(* Example 3-- nx=1.2, ny = 1.3, nz = 1.4, what is n in the (1,1,0) direction? *)

(* After identifying n = 1.4 as a solution, Fresnel's Eq becomes this *)
Solve[1/n^2 == (1/2)/(n^2-1.2^2) + (1/2)/(n^2-1.3^2)  ,n]

Plot[{1/n^2 ,(1/2)/(n^2-1.2^2) + (1/2)/(n^2-1.3^2)},{n,1.1,1.5}, PlotRange-> {-10,10}]

FindRoot[1/n^2 == (1/2)/(n^2-1.2^2) + (1/2)/(n^2-1.3^2) ,{n,1.25}]

(* Using the limit method to get both roots at once. Assume the direction is close to (1,1,0) but not quite exactly.
For example, let u = 1/sqrt(20001) * (100,100,1). *)

Solve[1/n^2 == (10000/20001)/(n^2-1.2^2) + (10000/20001)/(n^2-1.3^2) +  (1/20001)/(n^2-1.4^2) ,n]