Thanks so much for trying to bludgeon this into my head... I'm afraid I'm still quite confused. Please forgive me in advance...
The incident [itex]k_i[/itex] is at a real angle, say 70 degrees, to a TIR interface. If it's a glass prism ([itex]n_1 ≈ 1.52[/itex]) and an aqueous medium ([itex]n_1 ≈ 1.33[/itex]) this gives [itex]θ_t = Pi/2 - 0.367 i[/itex]. My first reaction was just to say [itex]θ' = Pi/2 - θ_t = 0.367 i [/itex], but this doesn't sit well with me... it feels like I'm being na´ve.
I get that the imaginary component is the only bit that can propagate in the direction n, because otherwise this would carry energy away from the TIR interface.
I'm not sure if this is what you were suggesting - but just because k' has to contain a real component parallel to n', why would this imply that the angle it forms with the interface has to have a non-zero real component?
I fear I have the wrong end of some stick. Please can you have a look at the attached figure as it may (or may not) clarify the nature of my problem. You can ignore the sphere...