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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.13986 |
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| _version_ | 1866909971605618688 |
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| author | Zeller, Claude Cordery, Robert |
| author_facet | Zeller, Claude Cordery, Robert |
| contents | A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-return statistics in bounded radiative transport: A Motzkin polynomial framework Zeller, Claude Cordery, Robert Optics A photon entering a scattering medium executes a three-dimensional random walk determined by the Henyey-Greenstein phase function. The photon either reaches the boundary for a first passage or is absorbed. Projecting the walk onto the axial direction produces a one-dimensional alternating process whose peaks and valleys correspond to changes in the sign of the projected step. This reduction preserves first-return and first-passage events and leads to a representation in terms of Motzkin-type polynomials. The analytical formulation is complete except for boundary-constrained return terms, which appear as high-order integrals. We treat these contributions with a single truncation factor determined from Monte Carlo simulations of first-return distributions over a wide range of anisotropy g and scattering steps ms. The resulting factor follows a Cauchy distribution. Incorporating it yields first-return probabilities in agreement with full three-dimensional Monte Carlo to within 2% for g<=0.7. The approach gives backscattering coefficients from phase-function integrals and provides an efficient alternative to full three-dimensional simulations for problems of radiative transport in semi-infinite media. |
| title | First-return statistics in bounded radiative transport: A Motzkin polynomial framework |
| topic | Optics |
| url | https://arxiv.org/abs/2512.13986 |