Transmittance of absorbing multilayer thin-films on an absorbing substrate

[DivGradCurl]

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

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

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

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

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 (n_3) and (\theta_3) 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.

 

[DivGradCurl]

PS: maybe it's simply 0, because of my semi-infinite assumption for an absorbing substrate.

 

[DivGradCurl]

I've found an explanation in this reference [3]. This is not exactly what is in the book, but it's what I think is correct. Please correct me if I am wrong.

In a system where the media are:

# 0 = a semi-infinite dielectric (transparent incident medium)
# 1 = the first thin film (absorbing)
.
.
.
# m = the last thin film (absorbing)
# m+1 = a semi-infinite substrate (absorbing)

I assume:

1. Oblique incidence
2. I already know the amplitude transmittance (t)
3. The coordinate system has the origin at the last interface, where +z points down.

Beyond the last interface, the transmittance is

T = \left| \frac{\hat{n}_{m+1} \cos \hat{\theta} _{m+1}}{n_0 \cos \hat{\theta} _0} \right| \left| t \right| ^2 \exp \left[ - \left( \frac{4\pi \, \kappa_{m+1}}{\lambda} \right) \frac{z}{\cos \theta _{m+1}} \right] \qquad \qquad \mbox{(TE)} \right|

T = \left| \frac{ (\cos \hat{\theta} _{m+1})/\hat{n}_{m+1}}{(\cos \hat{\theta} _0)/n_0} \right| \left| t \right| ^2 \exp \left[ - \left( \frac{4\pi \, \kappa_{m+1}}{\lambda} \right) \frac{z}{\cos \theta _{m+1}} \right] \qquad \qquad \mbox{(TM)} \right|

where (\kappa) is the imaginary part of the complex refractive index.

This means I can generalize the two transmittance expressions (from the first post) for the case of complex media [3]. I also introduce a decay, because the semi-infinite substrate is absorbing---according to the Lambert law of absorption (in its oblique version) [3]. So, the better way to put it is that the transmittance approaches 0 very fast.

[3] C. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication. West Sussex, England: John Wiley & Sons, 2007.

 

[DivGradCurl]

One last thought:

I've found yet another reference [4] with a slightly different expression for the case of a multilayer stack:

T = \Re \left\{ \frac{\hat{n}_{s} \cos \hat{\theta} _s}{n_a \cos \hat{\theta} _a} \right\} \left| t \right| ^2

where "s" and "a" stand for substrate and ambient media, respectively. Of course, this author tells the reader to carry out separate computations for TE and TM-polarized light, but that's really the only formula presented. It appears that the only difference between the two cases is the amplitude transmittance, which does not sound right for a TM calculation. I would expect something like:T = \Re \left\{ \frac{\cos \hat{\theta} _s / \hat{n}_{s}}{\cos \hat{\theta} _a / n_a} \right\} \left| t \right| ^2

Also, notice that he does not take the absolute value of the ratio; he uses the real part of the result instead. What confuses me even more is that he cites [2] as his reference, which does not present the same formalism.

This takes me back to my original question:

How do I compute the transmittance in (absorbing) stratified media? Any help is highly appreciated!

[2] M. Born and E. Wolf, Principles of Optics. Cambridge, UK: Cambridge University Press, 7th ed., 1999.
[4] D. L. Windt, IMD - Software for modeling the optical properties of multilayer films, Computers in Physics, 12, 360 (1998).