Outline of thin film theory

The reflection and transmission properties of multiple layers of materials with different refractive indices can be treated either as a ray tracing or as a boundary value problem (e.g., Wolter, 1956 Bom Wolf, 1975). The ray tracing method leads to summations where it is sometimes difficult to follow the phase relations, especially if several layers are to be treated. We follow closely the boundary value method reviewed by Wolter (1956). In effect this method is a generalization of the one-interface boundary problem that led to the formulation of the Fresnel equations in Section 1.6. 

Assume a stack of layers, as shown in Fig. 5.6.1. In each layer there exists a downward and an upward propagating field, indicated by the arrows A and R, respectively. The amplitude, phase, polarization, and direction of the arriving wave in the top layer, m, is assumed to be known as well as the material properties, jurci and r + i i, of the substances forming the stack of plane parallel layers. We are interested in the amplitudes and phases of the reflected, Rm, and transmitted wave, Aq. In the lowest medium, 0, only a transmitted wave is postulated. All waves are assumed to be plane with the Poynting vectors in the x-y plane, that is, = 0. To apply the boundary conditions to each interface the downward and the upward waves need to be calculated for each layer (y = 1,2. m). For layer j they are, respectively, 

In general, each vector equation stands for six equations, one for each component of E and H in the three coordinates, but the number can be reduced by proper choice of the coordinate system. In setting all (j)j equal in the above equations we tacitly imply the validity of the reflection law. As in the discussion of the Fresnel equations we solve the problem for two orthogonal, linearly polarized waves. Other states of polarization may be represented by superposition of these solutions. For the transverse E wave (TE) the electric vector is perpendicular to the plane of incidence, and only the z-component of E exists, = E . The same is true for the transverse H wave (TM), where only the = H component is present. 

This is simply a consequence of the prudent choice of coordinates. In addition to the z-components, we need the other x-components of H and E, respectively, in order to apply the boundary conditions. They can be found from the z-components with the help of Maxwell s equations and the definition of the curl operator. For the TE and TM waves the tangential components in the x-direction, using the generalized form, are 

In this section, fij is the relative permeability. Attheboundary of layer j and — 1, the tangential components of E and H must be continuous. 

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