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  1. Recreation of Statistical Simulations Regarding Multiple-Scatterer Induced Frequency Splitting in Whispering Gallery Microresonators Samuel Wood, under the supervision of Dr. Lan Yang Micro/Nano Photonics Lab Department of Electrical and Systems Engineering Multiple-particle-induced frequency splitting SN and changes in frequency splitting ΔSN Abstract I investigated and recreated statistical simulations of changes in the amount of frequency splitting from the one-by-one absorption of fixed sized particles onto a whispering gallery mode (WGM) microresonator’s mode volume. The WGM phenomenon was first observed in St. Paul’s Cathedral’s whispering gallery, where whispers can be heard across the structure from sound waves traveling along the almost perfectly circular walls. This phenomenon can be recreated with light by passing a laser into an optical resonators trapping the light inside for a significant number of revolutions until it is ejected. A mode volume refers to the space that a traverse mode travels through in the micro resonator. A traverse mode is a particular type of electromagnetic field pattern of radiation commonly seen in optical fibers and optical resonators. When particles are placed into the mode volume of a WGM microresonator mode splitting is induced and measured from observations of the exiting laser. These measurements may be used to accurately estimate the size and number of particles in the mode. The statistical simulations that I recreated were first completed by Lina He in a paper titled “Statistics of Multiple-Scatterer Induced Frequency Splitting in Whispering Gallery Microresonators and Microlasers.” The simulations involved finding distributions of multiple-particle induced frequency splitting by varying the number and single size of the particles in the mode volume of a microresonator called a microtoroid. Later, I simulated the same distributions, but instead used a uniform and normal distribution for the radius values which is more realistic. I concluded that the standard deviation and the maximum value of the multiple-particle induced frequency splitting are proportional to the polarizability of the particles. Figure 1. (a) Distributions of frequency splitting induced by nanoparticles of differing N while R = 50nm. (b) Distributions of frequency splitting induced by nanoparticles of differing R while N = 50. Distributions in each panel obtained by 10000 repeated trials. (a) (b) Figure 2. (a) Mean and standard deviation of the frequency splitting as a function of N while R = 50nm. (b) Mean and standard deviation of the frequency splitting as a function of R while N = 50. (c) Expectations of ΔSσ and ΔSMaxas a function of R while N = 200. (c) Methods Section The theoretical model used is the same model used in the original statistical simulations. The amount of mode splitting (spectral distance between the split modes) 2g and particle position (θi,ψi) on the microtoroid can be used to obtain the frequency splitting SNafter the absorption of N particles. Typically the equation for polarizability α is difficult to determine, but for the sake of simplicity, all particles are assumed to be perfectly spherical. This allows us to use the equation: There is a retardation effect caused by geometric depolarization that effects the polarizability of large particles (R > 50nm) but it can be ignored in the simulations because I only use particles sizes less than or equal to 50nm. The simulations were done using the computer program MATLABTMand equations (1) – (4) above. For all numerical simulations the WGM distribution f(θ) = f(r) is assumed to be standard normal and is obtained by inputting a uniform distribution of θ into a standard normal distribution function with μ = fmax= .36. The mode volume V = 280 μm3 and the set wavelength λ = 1550 nm. It is assumed that ψ has a uniform distribution from 0 to π and that θ has a uniform distribution from 0 to 2π/3.Lina He used a much more accurate way to determine the WGM distribution by creating a model of the microtoroid and distribution field using the program COMSOL MultiphysicsTM. The accuracy of my numbers is not ideal, but for simulation purposes we only need to recreate the correct changes in the distributions. (a) (b) Figure 3. Distributions of ΔSσ(a) and ΔSMax(b) from 10000 repeated trials with particles size R = 50nm. (1) (3) (2) Figure 4. Illustrations showing particles attached to a microtoroid surface (top) and the normalized field distribution on the resonator ruface (bottom.) (b) (a) Conclusion Increases in particle size results in an increase in both the standard deviation of frequencysplitting SN and the maximum change in frequency splitting in multiple-particle induced frequency splitting. Because particle size has a direct effect on polarizability it can be said that both the standard deviation and the maximum value of the multiple-particle induced frequency splitting are proportional to the polarizability of the particles. An increase in the number of particles deposited also results in an increase in the standard deviation of frequency splitting. It also leads to more accurate estimated values because of the smaller standard deviation of both ΔSσand ΔSMax. (4) Citations He, L. (2012). Whispering Gallery Mode Microresonators for Lasing and Single Nanoparticle Detection (Ph.D.). Washington University in St. Louis, United States -- Missouri. Retrieved from http://search.proquest.com.libproxy.wustl.edu/pqdtlocal1005748/docview/1011649317/abstract/13F49C32891564FE6DB/3?accountid=15159 He, L., Özdemir, Ş. K., Zhu, J., Monifi, F., Yılmaz, H., & Yang, L. (2013). Statistics of multiple-scatterer-induced frequency splitting in whispering gallery microresonators and microlasers. New Journal of Physics, 15(7), 073030. doi:10.1088/1367-2630/15/7/073030