__Summary:__
We need to determine the geometry (= round-trip length and RoC of curved mirrors) of the OMC cavities for the 40m BHD experiment. Sticking to the aLIGO design of a 4 mirror bowite cavity with 2 flat mirrors and 2 curved mirrors, with a ~4deg angle of incidence, we need to modify the parameters for the 40m slightly on account of **our different modulation frequencies**. I've setup some infrastructure to do this **analytically** - even if we end up doing this with Finesse, it is useful to have an analytic calculation to validate against (also not sure if Finesse can calculate HOMs up to order 20 in a reasonable time, I've only seen maxtem 8).
**Attachment #1:** Heatmap of the OMC transmission for the following fields:
- Carrier TEM00 is
**excluded**, but HOMs up to m+n=20 included for both the horizontal and vertical modes of the cavity.
- f1 and f2
*upper and lower* sidebands, up to m+n=20 HOMs for both the horizontal and vertical modes of the cavity, **including*** *TEM00.
- Power law decay assumed for the HoM content incident on the OMC - this will need to be refined
- The white region is where the cavity isn't geometrically stable.
- Green dashed line indicates a possible operating point, white dashed line indicates the aLIGO OMC operating point.
**On the basis of this modeling, we would benefit from choosing a better operating point than the aLIGO OMC geometric parameters.**
**Algorithm:**
- Compute the round-trip Gouy phase, , for the cavity.
- With the carrier TEM00 mode resonant, compute the round-trip propagation phase, , and the round-trip Gouy phase, for the mode of the field, with specifying the offset from the carrier frequency (positive for the upper sideband, negative for the lower sideband).
*For the aLIGO cavity geometry, the 40m modulation sidebands acquire ~20% more propagation phase than the aLIGO modulation sidebands*.
- Compute the OMC transmission for this round-trip phase (propagation + Gouy).
- Multiply the incident mode power (depending on the power law model assumed) by the cavity transmission.
- Sum all the fields.
__Next steps:__
*Refine the incident mode content (and power) assumption*. Right now, I have not accounted for the fact that the f2 sideband is resonant inside the SRC while the f1 sideband is not. Can we somehow measure this for the 40m? I don't see an easy way as it's probably power dependent?
- Make plots for the projection along the slices indicated by the dashed lines - which HOMs are close to resonating? Might give us some insight.
- What is the requriement on transmitted power w.r.t. shot noise? i.e. the colorbar needs to be translated to dBVac.
- If we were being really fancy, we could simultaneously also optimize for the cavity finesse and angle of incidence as well.
**Question for Koji:** how is the aLIGO OMC angle of incidence of ~4 degrees chosen? Presumably we want it to be as small as possible to minimize astigmatism, and also, we want the geometric layout on the OMC breadboard to be easy to work with, but was there a quantitative metric? Koji points out that the backscatter is also expected to get worse with smaller angles of incidence.
The code used for the ABCD matrix calcs have been uploaded to the BHD modeling GIT (but not the one for making this plot, yet, I need to clean it up a bit). Some design considerations have also been added to our laundry list on the 40m wiki. |