Reflectivity from a Multilayer Film

Next we consider the reflection from a system consisting of a large number of thin parallel layers of different refractive indices. A sample with a continuously varying 

The system now effectively reduces to one consisting of two interfaces the 0-1 interface and the virtual interface at the 1-2 boundary with the effective reflection coefficient f 2. The overall reflection coefficient r is then obtained as 

Although the above method of calculating the reflectivity can be extended to multilayer systems with any number of discrete layers, it becomes unwieldy as soon as the number of layers involved exceeds four or five. A computationally more efficient method is offered by the use of an optical transfer matrix. Here we simply summarize the method as described by Lekner.6 For any single layer j within the multilayer 

The reflection coefficient r from the surface of the 0-1 interface is then obtained, in terms of the matrix elements of M, by 

The reader may want to verify that for a system consisting of a single layer of film on a substrate (i.e., N = 2), as in the preceding section, (7.36) indeed leads to (7.28). 

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