[waveguide_3d] fixup and doc update

meanas/fdfd/waveguide_3d.py

@@ -157,19 +157,31 @@ def compute_source(
 
 
 def compute_overlap_e(
-        E: cfdfield,
+        E: cfdfield_t,
         wavenumber: complex,
         dxes: dx_lists_t,
         axis: int,
         polarity: int,
         slices: Sequence[slice],
+        omega: float,
         ) -> cfdfield_t:
     """
     Given an eigenmode obtained by `solve_mode`, calculates an overlap_e for the
     mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
     [assumes reflection symmetry].
 
-    TODO: add reference or derivation for compute_overlap_e
+    E x H_mode + E_mode x H
+    -> Ex Hmy - EyHmx + Emx Hy - Emy Hx  (Z-prop)
+    Ex we/B Emx + Ex i/B dy Hmz - Ey (-we/B Emy) - Ey i/B dx Hmz
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy Hmz - Ey dx Hmz)
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy (dx Emy - dy Emx) - Ey dx (dx Emy - dy Emx))
+    we/B (Ex Emx + Ey Emy) + i/B (Ex dy dx Emy - Ex dy dy Emx - Ey dx dx Emy -  Ey dx dy Emx)
+
+    Ex j/wu (-jB Emx - dx Emz) - Ey j/wu (dy Emz + jB Emy)
+    B/wu (Ex Emx + Ey Emy) - j/wu (Ex dx Emz + Ey dy Emz)
+
+
+    TODO: add reference
 
     Args:
         E: E-field of the mode
@@ -197,9 +209,8 @@ def compute_overlap_e(
     Etgt = numpy.zeros_like(Ee)
     Etgt[slices2] = Ee[slices2]
 
-    # note no sqrt() when normalizing below since we want to get 1.0 after overlapping with the
-    # original field, not the normalized one
-    Etgt /= (Etgt.conj() * Etgt).sum()    # type: ignore
+    # Note: We normalize so that (Etgt @ E.conj()) == 1, so (Etgt @ Etgt.conj) != 1
+    Etgt /= (Etgt.conj() * Etgt).sum()
     return cfdfield_t(Etgt)