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In the context of a physics experiment preparation I am trying to figure out if I can calculate theoretically the distance in red on the below schema in order to determine the waist of a laser beam.

enter image description here

This is called an Optical Parametric Oscillator (for context) in a ring configuration. It is composed of 2 curved mirrors of known radius $R$, 2 flat mirrors, and a non linear crystal represented by the stripped rectangle, of length $d$ (in blue). I know the total length of the cavity to be $L$.

With only these values at my disposal is it possible to find the length in red? I'm more than rusty on Thales theorem or simple trigonometry as a M2 physics student in quantum mechanics.

See answers for more details

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    $\begingroup$ if this is a setup for a physics experiment, you should know these distances. It is not possible to calculate them with only the data you gave. $\endgroup$
    – paulina
    Commented May 16 at 14:45
  • $\begingroup$ @paulina This is actually part of my work to determine these distances but I might had a chance to get some with this problem. Good to know thanks ! $\endgroup$
    – Chris Ze Third
    Commented May 16 at 15:41
  • $\begingroup$ Following the remark by @paulina, I don't think you will get answers as long as we haven't enough information : beside name L (for the horizontal upper line ?) please give names a,b,c etc. for the different distances ; for example, the distance between the 2 horizontal line segments deserves to have a name. $\endgroup$
    – Jean Marie
    Commented May 16 at 22:39
  • $\begingroup$ @JeanMarie I did not mention other distances because I do not know them. My question was really "is it possible with such limited information to use a trick to calculate this distance in red", if several are telling me it is not unless I have more information, I'll take their word for it. The articles I'm reviewing do not provide any more information than the total distance L, the radii of curvature and the crystal length. $\endgroup$
    – Chris Ze Third
    Commented May 17 at 6:55
  • $\begingroup$ You do not know them, all right, but this doesn't prevent to give names $a,b,c,d...$ to them in order to have something on which you can do algebraic calculations. $\endgroup$
    – Jean Marie
    Commented May 17 at 9:00

1 Answer 1

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So, you need at least one length: either $d_c$ the distance between the two concave mirrors, or $d_f$ the one between the flat mirrors. Let's fix $d_c$ in this case. I re-made the schema to be clearer (the green-blue rectangle is the non-linear crystal):

enter image description here

I first had to make the assumption that the ray coming from $O$ arrives on the edges of the crystal such that its image forms there as presented on the next figure: enter image description here

Then I use relationships on cosine to find the angle $\theta$, which then helps me identify the remaining distances (diagonals and $d_F$). But you also need to set the total length of the round-trip $L = d_c + d_F + 2 \times F_2 C_1$. If you have at least one length it's possible to find the remaining ones. Otherwise there seems to be no way around.

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    $\begingroup$ @amWhy this answer was posted by the same person who asked the question. $\endgroup$
    – paulina
    Commented May 17 at 23:01
  • $\begingroup$ @paulina I think I missed an episode $\endgroup$
    – Chris Ze Third
    Commented May 18 at 8:31

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