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Let's suppose I have a transmission curve, $T$. How could I calculate the resulting transmission $T_\text{final}$ if the light propagated trough a much longer distance in the same media and eventually escaped this media in vacuum?

The propagation distance is $d$. The refractive index of the media is $N = n + ik,\ n > 1$.

When I check transfer matrix theory, I don't see any mention on how to calculate from an existing $T$ curve what would happen if the light propagated for a longer distance and through potentially different medias as well.

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  • $\begingroup$ Maybe you need to use the Fresnel equations. The transfer matrix might be for multiple thin layered objects - lens with coatings. $\endgroup$ – Cinaed Simson Apr 12 at 22:55
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The transmission curves give the amount of light that isn't absorbed by a certain length of media $L$. The curve values are between 0 and 1 (or a percent that can be converted into a fraction by division by 100). So if you want the transmission for twice that length you square the curve. For example, if a meter of material transmits 0.4 = 40% at a particular wavelength, then 3 meters will transmit $(0.40)^3 =0.064$ or 6.4%.

Another example. If the transmission curve had 0.93 or 93% at the wavelength of interest but you want a distance that is 9.52 times longer, then the transmission curve for your longer distance will be $(0.93)^{9.52}$.

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