A hand calculation seemed to show that I had the wrong expressions for (Subscript[α, L]),  (Subscript[α, R]) in my polarization notes.  Here’s a check of the correction of those expressions:

ClearAll[a1, a2, b1, b2, c1, c2, alphaL, alphaR, phi]
$Assumptions = a1 > 0 && a2 > 0 && b1 > 0 &&  b2 > 0 && phi > 0;

c1 = a1 + I b1;
c2 = a2 + I b2;
alphaL = (c1 + I c2)/2
alphaR = Conjugate[(c1 -I c2)/2] // Simplify

alphaL (Cos[phi] + I Sin[phi]) + alphaR (Cos[phi] - I Sin[phi]) // Simplify