Jones vector related calculations for GA representation of plane wave.

e_1 I = e_{1123} = e_{23} = -e_{32} -> -e_2
F(x,t)=(1+e3)((α1e1 +α2e2)cos(ωt−βz)−(β1e1 +β2e2)sin(ωt−βz)).

c1 e1 + c2 e2

ClearAll[al, al1, al2, ar1, ar2, phi, e1, e2]
$Assumptions = (ar1 > 0) && (ar2 > 0) &&(al1 > 0) && (al2 > 0) && (phi > 0);
al = al1 - I al2;
ar = ar1 - I ar2;
e1 = {1,0,0};
e2 = {0,1,0};

ecomplex = al Exp[ -I phi] + ar Exp[ I phi]// ExpToTrig 
ereal = ecomplex // Re // Simplify;
eimag = ecomplex // Im// Simplify;

efield = e1 ereal - e2 eimag // Simplify

alpha1 = al1+ar1;
alpha2 = al2+ar2;
beta1 = -(-al2+ar2);
beta2 = -(al1-ar1);
format = {al1 -> (Subscript[α, L1]),al2 -> (Subscript[α, L2]),ar1 -> (Subscript[α, R2]),ar2 -> (Subscript[α, R1])};

(*c1 = alpha1 + I beta1 // Simplify
c2 = alpha2 + I beta2;
Row[{(c2 // Re) // Simplify," + ", (c2 // Im)I // Simplify}]*)

a ={{(Subscript[α, 1]), alpha1 },
{(Subscript[α, 2]), alpha2 },
{(Subscript[β, 1]), beta1 },
{(Subscript[β, 2]), beta2 }};

Grid[{#// First, " = ", #//Last (*(#//Last)/. format*)} & /@a ]/. format