Statistics of maximum photon penetration depth in a two-layer diffusive medium.

We present numerical results for the probability density function f(z) and for the mean value of photon maximum penetration depth zmax> in a two-layer diffusive medium. Both time domain and continuous wave regime are considered with several combinations of the optical properties (absorption coefficient, reduced scattering coefficient) of the two layers, and with different geometrical configurations (source detector distance, thickness of the upper layer). Practical considerations on the design of time domain and continuous wave systems are derived. The methods and the results are of interest for many research fields such as biomedical optics and advanced microscopy.

Application limits of the scaling relations for Monte Carlo simulations in diffuse optics. Part 1: theory.

Monte Carlo (MC) is a powerful tool to study photon migration in scattering media, yet quite time-consuming to solve inverse problems. To speed up MC-simulations, scaling relations can be applied to an existing initial MC-simulation to generate a new data-set with different optical properties. We named this approach trajectory-based since it uses the knowledge of the detected photon trajectories of the initial MC-simulation, in opposition to the slower photon-based approach, where a novel MC-simulation is rerun with new optical properties. We investigated the convergence and applicability limits of the scaling relations, both related to the likelihood that the sample of trajectories considered is representative also for the new optical properties. For absorption, the scaling relation contains smoothly converging Lambert-Beer factors, whereas for scattering it is the product of two quickly diverging factors, whose ratio, for NIRS cases, can easily reach ten orders of magnitude. We investigated such instability by studying the probability-distribution for the number of scattering events in trajectories of given length. We propose a convergence test of the scattering scaling relation based on the minimum-maximum number of scattering events in recorded trajectories. We also studied the dependence of MC-simulations on optical properties, most critical in inverse problems, finding that scattering derivatives are ascribed to small deviations in the distribution of scattering events from a Poisson distribution. This paper, which can also serve as a tutorial, helps to understand the physics of the scaling relations with the causes of their limitations and devise new strategies to deal with them.

Fluence rate directly derived from photon pathlengths: a tool for Monte Carlo simulations in biomedical optics.

In biomedical optics, the mean fluence rate of photons, assessed in a sub-volume of a propagating medium, is classically obtained in Monte Carlo simulations by taking into account the power deposited by the absorbed photons in the sub-volume. In the present contribution, we propose and analytically demonstrate an alternative method based on the assessment of the mean pathlength traveled by all the photons inside the sub-volume. Few practical examples of its applications are given. This method has the advantage of improving, in many cases, the statistics and the convergence of the Monte Carlo simulations. Further, it also works when the absorption coefficient is nil and for a non-constant spatial distribution of the absorption coefficient inside the sub-volume. The proposed approach is a re-visitation of a well-known method applied in radiation and nuclear physics in the context of radiative transfer, where it can be derived in a more natural manner.

On the mean path length invariance property for random walks of animals in open environment.

Random walks are common in nature and are at the basis of many different phenomena that span from neutrons and light scattering to the behaviour of animals. Despite the evident differences among all these phenomena, theory predicts that they all share a common fascinating feature known as Invariance Property (IP). In a nutshell, IP means that the mean length of the total path of a random walker inside a closed domain is fixed by the geometry and size of the medium. Such a property has been demonstrated to hold not only in optics, but recently also in the field of biology, by studying the movement of bacteria. However, the range of validity of such a universal property, strictly linked to the fulfilment of equilibrium conditions and to the statistical distributions of the steps of the random walkers, is not trivial and needs to be studied in different contexts, such as in the case of biological entities occupied in random foraging in an open environment. Hence, in this paper the IP in a virtual medium inside an open environment has been studied by using actual movements of animals recorded in nature. In particular, we analysed the behaviour of a grazer mollusc, the chiton Acanthopleura granulata. The results depart from those predicted by the IP when the dimension of the medium increases. Such findings are framed in both the condition of nonequilibrium of the walkers, which is typical of animals in nature, and the characteristics of actual animal movements.

Two-step verification method for Monte Carlo codes in biomedical optics applications.

