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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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Critical angle is just an indication that snell's law won't apply beyond this. Also, You'll get no solution for snell's equation if one side is greater than 1. It just means that intensity of refracted light has become zero.

The middle case in the picture is just a hypothetical condition of transition between two cases.

ALso TIR occurs because the angle of refraction reaches a 90-degree angle before the angle of incidence reaches a 90-degree angle. The only way for the angle of refraction to be greater than the angle of incidence is for light to bend away from the normal.

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