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I am trying to work out where equation 9 comes from in Martynov et al. (2016), who discuss the radiation pressure noise in the LIGO detector. $$L(f) = \frac{2}{cM\pi^2 f^2} \left(h \nu G_{-} P_{\rm arm}\right)^{1/2} K_{-}(f) ,$$ where $L(f)$ is the frequency dependent displacement noise in the arm length due to radiation pressure fluctuations, $f$ is the GW frequency, $P_{\rm arm}$ is the power in one of the Fabry Perot arm cavities, $\nu$ is the laser frequency and $K_{-}$ is the frequency-dependent amplitude gain factor for the cavity.

I tried to reproduce this by assuming there were $K_{-}^2 P_{\rm arm}/h\nu$ photons per second in the cavity, each with momentum $h\nu/c$. These exert a fluctuating force on the mirrors of $(K_{-}^2 P_{\rm arm}/h\nu)^{1/2} \times 2h\nu/c$ (assuming perfectly reflective mirrors). Then assuming the forces on the two Fabry Perot mirrors are in opposite directions and that the fluctuation is sinusoidal, I arrive at
$$L(f) = \frac{1}{cM\pi^2 f^2} \left(h \nu P_{\rm arm}\right)^{1/2} K_{-}(f) .$$

So I have two problems. One is where does the additional factor 2 come from in Martynov's equation? More importantly, where does $G_{-}$ come into this calculation? Martynov et al. describes this as the "Differential coupled cavity build up" and it has a value of 31.4. It is not defined, is only used in this equation and doesn't appear anywhere else in the paper (or as far as I can see, the rest of the internet!).

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    $\begingroup$ This question got me interested in looking at the paper you linked, but I too have trouble following their terminologies. I assume that the term "coupled cavity" refers to the fact that their Fabry-Perot arms are coupled with power recycling and signal recycling mirrors, and "differential" refers to the differential movement in the Fabry-Perot arms (to which the GW signal is sensitive). $\endgroup$
    – wcc
    Apr 14 '19 at 14:06
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TL;DR: The factor of 2 comes from the fact that there are 2 Fabry-Pérot cavities; where you used $K_-^2 P_{\mathrm{arm}}/h\nu$, you should have used $G_- P_{\mathrm{arm}}/h\nu$ because that's the number of photons hitting the mirrors; and the factor of $K_-$ is a transfer function between coupled oscillators (the fluctuating pressure noise and the fluctuating antisymmetric output).

You're right that this was not very well explained at all. In fairness to the Martynov et al. paper, its purpose was just a status report. But more broadly, it is very difficult to track down any good explanation of how LIGO works. The full calculation that's usually cited was done by Buonanno and Chen in a full quantum framework. It's pretty complicated, with a view towards modifications that are only now being incorporated into LIGO (e.g., squeezing), so it's not very intuitive for the simpler case that's relevant to the particular equation you cite. The closest thing I've found to an explanation of how Advanced LIGO works is in this paper, which has lots of details that might be enlightening. The following expresses the idea, but is a bit hand-wavy. (And I'm not at all a detector person; I just happened to be surrounded by postdocs including Buonanno and Chen during grad school, so I got a lot of exposure.)

Remember that we only directly care about the differential channel — the difference in lengths of the two Fabry-Pérot cavities. But the signal-recycling mirror couples them together in an antisymmetric way, so that the differential signal bounces around a few times before leaking out. Let's say we look at the behavior over some period of time $\Delta t \sim 1/f$. And suppose we have $N$ photons circulating in the combined signal-recycling and arm-cavity system, with shot noise giving us uncertainty in that number by about $\sqrt{N}$. But those same photons will typically bounce around quite a few times inside the system before leaving to hit the photodetector. So to get the number of times the mirrors are hit by photons in a beam of power $P_{\mathrm{arm}}$, you have to multiply by the number of times those photons bounce around — which is precisely what this $G_-$ factor is meant to represent. ($G$ is meant to evoke "gain".) So if you're looking for the shot noise in the number of photons hitting these mirrors, you need the number of photons times the number of times each photon hits: $G_- P_{\mathrm{arm}} / h\nu$. Since the shot noise in the number of hits is the square root of the number of hits, we get the factor $(G_- P_{\mathrm{arm}} / h\nu)^{1/2}$

The $K_-$ factor is something different. Remember that we're decomposing the signal into frequencies, and just looking at a particular $f$. It might be easier to imagine there's no such thing as shot noise, but you're pumping in fluctuations of the field at that frequency. Because the field is bouncing around so much while gradually leaking out to the photodetector, the number of photons leaking out doesn't instantly respond to your input fluctuations; that number wants to change at its own pace. So what you have is a set of coupled oscillators, driving one while measuring the response of another. You get a transfer function with poles near the natural period of the cavities. If you were to drive the fluctuations at a frequency that the cavities "liked" responding to, you would get large output at the same frequency. So if you want to know how the output field responds to the input — which is actually just natural fluctuations due to shot noise decomposed into frequencies — then you need to multiply by $K_-$.

Updating your derivation with these corrections gives the same result as Martynov et al.'s equation 9.

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  • $\begingroup$ Thansks for helping to enlighten me. I have to teach this stuff to undergraduates! So what gain is $G_$ ? It's not the power recycling gain or the main FP cavity gain, so is it just the gain attributable to the signal recycling FP? It isn't the number of times a photon bounces around in the system since this is very much larger than 31 (it's 270 in the main arms alone). The $K_$ factor I had just assumed was (when squared) just the factor you multiply $P_{arm}$ by to account for the fact that main arm FP response drops off at high frequencies? $\endgroup$
    – ProfRob
    Mar 9 '20 at 22:32
  • $\begingroup$ That $G_{\mathrm{arm}}=270$ number is already factored into the $P_{\mathrm{arm}}$ — being the increase in circulating power within the arm cavities over the power entering them. Specifically, the input power is multiplied by $G_{\mathrm{prc}}=38$ in the power-recycling cavity, divided by two when splitting into the two arm cavities, and only then multiplied by $G_{\mathrm{arm}}$ to give the total $P_{\mathrm{arm}}$. $\endgroup$
    – Mike
    Mar 12 '20 at 15:14
  • $\begingroup$ So you have some random excess/deficit of photons in each arm. We only care about the difference in the amount of light hitting the beamsplitter, so it's going toward the signal-recycling mirror. I believe that asymmetric part is modeled as light at the fundamental lase frequency, and is mostly anti-resonant in the signal-recycling cavity, and so is then reflected back into the interferometer for multiple trips, with $G_-$ the "gain" of this part of the process. But since gain conventionally describes power, and you're taking the square-root, you include $G_-$ inside that square-root. $\endgroup$
    – Mike
    Mar 12 '20 at 15:16
  • $\begingroup$ As for $K_-$, it's definitely the transfer function between the differential carrier field and the "combined signal recycling and arm cavity system", because you're relating the light pressure to the induced motion. IIRC, you get these results by folding the two main FP cavities onto each other, putting the SR mirror at the input, and treating the input test mass and the SR mirror as a compound mirror. You can calculate its reflectivity and transmissivity, then treat it as an FP with the compound mirror at one end. $\endgroup$
    – Mike
    Mar 12 '20 at 15:23

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