# EM Lasing Medium Green's Function

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.

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