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Message ID: 2512     Entry time: Thu Nov 5 11:20:45 2020
 Author: anchal Type: Summary Category: ECDL Subject: True PDH Error signal TF and including FSR effects in approximated models

If we use ECDL for auxiliary frequency in 40m and hope to stabilize it up to 1 MHz with digital compensation of PZT, it is important to take into account any phase effect of the nearby FSR at 3.97 MHz. This should ideally be included in the Input Mode Cleaner loop considerations as well. These effects would be more prominent in longer cavities like aLIGO and LISA where FSR is very low and should we attempt to stabilize a laser lock beyond cavity's FSR.

I did a no assumptions calculation for getting a general transfer function fo PDH error signal in units of [W/Hz] assuming 1 W of incident power. This calculation would soon be uploaded here. I'll put down here primary results.

For incident field on a Fabry-Perot cavity (with fsr of $\nu_{fsr}$), reflected electric field transfer function (unitless) is given by:

$\dpi{300} \tiny R(\omega) = \frac{-r_1 + (r_2^2 + t_2^2) r_2 e^{-i \omega/\nu_{FSR}}} {1 - r_1 r_2 e^{-i \omega/\nu_{FSR}}}$

Then, PDH error signal for a modulation frequency of $\Omega$ at a modulation index of $\Gamma$, in units of [W/Hz] (i.e. error signal power per Hz of error in laser frequency from cavity resonance) is given by:

$\dpi{300} \tiny H_{\nu 2P}(\omega) = -\frac{i \pi P_0 J_0(\Gamma)J_1(\Gamma)}{\omega} \Bigl(R(\Omega)R(\omega) - R(0)R(\Omega + \omega) + R^*(\Omega)R(\omega) - R(0)R^*(\Omega - \omega)\Bigr)$

after demodulation and low pass filtering. Note this transfer function is a complex quantity as it carries phase information of the transfer function too. The real signal is obtained by multiplying this signal at $\omega$ with $e^{i \omega t}$ and taking the real value of the product.

Having done this, we can see how in the real PDH error signal, there is a low pass at cavity pole, given by $- \frac{\nu_{fsr}}{2\pi}log(r_1 r_2)$ and a notch every fsr. The notch creates a zig-zag in the phase of the tranfer function and has a HWHM same as cavity pole. After this point, I just fitted a ZPK model to the transfer function to obtain a empirically derived model for PDH error signal transfer function. Apart from the cavity pole, this model needs to have resonance and antiresonance features present at each FSR with resonance having a linewidth of cavity pole while anti-resonance having a linewdth of $\pi/\nu_{fsr}$. Here's how the ZPK model would look like:

\dpi{200} \begin{aligned} z:&\, \pi/\nu_{fsr} \pm n \nu_{fsr} j \\ p:&\, f_p \pm n \nu_{fsr} j, f_p \end{aligned}

I've attached my notebook where I did the fitting analysis and the overlap plot of real PDH error signal TF and the modelled approximation.

 Attachment 1: PDHErrorTFModel.pdf  31 kB
 Attachment 2: PDHTFforACavity.ipynb.zip  3 kB
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