From the bimodal distribution to the quarter circle law

[A. Goetschy and A. D. Stone, Phys. Rev. Lett., 1304.5562, (2013)]

Almost thirty years ago, theoreticians predicted that the distribution of the transmission values of a multiple scattering sample should follow a 'bimodal distribution'.  Physically, that means that, in the diffusive regime, there is a large number of strongly reflected channels - the closed channels - and a small number of channels of transmission close to one - the open channels. The existence of these open channels regardless of the thickness of the medium is of big interest for researchers, especially for imaging or communication applications. Nevertheless, such channels have not yet been directly observed. A investigation on those channels requires a measurement of the entire transmission matrix of a lossless scattering medium. For practical reasons (open geometry, limited numerical aperture, noise...) one usually has access to a subpart of the total transmission matrix. In recent experimental measures of the transmission matrix in optics [S.M. Popoff et al., Phys. Rev. Lett., 104, 100601, (2010)] the distribution of the transmission values follows a 'quarter circle law', characteristic of totally uncorrelated systems. This means that the fraction of the transmission matrix measured shows no effect of the correlations at the origin of the bimodal distribution due to the loss of information. In this paper, A. Goetschy and D. Stone theoretically study the effect of the loss of information or the imperfect control on the statistics of the transmission matrix of a scattering system.

The system studied consists of a random scattering medium inside a waveguide. The size of the transmission matrix is finite due to the limited number of propagative modes in the waveguide. The distribution of the transmission values of such a system follows the bimodal distribution. To introduce the loss of information, the authors used projectors to randomly remove lines and/or columns of the transmission matrix. This corresponds to controlling/measuring only a part of the incoming/outgoing modes of the waveguide. For instance, when a portion m < 1 of both input and output modes are controlled, we see in figure 1 that the distribution of the transmission values quickly differs from the bimodal distribution.

Figure 1. Distribution of the transmission values of a scattering system for an imperfect control (m < 1) both in input and output. Image from [A. Goetschy and A. D. Stone, Phys. Rev. Lett., 1304.5562, (2013)].

In a second part, the authors study the effect of the imperfect control on the information capacity per channel. This value describes how much bits of information per second one can send, without noticeable error, in a telecommunication channel.

This paper provide a toolbox for experimentalists to understand the effect of imperfect control when working with scattering environments. This would be especially valuable for imaging and telecommunication applications.