Abstract
In this paper the eigenmodes of finite length microring resonator arrays have been systematically studied, both analytically using temporal coupled mode theory (CMT), and numerically using two-dimensional finite element method (FEM). The method for obtaining the values of parameters appearing in simplified CMT model using results of FEM calculations is presented. Calculations were carried out by COMSOL FEM packages for a wide range of distances between the rings. The obtained results reveal that the rotational degeneracy is preserved for a wide range of interrings distances. It is shown how the eigenvalue spectrum depends on the number of cavities in the system. The differences for the cases of odd and even numbers of rings, and its implications on actual applications, are discussed in details. The central branch appearing in odd-number arrays plays significant role for the delay-lines and optical buffering applications. Based on the first order perturbation theory, an analytical expressions for the eigenfrequencies of arbitrary (finite) length linear array of microring resonators are derived. The analytical expressions describing eigenfrequencies are useful for determining positions of the maxima in transfer characteristics in microring arrays with external buses.
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Bayer, M., Gutbrod, T., Reithmaier, J.P., Forchel, A.: Optical modes in photonic molecules. Phys. Rev. Lett. 81, 2583–2585 (1998)
Bogaerts, W., De Heyn, P., Van Vaerenbergh, T., De Vos, K., Selvaraja, S., Claes, T., Dumon, P., Bienstman, P., Van Thourhout, D., Baets, R.: Silicon microring resonators. Laser Photon. Rev. 6, 47–73 (2012)
Boriskina, S.V.: Photonic molecules and spectral engineering. In: Chremmos, I., Schwelb, O., Uzunoglu, N. (eds.) Photonic Microresonator Research and Applications. Springer, New York (2010)
COMSOL MULTIPHYSICS. COMSOL, INC., BURLINGTON, MA, USA (2008)
Chakravarty, S., Bhattacharya, P., Topol’ančik, J., Wu, Z.: Electrically injected quantum dot photonic crystal microcavity light emitters and microcavity arrays. J. Phys. D Appl. Phys. 40, 2683–2690 (2007)
Chen, Y., Blair, S.: Nonlinearity enhancement in finite coupled-resonator slow-light waveguides. Opt. Express 12, 3353–3366 (2004)
Chremmos, I., Schwelb, O., Uzunoglu, N.: Photonic Microresonator Research and Applications. Springer, New York (2010)
Chremmos, I., Uzunoglu, N.: Modes of the infinite square lattice of coupled microring resonators. J. Opt. Soc. Am. A 25, 3043–3050 (2008)
Donzella, V., Sherwali, A., Flueckiger, J., Grist, S.M., Fard, S.T., Chrostowski, L.: Design and fabrication of SOI micro-ring resonators based on sub-wavelength grating waveguides. Opt. Express 23, 4791–4803 (2015)
Escalante, J.M., Martnez, A., Laude, V.: Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal. J. Phys. D Appl. Phys. 46, 475301 (2013)
Feng, S., Lei, T., Chen, H., Cai, H., Luo, X., Poon, A.W.: Silicon photonics—from a microresonator perspective. Laser Photon. Rev. 6, 145–177 (2012)
Franchimon, E.F., Hiremath, K.R., Stoffer, R., Hammer, M.: Interaction of whispering gallery modes in integrated optical microring or microdisk circuits—hybrid coupled mode theory model. J. Opt. Soc. Am. B 30, 1048–1057 (2013)
Haus, H.A.: Waves and Fields in Optoelectronics. Prentice-Hall, Englewood Cliffs (1984)
Haus, H.A., Huang, W.: Coupled-mode theory. Proc. IEEE 19, 1505–1518 (1991)
Li, Q., Wang, T., Yikai, S., Yan, M., Qiu, M.: Coupled mode theory analysis of mode-splitting in coupled cavity system. Opt. Express 18, 8367–8382 (2010)
Little, B.E., Chu, S.T., Haus, H.A., Foresi, J., Laine, J.P.: Microring resonator channel dropping filters. IEEE J. Lightwave Technol. 15, 9981005 (1997)
