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When a ray of light coming from the medium $n_1$

$\mbox{ has an incident angle } \theta_1 = \theta_c$

where $\theta_c = \mbox{critical angle}$

what phenomenon are we observing? Is it called a partial internal reflection? In others, is only some of the light reflected back into the medium? What about the rest?

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It's an evanescent wave. Because that surface path is neither in medium $n_2$ nor medium $n_1$, it has neither propagating-wave character. It will exist as a kind of ripple, not truly a "ray", and because internal reflection is total, it doesn't carry energy away as a true refracted beam might.

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