Abstract

The existence of elegant Ince–Gaussian beams that constitute a third complete family of exact and biorthogonal elegant solutions of the paraxial wave equation is demonstrated. Their transverse structure is described by Ince polynomials with a complex argument. Elegant Ince–Gaussian beams constitute exact and continuous transition modes between elegant Laguerre–Gaussian and elegant Hermite–Gaussian beams. The expansion formulas among the three elegant families are derived.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2004

1986

1985

1977

1975

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

1973

1967

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

Arscott, F. M.

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, England, 1964).

Bandres, M. A.

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Felsen, L. B.

Fukumitsu, O.

Gutiérrez-Vega, J. C.

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

A. E. Siegman, J. Opt. Soc. Am. 63, 1093 (1973).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Takenaka, T.

Yokota, M.

Zauderer, E.

J. Math. Phys.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Proc. R. Soc. Edinburgh Sect. A

F. M. Arscott, Proc. R. Soc. Edinburgh Sect. A 67, 265 (1967).

Other

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, England, 1964).

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Figures (1)

Fig. 1
Fig. 1

Transverse amplitudes and phases of eLGBs, eIGBs, and eHGBs at z=zR. The eIGBs correspond to =2. eIGBs tend to eLGBs or eHGBs when 0 or , respectively.

Equations (15)

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eIG r=AzEξNηexp-cr2,
d2Edξ2- sinh 2ξdEdξ-a-p cosh 2ξE=0,
d2Ndη2+ sin 2ηdNdη+a-p cos 2ηN=0,
qAdAdq+p/2+1=0,
eIGp,mer,=Ceq0/qp/2+1Cpmiξ,×Cpmη,exp-cr2,
eIGp,mor,=Seq0/qp/2+1Spmiξ,×Spmη,exp-cr2,
L1h22ξ2+2η2+h2ξsinh 2ξ-ηsin 2η,
L1h*22ξ*2+2η*2-h*2sinh 2ξ*ξ*-sin 2η*η*.
eIGˆp,mer,=Ceq0*/q*-p/2Cpmiξ*,Cpmη*,,
eIGˆp,mor,=Seq0*/q*-p/2Spmiξ*,Spmη*,.
eHGnx,nyr=Dnx,nyq0/qnx+ny/2+1Hnxc1/2x×Hnyc1/2yexp-cr2,
eHGˆnx,nyr=Dnx,nyq0*/q*-nx+ny/2×Hnxc*1/2xHnyc*1/2y,
eLGn,le,or=Klq0/q2n+l+2/2c1/2rlLnlcr2×exp-cr2cos lϕsin lϕ,
eLGˆn,le,or=Klq0*/q*-2n+l/2c*1/2rl×Lnlc*r2cos lϕsin lϕ,
eLGn,lσ,eIGˆp,mσ=δσσδp,2n+liδo,σ-1n+l+p+m/2×1+δ0,ln+l!n!1/2Al+δo,σ/2σapm,

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