1. NEMSS-2008, Middletown Single-particle Rayleigh scattering of whispering gallery modes: split or not to split? Lev Deych, Joel Rubin Queens College-CUNY

2. NEMSS-2008, Middletown Acknowledgements Thanks go to Thomas Pertsch, Arkadi Chipouline, and Carsten Schimdt of the Friedrich Schiller University of Jena for their hospitality last summer, when part of this work was done Partial support for this work came from AFOSR grant FA9550-07-1-0391, and PCS-CUNY grants

3. NEMSS-2008, Middletown WGM in a single sphere Modes are characterized by angular (l), azimuthal (m), and radial (s) numbers. Poles of the scattering coefficients determine their frequencies and life- times, which are degenerate with respect to m.

4. NEMSS-2008, Middletown Fundamental modes Z Y X Fundamental modes are concentrated in the equatorial plane

5. NEMSS-2008, Middletown Fundamental modes and the coordinate system Y Z X Linear combination of VSH with A mode is fundamental only with respect to a given coordinate system Z Y X Single VSH with

6. NEMSS-2008, Middletown Double peak structure of the spectrum in single resonators Transmission through an optical fiber coupled to a silicon microdisk. M.Borselli,T.J.Johnson,and O.Painter, Opt.Express 13,1515 (2005). Near field spectrum showing the peak structure caused by coupling to the tip of the near field microscope itself. A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007).

7. NEMSS-2008, Middletown CW-CCW splitting – origin of the idea “We have observed that very high-Q Mie resonances in silica microspheres are split into doublets. This splitting is attributed to internal backscattering that couples the two degenerate whispering-gallery modes propagating in opposite directions along the sphere equator”

8. NEMSS-2008, Middletown CW-CCW splitting paradigm “… backscattering is observed as the splitting of initially degenerate WG mode resonances and the occurrence of characteristic mode doublets.” “mode splitting has been … explained as the result of the coupling between … degenerate clockwise and counterclockwise modes via back scattering.” M.L. Gorodetsky, et al. Opt. Soc. Am. B 17, 1051 (2000) A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007) Coupling coefficient

9. NEMSS-2008, Middletown Axial rotational symmetry and CW- CW degeneracy Why ? Abelian group: Only one-dimensional representations: no degeneracy! Typical answers: 1 Maxwell equations are 2 nd order – time reversal is not linked to complex conjugation Phys. Rev. A, 77, 013804 (2008), Dubetrand, et al In disks and ellipsoids full rotational symmetry is replaced by an axial rotational symmetry. Degeneracy with respect to m is lifted, but 2. Kramers degeneracy D.S. Weiss. Optics Letters, 20, 1835, (1995) Both answers are wrong

10. NEMSS-2008, Middletown Inversion symmetry and CW-CCW degeneracy for any angle  With the inversion, the group is non-Abelian and permits two-dimensional representations. due to inversion symmetry, not rotation

11. NEMSS-2008, Middletown Symmetry, CCW-CW coupling and Rayleigh scattering Sub-wavelength scatterers = dipole approximation for the scatterer = shape of the scatterer is not important, can be assumed to be spherical No axial rotation symmetry, but the inversion symmetry is still there = No coupling between cw and ccw modes in the dipole approximation = no lifting of degeneracy Z Y X For multiple scatterers (surface roughness) the same is true in the single scattering approximation

12. NEMSS-2008, Middletown Mie theory of scattering of WGM Model a scatterer as a sphere and solve the two-sphere scattering problem, using multi-sphere Mie formalism Excites a fundamental ccw WGM Z Y Scattered field Internal field

13. NEMSS-2008, Middletown Scattering coefficients Application of the Maxwell boundary conditions gives, for the scattering coefficients (neglecting cross-polarization coupling) X In the chosen coordinate system translation coefficients are diagonal in m Translation coefficients describe coupling between spheres

14. NEMSS-2008, Middletown Dipole approximation In the dipole approximation Now equation for the scattering coefficients can be solved exactly

15. NEMSS-2008, Middletown Convergence of the sum over l Translation coefficients grow with l, therefore there is an issue of convergence of the sum over l in the equation for scattering coefficients. For one obtains proving convergence To improve numerical convergence we introduce and present

16. NEMSS-2008, Middletown Single mode approximation and resonances A resonance at the original single sphere frequency, unmodified Two new frequencies for Weak resonances from terms with

17. NEMSS-2008, Middletown Approximate expressions for the shifted frequencies: Scattering induced resonances Effective polarizability of the scatterer is renormalized by higher l terms. This explains experimental fundings of Mazzei et al A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007).

18. NEMSS-2008, Middletown Rayleigh scattering of WGM A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007). To treat WGM’s scattering within a framework developed for plane waves leads to wrong results. Famous Rayleigh law for scattering cross section is replaced with law for WGMs. This change in scattering law is traced to changes in asymptotic behavior of Hankel function from

19. NEMSS-2008, Middletown Numerical results Exact numerical computation for TM39 mode. Terms with angular momentum up to 50 were included. The third peak is too weak to be seen here. Relative height of the peaks depends on the size of the scatterer and distance d. The result is the same when a cw mode is excited: degeneracy is not lifted Single-sphere resonance

20. NEMSS-2008, Middletown Conclusion An exact ab initio theory of Rayleigh (dipole) scattering of WGM of a sphere based on multisphere Mie theory is derived The picture of scattering based on coupling between cw and ccw modes is proven wrong. It is shown that one of peaks in the optical response corresponds to the single sphere resonance, while the other comes from excitation by the scatterer of WGM with azimuthal numbers Quadratic dependence of peak’s width versus shift is explained by renormalization of the effective polarizability due to interaction with high order modes