Measurement of Schnupp asymmetry

This was done by measuring the relative phase between the sidebands reflected from the two arms while the arm cavities are locked.

The Schnupp asymmetry is measured to be:   L_sa = 3.64 ± 0.32 cm

Description:

As a phase reference we use the zero crossing of the response function for the out-of-phase control signal for the single arm cavity lock [0]. The difference in the RD rotation phase of the response zero crossings indicates the phase difference in the sideband signals reflected from the arms. Assuming the asymmetry is less than half the RF modulation wavelength [1], the asymmetry is given by the following formula:

       \Delta \phi   c   1 
L_sa = ----------- ----- -
           360     f_RSB 2

We use a LSC digital lock-in to measure the response of the arm cavity at a single-frequency drive of it's end mirror.

[0] The locations of the zero crossings in the out-of-phase components of the response can be determined to higher precision than the maxima of the in-phase components.

[1] f_RSB = 55 MHz,     c/f_RSB/2 = 2.725 m

Procedure:

1. Lock/tune the Y arm only.
   
   • We use AS55_I to lock the arms.
2. Engage the LSC lock-in.
3. Tune the lock-in parameters:

lock-in freq: 103.1313 Hz
I/Q filters:  0.1 Hz low-pass
phase:        0 degrees

4. Set as input to the lock-in the out-of-phase quadrature from the control RFPD.  In this case AS55_Q->LOCKIN.
5. Drive the arm cavity end mirror by setting the LOCKIN->Y_arm element in the control matrix.
6. Note the "RD Rotation" phase between the demodulated signals from the control PD (AS55)
7. For some reasonable distribution of phases around the nominal "RD Rotation" value, measure the amplitude of the lock-in I output.
   • Assuming the Q output is nearly zero, it can be neglected.  In this case the Q amplitude was more than a factor of 10 less than the I amplitude.
   • Here we take 5 measurements, each separated by one over the measurement bandwidth (as determined by the lock-in low pass filter), in this case 10 seconds.  The figure above plots the mean of these measurements, and the error bars indicate the standard deviation.

The data and python data-taking and plotting scripts are attached.

Error Analysis:

To to determine the parameters of the response (which we know to be linear) we use a weighted linear least-squares fit to the data:

y = b X

where:

X_0j = 1
X_1j = x_j              # the measurement points
y = y_i                 # the response
b = (b₀, b₁)     # line parameters

The weighting is given by the inverse of the measurement covariance matrix. Since we assume the measurements are independent, the matrix is diagonal and W_ii = 1/\sigma_i² The
estimated parameter values are given by:

\beta  =  ( X^T W X )^-1 X^T W y  =  ( X'^T X' )^-1 X'^T y'

where X' = w X, y' = w y and w_ii = \sqrt{W_ii}.

The X' and y' are calculated from the data and passed into the lstsq routine. The output is \beta.

The error on the parameters is described by the covariance matrix M^\beta:

M^\beta = ( X^T W X)^-1 = ( X'^T X')^-1

with correlation coefficients \rho_ij = M^\beta_ij / \sigma_i / \sigma_j.

The x-axis crossing is then given by:

X(Y=0) = - \beta₁ / \beta₀

References:

Valera's LLO measurement
http://en.wikipedia.org/wiki/Weighted_least_squares
http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)#Weighted_linear_least_squares
http://en.wikipedia.org/wiki/Error_propagation

#!/usr/bin/env python

import sys
import os
import subprocess
import time
import pickle
from numpy import *
import nds
import matplotlib

#!/usr/bin/env python

import pickle
from numpy import *
import matplotlib
#matplotlib.use('AGG')
from matplotlib.pyplot import *

##################################################

(dp0
S'I'
p1
(dp2
cnumpy.core.multiarray
scalar
p3
(cnumpy
dtype
p4

(dp0
S'I'
p1
(dp2
cnumpy.core.multiarray
scalar
p3
(cnumpy
dtype
p4