Calculation the reflection coefficient of a mutlilayer material

Question

For our project we have to study an infrared filter. This filter is composed of glass and several layers (nanolayers of titanium oxide, silver and cupper deposited on one side of the glass).
Now we have to determine the reflection coefficient of the coating (i.e. the nanolayers alone) given that of the glass and that of the filter as a whole.

Now our supervisor claims that this is calculated by the formula: $R_{filter} = R_{coating}R_{glass}$. This formula seems a bit odd to us though, f.i. because it doesn't hold in general (one can imagine a material composed of two materials, where one has 100% reflectance and the other 50%, so this material should have 100% reflectance, while according to the formula it has 50%). I wonder if somebody maybe knows if this formula holds, and maybe can explain which assumptions one could make under which one can derive the formula.

Answer 1

Yes... your supervisors formula is incorrect. Smart people make mistakes -- it happens. If your supervisor isn't open to criticism, then I suggest that you give him or her a few simple examples that illustrate the problem, and then immediately present a solution. Your example of a mirror coated over a non-mirror is a great place to start.

In order to solve your multiple reflection problem, you can use the transfer-matrix method. Unfortunately you'll need to invert the result, because you want to calculate the properties of one layer given the properties of several, which is the opposite of what the method is usually set up to do. Whether or not you can do this depends on if you have enough information to separate the effects of the different layers.

Also, note that the reflectance of one layer is actually undefined, because reflection is a property of an interface between two materials, not a property of a single material. Water in water doesn't reflect, and air in air doesn't reflect, but there is reflection at air-water interfaces. Your coatings will reflect different amounts if they are in contact with one another as compared to if they're in contact with air.