 , q ˜ 0 ) = C γ σ , m ( i ξ 0 , ε ) C γ σ , m ( η 0 , ε ) exp ( i k r 0 2 2 q 0 ) ,
Ψ 1 ( r 1 ; ε , q 1 , q ˜ 1 ) = ( A + B q ˜ 0 A + B q ˜ 0 ) i γ 2 1 2 C γ σ , m ( i ξ 1 , ε ) C γ σ , m ( η 1 , ε ) GB ( r 1 , q 1 ) ,
x j = 2 ε χ j ( z ) cosh ( ξ j ) cos ( η j ) , y j = 2 ε χ j ( z ) sinh ( ξ j ) sin ( η j ) ,
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 ) = ( A + B q 0 ) 2 [ χ 0 2 i B k ( A + B q 0 ) ] ,

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