# Is there a formula to calculate the deviated angle of a ray passing through a prism?

I'm assuming the angle of incidence to be the number of degrees from a perpendicular line on the side of a prism the ray starts passing through... I'm trying to figure out the angle the ray would be off of a perpendicular line that intersected the side of the prism that the ray exited.

Is there a formula to calculate this?

Background: I'm trying to use anamorphic prisms to stretch an image by a specific ratio... I can do it by trial and error and pretty much get what I am looking for but I wanted to see what the math was behind what was happening... I'm using 2 prisms with a 20 degree angle and made of BK7.

• Answered by en.wikipedia.org/wiki/Prism#Deviation_angle_and_dispersion isn't it? Jun 22, 2015 at 22:11
• Thank you Rob... that's exactly what I needed... Don't know how I missed it when I was googling. Jun 23, 2015 at 13:27
• Jun 19, 2019 at 20:23

20° is quite a small angle. If you are also sending light into the prism at a small angle (say ≤ 20°) to the normal to the first surface), then a good approximation to the total angle, $$D,$$ of deviation is $$D=(n-1)A$$ in which $$A$$ is the prism angle (the angle between the faces through which the ray enters and leaves). $$A$$ must be small for the approximation to hold. $$n$$ is the refractive index of the material of the prism.
For example, consider a prism with $$A=20.00°$$ and $$n=1.500$$. For a ray incident on the first surface at 20° 'below' the normal, accurate (I hope) calculations based on Snell's law give $$D=10.26°,$$ whereas $$(n-1)A=10.00°$$.
• Then don't worry about the parallel case. Just calculate the path through the prism by the ray based on entering at 20 degrees from $n=1$ to $n=1.5$. Then you need to find the angle it hits the other face and there it goes from $n=1.5$ to $n=1$. Jun 22, 2015 at 20:49