Abstract

Expressions for the fields of TM and TE laser beams in free space that are rigorous solutions to Maxwell’s equations are given in a closed form. The electric and the magnetic fields are both expressed in terms of nonparaxial elegant Laguerre–Gaussian beams that are exact solutions of the Helmholtz equation. These solutions involve well-known functions, such as spherical Bessel and associated Legendre functions. Radially and azimuthally polarized beams of arbitrary order are considered, and the lowest-order radially polarized beam (TM01 beam) is investigated in detail.

© 2008 Optical Society of America

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2008

2007

2006

2003

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2000

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, Opt. Express 7, 77 (2000).
[CrossRef] [PubMed]

1999

1981

1975

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1959

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

April, A.

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Brown, T. G.

Davis, L. W.

Deng, D.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Kozawa, Y.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Patsakos, G.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Saghafi, S.

Salamin, Y. I.

Sato, S.

Sheppard, C. J. R.

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Youngworth, K. S.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Phys. Rev. Lett.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

B. Richards and E. Wolf, Proc. R. Soc. London, Ser. A 253, 358 (1959).
[CrossRef]

Other

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

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Figures (1)

Fig. 1
Fig. 1

Transverse variation of the electric energy density (in arbitrary units) in the beam waist ( z = 0 ) for a TM 01 beam, (a) k a = 20 , 50, 200 and (b) k a = 1 , 5, 10.

Equations (17)

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U ̃ p , m σ = 2 p + 2 K p , m ( k w o 2 ) 2 p + m + 2 s = 0 p ( p + m s + m ) ( 4 s + 2 m + 1 ) ( 2 s 1 ) ! ! ( 2 p + 2 s + 2 m + 1 ) ! ! ψ ̃ 2 s + m , m σ ,
V ̃ p , m σ = j 2 p + 2 K p , m ( k w o 2 ) 2 p + m + 2 s = 0 p ( p + m s + m ) ( 4 s + 2 m + 3 ) ( 2 s + 1 ) ! ! ( 2 p + 2 s + 2 m + 3 ) ! ! ψ ̃ 2 s + m + 1 , m σ .
U ̃ p , m e x = 1 w o [ ( p + 1 ) K p , m K p + 1 , m 1 U ̃ p + 1 , m 1 e K p , m K p , m + 1 U ̃ p , m + 1 e ] ,
U ̃ p , m e y = 1 w o [ ( p + 1 ) K p , m K p + 1 , m 1 U ̃ p + 1 , m 1 o + K p , m K p , m + 1 U ̃ p , m + 1 o ] ,
U ̃ p , m σ z = j k V ̃ p , m σ ,
V ̃ p , m σ z = j k [ U ̃ p , m σ 4 ( p + 1 ) k 2 w o 2 K p , m K p + 1 , m U ̃ p + 1 , m σ ] .
H x = H o 2 [ ( p + 1 ) K p , m K p + 1 , m 1 U ̃ p + 1 , m 1 o + K p , m K p , m + 1 U ̃ p , m + 1 o ] ,
H y = H o 2 [ ( p + 1 ) K p , m K p + 1 , m 1 U ̃ p + 1 , m 1 e K p , m K p , m + 1 U ̃ p , m + 1 e ] ,
E x = E o 2 [ ( p + 1 ) K p , m K p + 1 , m 1 V ̃ p + 1 , m 1 e K p , m K p , m + 1 V ̃ p , m + 1 e ] ,
E y = E o 2 [ ( p + 1 ) K p , m K p + 1 , m 1 V ̃ p + 1 , m 1 o + K p , m K p , m + 1 V ̃ p , m + 1 o ] ,
E z = E o 2 ( p + 1 ) j k w o K p , m K p + 1 , m U ̃ p + 1 , m e ,
H ϕ = H o K p , 0 K p , 1 ( U ̃ p , 1 o sin ϕ + U ̃ p , 1 e cos ϕ ) ,
E r = E o K p , 0 K p , 1 ( V ̃ p , 1 e cos ϕ + V ̃ p , 1 o sin ϕ ) ,
E z = E o 2 ( p + 1 ) j k w o K p , 0 K p + 1 , 0 U ̃ p + 1 , 0 e .
H ϕ = H o exp ( k a ) j 1 ( k R ̃ ) sin θ ̃ ,
E r = 1 2 j E o exp ( k a ) j 2 ( k R ̃ ) sin ( 2 θ ̃ ) ,
E z = 2 3 j E o exp ( k a ) [ j 0 ( k R ̃ ) + j 2 ( k R ̃ ) P 2 ( cos θ ̃ ) ] ,

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