Abstract

A general method is presented for the determination of the time-averaged power associated with the self-interaction and the mutual interaction of cylindrically symmetric complex-argument Laguerre–Gauss beams. The method is also applied for the determination of two useful moments of the time-averaged Poynting vector.

© 2006 Optical Society of America

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References

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  1. H. Kogelnik and T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  5. S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998).
    [CrossRef]
  6. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
    [CrossRef]
  7. M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Printing Office, 1965), p. 784, formula (22.9.15).

2002 (1)

2001 (1)

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
[CrossRef]

1998 (1)

S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998).
[CrossRef]

1985 (1)

1973 (1)

1966 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Printing Office, 1965), p. 784, formula (22.9.15).

Fukumitsu, O.

Kogelnik, H.

Li, T.

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
[CrossRef]

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998).
[CrossRef]

Seshadri, S. R.

Sheppard, C. J.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998).
[CrossRef]

Siegman, A. E.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Printing Office, 1965), p. 784, formula (22.9.15).

Takenaka, T.

Yokota, M.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

S. Saghafi and C. J. R. Sheppard, Opt. Commun. 153, 207 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, Opt. Commun. 191, 173 (2001).
[CrossRef]

Opt. Lett. (1)

Other (1)

M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions (U.S. Government Printing Office, 1965), p. 784, formula (22.9.15).

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Equations (37)

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S m ( ρ , z ) = exp ( i k z ) ( 1 ) m m ! 2 2 m q 2 ( m + 1 ) L m ( q 2 ρ n 2 ) exp ( q 2 ρ n 2 ) ,
q 2 = ( 1 + i z b ) 1 ,
ρ n = ρ w 0 .
0 d x L m ( c x ) L n ( c x ) exp ( c x ) = δ m n c ,
z ( ρ , z ) = ( 1 2 ) ω k Re [ S m * ( ρ , z ) S m ( ρ , z ) ] ,
z ( ρ , z ) = ( 1 2 ) ω k Re S m * ( ρ , z ) S n ( ρ , z ) .
z ( ρ , z ) = ω k ( 1 ) m + n m ! n ! 2 2 ( m + n ) 1 ( q * q ) 2 Re [ q * 2 m q 2 n L m ( q * 2 ρ n 2 ) L n ( q 2 ρ n 2 ) exp ( 2 ρ 2 w 2 ) ] ,
w = w 0 ( 1 + z 2 b 2 ) 1 2 = w 0 q q * .
a = q * 2 w 2 2 w 0 2 = 1 2 q 2 = 1 2 ( 1 + i z b )
a + a * = 1 .
P m , n = π ω b ( 1 ) m + n m ! n ! 2 2 ( m + n ) 1 ( q * q ) m + n Re [ q * ( m n ) q ( m n ) I m n ( 0 ) ] ,
I m n ( α ) = I n m * ( α ) = 0 L m ( a x ) L n ( a * x ) exp ( x ) x α d x
P m , n = π ω b ( 1 ) m + n 2 m + n 1 ( m + n ) ! .
ρ 2 = ( P m , m ) 1 0 d ρ 2 π ρ ρ Π z 2 ( ρ , z ) .
ρ 2 = w 0 2 2 ( 1 + z 2 b 2 ) I m m ( 1 ) I m m ( 0 ) .
ρ 2 = w 0 2 2 [ 1 + ( 2 m + 1 ) z 2 b 2 ] ,
ρ 4 = ( P m , m ) 1 0 d ρ 2 π ρ ρ Π z 4 ( ρ , z ) .
ρ 4 = w 0 4 4 ( 1 + z 2 b 2 ) 2 I m m ( 2 ) I m m ( 0 ) .
ρ 4 = w 0 4 2 [ 3 m 1 2 m 1 + 2 ( 3 m + 1 ) z 2 b 2 + ( m + 1 ) ( 2 m + 1 ) z 4 b 4 ] .
g ( x , t ) = ( 1 t ) 1 exp [ x t ( 1 t ) 1 ] ,
L m ( x ) = 1 m ! Lim t = 0 m g ( x , t ) t m .
I m n ( 0 ) = 1 m ! n ! Lim t = 0 Lim s = 0 m t m n s n 1 ( 1 t ) ( 1 s ) 0 d x exp [ x ( 1 + a t 1 t + a * s 1 s ) ] .
I m n ( 0 ) = 1 m ! n ! Lim t = 0 Lim s = 0 m t m n s n [ 1 ( a * t + a s ) ] 1 .
I m n ( 0 ) = 1 m ! n ! Lim t = 0 Lim s = 0 m t m n s n r = 0 r = l = 0 l = r r ! l ! ( r l ) ! ( a * t ) l ( a s ) r l .
I m n ( 0 ) = 1 m ! n ! Lim t = 0 Lim s = 0 r = 0 r = l = 0 l = r r ! a * l a r l t l m s r l n ( l m ) ! ( r l n ) ! .
I m n ( 0 ) = ( m + n ) ! a * m a n m ! n ! .
I m n ( 0 ) = I n m * ( 0 ) = ( m + n ) m ! n ! 2 m + n q * 2 m q 2 n .
I m m ( 1 ) = 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m 1 ( 1 t ) ( 1 s ) 0 d x x exp [ x ( 1 + a t 1 t + a * s 1 s ) ] .
I m m ( 1 ) = 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m ( 1 t ) ( 1 s ) [ 1 ( a * t + a s ) ] 2 .
I m m ( 1 ) = [ 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m ( 1 t s + s t ) r = 0 r = l = 0 l = r ( r + 1 ) ! l ! ( r l ) ! ( a * t ) l ( a s ) r l ] .
I m m ( 1 ) = 1 m ! m ! Lim t = 0 Lim s = 0 r = 0 r = l = 0 l = r ( r + 1 ) ! l ! ( r l ) ! a * l a r l [ m t l t m m s r l s m m t l + 1 t m m s r l s m m t l t m m s r l + 1 s m + m t l + 1 t m m s r l + 1 s m ] .
I m m ( 1 ) = 1 m ! m ! [ ( 2 m + 1 ) ! ( a * a ) m m ( 2 m ) ! a * ( m 1 ) a ( m 1 ) ( a + a * ) + m 2 ( 2 m 1 ) ! ( a * a ) m 1 ] .
I m m ( 1 ) = I m m ( 0 ) ( 2 m + 1 m 2 a * a ) .
I m m ( 2 ) = 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m 1 ( 1 t ) ( 1 s ) 0 d x x 2 exp [ x ( 1 + a t 1 t + a * s 1 s ) ] .
I m m ( 2 ) = 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m 2 ( 1 t ) 2 ( 1 s ) 2 [ 1 ( a * t + a s ) ] 3 .
I m m ( 2 ) = [ 1 m ! m ! Lim t = 0 Lim s = 0 m t m m s m ( 1 t s + s t ) 2 r = 0 r = l = 0 l = r ( r + 2 ) ! l ! ( r l ) ! ( a * t ) l ( a s ) r l ] .
I m m ( 2 ) = I m m ( 0 ) ( a * a ) 2 [ 2 ( m + 1 ) ( 2 m + 1 ) ( a * a ) 2 2 m 2 a * a + m ( m 1 ) 2 2 ( 2 m 1 ) ] .

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