Deterministic Approximations of Random Reflectors
Within classical optics, one may add microscopic "roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflec...
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| Format: | Article |
| Language: | en_US |
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American Mathematical Society (AMS)
2015
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| Online Access: | http://hdl.handle.net/1721.1/93154 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1826213182506532864 |
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| author | Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger |
| author_sort | Sheffield, Scott Roger |
| collection | MIT |
| description | Within classical optics, one may add microscopic "roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness).
We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way. |
| first_indexed | 2024-09-23T15:44:38Z |
| format | Article |
| id | mit-1721.1/93154 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:44:38Z |
| publishDate | 2015 |
| publisher | American Mathematical Society (AMS) |
| record_format | dspace |
| spelling | mit-1721.1/931542022-10-04T05:30:54Z Deterministic Approximations of Random Reflectors Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger Within classical optics, one may add microscopic "roughness'' to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness). We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way. National Science Foundation (U.S.) (Grant DMS 0645585) 2015-01-22T20:09:44Z 2015-01-22T20:09:44Z 2013-06 2012-04 Article http://purl.org/eprint/type/JournalArticle 0002-9947 1088-6850 http://hdl.handle.net/1721.1/93154 Angel, Omer, Krysztof Burdzy, and Scott Sheffield. "Deterministic Approximations of Random Reflectors." Trans. Amer. Math. Soc. 365 (2013): 6367-6383. © 2013 American Mathematical Society https://orcid.org/0000-0002-5951-4933 en_US http://www.ams.org/journals/tran/2013-365-12/S0002-9947-2013-05851-5/ Transactions of the American Mathematical Society Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Mathematical Society (AMS) American Mathematical Society |
| spellingShingle | Sheffield, Scott Roger Deterministic Approximations of Random Reflectors |
| title | Deterministic Approximations of Random Reflectors |
| title_full | Deterministic Approximations of Random Reflectors |
| title_fullStr | Deterministic Approximations of Random Reflectors |
| title_full_unstemmed | Deterministic Approximations of Random Reflectors |
| title_short | Deterministic Approximations of Random Reflectors |
| title_sort | deterministic approximations of random reflectors |
| url | http://hdl.handle.net/1721.1/93154 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT sheffieldscottroger deterministicapproximationsofrandomreflectors |