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Transmittance of absorbing multilayer thinfilms on an absorbing substrate 
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#1
Jan2109, 09:44 PM

P: 351

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].
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 semiinfinite incidence medium (dielectric), many thinfilms (absorbing), and a semiinfinite 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. 


#2
Jan2109, 09:50 PM

P: 351

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



#3
Jan2309, 08:58 AM

P: 351

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 semiinfinite dielectric (transparent incident medium) # 1 = the first thin film (absorbing) . . . # m = the last thin film (absorbing) # m+1 = a semiinfinite substrate (absorbing) I assume: 1. Oblique incidence 2. I already know the amplitude transmittance [tex](t)[/tex] 3. The coordinate system has the origin at the last interface, where +z points down. Beyond the last interface, the transmittance is [tex]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[/tex] [tex]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[/tex] where [tex](\kappa)[/tex] 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 semiinfinite substrate is absorbingaccording 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. 


#4
Jan2309, 04:23 PM

P: 351

Transmittance of absorbing multilayer thinfilms on an absorbing substrate
One last thought:
I've found yet another reference [4] with a slightly different expression for the case of a multilayer stack: [tex] T = \Re \left\{ \frac{\hat{n}_{s} \cos \hat{\theta} _s}{n_a \cos \hat{\theta} _a} \right\} \left t \right ^2 [/tex] 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 TMpolarized 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: [tex] T = \Re \left\{ \frac{\cos \hat{\theta} _s / \hat{n}_{s}}{\cos \hat{\theta} _a / n_a} \right\} \left t \right ^2 [/tex] 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). 


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