{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:44:34Z","timestamp":1760147074934,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T00:00:00Z","timestamp":1672704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Chemnitz University of Technology and by the Deutsche Forschungsgemeinschaft","award":["491193532"],"award-info":[{"award-number":["491193532"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Optical microcavity billiards are a paradigm of a mesoscopic model system for quantum chaos. We demonstrate the action and origin of ray-wave correspondence in real and phase space using far-field emission characteristics and Husimi functions. Whereas universality induced by the invariant-measure dominated far-field emission is known to be a feature shaping the properties of many lasing optical microcavities, the situation changes in the presence of sources that we discuss here. We investigate the source-induced dynamics and the resulting limits of universality while we find ray-picture results to remain a useful tool in order to understand the wave behaviour of optical microcavities with sources. We demonstrate the source-induced dynamics in phase space from the source ignition until a stationary regime is reached comparing results from ray, ray-with-phase, and wave simulations and explore ray\u2013wave correspondence.<\/jats:p>","DOI":"10.3390\/e25010095","type":"journal-article","created":{"date-parts":[[2023,1,3]],"date-time":"2023-01-03T02:51:39Z","timestamp":1672714299000},"page":"95","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Universality and beyond in Optical Microcavity Billiards with Source-Induced Dynamics"],"prefix":"10.3390","volume":"25","author":[{"given":"Lukas","family":"Seemann","sequence":"first","affiliation":[{"name":"Institute of Physics, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0596-6982","authenticated-orcid":false,"given":"Martina","family":"Hentschel","sequence":"additional","affiliation":[{"name":"Institute of Physics, Technische Universit\u00e4t Chemnitz, D-09107 Chemnitz, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"St\u00f6ckmann, H.J. (1999). Quantum Dots: An Introduction, Cambridge University Press.","DOI":"10.1017\/CBO9780511524622"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Haake, F. (2001). Quantum Signatures of Chaos, Springer.","DOI":"10.1007\/978-3-662-04506-0"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nakamura, K., and Harayama, T. (2004). Quantum Chaos and Quantum Dots, Oxford University Press.","DOI":"10.1093\/oso\/9780198525899.001.0001"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF02798790","article-title":"On the connection between quantization of nonintegrable systems and statistical theory of spectra","volume":"28","author":"Casati","year":"1980","journal-title":"Lett. Nuovo C. (1971\u20131985)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevLett.52.1","article-title":"Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws","volume":"52","author":"Bohigas","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","unstructured":"Mehta, M.L. (2004). Random Matrix Theory, Elsevier."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Imry, Y. (1997). Introduction to Mesoscopic Physics, Oxford University Press.","DOI":"10.1093\/oso\/9780195101676.001.0001"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Vahala, K. (2004). Optical Microcavities, World Scientific.","DOI":"10.1142\/9789812565730"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1038\/385045a0","article-title":"Ray and wave chaos in asymmetric resonant optical cavities","volume":"385","author":"Stone","year":"1997","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1556","DOI":"10.1126\/science.280.5369.1556","article-title":"High-Power Directional Emission from Microlasers with Chaotic Resonators","volume":"280","author":"Gmachl","year":"1998","journal-title":"Science"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"014102","DOI":"10.1103\/PhysRevLett.88.014102","article-title":"Light scarring in an optical fiber","volume":"88","author":"Doya","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1088\/1367-2630\/8\/3\/046","article-title":"Directed emission from a dielectric microwave billiard with quadrupolar shape","volume":"8","author":"Kuhl","year":"2006","journal-title":"New J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"174103","DOI":"10.1103\/PhysRevLett.100.174103","article-title":"Dynamical Tunneling in Mushroom Billiards","volume":"100","author":"Ketzmerick","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"024101","DOI":"10.1103\/PhysRevLett.108.024101","article-title":"PT Symmetry and Spontaneous Symmetry Breaking in a Microwave Billiard","volume":"108","author":"Bittner","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"203901","DOI":"10.1103\/PhysRevLett.127.203901","article-title":"M\u00f6bius Strip Microlasers: A Testbed for Non-Euclidean Photonics","volume":"127","author":"Song","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"155436","DOI":"10.1103\/PhysRevB.104.155436","article-title":"Dirac fermion optics and directed emission from single- and bilayer graphene cavities","volume":"104","author":"Schrepfer","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1103\/RevModPhys.87.61","article-title":"Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics","volume":"87","author":"Cao","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"101107","DOI":"10.1063\/1.4914498","article-title":"Unidirectional light emission from low-index polymer microlasers","volume":"106","author":"Schermer","year":"2015","journal-title":"Appl. Phys. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"033901","DOI":"10.1103\/PhysRevLett.100.033901","article-title":"Combining Directional Light Output and Ultralow Loss in Deformed Microdisks","volume":"100","author":"Wiersig","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"041807","DOI":"10.1103\/PhysRevA.80.041807","article-title":"Chaotic microcavity laser with high quality factor and unidirectional output","volume":"80","author":"Song","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"141107","DOI":"10.1063\/1.3242014","article-title":"Lasing characteristics of a Lima\u00e7on-shaped microcavity laser","volume":"95","author":"Yi","year":"2009","journal-title":"Appl. Phys. