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!).

  • $\begingroup$ 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 at 14:06

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