Broadening of sodium lines by a prism upon change of incidence angle

Question

How angle of incidence affect the lines of sodium when we are observing refracted rays from prism ? Or is it the deviation of rays that makes the lines broader.

Let me put it in a better way , when doing the experiment of measuring angle of deviation with a spectrometer , I observed the refracted rays from prism, now if the tip of the prism moves toward the collimator (decrease in angle of incidence) then the refracted lines broadens and if it moves towards the telescope (increase in angle of incidence) the lines get narrower. Now, we know that there is a unique angle of deviation for each color, and if we go either left and right from the corresponding angle of incidence of minimum deviation angle in the $i-\delta$ curve, then deviation($\delta$) increases. So, my common sense tells me that the lines will be broader no matter which way the tip is going towards(the collimator or the telescope), but I myself observed that if $i$ increases lines get narrower and if $i$ decreases lines are getting broader. Is there anything that I am missing or is this related to some other phenomena ?

Answer 1

Snell's law says, differentiated for a small opening angle $d\theta_{\text{in}},$

$$d\theta_{\text{out}} = d (\sin^{-1}(n \sin \theta_{\text{in}})) = \frac{n \cos (\theta )}{\sqrt{1-n^2 \sin ^2(\theta_{\text{in} )}} }\ d \theta_{\text{in}}$$ with opening angle $d\theta_{\text{out}}$ going to infinity near the limiting angle of total reflection.