8 Interferometer Matrix with Hermite–Gauss Modes

In the plane-wave analysis Section 1.4, a laser beam is described in general by the sum of various frequency components of its electric field

∑ ( ) E(t,z) = aj exp i(ωjt − kjz ) . (160 ) j
Here we include the geometric shape of the beam by describing each frequency component as a sum of Hermite–Gauss modes:
∑ ∑ E (t,x,y,z) = ajnm unm (x,y ) exp (i(ωjt − kjz )). (161 ) j n,m
The shape of such a beam does not change along the z-axis (in the paraxial approximation). More precisely, the spot size and the position of the maximum intensity with respect to the z-axis may change, but the relative intensity distribution across the beam does not change its shape. Each part of the sum may be treated as an independent field that can be described using the equation for plane-waves with only two exceptions:

The Gouy phase shift can be included in the simulation in several ways. For example, for reasons of flexibility the Gouy phase has been included in Finesse as a phase shift of the component space.

 8.1 Coupling of Hermite–Gauss modes
 8.2 Coupling coefficients for Hermite–Gauss modes
 8.3 Finesse examples
  8.3.1 Beam parameter
  8.3.2 Mode cleaner
  8.3.3 LG33 mode

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