Path Length of Light in Opaque Media

Simulation results for light paths in circular areas with different degrees of opacity. Light comes in from the left with many different injection angles. (Source: R. Pierret & R. Savo)

What happens when light passes through a glass of milk? It enters the liquid, is scattered unpre­dictably at countless tiny particles and exits the glass again. This effect makes milk appear white. The specific paths that the incident light beam takes depends, however, on the opacity of the liquid: A transparent substance will allow the light to travel through on a straight line, in a turbid substance the light will be scattered numerous times, travelling on more compli­cated zig-zag trajec­tories. But astoni­shingly, the average total distance covered by the light inside the substance is always the same.

Stefan Rotter and his team at the Tech­nical University of Vienna, Austria, predicted this counter-intui­tive result together with French colleagues three years ago. Now he and his colla­borators from Paris verified this theory in an experiment. “We can get a simplified idea of this pheno­menon when we imagine light as a stream of tiny particles,” says Rotter. “The trajectories of the photons in the liquid depend on the number of obstacles they encounter.” In a clear, completely transparent liquid, the particles travel along straight lines, until they leave the liquid on the opposite side. In an opaque liquid, however, the trajec­tories are more compli­cated. The beam of light is scattered frequently along its way, it changes its direc­tion many times, and it can only reach the opposite side after covering a long distance inside the opaque substance.

But in a turbid liquid, there are also many photons, which will never reach the opposite side. They do not completely traverse the liquid, but just pene­trate a little below the surface and after a few scat­tering events they exit the liquid again, so their trajec­tories are rather short. “It can be shown mathema­tically that, rather surpri­singly, these two effects exactly balance”, says Rotter. “The average path length inside the liquid is thus always the same – inde­pendent of whether the liquid is transparent or opaque.” At second glance, the situation is a bit more compli­cated: “We have to take into account that light travels through the liquid as a wave rather than as a particle along a specific trajec­tory”, says Rotter. “This makes it more challenging to come up with a mathe­matical descrip­tion, but as it turns out, this does not change the main result. Also if we consider the wave proper­ties of light the mean length associated with light pene­trating the liquid always stays the same, irrespective of how strongly the wave is scattered inside the medium.”

The theoretical calcu­lations describing this counter­intuitive behaviour have already been published three years ago in a joint publi­cation by Stefan Rotter’s team and his colleagues from Paris. Now the same research groups managed to verify the remarkable result in an expe­riment. Test tubes were filled with water, which was then obfuscated with nano­particles. As more nano­particles are added, the light is scattered more strongly and the liquid appears more turbid. “When light is sent through the liquid, the way it is scattered changes conti­nuously, because the nano­particles keep moving in the liquid”, says Stefan Rotter. “This leads to a charac­teristic sparkling effect on the tubes’ outer surface. When this effect is measured and analysed carefully, it can be used to deduce the path­length of the light wave inside the liquid.” And indeed: irre­spective of the number of nano­particles, no matter whether the light was sent through an almost per­fectly transparent sample or a milk-like liquid, the average path length of light was observed to be always the same.

This result helps to under­stand the propa­gation of waves in disordered media. There are many possible appli­cations for this: “It is a uni­versal law, which in principle holds for any kind of wave”, says Rotter. “The same rules that apply to light in an opaque liquid also hold for sound waves, scattered at tiny objects in air or even gravity waves, travelling through a galaxy. The basic physics is always the same.” (Source: TU Vienna)

Reference: R. Savo et al.: Observation of mean path length invariance in light-scattering media, Science, online 09 November 2017; DOI: 10.1126/science.aan4054

Link: Nonlinear Dynamics & Complex Scattering, Inst. for Theoretical Physics, Vienna Univ. of Technology, Vienna, Austria

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