Browse > Article
http://dx.doi.org/10.3807/COPP.2021.5.1.016

Shannon Entropy as an Indicator of the Spatial Resolutions of the Morphologies of the Mode Patterns in an Optical Resonator  

Park, Kyu-Won (Department of Physics and Astronomy & Institute of Applied Physics, Seoul National University)
Kim, Jinuk (Department of Physics and Astronomy & Institute of Applied Physics, Seoul National University)
Moon, Songky (Faculty of Liberal Education, Seoul National University)
Publication Information
Current Optics and Photonics / v.5, no.1, 2021 , pp. 16-22 More about this Journal
Abstract
We present the Shannon entropy as an indicator of the spatial resolutions of the morphologies of the resonance mode patterns in an optical resonator. We obtain each optimized number of mesh points, one of minimum size and the other of maximum one. The optimized mesh-point number of minimum size is determined by the identifiable quantum number through a chi-squared test, whereas the saturation of the difference between Shannon entropies corresponds to the other mesh-point number of maximum size. We also show that the optimized minimum mesh-point increases as the (real) wave number increases and approximates the proportionality constant between them.
Keywords
Shannon entropy; Optical resonator; Boundary element method; Spatial resolution;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. U. Nockel, and A. D. Stone, "Ray and wave chaos in asymmetric resonant optical cavities," Nature 385, 45-47 (1997).   DOI
2 C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, "High-power directional emission from microlasers with chaotic resonators," Science 280, 1556-1564 (1998).   DOI
3 S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, K. An, J.-B. Shim, H.-W. Lee, and S. W. Kim, "Universal output directionality of single modes in a deformed microcavity," Phys. Rev. A 75, 011802 (2007).   DOI
4 Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, "Directional laser emission from a wavelength-scale chaotic microcavity," Phys. Rev. Lett. 105, 103902 (2010).   DOI
5 X. F. Jiang, Y. F. Xiao, C. L. Zou, L. He, C. H. Dong, B. B. Li, Y. Li, F. W. Sun, L. Yang, and Q. Gong, "Highly unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities," Adv. Mater. 24, OP260-OP264 (2012).
6 S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralowthreshold Raman laser using a spherical dielectric microcavity," Nature 415, 621-623 (2002).   DOI
7 J.-W. Ryu, and M. Hentschel, "Ray model and ray-wave correspondence in coupled optical microdisk," Phys. Rev. A 82, 033824 (2010).   DOI
8 M. R. Foreman, D. Keng, E. Treasurer, J. R. Lopez, and S. Arnold, "Whispering gallery mode single nanoparticle detection and sizing: the validity of the dipole approximation," Opt. Lett. 42, 963-966 (2017).   DOI
9 T. Nobis, E. M. Kaidashev, A. Rahm, M. Lorenz, and M. Grundmann, "Whispering gallery modes in nanosized dielectric resonators with hexagonal cross section," Phys. Rev. Lett. 93, 103903 (2004).   DOI
10 S. Shinohara, M. Hentschel, J. Wiersig, T. Sasaki, and T. Harayama, "Ray-wave correspondence in limacon-shaped semiconductor microcavities," Phys. Rev. A 80, 031801 (2009).   DOI
11 J. Kullig, and J. Wiersig, "Q spoiling in deformed optical microdisks due to resonance-assisted tunneling," Phys. Rev. E 94, 022202 (2016).   DOI
12 S. Lock, A. Backer, R. Ketzmerick, and P. Schlagheck, "Regular-to-chaotic tunneling rates: from the quantum to the semiclassical regime," Phys. Rev. Lett. 104, 114101 (2010).   DOI
13 X. Jiang, L. Shao, S.-X. Zhang, X. Yi, J. Wiersig, L. Wang, Q. Gong, M. Loncar, L. Yang, and Y.-F. Xiao, "Chaos-assisted broadband momentum transformation in optical microresonators," Science 358, 344-347 (2017).   DOI
14 W. D. Heiss, "Repulsion of resonance states and exceptional points," Phys. Rev. E 61, 929 (2000).   DOI
15 S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, "Observation of an exceptional point in a chaotic optical microcavity," Phys. Rev. Lett. 103, 134101 (2009).   DOI
16 L. Wang, D. Lippolis, Z. Y. Li, X.-F. Jiang, Q. Gong, and Y.-F. Xiao, "Statistics of chaotic resonances in an optical microcavity," Phys. Rev. E 93, 040201 (2016).   DOI
17 K.-W. Park, S. Moo, Y. Shin, J. Kim, K. Jeong, and K. An, "Shannon entropy and avoided crossings in closed and open quantum billiards," Phys. Rev. E 97, 062205 (2018).   DOI
18 C. E. Shannon, "A mathematical theory of communication," Bell Syst. Tech. J. 27, 379-423 (1948).   DOI
19 E.T. Jaynes, "Information theory and statistical mechanics," Phys. Rev. 106, 620 (1957).   DOI
20 J. W. Godden, F. L. Stahura, and J. Bajorath, "Variability of molecular descriptors in compound databases revealed by shannon entropy calculations," J. Chem. Inf. Comput. Sci. 40, 796-800 (2000).   DOI
21 B. J. Strait, and T. G. Dewey, "The shannon information entropy of protein sequences," Biophys. J. 71, 148-155 (1996).   DOI
22 H. Zenil, S. Hernández-Orozco, N. A. Kiani, F. Soler-Toscano, A. Rueda-Toicen, and J. Tegner, "A decomposition method for global evaluation of shannon entropy and local estimations of algorithmic complexity," Entropy 20, 605 (2018).   DOI
23 F. Evers, and A. D. Mirlin, "Fluctuations of the inverse participation ratio at the Anderson transition," Phys. Rev. Lett. 84, 3690 (2000).   DOI
24 R. A. Kullback, and R. A. Leibler, "On information and sufficiency," Ann. Math. Stat. 22, 79-86 (1951).   DOI
25 J. N. Kapur, and H. K. Kesavan, "Entropy Optimization Principles and Their Applications," in Entropy and energy dissipation in water resources, V. P. Singh, M. Fiorentino. Ed. (Springer, Dordrecht, Nederlands. 1992), pp. 3-20.
