EDIT (ZK): All the plots here were generated using my MATLAB cavity modeling tool, ArbCav. The utility description is below. The higher-order mode resonance plots are direct outputs of the function. The overlap plots were made by modifying the function to output a list of all HOM resonant frequencies, and then plotting the closest one as a function of cavity length. This was done for various values of highest mode order to consider, as described in the original entry.
Description:
This function calculates information about an arbitrary optical cavity. It can plot the cavity geometry, calculate the transmission/reflection spectrum, and generate the higher-order mode spectrum for the carrier and up to 2 sets of sidebands.
The code accepts any number of mirrors with any radius of curvature and transmission, and includes any astigmatic effects in its output.
As opposed to the previous version, which converted a limited number of cavity shapes into linear cavities before performing the calculation, this version explicitly propagates the gouy phase of the beam around each leg of the cavity, and is therefore truly able to handle an arbitrary geometry.
----------------Original Post----------------
I expressed concern that arbitrarily choosing some maximum HOM order above which not to consider makes us vulnerable to sitting directly on a slightly-higher-order mode. At first, I figured the best way around this is to apply an appropriate weighting function to the computed HOM frequency spacing. Since this will also have some arbitrariness to it, I have decided to do it in a more straightforward way. Namely, look at the spacing for different values of the maximum mode number, n_{max}, and then use this extra information to better select the length.
Assumptions:
- The curved mirror RoC is the design value of 2.50±0.025 m
- The ±9 MHz sidebands will have ~1% the power of the other fields at the dark port. Accordingly, as in Sam's note, their calculated spacing is artificially increased by 10 linewidths.
- The opening angle of 4º is FIXED, and the total length is scaled accordingly
Below are the spacing plots for the bowtie (flat-flat-curved-curved) and non-bowtie (flat-curved-flat-curved) configurations. Points on each line should be read out as "there are are no modes of order N or lower within [Y value] linewidths of the carrier TEM_{00} transmission", where N is the n_{max} appropriate for that trace. Intuitively, as more orders are included, the maxima go down, because more orders are added to the calculation.
*All calculations are done using my cavity simulation function, ArbCav. The mode spacing is calculated for each particular geometry by explicitly propagating the gouy phase through each leg of the cavity, rather than by finding an equivalent linear cavity*
Since achievable HOM rejection is only one of the criteria that should be used to choose between the two topologies, the idea is to pick one length solution for EACH topology. Basically, one maximum should be chosen for each plot, based on how how high an order we care about.
Bowtie
For the bowtie, the n_{max} = 20 maximum at L = 1.145 m is attractive, because there are no n < 20 modes within 5 linewidths, and no n < 25 modes within ~4.5 linewidths. However, this means that there are also n < 10 modes within 5 linewidths, while they could be pushed (BLUE line) to ~8.5 linewidths at the expense of proximity to n > 15 modes.
Therefore, it's probably best to pick something between the red and green maxima: 1.145 m < L < 1.152 m.
By manually inspecting the HOM spectrum for n_{max} = 20, it seems that L = 1.150 m is the best choice. In the HOM zoom plot below and the one to follow, the legend is as follows
- BLUE: Carrier
- GREEN: +9 MHz
- RED: -9 MHz
- CYAN: +45 MHz
- BLACK: -45 MHz
Non-bowtie
Following the same logic as above, the most obvious choice for the non-bowtie is somewhere between the red maximum at 1.241 m and the magenta maximum at 1.248 m. This still allows for reasonable suppression of the n < 10 modes without sacrificing the n < 15 mode suppression completely.
Upon inspection, I suggest L = 1.246 m
I reiterate that these calculations are taking into account modes of up to n ~ 20. If there is a reason we really only care about a lower order than this, then we can do better. Otherwise, this is a nice compromise between full low-order mode isolation and not sitting directly on slightly higher modes.
RoC dependence
One complication that arises is that all of these are highly dependent on the actual RoC of the mirrors. Unfortunately, even the quoted tolerance of ±1% makes a difference. Below is a rendering of the RED traces (n_{max} = 20) in the first two plots, but for R varying by ±2% (i.e., for R = 2.45 m, 2.50 m, 2.55 m).
The case for the non-bowtie only superficially seems better; the important spacing is the large one between the three highest peaks centered around 1.24 m.
Also unfortunately, this strong dependence is also true for the lowest-order modes. Below is the same two plots, but for the BLUE (n_{max} = 10) lines in the first plots.
Therefore, it is prudent not to pick a specific length until the precise RoC of the mirrors is measured.
Conclusion
Assuming the validity of looking at modes between 10 < n < 20, and that the curved mirror RoC is the design value of 2.50 m, the recommended lengths for each case are:
- Bowtie: L_{RT} = 1.150 m
- Non-bowtie: L_{RT} = 1.246 m
HOWEVER, variation within the design tolerance of the mirror RoC will change these numbers appreciably, and so the RoC should be measured before a length is firmly chosen. |