2.1 Laguerre–Gauss beams

Because we are developing models based on axisymmetry, we pay special attention to cylindrical coordinates, and consequently to the Laguerre–Gauss family of modes. If z is the coordinate along the optical axis, r the radial coordinate and φ the azimuthal, a Laguerre–Gauss mode (LGn,m) of parameter w has the following complex amplitude at z (the wavelength is λ and k ≡ 2 π∕λ):
{ cos(n φ)} Ψ (mn)(r,φ, z) = R(mn)(r)1∕2 eikzei(2m+n+1 )G(z), (2.1 ) sin(n φ)
where G(z) is the Gouy phase,
G (z ) = tan −1(z∕zR), (2.2 )
2 zR ≡ πw 0∕λ is the Rayleigh parameter, and ∘ ----------2- w = w0 1 + (z∕zR ) is the beam width at z. Parameter w0 represents the minimum width (waist) of the beam, localized at abscissa z = 0. The normalized radial function R (mn)(r) is given by
( 2)n ( 2)2 ( 2) R (n)(r,φ, z) = ------4---------m!---- 2r-- L(n) 2r-- exp − 2r-- . (2.3 ) m (1 + δn,0)πw2 (n + m )! w2 m w2 w2
(n) L m (X ) are the generalized Laguerre polynomials. Figure 1View Image shows the intensity pattern of a nonaxisymmetric Laguerre–Gauss mode of cosine angular parity. Obviously the origin of angles is arbitrary, so that we can replace the cosine by a sine, and even combine a cosine mode with a sine mode to obtain a mode having an axisymmetric intensity. For such an axisymmetric (in intensity) mode, the normalization is slightly different, so that
2 m! ( 2r2)n ( 2r2 )2 ( 2r2 ) R(mn)(r,φ,z) = ---2 --------- --2- L (nm) --2- exp − --2- . (2.4 ) πw (n + m )! w w w
See Figure 2View Image for the intensity pattern of such a readout beam.
View Image

Figure 1: Intensity distribution in an LG5,5 mode of width parameter w = 3.5 cm. Dashed circle: edge of a mirror of radius 17.5 cm.
View Image

Figure 2: Intensity distribution in an axisymmetric LG5,5 mode of width parameter w = 3.5 cm. Dashed circle: edge of a mirror of radius 17.5 cm.

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