# Gaussian beam propagation in a medium

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 $$w(z)$$.