Abstract

Closed-form nonparaxial expressions for optical beams are useful to calculate the fields produced by tightly focused laser beams. Such expressions for elegant Laguerre–Gaussian (eLG) beams that are exact solutions of the Helmholtz equation are introduced. These solutions are expressed as linear combinations of a finite number of analytic functions that involve spherical Bessel functions and associated Legendre functions of complex arguments. In the paraxial limit, the expressions proposed have the property to reduce to the well-known eLG beams.

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References

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2004 (3)

2002 (1)

2000 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

1990 (1)

1988 (1)

1986 (1)

P. A. Bobbert and J. Vlieger, Physica A 137, 209 (1986).
[CrossRef]

Bandres, M. A.

Barrett, H. H.

Bobbert, P. A.

P. A. Bobbert and J. Vlieger, Physica A 137, 209 (1986).
[CrossRef]

Chavez-Cerda, S.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Gutiérrez-Véga, J. C.

Kraus, H. G.

Landesman, T. B.

Ludlow, I. K.

Rodrigez-Morales, G.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Seshadri, S. R.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Siegman, A. E.

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

Ulanowski, Z.

Vlieger, J.

P. A. Bobbert and J. Vlieger, Physica A 137, 209 (1986).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (5)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Physica A (1)

P. A. Bobbert and J. Vlieger, Physica A 137, 209 (1986).
[CrossRef]

Other (2)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

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

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Equations (13)

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u ̃ p , m e = K p , m ( j z R q ̃ ( z ) ) p + m + 1 r m w o m L p m ( j k r 2 2 q ̃ ( z ) ) exp [ j k ( r 2 2 q ̃ ( z ) + z ) ] cos ( m ϕ ) ,
ψ ̃ n , m e = exp ( k a ) j n ( k R ̃ ) P n m ( cos θ ̃ ) cos ( m ϕ ) ,
ψ ̃ n , m e = ( 1 ) n m exp ( k a ) k cos ( m ϕ ) 0 P n m ( k z k ) cos [ k z ( z + j a ) + ( n m ) π 2 ] k z J m ( k r r ) k r d k r ,
U ̃ p , m e = U o cos ( m ϕ ) Θ m ( r 2 ) p ψ ̃ 0 , 0 ,
U ̃ p , m e = ( 1 ) p + m U o exp ( k a ) k cos ( m ϕ ) 0 k r 2 p + m cos [ k z ( z + j a ) ] k z J m ( k r r ) k r d k r .
k r 2 p + m = k 2 p + m ( 2 p ) ! ! s = 0 p ( 1 ) s ( p + m s + m ) ( 4 s + 2 m + 1 ) ( 2 s 1 ) ! ! ( 2 p + 2 s + 2 m + 1 ) ! ! P 2 s + m m ( k z k ) .
U ̃ p , m σ = ( 1 ) p + m k 2 p + m U o ( 2 p ) ! ! 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 .
k r 2 p + m = k 2 p + m + 1 ( 2 p ) ! ! s = 0 p ( 1 ) s ( p + m s + m ) ( 4 s + 2 m + 3 ) ( 2 s + 1 ) ! ! ( 2 p + 2 s + 2 m + 3 ) ! ! 1 k z P 2 s + m + 1 m ( k z k ) .
V ̃ p , m σ = ( 1 ) p + m k 2 p + m V o ( 2 p ) ! ! 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 .
lim k a 1 { 2 k exp ( k a ) cos [ k z ( z + j a ) ] k z } = exp ( j q ̃ ( z ) 2 k k r 2 j k z ) .
lim k a 1 U ̃ p , m e = ( 1 ) p + m U o exp ( j k z ) 2 k 2 cos ( m ϕ ) 0 k r 2 p + m exp ( j q ̃ ( z ) 2 k k r 2 ) J m ( k r r ) k r d k r .
lim k a 1 U ̃ p , m e = ( 1 ) p + m U o 2 p p ! 2 k 2 ( j k q ̃ ( z ) ) p + m + 1 r m L p m ( j k r 2 2 q ̃ ( z ) ) exp [ j k ( r 2 2 q ̃ ( z ) + z ) ] cos ( m ϕ ) .
U o = j V o ( 1 ) p + m K p , m k 2 w o 2 p ! ( w o 2 ) 2 p + m .

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