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 Wed Jun 9 11:46:01 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF measurement Thu Jun 10 14:01:36 2021, Anchal, Summary, AUX, Xend Green Laser PDH OLTF measurement loop algebra Mon Jun 14 18:57:49 2021, Anchal, Update, AUX, Xend is unbearably hot. Green laser is loosing lock in 10's of seconds Tue Jun 15 15:26:43 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF measurement loop algebra, excitation at control point Fri Jun 18 10:07:23 2021, Anchal, Paco, Summary, AUX, Xend Green Laser PDH OLTF with coherence
Message ID: 16197     Entry time: Thu Jun 10 14:01:36 2021     In reply to: 16194     Reply to this: 16200   16202
 Author: Anchal Type: Summary Category: AUX Subject: Xend Green Laser PDH OLTF measurement loop algebra

Attachment 1 shows the closed loop of Xend Green laser Arm PDH lock loop. Free running laser noise gets injected at laser head after the PZT actuation as $\eta$. The PDH error signal at output of miser is fed to a gain 1 SR560 used as summing junction here. Used in 'A-B mode', the B port is used for sending in excitation $\nu_e e^{st}$ where $s = i\omega$.

We have access to three ports for measurement, marked $\alpha$ at output of mixer, $\beta$ at output of SR560, and $\gamma$ at PZT out monitor port in uPDH box. From loop algebra, we get following:

$\large \left[ (\alpha - \nu_e) K(s)A(s) + \eta \right ]C(s)D(s) = \alpha$

$\large \Rightarrow (\alpha - \nu_e) G_{OL}(s) + \eta C(s)D(s) = \alpha$, where $\large G_{OL}(s) = C(s) D(s) K(s) A(s)$ is the open loop transfer function of the loop.

$\large \Rightarrow \alpha = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{G_{OL}(s)}{1 - G_{OL}(s)}$

$\large \Rightarrow \beta = \eta \frac{C(s) D(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{1}{1 - G_{OL}(s)}$

$\large \Rightarrow \gamma = \eta \frac{1}{K(s)} \frac{G_{OL}(s)}{1 - G_{OL}(s)} \quad -\quad \nu_e\frac{K(s)}{1 - G_{OL}(s)}$

So measurement of $\large G_{OL}(s)$ can be done in following two ways (not a complete set):

1. $\large G_{OL}(s) \approx \frac{\alpha}{\beta} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{1 - \frac{\eta C(s)D(s)}{\nu_e}}$, if excitation amplitude is large enough such that $\large \frac{\eta C(s)D(s)}{\nu_e} \ll 1$over all frequencies.
• In this method however, note that SR785 would be taking ratio of unsuppresed excitation at $\large \alpha$ with suppressed excitation at $\large \beta.$
• If the closed loop gain (suppression) $\large 1/(1 - G_{OL}(s))$is too much, the excitation signal might drop below noise floor of SR785 while measuring $\large \beta$.
• This would then appear as a flat response in the transfer function.
• This happened with us when we tried to measure this transfer function using this method. Below few hundered Hz, the measurement will become flat at around 40 dB.
• Increasing the excitation amplitude where suppression is large should ideally work. We even tried to use Auto level reference option in SR785.
• But the PDH loop gets unlocked as soon as we put exciation above 35 mV at this point in this loop.
2. $\large \frac{G_{OL}(s)}{K(s)} \approx \frac{\alpha}{\gamma} = \frac{G_{OL}(s) - \frac{\eta C(s)D(s)}{\nu_e}}{K(s)\left(1 - \frac{\eta C(s)D(s)}{\nu_e}\right )}$, if excitation amplitude is large enough such that $\large \frac{\eta C(s)D(s)}{\nu_e} \ll 1$over all frequencies.
• In this method, channel 1 (denominator) on SR785 would remain high in amplitude throughout the measurement avoiding the above issue of suppression below noise floor.
• We can easily measure the feedback transfer funciton $\large K(s)$ with the loop open. Then multiplying the two measurements should give us estimate of open loop transfer function.
• This is waht we did in 16194. But we still could not increase the excitation amplitude beyond 35 mV at injection point and got a noisy measurement.
• We checked yesterday coherence of excitation signal with the three measurment points $\large \alpha, \beta, \gamma$ and it was 1 throughout the frequency region of measurement for excitation amplitudes above 20 mV.
• So as of now, we are not sure why our signal to noise was so poor in lower frequency measurement.
 Attachment 1: AUX_PDH_LOOP.pdf  367 kB  Uploaded Thu Jun 10 15:01:43 2021
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