Numerical simulation of diffraction grating alignment and phase noise

Optical simulations within and beyond theparaxial limit1Daniel Brown, Charlotte Bond and Andreas FreiseUniversity of Birmingham

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We need to know how to accurately calculate how distortions of optical elements effect the beam

Simulating realistic optics2Surface and bulk distortionsFinite element sizesThermal effectsManufacturing errorsMirror maps

Gaussian beam Higher order modes like Laguerre-Gaussian (LG) beams

Beam clipping

Ideal beamFinal distorted beam2Methods to simulate lightFast-Fourier Transform (FFT) methodsSolving scalar wave diffraction integrals with FFTs

Modal methodRepresent beam in some basis set, usually eigenfunctions of the system

Rigorous SimulationsWhat to use when the above breakdown?Resort to solving Maxwell equations properly3FFT Methodsinput fieldmirror surfaceoutput fieldThe 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.An FFT method commonly refers to solving the scalar diffraction integrals using Fast Fourier Transforms

Can quickly propagate beam through complicated distortions, useful when studying non-eigenmode problems

Quasi-time domain, cavity simulations can require computing multiple round trips to find steady state

4Diffraction mathematically based on Greens theorem, then making many approximations to make it solvable

Scalar Diffraction5

[1] Introduction to Fourier Optics, Goodman[2] Shen 2006, Fast-Fourier-transform based numerical integration method for the RayleighSommerfeld diffraction formula[3] Nascov 2009, Fast computation algorithm for the RayleighSommerfeld diffraction formula using a type of scaled convolution

Helmholtz-Kirchhoff integral equationRayleigh-Sommerfeld (RS) integral equationFresnel DiffractionFraunhofer DiffractionNumber of approximationsSolve withFFTs [1][2][3]Solve withNumericalIntegrationParaxialApproximations5Scalar Diffraction6

[1] Harvey 2006, Non-paraxial scalar treatment of sinusoidal phase gratingsSmooth surface, small h/dRough surface, large h/dFresnel and Fraunhofer conditions7Fresnel DiffractionFraunhofer DiffractionConditions on distance from aperture [1]Conditions on distance from beam axisIf 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!

[1] Angular criterion to distinguish between Fraunhofer and Fresnel diffraction, Medina 2004Testing difference between Rayleigh-Sommerfeld and Fresnel diffraction

Small cavitySmall mirrors and shortLarger angle involved

LIGO arm cavityBig mirrors and longSmall angles involved

Khalili CavityBig mirrors and shortLarger angles involvedExamples8

L=0.1m

L=4km

L=3m34cm34cm2.5cmHow do the conditions look for each example

Examples9

L=0.1m

L=4km

L=3m34cm34cm2.5cmMax angleRound trip beam power difference between Rayleigh-Sommerfeld and Fresnel FFTExamples10

L=0.1m

L=4km

L=3m34cm34cm2.5cmCavity field of viewPower differenceLargest power difference only seen when beam size is very large!But other things do not work

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FFT Aliasing with tilted surfaces12

12Modal models

nmModal model only deals with paraxial beams and small distortions, what we would expect in our optical systems 13Complicated light fields can be described by a sum of modes

In theory need infinite modes, in practice can describe very complicated ilght fields with limited modes (maxtem