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Message ID: 179     Entry time: Fri Feb 28 19:50:11 2014
Author: Koji 
Type: General 
Category: General 
Subject: Misalignment ABCD matrix for the aLIGO OMC 

Relationship between mirror misalignment and cavity mode shift was calculated.
The technique described in T0900647 by Sam Waldman was used.

The angles and displacement of the mirrors and beams are defined in the attached figure.

x1 = 0.893134 α + 1.10676 β + 1.32252 γ + 1.24619 δ
𝛳1 = 0.75864 α - 0.75864 β - 0.271075 γ + 0.271075 δ

x2 = 1.10676 α + 0.893134 β + 1.24619 γ + 1.32252 δ
𝛳2 = 0.75864 α + 1.24136 β - 0.271075 γ + 0.271075 δ

x3 = 1.32252 α + 1.24619 β + 1.1691 γ + 1.39962 δ
𝛳3 = -0.271075 α + 0.271075 β + 0.818668 γ - 0.818668 δ

x4 = 1.24619 α + 1.32252 β + 1.39962 γ + 1.1691 δ
𝛳4 = -1.24136 α - 0.75864 β - 0.271075 γ + 0.271075 δ

Assuming the flat mirrors are fixed:
If I want to move the x3 mirror up by 1mm without moving x4, the solution is
γ = -0.00197 mrad
δ = +0.00236 mrad

This yields:
x1 = +0.33mm, x2=+0.66mm, x3 = +1mm, x4 = 0mm

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misalignment.pdf
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