# Radiation Pressure Noise in Gravitational Wave Detection

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!).

• 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). – wcc Apr 14 at 14:06