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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?

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Indeed, an evanescent wave only has a wavevector real part parallel to the interface.

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An evanescent wave is a superposition of the incident (penetrated) wave and the medium reaction to this incident wave (a radiated everywhere wave). The interference is "destructive" in the medium direction and "constructive in the opposite direction.

You must consider all sources of a wave to understand its behaviour.

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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.

The existence of such waves are a consequence of the Fresnel equations, which describe what happens at a plane interface between two different effective media, solely characterised by their (complex) indices of refraction. The phenomenon is unrelated to interference.

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