Analysis of BS pitch correction for tunnel tilt

JGW-T1605630-v1 - 2016/09/08 - Mark Barton

Preliminaries

The support files RotationsXYZ.nb and RotationsXYZ.m should be installed in one of the directories on the Mathematica path ($Path).

Needs["RotationsXYZ`"]

?RotationMatrixXYZBS

Data

Calculation

On the assembly frame the beamsplitting face of the BS faces -Y:

BSfaceframe = {-1,0,0}; BSfaceframe//TableForm

In an ideal IFO the BS needs to be at -45° yaw to this

BSangle = RotationMatrixXYZBS[-Pi/4,0,0];BSangle//TableForm

The BS will be rotated by this 45° during installation, putting the BS face in the ideal direction.

BSfaceideal = BSangle.BSfaceframe; BSfaceideal//TableForm

However the Y arm is tilted down by 1/300 rad (negative pitch):

Yarmangle = RotationMatrixXYZBS[0,-1/300,0];Yarmangle//TableForm

And the X arm is tilted up by 1/300 rad (negative roll):

Xarmangle = RotationMatrixXYZBS[0,0,-1/300];Xarmangle//TableForm

So the desired final orientation is

Yarmangle.Xarmangle.BSangle.BSfaceframe //N//TableForm

The required pitch correction on the assembly frame then is -Sqrt[2]/300

pitchcorr = RotationMatrixXYZBS[0,-Sqrt[2]/300.,0];pitchcorr//TableForm

Applying just pitch correction and the 45° gives the right vertical angle.

BSangle.pitchcorr.BSfaceframe //N//TableForm

Conclusion

The beamsplitting face of the BS should be set to pitch down by Sqrt[2]/300 = 4.71405 mrad on the assembly frame.