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Message ID: 1992     Entry time: Fri Jul 31 15:21:19 2015
Author: Alessandra 
Type: Misc 
Category: Seismometer 
Subject: QPD calibration 

QPD calibration


In a previous entry I described the setup used to build the QPD calibration curve and the method used to take measurements.

Here I show the results of our measurements and calculate the Volts/displacement ratio and the laser's beam radius.


X axis calibration


  • In the following plot the X axis QPD calibration curve is shown and a fit in the linear region is made:


The errorbars on the plot where estimated by looking at the fluctuations of the voltage output.


The fit in the linear region with the function









where a gives us the Volts/displacement ratio.


  • The following plot shows the fit of the QPD calibration curve with the error function




The fit with the function:






a_1=(0.252\pm 0.007)\hspace{0.2cm}V\\ b_1 = (1.961\pm 0.009)\hspace{0.2cm} V\\ c_1 =(1.236\pm0.007) \hspace{0.2cm}\tfrac{1}{mm}\\ d_1 =-5.94\pm 0.04

​a_1=0.2517 \hspace{0.2cm}V\\ b_1 = -1.961\hspace{0.2cm} V\\ c_1 =-1.236 \hspace{0.2cm}\tfrac{1}{mm}\\ d_1 =5.939​a_1=0.2517 \hspace{0.2cm}V\\ b_1 = -1.961\hspace{0.2cm} V\\ c_1 =-1.236 \hspace{0.2cm}\tfrac{1}{mm}\\ d_1 =5.939

The QPD's X output is a voltage given by:




where \alpha is a constant, while P_{right}-P_{left} is the difference between the power on the right and left side of the QPD. Thus:


\Delta{V}=\alpha P_{0}\cdot{erf\left(\frac{\sqrt{2}(x-x_{0})}{w_x} \right )}

where P_0 is the total power transmitted by the beam, x is the distance from the beam's center to the QPD center, x_{0} is an offset and w_x is the beam radius.


From the expression above and the fit's results we obtain:

w_x=1.14 \hspace{0.2 cm} mm\\ x_0=4.81 \hspace{0.2cm} mm


Y axis calibration


  • Linear region fit


The fit with the function:






p=(2.89 \pm 0.03)\hspace{0.2cm}V/mm

q=(-28.647\pm0.003)\hspace{0.2 cm} V


where p gives us the Volts/displacement ratio.


  • Error function fit

The fit with the function:






a_2=(0.041\pm 0.007)\hspace{0.2cm}V\\ b_2 = (2.005\pm 0.008)\hspace{0.2cm} V\\ c_2 =(1.311\pm 0.008)\hspace{0.2cm}\tfrac{1}{mm}\\ d_2 =-13.02\pm 0.08


The expression for the QPD's Y output is analogous to the one for the X axis:


\Delta{V}=\alpha P_{0}\cdot{erf\left(\frac{\sqrt{2}(y-y_{0})}{w_y} \right )}


From the fit results we obtain:


w_y=1.08 \hspace{0.2 cm} mm\\ y_0=9.93 \hspace{0.2cm} mm


  • Observation: Both w_x and w_y are compatible with the beam spot size we observed in laboratory.


The Volts/displacement ratios obtained here will be used to measure the resonant frequencies of the rhomboid motion.

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