Wavefront of reflected and transmitted waves from an incident plane wave

Question

I am going back over an elementary example of a plane wave at an interface in Optics by Hecht. Now I am questioning something that I never realized.

At the beginning of the problem, we state that the reflected and transmitted waves can be written as

$ \vec{E_r} \cos (\vec{k_r} \cdot \vec{r} - \omega_r t +\varepsilon_r) $

$\vec{E_t} \cos (\vec{k_t} \cdot \vec{r} - \omega_t t +\varepsilon_t) $

My question is: aren't we making a huge assumption that the wavefront of the reflected and transmitted wave will have the same profile as that of the incident wave? Is there a more rigorous way to determine what the wavefront profiles of the reflected and transmitted waves should be?

Answer 1

We are. In essence, we're just coming up with a convenient Ansatz such that the final fields will be a solution to the Maxwell equations everywhere that we find useful. In particular, the need to match the boundary conditions to the incident wave at the surface then forces a plane-wave form for the reflected and transmitted fields if you want a solution both at the surface and away from it.