# Why don't evanescent waves give rise to electromagnetic waves?

I'm reading about evanescent waves for the first time. I understand that even thought no electromagnetic wave is transmitted across the boundary, an electric field is transmitted which decays exponentially into the material.

As far as I understand this is still an oscillating electric field. Can anyone explain, therefore, why it doesn't generate a magnetic field and give rise to an electromagnetic wave?

Or is there an electromagnetic wave moving parallel to the boundary but people just talk about the E field?

A propagating wave obeys $$\omega^2 = k_x^2+k_y^2+k_z^2$$ with $$k_i^2>=0$$. For an evanescent wave at least one wave vector component has $$k_i^2<0$$. In this direction the wave amplitude decays exponentially, while it propagates in the other direction(s). Such solutions of the wave equation are only possible near an interface. As an example, total internal reflection at a glass-vacuum or glass-air interface is accompanied by an evanescent wave outside the glass.