# EM Lasing Medium Green's Function

**Electricity and Magnetism**Level pending

Consider electromagnetic waves polarized in the direction propagating in one dimension in an active laser medium with current density \(J(x)\). Maxwell's equation for the \(z\)-component of the electric field of these waves then reads:

\[ \large \left(\dfrac{d^2}{dx^2} - \kappa^2 \right) E_z (x) = J(x),\]

for \(\kappa\) some constant. Suppose there is a source at \(x= 0\) and that there is a conducting mirror far away so that effectively \(E_z (0) = 1\) and \(E_z (\infty) = 0\) are the boundary conditions. Find the Green's function \(G(x,y)\) *for \(x<y\)* that will allow for determination of the electric field anywhere in space.