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<p>“Illuminance is a kind of ‘probability distribution’.” This claim appears to be absurd. However, from a statistical standpoint, individual randomness does not conflict with the determinacy of the totality. Here, as a geometric-optics example, we investigate the behavior of a random ray passing through a rotationally symmetric gradient-index (GRIN) focusing lens, and formulate a theorem to determine the probability density of its position. Choosing the rotational (principal) axis as the <i>z</i>-axis, a random ray’s position in 3D space is characterized by a random vector (<i>X</i>(<i>z</i>), <i>Y</i>(<i>z</i>)), which can also be seen as the distribution of a 2D stochastic process, with propagation distance <i>z</i> as the “time” parameter. Statistically, as the number of random parallel incident rays approaches infinity, they can reconstruct the true distribution of illuminance (which is proportional to the ray density) of a focusing field, aligning with a deterministic distribution law. The analytical model is tested using a Luneburg lens and a half Maxwell fish-eye, with results matching well with numerical simulations from Huygens-Fresnel Principle. Our study opens up a precedent of employing stochastic models to analyze 3D light fields within macro-scale focusing and image systems.</p>

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Stochastic model for calculating three-dimensional illuminance distribution in a rotationally symmetric gradient-index focusing system

  • 期刊论文

  • 发表日期：

    2025年11月28日

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  • Wanguo Liu

摘要

“Illuminance is a kind of ‘probability distribution’.” This claim appears to be absurd. However, from a statistical standpoint, individual randomness does not conflict with the determinacy of the totality. Here, as a geometric-optics example, we investigate the behavior of a random ray passing through a rotationally symmetric gradient-index (GRIN) focusing lens, and formulate a theorem to determine the probability density of its position. Choosing the rotational (principal) axis as the z-axis, a random ray’s position in 3D space is characterized by a random vector (X(z), Y(z)), which can also be seen as the distribution of a 2D stochastic process, with propagation distance z as the “time” parameter. Statistically, as the number of random parallel incident rays approaches infinity, they can reconstruct the true distribution of illuminance (which is proportional to the ray density) of a focusing field, aligning with a deterministic distribution law. The analytical model is tested using a Luneburg lens and a half Maxwell fish-eye, with results matching well with numerical simulations from Huygens-Fresnel Principle. Our study opens up a precedent of employing stochastic models to analyze 3D light fields within macro-scale focusing and image systems.