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where ms is EOM phase slope (15 mrad/V) and V_EOM is the applied actuation voltage. As you wrote, that leads to
![\delta \tilde f(f) = \frac{m_s}{2\pi} (-i 2 \pi f) \tilde V_{EOM}(f) = -i m_s f\tilde V_\textrm{EOM}(f) \quad [\textrm{Hz}/\textrm{V}]](https://latex.codecogs.com/gif.latex?%5Cdpi%7B100%7D%20%5Cdelta%20%5Ctilde%20f%28f%29%20%3D%20%5Cfrac%7Bm_s%7D%7B2%5Cpi%7D%20%28-i%202%20%5Cpi%20f%29%20%5Ctilde%20V_%7BEOM%7D%28f%29%20%3D%20-i%20m_s%20f%5Ctilde%20V_%5Ctextrm%7BEOM%7D%28f%29%20%5Cquad%20%5B%5Ctextrm%7BHz%7D/%5Ctextrm%7BV%7D%5D)
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This actually has units of Hz/Hz. Note, that  is just Fourier transform of frequency actuation, so it is unitless. I used this to get the transfer function of EOM actuation, from applied signal in V to resulting actuation in Hz, which is the prefactor of  above having units of Hz/V.
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If we assume that the EOM is taking 100% of the load for canceling laser frequency noise at a given frequency then it follows that the applied voltage to exactly cancel laser frequency noise is
![\delta \tilde V_\textrm{EOM}(f) = \frac{S^\textrm{Laser}_f (f)}{\delta \tilde f(f)} = i \frac{10^4}{m_s} \frac{1}{f^2}\quad [\textrm{V}/\sqrt{\textrm{Hz}}]](https://latex.codecogs.com/gif.latex?%5Cdpi%7B100%7D%20%5Cdelta%20%5Ctilde%20V_%5Ctextrm%7BEOM%7D%28f%29%20%3D%20%5Cfrac%7BS%5E%5Ctextrm%7BLaser%7D_f%20%28f%29%7D%7B%5Cdelta%20%5Ctilde%20f%28f%29%7D%20%3D%20i%20%5Cfrac%7B10%5E4%7D%7Bm_s%7D%20%5Cfrac%7B1%7D%7Bf%5E2%7D%5Cquad%20%5B%5Ctextrm%7BV%7D/%5Csqrt%7B%5Ctextrm%7BHz%7D%7D%5D)
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So, if EOM is taking the whole load of frequency noise, actuation signal ASD would be:
![\delta \tilde{V_{EOM}}(f) = \frac{TF_{EOMpath}}{-\iota m_s f} S_f^{laser}(f) \quad \left[ V/\sqrt{Hz} \right]](https://latex.codecogs.com/gif.latex?%5Cdelta%20%5Ctilde%7BV_%7BEOM%7D%7D%28f%29%20%3D%20%5Cfrac%7BTF_%7BEOMpath%7D%7D%7B-%5Ciota%20m_s%20f%7D%20S_f%5E%7Blaser%7D%28f%29%20%5Cquad%20%5Cleft%5B%20V/%5Csqrt%7BHz%7D%20%5Cright%5D)
But obviously, this is wrong because this just assume that we are not feeding back the actuation signal at all. So instead, I assumed that if everything is 'ideally' working and we actually have the frequency noise suppressed by a teamwork of PZT and EOM, the incoming noise signal to EOM path's transfer function is:
![\frac{1}{1+TF_{PZTpath}(f) + TF_{EOMpath}(f)} S_f^{laser}\quad \left[ Hz/\sqrt{Hz}\right]](https://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B1+TF_%7BPZTpath%7D%28f%29%20+%20TF_%7BEOMpath%7D%28f%29%7D%20S_f%5E%7Blaser%7D%5Cquad%20%5Cleft%5B%20Hz/%5Csqrt%7BHz%7D%5Cright%5D)
So, the actuation signal generated for EOM by EOM path's transfer function in an ideally working loop is:
![\delta\tilde{V_{EOM}}(f )=\frac{TF_{EOMpath}(f)}{-\iota m_sf}\frac{1}{1-TF_{PZTpath}(f) - TF_{EOMpath}(f)} S_f^{laser}\quad \left[ V/\sqrt{Hz}\right]](https://latex.codecogs.com/gif.latex?%5Cdelta%5Ctilde%7BV_%7BEOM%7D%7D%28f%20%29%3D%5Cfrac%7BTF_%7BEOMpath%7D%28f%29%7D%7B-%5Ciota%20m_sf%7D%5Cfrac%7B1%7D%7B1-TF_%7BPZTpath%7D%28f%29%20-%20TF_%7BEOMpath%7D%28f%29%7D%20S_f%5E%7Blaser%7D%5Cquad%20%5Cleft%5B%20V/%5Csqrt%7BHz%7D%5Cright%5D)
The error in the last plot in PSL:2305. is that I forgot to divide by  since transfer functions in the code are from Hz to Hz. So I think this is the real error.
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One thing to note is that the above estimates are an upper bound assuming that the EOM is taking all of the load down to that frequency point and that the PZT path isn't fighting or out of phase with EOM. To correctly compute the load on the EOM you are going to have to break down the EOM only portion of the loop from the laser frequency to the point of voltage injected into the EOM. This can be done by effectively nesting the PZT loop into the round trip gain in a way similar to that described in Josh Smith's Thesis section 2.6.2. Finding the actuation signal should be similar to finding the PLL actuation signal, at this point in the loop it is the G/(1-G)/A copy of the sensor noise. In the high gain regime the applied EOM control signal should just be the laser frequency divided by the EOM frequency slope. Of course you can compute for G_EOM OLG to get a true value with a bunch of algebra.
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But on reading section 2.6.2 of Josh Smith's Thesis (which btw has a typo in Eq. 2.33 and 2.34), I did the thing of nesting the PZT path with the plant. So as per Eq. 2.31:
is the transfer fucntion of Plant for the simplified loop with just EOM as the actuator. Then the actuation signal ASD would be (note  is free running laser frequency noise ASD):
which turns out to:
Which is exactly the same as above. But still my model has this error. I'll fix it and post it soon. For readers in the future, the last set of equations are the only correct equations to the best of my knowledge. |