As given in Equation (1), r01 is the Fresnel coefficient between air and film and r12 between film and substrate. Finally, the reflectance, R(��), is given by:R(��)=r?r*(3)where r* indicates the complex conjugate of r.Figure 1 illustrates a two-layer system. In this case, two films (n1 and n2) are deposited successively on an absorbing substrate (n3). The whole system is surrounded by air (n0). To calculate the reflection coefficient for this system, Equation (2) can be extended as follows [24]:r=r01+r12e?jd1+r23e?jd+r01r12r23e?jd21+r01r12e?jd1+r12r23e?jd2+r01r23e?jd(4)where dl = 4��nldl/�� and d = d1 + d2. Again, by using Equation (3), the reflectance, R(��), for a two-layer system with an absorbing substrate can be obtained.Figure 1.Two-layer system surrounded by air (n0, k0).3.

?Model ExtensionEquation (4) describes a system with ideal interfaces between layers. However, in practice, irregular interfaces affect the reflectance and must be considered in the model. One approach to model interfaces is based on the effective media approximation (EMA) [11]. By EMA, the inhomogeneous interfaces between layers are replaced by fictitious homogeneous layers, which are incorporated as such in the model [25]. Another approach proposes to modify the Fresnel coefficients in order to reproduce the effect of the interfaces on the reflectance [12]. In this case, the Fresnel coefficients, rlm, are altered by multiplying them with a function, f(gl), where gl assigns a thickness to the interface, sl, proportional to its grade of inhomogeneity.

The modified Fresnel coefficients are defined as follows:r��lm=rlm?f(gl)(5)Introducing Equation (5) in Equation (4), the modified reflection coefficient, ?, is obtained. This approach yields a simpler and faster solution than EMA, which makes it advantageous for our application. The form of f(gl) depends on the considered interface model. As explained in [12], f(gl) could be ideally defined if the exact three dimensional structure of the interface was known. In general, however, such detailed knowledge of the interface is unavailable, and it is more reasonable to model the interface profile using an analytical function. Dacomitinib Four different interface functions are presented in [12].The principal causes of interface inhomogeneities are: the roughness of a layer surface and the mix of materials originated when two layers came in contact.

In the case of polymer electronics, the substrate surface is smooth and does not mix with the first applied layer. Therefore, we can consider the interface, s2 (Figure 1), as ideal. On the contrary, we cannot discard the presence of roughness and material mix on the interface, s1, between the first and second layer. In this case, f(g1) must model both kinds of inhomogeneities [16]. Finally, the reflection coefficient of the interface, s0, between air and the first layer is modified only by the surface roughness.