SIGNIFICANCE: Code verification is an unavoidable step prior to using a Monte Carlo (MC) code. Indeed, in biomedical optics, a widespread verification procedure for MC codes is still missing. Analytical benchmarks that can be easily used for the verification of different MC routines offer an important resource. AIM: We aim to provide a two-step verification procedure for MC codes enabling the two main tasks of an MC simulator: (1) the generation of photons' trajectories and (2) the intersections of trajectories with boundaries separating the regions with different optical properties. The proposed method is purely based on elementary analytical benchmarks, therefore, the correctness of an MC code can be assessed with a one-sample t-test. APPROACH: The two-step verification is based on the following two analytical benchmarks: (1) the exact analytical formulas for the statistical moments of the spatial coordinates where the scattering events occur in an infinite medium and (2) the exact invariant solutions of the radiative transfer equation for radiance, fluence rate, and mean path length in media subjected to a Lambertian illumination. RESULTS: We carried out a wide set of comparisons between MC results and the two analytical benchmarks for a wide range of optical properties (from non-scattering to highly scattering media, with different types of scattering functions) in an infinite non-absorbing medium (step 1) and in a non-absorbing slab (step 2). The deviations between MC results and exact analytical values are usually within two standard errors (i.e., t-tests not rejected at a 5% level of significance). The comparisons show that the accuracy of the verification increases with the number of simulated trajectories so that, in principle, an arbitrary accuracy can be obtained. CONCLUSIONS: Given the simplicity of the verification method proposed, we envision that it can be widely used in the field of biomedical optics.

Direct Measurement of the Reduced Scattering Coefficient by a Calibrated Random Laser Sensor.

The research in optical sensors has been largely encouraged by the demand for low-cost and less or non-invasive new detection strategies. The invention of the random laser has opened a new frontier in optics, providing also the opportunity to explore new possibilities in the field of sensing, besides several different and peculiar phenomena. The main advantage in exploiting the physical principle of the random laser in optical sensors is due to the presence of the stimulated emission mechanism, which allows amplification and spectral modification of the signal. Here, we present a step forward in the exploitation of this optical phenomenon by a revisitation of a previous experimental setup, as well as the measurement method, in particular to mitigate the instability of the results due to shot-to-shot pump energy fluctuations. In particular, the main novelties of the setup are the use of optical fibers, a reference sensor, and a peristaltic pump. These improvements are devoted to: eliminating optical beam alignment issues; improving portability; mitigating the variation in pump energy and gain medium performances over time; realizing an easy and rapid change of the sensed medium. The results showed that such a setup can be considered a prototype for a portable device for directly measuring the scattering of liquid samples, without resorting to complicated numerical or analytic inversion procedures of the measured data, once the suitable calibration of the system is performed.

Verification method of Monte Carlo codes for transport processes with arbitrary accuracy.

In this work, we present a robust and powerful method for the verification, with arbitrary accuracy, of Monte Carlo codes for simulating random walks in complex media. Such random walks are typical of photon propagation in turbid media, scattering of particles, i.e., neutrons in a nuclear reactor or animal/humans' migration. Among the numerous applications, Monte Carlo method is also considered a gold standard for numerically "solving" the scalar radiative transport equation even in complex geometries and distributions of the optical properties. In this work, we apply the verification method to a Monte Carlo code which is a forward problem solver extensively used for typical applications in the field of tissue optics. The method is based on the well-known law of average path length invariance when the entrance of the entities/particles in a medium obeys to a simple cosine law, i.e., Lambertian entrance, and annihilation of particles inside the medium is absent. By using this law we achieve two important points: (1) the invariance of the average path length guarantees that the expected value is known regardless of the complexity of the medium; (2) the accuracy of a Monte Carlo code can be assessed by simple statistical tests. We will show that we can reach an arbitrary accuracy of the estimated average pathlength as the number of simulated trajectories increases. The method can be applied in complete generality versus the scattering and geometrical properties of the medium, as well as in presence of refractive index mismatches in the optical case. In particular, this verification method is reliable to detect inaccuracies in the treatment of boundaries of finite media. The results presented in this paper, obtained by a standard computer machine, show a verification of our Monte Carlo code up to the sixth decimal digit. We discuss how this method can provide a fundamental tool for the verification of Monte Carlo codes in the geometry of interest, without resorting to simpler geometries and uniform distribution of the scattering properties.

