analytic solution?

[Michael Koch]

Hi all,

I'd like to know if there exists an anlytic solution for this problem.
A beam starts at height = 0 with angle alpha, and passes through air of thickness A and then through a glass window of thickness D. When exiting the window, the beam's height is C.
A, D, C and the index of refraction are known. Is there an analytic solution for angle alpha? I think it doesn't exist, but maybe I'm overlooking something.

Loading Image...

Thanks,
Michael

[Lu Wei]

Post by Michael Koch
Hi all,
I'd like to know if there exists an anlytic solution for this problem.
A beam starts at height = 0 with angle alpha, and passes through air of thickness A and then through a glass window of thickness D. When exiting the window, the beam's height is C.
A, D, C and the index of refraction are known. Is there an analytic solution for angle alpha? I think it doesn't exist, but maybe I'm overlooking something.
http://www.astro-electronic.de/Brechung.jpg

Yes there is.
To reduce the use of brackets, post notation is used afterwards.
As in your draft, we have
α tan= B/A, β tan= (C-B)/D,
and
α sin= n * β sin (eq.1)
Since
α sin ^2 = 1-α cos ^2= 1- 1/(1+ α tan ^2)
Square eq.1 and substitute tangent terms, an equation about B, which is
up to the degree of 4, could be obtained. The general analytic root of
such an equation is given by:
https://en.wikipedia.org/wiki/Quartic_function#General_formula_for_roots
The general radical form is formidable, yet maybe this equation belongs
to those which could be transformed to a simpler case.
After B is solved, Then α = (B/A) atan.

[Michael Koch]

Post by Lu Wei
The general radical form is formidable, yet maybe this equation belongs
to those which could be transformed to a simpler case.

unfortunately it doesn't look like one of the simpler cases...

Michael

[Lu Wei]

Post by Michael Koch

Post by Lu Wei
The general radical form is formidable, yet maybe this equation belongs
to those which could be transformed to a simpler case.

unfortunately it doesn't look like one of the simpler cases...

Let computer do the calculation then. Maybe not what you expected though.