Waals interactions in layered planar systems

The electromagnetic surface modes in each region still have the form f(z) = Aepz + Be [Eq. (L3.ll)] with the A and B coefficients restricted in the semi-infinite spaces to the left, Bl = 0, and to the right AR = 0 [Eqs. (L3.12)]. As before, these restrictions ensure that we are looking at modes associated with the surfaces. For writing efficiency the procedure is described for the electric-field boundary conditions only. 

As in the derivation of the Lifshitz expression, at each interface at a position Z1/i+i between adjacent materials, i and i + 1, the electric-field boundary conditions for continuous Ex, Ey, and eEz create a connection between Ai( Ai+i and Bi( B1+i15  

This pair of equations can be written in matrix form, 

Aside from a multiplicative factor [(si+iPi + eiPi+i) /(2ei+ipi+i)], matrix Mi+1/j has the form16 

There is an equivalent transition matrix for magnetic modes with 

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L3.4. Derivation of van der Waals interactions in layered planar systems, 292... 

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