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

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

Lev Deych, Joel Rubin

Queens College-CUNY

NEMSS-2008, Middletown

- 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

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

Z

Y

X

Fundamental modes are concentrated in the equatorial plane

NEMSS-2008, Middletown

Y

Z

Z

Y

X

X

A mode is fundamental only with respect to a given coordinate system

Linear combination of VSH with

Single VSH with

NEMSS-2008, Middletown

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

“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

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

In disks and ellipsoids full rotational symmetry is replaced by an axial rotational symmetry. Degeneracy with respect to m is lifted, but

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

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

Z

Y

X

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

Z

Y

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

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

In the dipole approximation

Now equation for the scattering coefficients can be solved exactly

NEMSS-2008, Middletown

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

A resonance at the original single sphere frequency, unmodified

Two new frequencies for

Weak resonances from terms with

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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

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

NEMSS-2008, Middletown

Single-sphere resonance

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

- 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