Nick Hutzler and Eric Hudson proposed that conversion of 775 nm light into 1550 nm photons in our last output filter cavity would pose a problem on our SNSPD since we are shining 2*10^14 photons/second within the cavity and trying to keep this noise source to ~10^-4 Hz to be safe, 18 orders of magnitude of isolation. I propose the following rough model for the number of photons incident on the SNSPD from conversion of 775 nm light to 1550 nm photons.

\[ \dot{N}_{\text{formerly 775nm light on SNSPD}} = \int_0^\infty d\lambda \frac{\lambda}{hc} F(\lambda) P_{\text{775 nm light}} M \big[\text{Absorb}_{\text{HR Coating}} \cdot C(\lambda; \text{HR coating} ; \text{775 nm}) +  \text{Trans}_{\text{HR Coating}} \cdot \text{Absorb}_{\text{Mirror Substrate}} \cdot C(\lambda; \text{Mirror Substrate} ; \text{775 nm}) \big]  \]

Here, \lambda is wavelength, h is Planck's constant, c is the speed of light, F(\lambda) is the filter in wavelength in front of the SNSPD, P is intercavity power, M is the mode matching between the converted light and the fiber to the SNSPD, Absorb is the fractional power absorbed, Trans is the fractional power transmitted, and C is the fractional power distribution over wavelength of emitted light from a given power absorbed. Its integral over wavelength is unitless and should integrate to less than or equal to 1 by energy conservation.

We can assign some numbers:

\lambda = 1550 nm; P = 5 mW, Absorb_{HR Coating} = 10^-5, Trans_{HR Coating} = 10^-6, Absorb_{Mirror Substrate} = 10^-3. For florescence, we can roughly model M as a fraction of the area of the fiberoptic cable over the surface of the sphere the fiber's distance from the mirror

\[ M \approx \frac{\pi~(100~\mu\text{m})^2}{4\pi~(0.2~\text{m})^2} \]

M is more complicated than this and maybe much more complicated for 2nd order Raman scattering, another process. 

We can assume F(\lambda) is a bandpass filter around 1550 nm with a width of 3 nm and assume C is constant in this range. Given the very small fraction of power transmitted by the HR coating and absorbed by the substrate and that half of the HR coating is also made of SiO2, we can ignore the substrate's contribution. We then get that 

\[ \dot{N}_{\text{formerly 775nm light on SNSPD}} = 24 kHz \cdot 3~\text{nm} \cdot C(1550~\text{nm}; \text{HR coating} ; \text{775 nm}) \]

To meet our 10^-4 Hz "requirement", less than 4 parts in 10^9 of the absorbed power on the HR coating can be emmited in this 3 nm range about 1550 nm. This feels a bit marginal.

Raman scattering involves the exchange of energy between the photon and the material, so I imagine light is preferentially scattered away from its incident angle. This would help us if the 775 nm and 1550 nm intercavity beams are co-circulating in the same direction. If the beams were circulating in opposite directions, I am not sure of the effects.

In ACME (see figures 4.2, 4.6, and appendix C), a 1 W excitation laser at 1090 nm was used and around 15 * 10^4 photons/s were observed in a 10 nm wide band around 690 nm. This photon level was not attenuated by adding additional filters, making it seem like this light was indeed around 690 nm. Factoring in a 10% light collection and 10% quantum efficiency of the ACME photodetector, Nick estimates 10^7 photons/s are generated in this band by the 1090 nm laser. In Nick's thesis (section A.2.2), they assume 4% of the light is assumed by the ITO coating.

Making the big assumption that the density of photons converted into different wavelengths is the same from this process as ours (even though we are looking at a wavelength increase), this would yield C to be 10^-11 /nm. This would make this noise source subdominant, giving an photon rate of 10^-7 Hz. Nevertheless, there are a lot of assumptions that go into this estimate.