WGMode : A Matlab toolbox for whispering gallery modes volume computation in spherical optical micro-resonators

Abstract

We present a Matlab toolbox aimed at studying Whispering Gallery Modes (WGM) in spherical optical micro-resonators. Whispering gallery modes are specific resonances of an optical wave trapped inside a dielectric cavity with smooth edges due to continuous total internal reflection of light. The novelty of the work stands in the implementation of analytical formulas to compute the volume of a WGM in a spherical cavity while only approximate asymptotic expressions or purely numerical computations based on the Finite Element Method were previously available. The toolbox also provides Matlab programs to compute and visualize WGM. Program summaryProgram Title: WGModeProgram Files doi:http://dx.doi.org/10.17632/f9k6rgfkmy.1Licensing provisions: LGPLProgramming language:Matlab languageNature of problem: The WGMode toolbox has been developed to compute the volume of a Whispering Gallery Mode (WGM) in a spherical optical micro-resonator [1]. The volume of a WGM is defined as the integral over the whole space of the energy density normalized by its maximum value [2]. Together with the quality factor, the mode volume is used to characterize WGM. The WGMode toolbox gathers together several Matlab scripts aimed at studying WGM in a spherical optical micro-resonator.Solution method: The volume of a WGM is defined as the integral over the whole space of the energy density normalized by its maximum value. For a spherical dielectric cavity, an analytical expression of the electromagnetic field for a WGM can be computed analytically from Maxwell’s equations using Hansen’s method [2]. The expression of the mode field involves Vector Spherical Harmonics and requires the finding of roots of a non-linear equation with Bessel’s functions. From the analytical expression of the mode field, the mode volume integral can be computed analytically. These formulas are implemented in the WGMode toolbox to compute the volume of a WGM.Additional comments including restrictions and unusual features: The WGMode toolbox has been tested with Matlab version 8.1 (R2013a), version 9 (R2016a), version 9.4 (R2018a) The WGMode toolbox also works under GNU Octave software (tested with version 4.0.0). However, under GNU Octave the scripts pltmod and volmod written to display the mode field (electric field and magnetic induction parts) and to compute the mode volume provide the results after an excessive time due to the computation of the Vector Spherical Harmonics that is not efficient under Octave.[1] J. Ward, O. Benson, WGM micro-resonators: sensing, lasing and fundamental optics with microspheres, Laser & Photonics Reviews 5 (4) (2011) 553–570.[2] A. N. Oraevsky, Whispering-gallery waves, Quantum Electronics 32 (2002) 377–400.