Single-particle Rayleigh scattering of whispering gallery modes: split or not to split?

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

2. 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 NEMSS-2008, Middletown

3. 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. NEMSS-2008, Middletown

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

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

6. Double peak structure of the spectrum in single resonators 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). Transmission through an optical fiber coupled to a silicon microdisk. M.Borselli,T.J.Johnson,andO.Painter,Opt.Express13,1515 (2005). NEMSS-2008, Middletown

7. 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” NEMSS-2008, Middletown

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

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

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

11. Z Y X 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 For multiple scatterers (surface roughness) the same is true in the single scattering approximation NEMSS-2008, Middletown

12. Z Y 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 Scattered field Internal field NEMSS-2008, Middletown

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

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

15. 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 proving convergence one obtains To improve numerical convergence we introduce and present NEMSS-2008, Middletown

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

17. Scattering induced resonances Approximate expressions for the shifted frequencies: 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). NEMSS-2008, Middletown

18. 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 Rayleigh scattering of WGM NEMSS-2008, Middletown

19. Single-sphere resonance 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 NEMSS-2008, Middletown

20. 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 NEMSS-2008, Middletown