Little, B.E., Foresi, J.S., Steinmeyer, G., Thoen, E.R., Chu, S.T., Haus, H.A., Ippen, E.P., Kimerling, L.C., Greene, W.: 1998 Ultra-compact \({\rm Si-SiO}\) microring resonator optical channel dropping filters. IEEE Photon. Technol. Lett. 10, 549–551 (1998)
Liu, H.C., Yariv, A.: Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs). Opt. Express 19, 17653–17668 (2011)
Manolatou, C., Khan, M.J., Fan, S., Villeneuve, P.R., Haus, H.A., Joannopoulos, J.D.: Coupling of Modes Analysis of Resonant Channel Add-Drop Filters. IEEE J. Quantum Electron. 35, 1322–1331 (1999)
McIsaac, C.P.R.: Symmetry-induced modal characteristics of uniform waveguides—II. Theory. IEEE Trans. Microw. Theory Tech. MTT–23, 429–433 (1975)
McIsaac, P.R.: Symmetry-induced modal characteristics of uniform waveguides 1: summary of results. IEEE Trans. Microw. Theory Tech. MTT–23, 421–429 (1975)
Moller, B.M., Woggon, U.: Band formation in coupled-resonator slow-wave structures. Opt. Express 15, 17362–17370 (2007)
Morichetti, F., Ferrari, C., Canciamilla, A., Melloni, A.: The rst decade of coupled resonator optical waveguides—bringing slow light to applications. Laser Photon. Rev. 6, 74–96 (2012)
Poon, J.K.S., Scheuer, J., Mookherjea, S., Paloczi, G.T., Huang, Y., Yariv, A.: Matrix analysis of microring coupled-resonator optical waveguides. Opt. Express 12, 90–103 (2004)
Poon, J.K.S., Scheuer, J., Xu, Y., Yariv, A.: Designing coupled-resonator optical waveguide delay lines. J. Opt. Soc. Am. B 21, 1665–1673 (2004)
Poon, J.K.S., Yariv, A.: Active coupled-resonator optical waveguides. I. Gain enhancement and noise. J. Opt. Soc. Am. B 24, 2378–2388 (2007)
Popović, M.A., Manolatou, C., Watts, M.R.: Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters. Opt. Express 14, 1208–1222 (2006)
Radjenović, B., Radmilović-Radjenović, M.: Excitation of confined modes in silicon slotted waveguides and microring resonators for sensing purposes. IEEE Sens. J. 14, 1412–1417 (2014)
Smotrova, E.I., Nosich, A.I., Benson, T.M., Sewell, P.: Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing. IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006)
Stefanou, N., Modinos, A.: Impurity bands in photonic insulators. Phys. Rev. B. 57, 12127–12133 (1998)
Sutton, A.P., Finnis, M.W., Pettifor, D.G., Ohta, Y.: The tight-binding bond model. J. Phys. C Solid State Phys. 21, 35 (1988)
Xia, F., Šekarić, L., Vlasov, Y.: Ultracompact optical buffers on a chip. Nat. Photon. 1, 65–71 (2007)
Xiao, S., Khan, M.H., Shen, H., Qi, M.: Silicon-on-insulator microring add-drop filters with free spectral ranges over \({\rm 30\, nm}\). J. Lightwave Technol. 26, 228–236 (2008)
Yang, Y.D., Huang, Y.Z.: Symmetry analysis and numerical simulation of mode characteristics for equilateral-polygonal optical microresonators. Phys. Rev. A 76, 023822 (2007)
Yang, Y.D., Huang, Y.Z.: Mode characteristics and directional emission for square microcavity lasers. J. Phys. D Appl. Phys. 49, 253001 (2016). (18pp)
Yariv, A., Xu, Y., Lee, R.K., Scherer, A.: Coupled resonator optical waveguides: a proposal and analysis. Opt. Lett. 24, 711–713 (1999)
Yueh, W.C.: Eigenvalues of several tridiagonal matrices. Appl. Math. E Notes 5, 66–74 (2005)
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The work has also been carried out under Ministry of Education and Science Republic of Serbia O171037 and 151005B projects.
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Radjenović, B., Radmilović-Radjenović, M. & Beličev, P. Eigenmodes of finite length silicon-on-insulator microring resonator arrays. Opt Quant Electron 49, 149 (2017). https://doi.org/10.1007/s11082-017-0984-9
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DOI: https://doi.org/10.1007/s11082-017-0984-9