Accurate and efficient solutions of the three dimensional radiative transport equation were derived in all domains for the case of layered scattering media. Index mismatched boundary conditions based on Fresnel’s equations were implemented. Arbitrary rotationally symmetric phase functions can be applied to characterize the scattering in the turbid media. Solutions were derived for an obliquely incident beam having arbitrary spatial profiles. The derived solutions were successfully validated with Monte Carlo simulations and partly compared with analytical solutions of the diffusion equation.

In this paper, we consider the time-dependent Schrödinger equation: $i\frac{\partial \psi (x,t)}{\partial t}=\frac{1}{2}{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}\psi (x,t)+V\left(x\right)\psi (x,t),$ $x\in \mathbb{R},t>0$ with the Riesz space-fractional derivative of order $0<\alpha \le 2$ in the presence of the linear potential $V\left(x\right)=\beta x$ . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of $\alpha =1$ , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential $V\left(\mathit{\rho }\right)=\mathbf{F}·\mathit{\rho }$ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach.

In this paper, we derive some explicit analytical solutions to the $P_3$ equations for the slab geometry that is illuminated by a collimated plane source. The resulting expressions for the total reflectance and transmittance are compared with the corresponding transport theory solution predicted by the Monte Carlo method. Further, for the special case of a non-absorbing anisotropically scattering slab, simple and accurate expressions in the $P_1$ approximation are obtained, yielding for optically thick slabs, the typical behavior of Ohm’s law. In view of the time domain, we present an alternative method to the classical frequency-domain approach avoiding the use of complex numbers.

In this paper, some explicit analytical solutions for single-scattered radiance in a half-space medium under consideration of a reflecting boundary are derived. We consider both a unidirectional beam source as well as an isotropic point source. In addition to direct applications within optical tomography and computer graphics, the obtained solutions are also needed when solving the radiative transport equation after the separation of the unscattered and single-scattered contribution. Comparisons between the derived analytical solutions and the Monte Carlo method display excellent agreement.

In the context of autonomous driving, the augmentation of existing data through simulations provides an elegant solution to the challenge of capturing the full range of adverse weather conditions in training datasets. However, existing physics-based augmentation models typically rely on single scattering approximations to predict light propagation under unfavorable conditions, such as fog. This can prevent the reproduction of important signal characteristics encountered in a real-world environment. Consequently, in this work, Monte Carlo simulations are employed to assess the relevance of multiple-scattered light to the detected LiDAR signal in different types of fog, with scattering phase functions calculated from Mie theory considering real particle size distributions. Bidirectional path tracing is used within the self-developed GPU-accelerated Monte Carlo software to compensate for the unfavorable photon statistics associated with the limited detection aperture of the LiDAR geometry. To validate the Monte Carlo software, an analytical solution of the radiative transfer equation for the time-resolved radiance in terms of scattering orders is derived, thereby providing an explicit representation of the double-scattered contributions. The results of the simulations demonstrate that the shape of the detected signal can be significantly impacted by multiple-scattered light, depending on LiDAR geometry and visibility. In particular, double-scattered light can dominate the overall signal at low visibilities. This indicates that considering higher scattering orders is essential for improving AI-based perception models.

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.