7.6 The Gaussian beam parameter

For a more compact description of the interaction of Gaussian modes with optical components we will make use of the Gaussian beam parameter q [34Jump To The Next Citation Point]. The beam parameter is a complex quantity defined as
1 1 λ ---- = ------ − i-------. (133 ) q(z) RC (z) πw2 (z)
It can also be written as
q (z ) = izR + z − z0 = q0 + z − z0 and q0 = izR. (134 )
Using this parameter, Equation (119View Equation) can be rewritten as
( ) 1 x2 + y2 u (x,y,z) = ---- exp − ik-------- . (135 ) q(z) 2q(z)
Other parameters, like the beam size and radius of curvature, can also be written in terms of the beam parameter q:
2 λ--|q|2-- w (z) = π ℑ {q} , (136 )
ℑ {q} λ w20 = -------, (137 ) π
z = ℑ {q} (138 ) R
and
2 R (z) = --|q|--. (139 ) C ℜ {q}
The Hermite–Gauss modes can also be written using the Gaussian beam parameter as7
u (x,y,z ) = u (x,z)u (y, z) with nm n( m)1 ∕2 ( )1∕2 ( )n ∕2 ( √- ) ( ) (140 ) un (x,z) = (2)1∕4 -n1--- -q0 q0∗ q∗(z) Hn -2x- exp − i kx2 . π 2 n!w0 q(z) q0 q(z) w(z) 2q(z)

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