some questions about photonic band calculation ..

the origin reference paper is:
Phys. Rev. B 60, 2610?2618 (1999)
Scattering-matrix treatment of patterned multilayer photonic structures

i try to reproduce its results using matlab.
the first problem is :
no matter which matlab function 'eig' i use, eig(M) or eig(M,

,"qz')(because M is an asymmetric matrix), to sove the equation 3.7 or 3.8,the resultant eigenvectors automatically satisfy the orthogonality relationship

,rather than equation (3.9)

.



another problem is how to rescale the frequency

and lattice period

?

?

?

?
（the parameters at right hand side of equations are all in SI （international system of units））
and

rescales to 1?


help please