Random laser based method for direct measurement of scattering properties.

Optical sensing is a very important method for investigating different kinds of samples. Recently, we proposed a new kind of optical sensor based on random lasing [ Sci. Rep.6, 35225 (2016)], that couples the advantages of stimulated emission in detecting small variations on scattering properties of a sensed material, to the needs of no alteration of the sample under investigation. Here, we present a method to achieve a quantitative measurement of the scattering properties of a material. The results on samples of calibrated microspheres show a dependence of the peak intensity of the emission spectrum on the transport mean free path of the light within the sample, whatever the dimension (down to ≈100 nm of particle diameter) and the concentration of scatterers dispersed in the sensed material. A direct and fast measurement of the scattering properties is obtained by calibration with a well-known and inexpensive reference medium.

Precursor propagation in inhomogeneous broadened media: experimental and numerical simulation.

We report experimental results on the propagation temporal characteristics of the precursor in an inhomogeneous sample. The transient behavior of a step-like pulse in an atomic hot medium is two orders of magnitude faster than the radiative broadened case up to now presented in the literature. Moreover, we show the dependence on the resonant or nonresonant condition. Numerical simulations compare favorable to experimental results.

Robustness of replica symmetry breaking phenomenology in random laser.

Random lasers are optical sources where light is amplified by stimulated emission along random paths through an amplifying scattering medium. Connections between their physics and the one of quenched disordered nonlinear systems, notably spin glasses, have been recently suggested. Here we report a first experimental study of correlations of spectral fluctuations intensity in a random laser medium where the scatterers displacement significantly changes among consecutive shots. Remarkably, our results reveal that the replica symmetry breaking (RSB) phenomenology is robust with respect to an averaging over different realizations of the disorder. Moreover, besides opening new intriguing questions about the understanding of such a phenomenon, this work aims to clarify the connection between the RSB with the onset of the Lévy regime, i.e. the fluctuations regime that is a peculiar feature of the random lasing under critical conditions. Our results suggest that the former occurs independently of the latter and then the RSB phenomenology is a generic feature linked to the random laser threshold.

A new class of optical sensors: a random laser based device.

In a random laser the optical feedback is provided by scattering rather than by an optical cavity. Then, since its emission characteristics are very susceptible to the scattering details, it is a natural candidate for making active sensors to use as a diagnostic tool for disordered media like biological samples. However, the methods reported up to now, requiring the injection of toxic substances in the sample, have the drawback of altering the physical-chemical composition of the medium and are not suitable for in-vivo measurements. Here we present a random laser based sensor that overcomes these problems by keeping gain and diffusion separated. We provide an experimental characterisation of the sensor by using a reference diffusive liquid phantom and we show that, compared to a passive method, this sensor takes advantage of the gain and spectral properties of the random laser principle.

Recovering the propagation delay of an optical pulse.

Causality and special relativity pose an upper limit to the amount of advance that an optical pulse can acquire during a superluminal propagation. Such a limit can be circumvented if the pulse, before entering the superluminal medium, is retarded by letting it propagate under normal dispersion. We present an experimental evidence of this fact by showing that a laser pulse propagating in an atomic vapor, quasi resonant with an inverted transition and in conditions of anomalous dispersion, moves faster if it is previously retarded in a cell containing the same medium with no population inversion. Optical transmission lines often need an amplification stage to overcome the signal attenuation and the unavoidable delay respect to propagation at c; in this paper we tailor such stage to provide also an optical controlled recover of such delay. We believe that our results can open exciting prospects for real-life optical data processing and communication.