Abstract

Based on the vector angular spectrum of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Laguerre–Gaussian beam has been presented in the far field. According to the analytical electromagnetic representations of the TE and TM terms, the energy flux distributions of the TE term, the TM term, and the whole beam are investigated in the far field, respectively. The formulas obtained are applicable not only to the paraxial case, but also to the nonparaxial case. The physical pictures of Laguerre–Gaussian beams are well illustrated from the vectorial structure, which may provide a new approach to manipulate laser beams.

© 2006 Optical Society of America

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References

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2006 (1)

2005 (1)

2004 (2)

2002 (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

2001 (1)

1996 (1)

1972 (1)

1966 (2)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

D. R. Rhodes, IEEE Trans. Antennas Propag. 14, 676 (1966).
[CrossRef]

Bandres, M. A.

Bosch, S.

Carnicer, A.

Carter, W. H.

Chen, J.

Gan, X.

Gu, M.

Guo, H.

Gutiérrez-Vega, J. C.

Jia, B.

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martínez-Herrero, R.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, J. Opt. Soc. Am. A 18, 1678 (2001).
[CrossRef]

Mejías, P. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, J. Opt. Soc. Am. A 18, 1678 (2001).
[CrossRef]

Miyaji, G.

Miyanaga, N.

Movilla, J. M.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

Nakatsuka, M.

Piquero, G.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

Rhodes, D. R.

D. R. Rhodes, IEEE Trans. Antennas Propag. 14, 676 (1966).
[CrossRef]

Sueda, K.

Török, P.

Varga, P.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Zhuang, S.

IEEE Trans. Antennas Propag. (1)

D. R. Rhodes, IEEE Trans. Antennas Propag. 14, 676 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (3)

Opt. Lett. (2)

Proc. IEEE (1)

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Prog. Quantum Electron. (1)

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, Prog. Quantum Electron. 26, 65 (2002).
[CrossRef]

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (3)

Fig. 1
Fig. 1

Energy flux distribution of Laguerre–Gaussian beam with cos m φ . Reference plane, z = 500 λ . w 0 = 5 λ , m = 0 , and n = 2 . (a) TE term, (b) TM term, (c) whole beam.

Fig. 2
Fig. 2

Energy flux distribution of Laguerre–Gaussian beam with cos m φ . Reference plane, z = 500 λ . w 0 = 5 λ , m = 2 , and n = 0 . (a) TE term, (b) TM term, (c) whole beam.

Fig. 3
Fig. 3

Energy flux distribution of Laguerre–Gaussian beam with sin m φ . Reference plane, z = 500 λ . w 0 = 5 λ , m = 2 and n = 0 . (a) TE term, (b) TM term, (c) whole beam.

Equations (16)

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( E x ( x , y , 0 ) E y ( x , y , 0 ) ) = ( ( 2 ρ w 0 ) m L n m ( 2 ρ 2 w 0 2 ) exp ( ρ 2 w 0 2 ) cos m φ 0 ) ,
E ( r ) = 0 0 A x ( p , q , γ ) ( i p γ k ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
A x ( p , q , γ ) = 1 λ 2 E x ( x , y , 0 ) exp [ i k ( p x + q y ) ] d x d y = i 2 n m 4 π f 2 ( p 2 + q 2 2 f ) m L n m ( p 2 + q 2 2 f 2 ) exp ( p 2 + q 2 4 f 2 ) cos m θ ,
E ( r ) = E TE ( r ) + E TM ( r ) ,
E TE ( r ) = 0 0 q p 2 + q 2 A x ( p , q , γ ) ( q i p j ) exp [ i k ( p x + q y + γ z ) ] d p d q ,
E TM ( r ) = 0 0 p γ ( p 2 + q 2 ) A x ( p , q , γ ) [ p γ i + q γ j ( p 2 + q 2 ) k ] exp [ i k ( p x + q y + γ z ) ] d p d q .
H ( r ) = H TE ( r ) + H TM ( r ) ,
H TE ( r ) = ϵ μ 0 0 q p 2 + q 2 A x ( p , q , γ ) [ p γ i + q γ j ( p 2 + q 2 ) k ] exp [ i k ( p x + q y + γ z ) ] d p d q ,
H TM ( r ) = ϵ μ 0 0 p γ ( p 2 + q 2 ) A x ( p , q , γ ) ( q i p j ) exp [ i k ( p x + q y + γ z ) ] d p d q .
E TE ( r ) = i 2 n m + 1 z r y z r 2 ρ 2 ( ρ 2 f r ) m L n m ( ρ 2 2 f 2 r 2 ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( y i x j ) cos m φ ,
H TE ( r ) = ϵ μ i 2 n m + 1 z r y z r 3 ρ 2 ( ρ 2 f r ) m L n m ( ρ 2 2 f 2 r 2 ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( x z i + y z j ρ 2 k ) cos m φ ,
E TM ( r ) = i 2 n m + 1 z r x r 2 ρ 2 ( ρ 2 f r ) m L n m ( ρ 2 2 f 2 r 2 ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( x z i + y z j ρ 2 k ) cos m φ ,
H TM ( r ) = ϵ μ i 2 n m + 1 z r x r ρ 2 ( ρ 2 f r ) m L n m ( ρ 2 2 f 2 r 2 ) exp ( ρ 2 4 f 2 r 2 + i k r ) ( y i x j ) cos m φ .
S z TE = ϵ μ y 2 z r 2 z 3 r 5 ρ 2 ( ρ 2 2 f 2 r 2 ) m [ L n m ( ρ 2 2 f 2 r 2 ) ] 2 exp ( ρ 2 2 f 2 r 2 ) cos 2 m φ ,
S z TM = ϵ μ x 2 z r 2 z r 3 ρ 2 ( ρ 2 2 f 2 r 2 ) m [ L n m ( ρ 2 2 f 2 r 2 ) ] 2 exp ( ρ 2 2 f 2 r 2 ) cos 2 m φ .
S z = S z TE + S z TM = ϵ μ z r 2 z r 3 ρ 2 ( z 2 r 2 y 2 + x 2 ) ( ρ 2 2 f 2 r 2 ) m [ L n m ( ρ 2 2 f 2 r 2 ) ] 2 exp ( ρ 2 2 f 2 r 2 ) cos 2 m φ .

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