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"031801","DOI":"10.1103\/PhysRevA.80.031801","article-title":"Ray-wave correspondence in lima\u00e7on-shaped semiconductor microcavities","volume":"80","author":"Shinohara","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"251101","DOI":"10.1063\/1.3153276","article-title":"Directional emission and universal far-field behavior from semiconductor lasers with Lima\u00e7on-shaped microcavity","volume":"94","author":"Yan","year":"2009","journal-title":"Appl. Phys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"125018","DOI":"10.1088\/1367-2630\/11\/12\/125018","article-title":"Deformed microcavity quantum cascade lasers with directional emission","volume":"11","author":"Wang","year":"2009","journal-title":"New J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"021116","DOI":"10.1063\/1.4733726","article-title":"Directional whispering gallery mode emission from Lima\u00e7on-shaped electrically pumped quantum dot micropillar lasers","volume":"101","author":"Albert","year":"2012","journal-title":"Appl. Phys. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"061801","DOI":"10.1103\/PhysRevA.72.061801","article-title":"Scarred resonances and steady probability distribution in a chaotic microcavity","volume":"72","author":"Lee","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"You, M., Sakakibara, D., Makino, K., Morishita, Y., Matsumura, K., Kawashima, Y., Yoshikawa, M., Tonosaki, M., Kanno, K., and Uchida, A. (2022). Universal Single-Mode Lasing in Fully Chaotic Billiard Lasers. Entropy, 24.","DOI":"10.3390\/e24111648"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"224101","DOI":"10.1103\/PhysRevLett.99.224101","article-title":"Selective amplification of scars in a chaotic optical fiber","volume":"99","author":"Michel","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1364\/OL.34.000163","article-title":"Designing and understanding directional emission from spiral microlasers","volume":"34","author":"Hentschel","year":"2009","journal-title":"Opt. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.cpc.2009.11.008","article-title":"MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method","volume":"181","author":"Oskooi","year":"2010","journal-title":"Comput. Phys. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1209\/epl\/i2003-00421-1","article-title":"Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond","volume":"62","author":"Hentschel","year":"2003","journal-title":"Europhys. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2413","DOI":"10.1088\/0305-4470\/17\/12\/013","article-title":"Semiclassical level spacings when regular and chaotic orbits coexist","volume":"17","author":"Berry","year":"1984","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"8059","DOI":"10.1088\/0305-4470\/27\/24\/017","article-title":"Semiclassical energy level statistics in the transition region between integrability and chaos: Transition from Brody-like to Berry-Robnik behaviour","volume":"27","author":"Prosen","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1364\/OL.27.000007","article-title":"Deviation from Snell\u2019s law for beams transmitted near the critical angle: Application to microcavity lasers","volume":"27","author":"Tureci","year":"2002","journal-title":"Opt. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"094102","DOI":"10.1103\/PhysRevLett.88.094102","article-title":"Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators","volume":"88","author":"Rex","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"045603","DOI":"10.1103\/PhysRevE.65.045603","article-title":"Fresnel laws at curved dielectric interfaces of microresonators","volume":"65","author":"Hentschel","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"243903","DOI":"10.1103\/PhysRevLett.96.243903","article-title":"Correcting Ray Optics at Curved Dielectric Microresonator Interfaces: Phase-Space Unification of Fresnel Filtering and the Goos-H\u00e4nchen Shift","volume":"96","author":"Schomerus","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"016201","DOI":"10.1103\/PhysRevE.78.016201","article-title":"Goos-H\u00e4nchen shift and localization of optical modes in deformed microcavities","volume":"78","author":"Unterhinninghofen","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"042916","DOI":"10.1103\/PhysRevE.92.042916","article-title":"Ray-wave correspondence in chaotic dielectric billiards","volume":"92","author":"Harayama","year":"2015","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"64001","DOI":"10.1209\/0295-5075\/107\/64001","article-title":"Curvature dependence of semiclassical corrections to ray optics: How Goos-H\u00e4nchen shift and Fresnel filtering deviate from the planar case result","volume":"107","author":"Kreismann","year":"2014","journal-title":"EPL"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"125603","DOI":"10.1088\/2040-8986\/aa93c2","article-title":"Consequences of a wave-correction extended ray dynamics for integrable and chaotic optical microcavities","volume":"19","author":"Hentschel","year":"2017","journal-title":"J. Opt."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1002\/andp.19474360704","article-title":"Ein neuer und fundamentaler Versuch zur Totalreflexion","volume":"436","author":"Goos","year":"1947","journal-title":"Ann. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1002\/andp.19494400312","article-title":"Neumessung des Strahlversetzungseffektes bei Totalreflexion","volume":"440","author":"Goos","year":"1949","journal-title":"Ann. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/95\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:56:58Z","timestamp":1760119018000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/95"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,3]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010095"],"URL":"https:\/\/doi.org\/10.3390\/e25010095","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,1,3]]}}}