ABCD matrix formalism for concave mirror

AI Thread Summary
The discussion focuses on calculating the ABCD matrix for a thick concave mirror in a Fabry Perot cavity setup, where light enters from the plane side. The proposed approach involves treating the mirror as a thick slab with a specific index of refraction and thickness, combined with a concave lens representation. Participants debate the relevance of the mirror's properties, particularly the impact of its coating on light behavior and how it affects beam parameters like radius and waist. Clarifications are made regarding the importance of the mirror in mode matching the laser beam as it transitions from outside to inside the cavity. The conversation highlights the complexities of using the ABCD matrix formalism in this context, especially concerning the effects of coatings and reflections.
kelly0303
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Hello! I need to calculate the ABCD matrix for a thick concave mirror, in the situation in which the light comes from the plane side of the mirror, and it is the concave part that is coated (for reference, I have a Fabry Perot cavity with 2 concave mirrors, and I want to mode match the laser coming from outside). The mirror has a radius of curvature R, a thickness d and an index of refraction n. The way I was thinking to do it, was to treat the mirror as a thick slab of index of refraction n and thickness d, which has the ABCD matrix:
$$
\begin{pmatrix}
1 & d/n\\
0 & 1
\end{pmatrix}
$$
plus a concave lens of focal length ##-R/2##, with R here being positive, so f being negative (I assumed it is concave, as the light coming from the direction I need here would diverge upon reflection) which has the ABCD matrix:
$$
\begin{pmatrix}
1 & 0\\
2/R & 1
\end{pmatrix}
$$
Overall, the total ABCD matrix would be:
$$
\begin{pmatrix}
1 & 0\\
2/R & 1
\end{pmatrix} \times
\begin{pmatrix}
1 & d/n\\
0 & 1
\end{pmatrix} =
\begin{pmatrix}
1 & d/n\\
2/R & 1+\frac{2d}{nR}
\end{pmatrix}
$$
Is this right? Can someone tell me how to fix it, in case it is wrong? Thank you!
 
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DaveE said:
Just to clarify your question. Since you are only interested in the light that enters the cavity, the mirror is irrelevant, you only care about the light that gets through it. Is that correct?

If so isn't this just a thick lens model?

This guys web page looks useful, in either case:
https://www.sjsu.edu/faculty/beyersdorf/Archive/Phys158F06/10-12 Thick Lenses and the ABCD formalism.pdf
I am not sure what you mean. The laser light (assuming a gaussian beam profile) has a given radius and waist outside the cavity, and a different one inside the cavity. In order to get the full power amplification inside the cavity, I need to match the radius and waist of the laser after it passes through the thick mirror (from outside to inside the cavity) with the radius and waist of the beam that is developing inside the cavity. The mirror does changes these parameters of the laser beam from outside to inside. Why would it be irrelevant?
 
kelly0303 said:
Why would it be irrelevant?
I just meant the partially reflective coating.
 
DaveE said:
I just meant the partially reflective coating.
I am actually not sure how coating works. The mirror is coated on the concave part, and it has a reflectivity R. Is the reflectivity on the other side on the curvature also R or that is acting just like a piece of glass, without any reflection?
 
kelly0303 said:
I am actually not sure how coating works. The mirror is coated on the concave part, and it has a reflectivity R. Is the reflectivity on the other side on the curvature also R or that is acting just like a piece of glass, without any reflection?
I wouldn't think it matters from a ray tracing perspective. Either the ray gets through, or it doesn't. Then you want to do the matching for the ones that get through.

Even if the coating is sensitive to the angle of incidence (like a Bragg reflector), it will change the probability distribution (or intensity) for each ray, but not the position or angle. So I don't think a simple ABCD matrix is the right tool for those problems. However, this is way beyond my knowledge of optics.
 
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