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

In summary, the conversation discusses finding the transmittance (T) for a system consisting of a semi-infinite incidence medium, multiple thin-films, and a semi-infinite substrate, where the amplitude transmittance (t) is known. The literature provides different expressions for T, including the use of complex refractive indices and angles, as well as the Lambert law of absorption. Another reference suggests a different formula using the real part of the ratio of refractive indices. The question remains on how to accurately compute T in absorbing stratified media.
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.

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

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.

Last edited:
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).

## What is the purpose of studying transmittance of multilayer thin-films on an absorbing substrate?

The purpose of studying transmittance of multilayer thin-films on an absorbing substrate is to understand how different layers of absorbing materials interact with light, and how this affects the overall transmittance of the film. This information is important in fields such as optics, materials science, and engineering.

## How is transmittance of multilayer thin-films on an absorbing substrate measured?

Transmittance of multilayer thin-films on an absorbing substrate is typically measured using spectrophotometry, which involves passing a beam of light through the film and measuring the intensity of light that passes through the other side. This data can then be used to calculate the transmittance of the film.

## What factors can affect the transmittance of multilayer thin-films on an absorbing substrate?

The transmittance of multilayer thin-films on an absorbing substrate can be affected by a variety of factors, including the thickness and composition of each layer, the angle of incident light, and the wavelength of light used. Additionally, the properties of the absorbing substrate, such as its thickness and absorption coefficient, can also impact transmittance.

## What is the difference between transmittance and absorbance?

Transmittance and absorbance are two related measures of how much light passes through a material. Transmittance is the ratio of transmitted light to incident light, while absorbance is a logarithmic measure of how much light is absorbed by the material. In other words, transmittance measures the amount of light that passes through a material, while absorbance measures the amount of light that is lost due to absorption.

## How can the transmittance of multilayer thin-films on an absorbing substrate be optimized?

The transmittance of multilayer thin-films on an absorbing substrate can be optimized by carefully selecting the materials and thicknesses of each layer, as well as the angle and wavelength of incident light. Numerical simulations and experiments can also be used to optimize transmittance and improve the performance of these films for specific applications.

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