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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.00173 |
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| _version_ | 1866917367879041024 |
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| author | Zeller, C Cordery, R |
| author_facet | Zeller, C Cordery, R |
| contents | We study first-return statistics for photons undergoing three-dimensional Henyey-Greenstein scattering in a semi-infinite medium. In previous work, we showed that one-dimensional first-passage probabilities can be expanded using Catalan and Motzkin generating functions. Extending this framework to three dimensions requires introducing a Boundary Truncation Factor (BTF), which accounts for the restricted angular phase space imposed by the boundary.
Extensive Monte Carlo simulations are used to determine the BTF empirically as a function of scattering order and anisotropy. For moderate anisotropy, the BTF is accurately described by a Cauchy kernel, with parameters depending only on the Henyey-Greenstein asymmetry factor. This closed-form expression reproduces Monte Carlo results with 1-2% accuracy over a broad range of scattering orders.
At higher anisotropy, systematic deviations from the Cauchy form are observed. These deviations can be reduced using a one-parameter generalized kernel. We further extend the framework to oblique incidence by replacing the normal-incidence return probability with a Legendre-series formulation. The BTF parameters and Motzkin counting machinery remain independent of the incidence angle, so only the anchor point of the algorithm changes.
The resulting framework provides a computationally efficient mapping from three-dimensional anisotropic transport at arbitrary incidence to one-dimensional combinatorial first-passage theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00173 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | First-Return Statistics in Henyey-Greenstein Scattering: Colored Motzkin Polynomials and the Cauchy Kernel Zeller, C Cordery, R Optics We study first-return statistics for photons undergoing three-dimensional Henyey-Greenstein scattering in a semi-infinite medium. In previous work, we showed that one-dimensional first-passage probabilities can be expanded using Catalan and Motzkin generating functions. Extending this framework to three dimensions requires introducing a Boundary Truncation Factor (BTF), which accounts for the restricted angular phase space imposed by the boundary. Extensive Monte Carlo simulations are used to determine the BTF empirically as a function of scattering order and anisotropy. For moderate anisotropy, the BTF is accurately described by a Cauchy kernel, with parameters depending only on the Henyey-Greenstein asymmetry factor. This closed-form expression reproduces Monte Carlo results with 1-2% accuracy over a broad range of scattering orders. At higher anisotropy, systematic deviations from the Cauchy form are observed. These deviations can be reduced using a one-parameter generalized kernel. We further extend the framework to oblique incidence by replacing the normal-incidence return probability with a Legendre-series formulation. The BTF parameters and Motzkin counting machinery remain independent of the incidence angle, so only the anchor point of the algorithm changes. The resulting framework provides a computationally efficient mapping from three-dimensional anisotropic transport at arbitrary incidence to one-dimensional combinatorial first-passage theory. |
| title | First-Return Statistics in Henyey-Greenstein Scattering: Colored Motzkin Polynomials and the Cauchy Kernel |
| topic | Optics |
| url | https://arxiv.org/abs/2601.00173 |