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As part of my master thesis, my task is to find the optimal parameters to set-up a bow-tie optical parametric oscillator (OPO) for squeezed states generation. I'm currently looking at the possible parameters I can evaluate and I have two very good references on the topic: the first one is the chapter 2.5 of "Fundamentals of Photonics / Quantum Electronics" by Franz X. Kaertner (2006) about ABCD matrices, and the second is exactly what I am trying to look at, "Miniaturization of an Optical Parametric Oscillator with a Bow-Tie Configuration for Broadening a Spectrum of Squeezed Light" by Genta Masada, who gives all the formulas I need for my work.

ABCD matrix is a concept very new to me, and if its construction is not so difficult, I can't see how G. Masada retrieves the expression of waist at the center of the non-linear crystal and the intermediate between the two flat mirrors, respectively notes $w_1$ and $w_2$ as a function of the matrix elements A, B, C and D. The lectures notes by F. X. Kaertner only gives the waist (or "the variations of the beam spot size") as: $$ w(z) = w_0 \left[1 + \left( \frac{z}{z_R}\right)^2\right]^{1/2}$$

with $w_0$ related to the Rayleigh length by $z_R = \pi \frac{w_0^2}{\lambda}$. Kaertner never uses ABCD matrix to express the waist and Masada does not provide calculation details to connect the two (the latter only gives $w_1$ which corresponds to the $w_0$ of Kaertner):

$$ w_2 = w_1 \left[A^2 + (\frac{B}{z_R})^2\right]^{1/2} $$

Similarly, Masada manages to connect the Rayleigh length to the ABCD matrix, but how?

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  • $\begingroup$ The RP-Photonics Encyclopedia is a good place to start for this kind of thing. For example, see ABCD Matrix and Resonator Design. If you are buying components, talking to engineers is also good. See the Buyer's Guide. For example, Where to Buy Optical Resonators $\endgroup$
    – mmesser314
    Commented May 24 at 14:11
  • $\begingroup$ @mmesser314 I'm really only interested in finding adequate values to match a cavity bandwidth of 6 to 10 MHz (it corresponds to the D2 transition line of Rubidium atom). I already know about many aspects of cavities and non linear crystal (I wrote a 20 pages document about for my supervisors). So it's really a matter of plugging some values to find optimal waist, correct numbers to obtain the right bandwidth and so on. But I can't verify the formulas I gave because I don't understand how to connect ABCD matrix and waist. $\endgroup$
    – Chris Ze Third
    Commented May 24 at 20:33
  • $\begingroup$ If I may, a lot of what you wrote in terms of your education (ie providing the adequate bibliography) comes from your supervisor advising you. 1- You identify your problem (as in above) and communicate it with him/her. He/she should then give you appropriate bibliography for you to figure out this problem. Here some bibliography ABCD formalism, Matrices, waist in astigmatic cavities. $\endgroup$
    – José Andrade
    Commented May 25 at 13:30
  • $\begingroup$ Do not forget that such a cavity is an astigmatic cavity and you need to do ABCD analysis for the tangential and sagittal axes independently. $\endgroup$
    – José Andrade
    Commented May 25 at 13:32
  • $\begingroup$ @JoséAndrade It's not something they really looked into so I'm the one finding ressources. I just need to understand how you manage to express the Rayleigh length with respect the the ABCD matrix coefficient, all I find is connect the q-parameter to the waist but it's the snake biting its own tail $\endgroup$
    – Chris Ze Third
    Commented May 27 at 12:42

1 Answer 1

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The formulae given in the reference I cited in my question are not correct, or at the very least inaccurate. In chapter 5 of Svelto's "Principles of Lasers" (fifth edition) (section 5.5) , I recover indeed the q-parameter as a function of a quotient of the A, B, C, D coefficients. However this quotient is not exactly the same depending on the mirror of reference. In front of the first mirror:

$$ q_1 = j \sqrt{-\frac{BD}{AC}} $$

With A,B,C,D the matrix coefficients of a single pass. The cavity is indeed symmetrical so one only considers a single pass matrix and the opposite direction matrix to simplify the calculations. In front of mirror 2 we then have:

$$ q_2 =j \sqrt{-\frac{AB}{CD}} $$

The waist is then simply expressed using the fact $q_1$ and $q_2$ are pure imaginary numbers and the general formula $q = j\frac{\pi w^2}{\lambda}$.

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