26 R. L. Plackett, "Karl pearson and the chi-squared test," Int. Stat. Rev. 51, 59-72 (1983).   DOI
27 N. C. Murphy, R. Wortis, and W. Atkinson, "Generalized inverse participation ratio as a possible measure of localization for interacting systems," Phys. Rev. B 83, 184206 (2011).   DOI
28 J. Zhu, S. K. Ozdemir, Y. F. Xiao, L. Li, L. He, D.-R. Chen and L. Yang, "On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator," Nat. Photonics 4, 46-49 (2009).   DOI
29 S. Anders, W. Schrenk, E. Gornik, and G. Strasser, "Room-temperature operation of electrically pumped quantum-cascade microcylinder lasers," Appl. Phys. Lett. 80, 4094-4096 (2002).   DOI
30 I. Teraoka, and S. Arnold, "Enhancing the sensitivity of a whispering-gallery mode microsphere sensor by a high-refractive-index surface layer," J. Opt. Soc. Am. B 23, 1434-1441 (2006).   DOI
31 F. Bo, J. Wang, J. Cui, S. K. Ozdemir, Y. Kong, G. Zhang, J. Xu, and L. Yang, "Lithium-niobate-silica hybrid whispering-gallery-mode resonators," Adv. Mater. 27, 8075-8081 (2015).   DOI
32 J.-W. Ryu, S.-Y, Lee, and S.W. Kim, "Coupled nonidentical microdisks: avoided crossing of energy levels and unidirectional far-field emission," Phys. Rev. A 79, 053858 (2009).   DOI
33 E. J. Heller, "Bound-state eigenfunctions of classically chaotic hamiltonian systems: scars of periodic orbits," Phys. Rev. Lett. 53, 1515 (1984).   DOI
34 S.-B. Lee, J.-H. Lee, J.-S. Chang, H-. J. Moon, S. W. Kim, and K. An, "Observation of scarred modes in asymmetrically deformed microcylinder lasers," Phys. Rev. Lett. 88, 033903 (2002).   DOI
35 J. Wiersig, "Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities," Phys. Rev. Lett. 97, 253901 (2006).   DOI
36 J. Wiersig, "Boundary element method for resonances in dielectric microcavities," J. Opt. A 5, 53 (2002).   DOI
37 R. González-Ferez, and J. S. Dehesa, "Shannon entropy as an indicator of atomic avoided crossings in strong parallel magnetic and electric fields," Phys. Rev. Lett. 91, 113001 (2003).   DOI
38 Y. L. He, Y. Chen, J. N. Han, Z. B. Zhu, G. X. Xiang, H. D. Liu, B. H. Ma, and D. C. He, "Shannon entropy as an indicator of atomic avoided crossings for Rydberg potassium atoms interacting with a static electric field," Eur. Phys. J. D 69, 283 (2015).   DOI
39 F. Arranz, R.Benito, and F. Borondo, "Shannon entropy at avoided crossing in the quantum transition from order to chaos," Phys. Rev. E 99, 062209 (2019).   DOI
40 A. M. Zhang, and Y. L. Liu, "Improved three-dimensional bubble dynamics model based on boundary element method," J. Comput. Phys. 294, 208-223 (2015).   DOI
41 Z. Liu, S. Sun, A. H. D. Cheng, and Y. Wang, "A fast multipole accelerated indirect boundary element method for broadband scattering of elastic waves in a fluid-saturated poroelastic domain," Int. J. Numer. Anal. Methods Geomech. 42, 2133-2160 (2018).   DOI
42 M. E. Gruber, and T. F. Eibert, "A hybrid Ewald-spectral cavity greens function boundary element method with spectral domain acceleration for modeling of over-moded cavities," IEEE Trans. Antennas Propag. 63, 2627-2635 (2015).   DOI
43 I. Rotter, "A non-Hermitian Hamilton operator and the physics of open quantum systems," J. Phys. A: Math. Theor. 42, 153001 (2009).   DOI
44 K.-W. Park, S. Moon, H. Jeong, J. Kim, and K. Jeong, "Non-hermiticity and conservation of orthogonal relation in dielectric microcavity," J. Phys. Commun. 2, 075007 (2018).   DOI