Fig. 2: Schematic figure illustrating the theory of the statistics of first-arriving backscattered photons from a time-gated window.
The left subplot shows a time-gated window divided to m sublayers, each contains ni particles responsible for light scattering where i is the sublayer number. The zoom-in sublayer i in the middle indicates that each particle therein has a probability pi,j to generate a detectable backscattered photon in response to a single pulse, where i and j are sublayer and particle indices. The intermediate variable, Ii, represents the probability to generate a detectable backscattered photon from sublayer i, while Pi defined in Eq. (1) is the probability of observing a first-arriving backscattered photon from sublayer i. Pi is the product of the probability of not receiving backscattered photons before sublayer i and the probability of receiving backscattered photons at sublayer i. Pi can also be interpreted geometrically in the right subplot as the ratio of the hatched area to the whole bar area (i.e., light yellow + hatched blue + light blue). The fraction of the yellow area represents the probability of receiving first-arriving photons before that sublayer, i.e., \(\mathop{\sum }\nolimits_{k = 1}^{i-1}{P}_{k}\).
