An Algebraic Approach to Bifurcations in Kerr Ring and Fabry-Perot Resonators
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914250203594752 |
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| author | Mazo-Vasquez, Juan Diego Gohsrich, Julius T. Kunst, Flore K. Hill, Lewis |
| author_facet | Mazo-Vasquez, Juan Diego Gohsrich, Julius T. Kunst, Flore K. Hill, Lewis |
| contents | Nonlinear phenomena such as optical bistability and spontaneous symmetry breaking play a central role in Kerr resonators, and are increasingly exploited in photonic integrated circuits for all-optical information processing. In this work, we present an analytical framework allowing to find the stationary states and their bifurcations for the propagating fields in Kerr ring and Fabry-Perot resonators, which can be generalized to other nonlinear systems. Using tools from nonlinear algebra, namely, polynomial resultants and Groebner bases, we derive compact polynomial expressions describing the system's full solution in both intensity and amplitude representations. The bifurcations follow directly from these expressions, and are additionally characterized as exceptional points of an auxiliary linear non-Hermitian system. Together, these results unify optical bistability and spontaneous symmetry breaking within a single analytical framework, and offer a route toward improved control of nonlinear optical systems, and the design of photonic devices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Algebraic Approach to Bifurcations in Kerr Ring and Fabry-Perot Resonators Mazo-Vasquez, Juan Diego Gohsrich, Julius T. Kunst, Flore K. Hill, Lewis Optics Mathematical Physics Nonlinear phenomena such as optical bistability and spontaneous symmetry breaking play a central role in Kerr resonators, and are increasingly exploited in photonic integrated circuits for all-optical information processing. In this work, we present an analytical framework allowing to find the stationary states and their bifurcations for the propagating fields in Kerr ring and Fabry-Perot resonators, which can be generalized to other nonlinear systems. Using tools from nonlinear algebra, namely, polynomial resultants and Groebner bases, we derive compact polynomial expressions describing the system's full solution in both intensity and amplitude representations. The bifurcations follow directly from these expressions, and are additionally characterized as exceptional points of an auxiliary linear non-Hermitian system. Together, these results unify optical bistability and spontaneous symmetry breaking within a single analytical framework, and offer a route toward improved control of nonlinear optical systems, and the design of photonic devices. |
| title | An Algebraic Approach to Bifurcations in Kerr Ring and Fabry-Perot Resonators |
| topic | Optics Mathematical Physics |
| url | https://arxiv.org/abs/2512.14168 |