Effective Refractive Index - Should be simple

[nathangrand]

Imagine a dielectric made of of alternating layers of widths A and B and refractive indices (a*) and (b*). Find the effective refractive index, N

So in general: c/n = wavelength x frequency = phase speed

My thinking was find the total time taken for the wave to propagate through the distance A+B and work out the refractive index from this.

So,

Time=distance/speed = (A+B)/(c/N) = (A/(c/a*)) +(B/(c/b*))

But this gives me the wrong value for N

The answer I'm looking for is N^2= [A(a*)^2 + B(b*)^2]/(A+B)

Where am I going wrong?

 

[nathangrand]

Is it that I need to use the optical path length..ie refractive index x physical length in my calculations? That will make it work I think but can someone explain why...