[Umran]

The goal is to mode match the two cavities in the last picture, changing the setup so that the curved mirror is M3 (Roc=-1.6) and the light from the first cavity is coming out of M4 and going into M5 of the second cavity. The solutions are summarized below starting from no lenses and then including potential solutions with 1 lens.

The beam we are considering is 1550 nm. The beam is adjusted to be resonant with the cavity using Finesse, which corresponds to a waist of 584.03 um at M1. The beam propagation is summarized in the first attached picture. The solution with no Lens is to match the distance between the two cavities to the distance between M3 and M4. After coming out of M3, the beam has a waist upon propagating a distance equal to twice the distance from M3 and M4, which is 0.589 m after exiting out of M4. Thus, the inter-cavity space adjusted to this produces a waist at M5 with size 584.03 nm the same as what we observe for the beam entering the 1st cavity. The profile for the second cavity with the inter-cavity distance set to 0.589 m is given in the 2nd picture.

There are 2 different lens solutions with a minimum 99% mode overlap on JamMt. The first one of these solutions is the 2000 mm lens (All the lenses were taken from this website: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=3281&pn=LA1951-C). For this solution, the best among 11 potential configurations was placing the lens at 0.596 m. The resulting beam has a waist of 523.088 um at 0.997 m. When these parameters were directly input into finesse, I did not get the same resulting beam waist or position. Thus, I adjusted the distance where I put the lens and the total inter cavity length to get the best possible mode matching (minimizing mismatch and making the beam width as close to 584.03 um as possible at M5). The resulting configuration had the 2000 mm lens at 0.7246 m and M5 at 0.846 m after M4. This ensured that the beam at M5 was at the waist and this waist was 584.04 um. This is shown in the 3rd and 4th picture.

The second solution is the 2500 mm lens. In JamMt, the best out of 49 solutions given is obtained by putting the 2500 mm lens at a distance of 0.755 from M4. The resulting beam has a waist of 574.377 um at 0.998 m. Inputting this exactly into Finesse does not give us the best result with the beam waist and position mismatched between the two cavities. However, tweaking this to put the lens at 0.6988 m and M5 at a distance of 0.795 m after M4 achieves the desired beam waist of 584.04 um (only 0.01 um off) at M5. This also keeps the mismatches below 1%. I am not entirely sure how these mismatches are calculated, so I will need to read more about this. This is shown in the 5th and 6th picture.

Since there seems to be a discrepancy between JamMt and Finesse in terms of distances, there might be more solutions. I will see if I can lower the mode matching percentage on JamMt and get different lens solutions and then input them into finesse to finetune an initially bad mode matching into a better one.