Little help with optics - [ABCD matrices and Gaussian beams]?

[u/Marvellover13]

I'm practicing for an upcoming exam and needed some help with this question I had yesterday:

In the question, we have a Gaussian beam with it's waist at what we would consider or axis origin, at a distance 'a' to the right of it lies the flat side of a lens with refracting index 'n' which continues for distance d, and then there's a curved side of the lens with curvature |R| which is convex.

We were asked 3 things:

1. Find the ABCD matrix of the system for a light ray starting at the waist of the beam that travels right, and then, after passing the lens, continues a distance Z.

2. Find the radius and size of the spot immediately after the lens.

3. What's the divergence angle of the beam right after the lens?

Q1 was easy enough; it was just multiplying the matrices and noticing the curvature needs to be negative (-R).

Q2 was a little more involved since it felt weird finding q_2(z), w(z), R(z) where we set Z=0 (notice this is the capital Z and not the real z, which is 0 at the beam waist - it's just a poor choice of parameters from the side of the test writer IMO), it's hard to write it all here as i used LaTex to write and we can't upload pics here but i got the standard w(z) form where instead of z there was the distance a+d/n, R(z) was more algebrically involved and not as clean.

But in Q3, I have absolutely no idea how to find the divergence angle, as it's supposed to be atan[w(z_R)/z], but here our z is 0, which breaks this calculation even at the limit since w(z_R) is finite.

I only need some help with the last question, which I don't understand how to solve.

[u/UVlight1]

It’s been a little while, and the question seems worded a little odd to me. But the way I would look at it would be 1) you start out with a Gaussian beam. You know the characteristics. 2) the matrix operations give you a new beam. That beam is described including the divergence. The first beam you defined a coordinate where z was at the beam waist. The new beam after going through the beam, can be thought of as a new Gaussian beam. Its beam waist location is not the same as where the first beams waist was. So if you want to know the divergence of the new beam, just think of it as a beam and give the divergence. If you want it a particular location, then compute that value with respect to the new beam waist.

This is easier to visualize if you draw solid lines as the beam leaves the surface of the lens to the right, then draw dotted lines to the left of the lens surface to show how the beam would be, if you didn’t know there was a lens there at all.

[u/Marvellover13 → u/UVlight1]

If I understand you correctly what you're saying is that I should act as if the beam after the lens is a different beam, and then using the equations of q1 and q2 with the characteristics of the beam I found earlier, find the "virtual beam waist"?

[u/UVlight1 → u/Marvellover13]

Pretty much. Once you have the new beam, it is completely defined. If you know where the beam waist is, then at any distance from that beam waist you can find whatever parameter you want for the beam.

It’s been so long I don’t remember all the equations, q1,q2 etc.

But after you have done the matrix math, the new beam would have stuff from that in the Gaussian beam equation. Then you can match up the terms that correspond to what you want to find.