Abstract

For the azimuthally varying complex-argument Laguerre–Gauss beams, the Fourier integral representation is used to obtain the time-averaged power and the transport equation for the mean-squared beam width. From the coefficients in the transport equation, two propagation parameters are derived and compared with the previous treatments.

© 2005 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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  5. M. A. Bandres, J. C. Gutierrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29, 2213–2215 (2004).
    [Crossref] [PubMed]
  6. S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2004 (1)

1998 (1)

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

1986 (1)

1985 (1)

1973 (1)

1966 (1)

Bandres, M. A.

Fukumitsu, O.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gutierrez-Vega, J. C.

Kogelnik, H.

Li, T.

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[Crossref]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1960).
[Crossref]

Takenaka, T.

Yokota, M.

Zauderer, E.

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Equations (31)

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S n , m ( ρ , ϕ , z ) = exp ( i k z ) trig ( m ϕ ) g n , m ( ρ , z ) ,
g n , m ( ρ , z ) = ( 1 ) n 2 2 n + | m | / 2 n ! q 2 ( n + 1 + | m | / 2 ) × ( q ρ w 0 ) | m | L n | m | ( q 2 ρ 2 w 0 2 ) exp ( q 2 ρ 2 w 0 2 ) ,
q 2 = ( 1 + i z b ) 1 ,
b = 1 2 k w 0 2 ,
g n , m ( ρ , z ) = 0 d η η J | m | ( ρ η ) n , m ( η , z ) ,
n , m ( η , z ) = 0 d ρ ρ J | m | ( ρ η ) g n , m ( ρ , z ) ,
n , m ( η , z ) = ( 1 ) n 2 | m | / 2 ( η w 0 ) 2 ( n + | m | / 2 ) × w 0 2 2 exp ( w 0 2 η 2 4 q 2 ) ,
0 d ρ ρ J | m | ( ρ η ) J | m | ( ρ η 2 ) = δ ( η η 2 ) / η 2 ,
z ( ρ , ϕ , z ) = 1 2 ω k S n , m * ( ρ , ϕ , z ) S n , m ( ρ , ϕ , z ) .
P n , m = 0 2 π d ϕ 0 d ρ ρ 1 2 ω k cos 2 m ϕ g n , m * ( ρ , z ) g n , m ( ρ , z ) = 1 2 ω k ɛ m π 0 d ρ ρ g n , m * ( ρ , z ) g n , m ( ρ , z ) ,
P n , m = 1 2 ω k ɛ m π 0 d η η n , m * ( η , z ) 0 d η 2 η 2 n , m ( η 2 , z ) × 0 d ρ ρ J | m | ( ρ η ) J | m | ( ρ η 2 ) = 1 2 ω k ɛ m π 0 d η η n , m * ( η , z ) n , m ( η , z ) .
P n , m = 1 2 ω k ɛ m π 2 | m | w 0 4 4 0 d η η ( η w 0 ) 4 ( n + | m | / 2 ) × exp ( w 0 2 η 2 2 ) .
w 0 η / 2 = s 1 / 2
P n , m = 1 2 ω k ɛ m π 2 2 n w 0 2 4 0 d s s 2 n + | m | exp ( s ) = π ω b ɛ m 4 2 2 n ( 2 n + | m | ) !
ρ 2 = 1 P n , m 0 2 π d ϕ 0 d ρ ρ ρ 2 z ( ρ , ϕ , z ) .
ρ 2 = ω k ɛ m π 2 P n , m 0 d η η n , m * ( η , z ) 0 d η 2 η 2 n , m ( η 2 , z ) × 0 d ρ ρ ρ 2 J | m | ( ρ η ) J | m | ( ρ η 2 ) .
[ 1 η η ( η η ) + ρ 2 m 2 η 2 ] J | m | ( η ρ ) = 0 .
[ 1 η η ( η η ) m 2 η 2 ] ,
0 d ρ ρ ρ 2 J | m | ( ρ η ) J | m | ( ρ η 2 ) = [ 1 η η ( η η ) + m 2 η 2 ] δ ( η η 2 ) η 2 .
ρ 2 = ω k ɛ m π m 2 2 P n , m 0 d η 1 η n , m * ( η , z ) n , m ( η , z ) ω k ɛ m π 2 P n , m 0 d η n , m * ( η , z ) η ( η η ) × n , m ( η , z ) .
ρ 2 = ω k ɛ m π m 2 2 P n , m 0 d η 1 η n , m * ( η , z ) n , m ( η , z ) + ω k ɛ m π 2 P n , m 0 d η η η n , m * ( η , z ) η n , m ( η , z ) .
ρ 2 = ω k ɛ m π m 2 2 P n , m 2 | m | w 0 4 4 0 d η η ( η w 0 ) 4 n + 2 | m | × exp ( w 0 2 η 2 2 ) + ω k ɛ m π 2 P n , m 2 | m | w 0 6 4 × 0 d η η [ ( 2 n + | m | ) 2 ( η w 0 ) 4 n + 2 | m | 2 ( 2 n + | m | ) ( η w 0 ) 4 n + 2 | m | + 1 4 ( η w 0 ) 4 n + 2 | m | + 2 ( 1 + z 2 b 2 ) ] exp ( w 0 2 η 2 2 ) .
ρ 2 = ω k ɛ m π m 2 2 P n , m 2 2 n w 0 4 8 0 d s s 2 n + | m | 1 exp ( s ) + ω k ɛ m π 2 P n , m 2 2 n w 0 4 8 0 d s [ ( 2 n + | m | ) 2 × s 2 n + | m | 1 2 ( 2 n + | m | ) s 2 n + | m | + s 2 n + | m | + 1 × ( 1 + z 2 b 2 ) ] exp ( s ) = ω k ɛ m π 2 P n , m 2 2 n w 0 4 8 [ m 2 ( 2 n + | m | 1 ) ! ( 2 n + | m | ) ( 2 n + | m | ) ! + ( 2 n + | m | + 1 ) ! × ( 1 + z 2 b 2 ) ] .
ρ 2 = w 0 2 2 [ m 2 2 n + | m | + 1 + ( 2 n + | m | + 1 ) z 2 b 2 ] .
ρ 2 = a 0 + a 2 z 2 k 2 ,
a 0 = w 0 2 2 ( m 2 2 n + | m | + 1 ) ,
a 2 = ( 2 n + | m | + 1 ) 2 / w 0 2 .
M 2 = ( a 0 a 2 ) 1 / 2 = [ ( m 2 2 n + | m | + 1 ) ( 2 n + | m | + 1 ) ] 1 / 2 .
b R = k ( a 0 / a 2 ) 1 / 2 .
a 0 = M 2 b R / k and a 2 = k M 2 / b R .
ρ 2 = M 2 b R k ( 1 + z 2 b R 2 ) ,

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