using those angles in your first diagram, the refractive index (based upon the same values of air as the previous post) would be 1.2999 (to 4 decimal places for both 350 nm to 750 nm). So it is closer, but still not exact, (which I am guessing you are looking for, but for which, if correct, i would not expect) with the best match occurring at longer wavelengths (i.e. closer to the 700 nm visible limit). Recall for 700 nm, the refractive index of ice at -40 C is 1.3069, compared to the above computed value of 1.2999. This is about 0.007 out at the 700 nm limit.
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