# Transfer matrix

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.

• Maybe you need to use the Fresnel equations. The transfer matrix might be for multiple thin layered objects - lens with coatings. – Cinaed Simson Apr 12 at 22:55

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 $$(0.93)^{9.52}$$.