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Suppose we have a gaussian beam in a medium with $n>1$ with a waist $w_0$ centered at $z = 0$. After a distance of $d_{med}$ the beam leaves the medium and travels through air. To compute the change of the complex beam parameter $q(z)$ one can apply the appropriate ABCD-matrices.

Inside the medium $q(z) = z + i\ z_R$ with $z_R=\frac{\pi w_0^2 n}{\lambda_{vac}}$. The matrix to describe the beam propagation consists of:

$(1)$ propagation in medium for $d_{med}$: $m_1 = \left( \begin{array}{cc} 1 & d_{med} \\ 0 & 1 \\ \end{array} \right) $

$(2)$ refraction at medium/air boundary: $m_2 = \left( \begin{array}{cc} 1 & 0 \\ 0 & n \\ \end{array} \right) $

$(3)$ propagation in air for $d_{air}$: $m_3 = \left( \begin{array}{cc} 1 & d_{air} \\ 0 & 1 \\ \end{array} \right) $

Thus $M = m_3 \times m_2 \times m_1 =\left( \begin{array}{cc} 1 & d_{med} + n \ d_{air} \\ 0 & 1 \\ \end{array} \right)$

and from this $A = 1,\ B = d_{med} + n d_{air},\ c = 0$ and $d = n$.

So one ends up with $q=\frac{A q_0 + B}{C q_0 +D} = \frac{q_0 + d_{med} + n d_{air}}{n}$ with $q_0 = i\ z_R$.

If now $d_{air}\rightarrow 0$, i.e. one approaches the boundary from the right, q is given given by $ \frac{q_0 + d_{med}}{n}$, in contrast to $q=q_0 + d_{med}$ when approaching from within the medium.

Naturally I would say one also needs to change $q_0$ and $d_{med}$, i.e. $d_{med}\rightarrow n\ d_{med}$ and $q_0\rightarrow n\ q_0$ since $\lambda = \lambda_{vac} /n \rightarrow \lambda_{vac}$, but I think of $q_0$ as some constant, which is determined by the mediums properties. So I am missing a physical reason for this operation.

And if this is not the case, although I could live with a discontinuity in $R(z)$, I am troubled with the idea of a sudden jump in the beam radius $w(z)$.

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The change in q is due to refraction and you point this out nicely. I did exactly the same thing and hit the same problem. I believe the issue with this calculation is that the Gaussian beam with its curvature has the same absolute position for the minimum waist. When looking from the second medium the beam should appear as if it comes from a position a factor n closer or further from the interface. This is not included in your calculation.

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