This question stems from what I've seen as the definition of a wave vector and from not being able to reconcile how, when the incident medium is lossy, the parallel component can always be real. Let's define a wave vector $k$:


Let's say that $k_x$ is the parallel component, $\omega n\sin\theta$, and $k_z$ is the perpendicular component, $\omega n\cos\theta$. The parallel component is said to be the same everywhere to satisfy the boundary conditions which leads us to Snell's Law:


The terms with subscript $0$ are the refractive index of the incident medium and the incident angle, respectively. Now, if $n_0$ is made to be complex to indicate loss, then the parallel component is not real.

Mathematically, this leads to a complex perpendicular component because $k_z=((\omega n)^2-k^2_x)^{1/2}$. This becomes a problem when light is exiting a lossy medium into a dielectric. For illustration purposes, let's say we have a sufficiently large piece of amber where light is not completely attenuated by the time it is transmitted into air. At that point, we can say the incident medium is lossy, but how is the parallel component of the wave vector kept real such that the perpendicular component does not anymore indicate loss when it passes into the air?

Also, I've seen from many sources that the parallel component is, by definition, real, and I wish to satisfy that condition in the equations I am using. This is another point that lead me to my question.



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