I have found a general expression for the amplitude transmittance [tex](t)[/tex] of multilayer film stacks in the literature [1], but the author does not explain how to obtain the transmittance [tex](T)[/tex]. I looked up other references, and the closest I could find was the description of "an absorbing film on a transparent substrate" [2].(adsbygoogle = window.adsbygoogle || []).push({});

On page 756 of [2] there are expressions for transmittance:

[tex]T = \frac{n_3 \cos \theta _3}{n_1 \cos \theta _1} \left| t \right| ^2 \qquad \qquad \mbox{(TE)}[/tex]

[tex]T = \frac{(\cos \theta _3) / n_3}{(\cos \theta _1)/ n_1} \left| t \right| ^2 \qquad \qquad \mbox{(TM)}[/tex]

In other words, I'm trying to find the transmittance (using the amplitude transmittance value I already know) for a system that consists of a semi-infinite incidence medium (dielectric), many thin-films (absorbing), and a semi-infinite substrate (absorbing). In comparison, the reflectance is easy to find, because you just multiply the reflectivity by its complex conjugate; this is not the case. If you use the expressions above, replacing [tex](n_3)[/tex] and [tex](\theta_3)[/tex] by the substrate complex refractive index and the complex angle on the exit side, respectively, the results will be complex as well.

Any ideas? Thanks.

[1] J. Eastman,Surface scattering in optical interference coatings. PhD thesis, University of Rochester, 1974.

[2] M. Born and E. Wolf,Principles of Optics. Cambridge, UK: Cambridge University Press, 7th ed., 1999.

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# Transmittance of absorbing multilayer thin-films on an absorbing substrate

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