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PreprintCC BY 4.0HAL

Shape sensitivity of a reflection problem in a ray tracing framework

  • Samuel Amstutz
  • Benjamin Aymard
  • Audric Drogoul
  • Luca Gorini
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Abstract

We investigate the sensitivity of a reflection problem with respect to shape perturbations of the reflecting surface. A geometrical optics framework is assumed, in which the incident light rays are described by random variables. The rays are propagated through a ray tracing operator obtained by composing a projection onto the mirror, a specular reflection and a projection onto the focal plane. Following Hadamard's boundary variation method, we prove the Fréchet differentiability with respect to the deformation field of a generic class of cost functions depending on the impact points and directions on the focal plane, and we derive an explicit expression of the shape derivative. The derivative formulas are validated numerically, and two-dimensional imaging and non-imaging optical design problems are treated.

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Cite

BibTeXRIS
BibTeX
@misc{amstutz2026shape,
  title = {{Shape sensitivity of a reflection problem in a ray tracing framework}},
  author = {Samuel Amstutz and Benjamin Aymard and Audric Drogoul and Luca Gorini},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05734819v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Shape sensitivity of a reflection problem in a ray tracing framework
AU  - Samuel Amstutz
AU  - Benjamin Aymard
AU  - Audric Drogoul
AU  - Luca Gorini
PY  - 2026
DA  - 2026/09/01
PB  - HAL
UR  - https://hal.science/hal-05734819v1
AB  - We investigate the sensitivity of a reflection problem with respect to shape perturbations of the reflecting surface. A geometrical optics framework is assumed, in which the incident light rays are described by random variables. The rays are propagated through a ray tracing operator obtained by composing a projection onto the mirror, a specular reflection and a projection onto the focal plane. Following Hadamard's boundary variation method, we prove the Fréchet differentiability with respect to the deformation field of a generic class of cost functions depending on the impact points and directions on the focal plane, and we derive an explicit expression of the shape derivative. The derivative formulas are validated numerically, and two-dimensional imaging and non-imaging optical design problems are treated.
ER  -