Abstract
The Snell law in vector form is stated and proved using wavefronts. Surfaces for uniform refraction both in the near field and far field cases are calculated.
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Notes
- 1.
Since the refraction angle depends on the frequency of the radiation, we assume radiation is monochromatic.
- 2.
If \(\theta _1>\theta _c\), then the phenomenon of total internal reflection occurs, see Fig. 12.1c.
- 3.
See the paper [39] where it is mentioned that the demonstration of the optical properties of ovals is given in [44, Book I, Propositions XCVII and XCVIII, pp. 247-248]. See also [34, Chapter VI]. Also the paper [39] contains generalized ovals having more than two foci; see also [40] for more detailed results.
References
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Maxwell, J.C.: On the description of oval curves, and those having a plurality of foci; with remarks from Prof. Forbes;. Proc. R. Soc. Edin. II, 1–3 (1846). https://archive.org/details/scientificpapers01maxw
Maxwell, J.C.: In: Harman, P.M. (eds.) The Scientific Letters and Papers of James Clerk Maxwell, vol. 1, pp. 1846–1862. Cambridge University Press, Cambridge (1990)
Newton, S.I.: Newton’s Principia, the Mathematical Principles of Natural Philosophy. First American Edition 1846 edn., Daniel Adee, New York (1687). https://archive.org/details/newtonspmathema00newtrich.
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Gutiérrez, C.E. (2023). Snell’s Law of Refraction. In: Optimal Transport and Applications to Geometric Optics. SpringerBriefs on PDEs and Data Science. Springer, Singapore. https://doi.org/10.1007/978-981-99-4867-3_12
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DOI: https://doi.org/10.1007/978-981-99-4867-3_12
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