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Optical simulations within and beyond the paraxial limit. Daniel Brown, Charlotte Bond and Andreas Freise University of Birmingham. Simulating realistic optics. We need to know how to accurately calculate how distortions of optical elements effect the beam. Ideal beam.
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Daniel Brown, Charlotte Bond and Andreas Freise
University of Birmingham
Surface and bulk distortions
Final distorted beam
Finite element sizes
The effect of a mirror surface is computed by multiplying a grid of complex numbers describing the input field by a grid describing a function of the mirror surface.
Helmholtz-Kirchhoff integral equation
Difference is in approximation of and , which leads to limitations in accuracy at wider angles and proximity to aperture
Rayleigh-Sommerfeld (RS) integral equation
Number of approximations
Plots showing the 1st order diffraction efficiency of a sinusoidal gratings 
Smooth surface, small h/d
Rough surface, large h/d
Conditions on distance from aperture 
Conditions on distance from beam axis
Using with the upper limit, we can find the max angle for a given distance to a plane
If you are too close to the aperture or looking at a point too far from the beam axis, you should be using Rayleigh-Sommerfeld Diffraction!
 Angular criterion to distinguish between Fraunhofer and Fresnel diffraction, Medina 2004
For Fresnel diffraction to be (safely) valid :
Largest power difference only seen when beam size is very large!
Cavity field of view
In this example:
is the grating period
 Winkler 94, Light scattering described in the mode picture
80ppm is the power in the 1st orders
As spatial frequency becomes higher, higher diffraction orders appear which modal model can’t handle. Only 0th order is accurately modeled
For low spatial frequency distortions only a few modes are needed
Low spatial frequency compared to beam size
High spatial frequency compared to beam size
Modes struggle with complicated beam distortions, requires many modes
 Phase and alignment noise in grating interferometers, Freise et al 2007
 Invariance of waveguide grating mirrors to lateral beam displacement, Freidrich et al 2010
Taflove, Allen and Hagness, Susan C. Computational Electrodynamics: The Finite-Difference. Time-Domain Method, Third Edition
Power flow measured across this boundary
Incident field injected along TFSF boundary
Only beam reflected from grating here
s – waveguide depth, g – grating depth
 High reflectivity grating waveguide coatings for 1064 nm, Bunkowski2006