Abstract

We report an experimental observation of strong variations of quality factor and mode splitting among whispering-gallery modes with the same radial order and different azimuthal orders in a scattering-limited microdisk resonator. A theoretical analysis based on the statistical properties of the surface roughness reveals that mode splittings for different azimuthal orders are uncorrelated, and variations of mode splitting and quality factor among the same radial mode family are possible. Simulation results agree well with the experimental observations.

© 2012 Optical Society of America

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References

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  1. M. Borselli, T. J. Johnson, and O. Painter, Opt. Express 13, 1515 (2005).
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    [CrossRef]

2009 (1)

2007 (2)

2005 (1)

1996 (1)

1995 (1)

Adibi, A.

Borselli, M.

Chu, S. T.

Hare, J.

Haroche, S.

Johnson, T. J.

Lefevre-Seguin, V.

Li, Q.

Little, B. E.

Painter, O.

Raimond, J. M.

Sandoghdar, V.

Soltani, M.

Weiss, D. S.

Yegnanarayanan, S.

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Figures (3)

Fig. 1.
Fig. 1.

(a) SEM of a 10 μm radius microdisk resonator; (b) transmission of the microdisk resonator at low input power (solid black) and fitting curves for the second-order (dotted red) and fifth-order (dotted green) radial modes; the inset shows the group index comparisons between the finite element method (FEM) simulation (solid black) for the lowest five radial TE modes and the measured ones (with the red and green squares corresponding to the resonances marked by the dotted red and green curves, respectively) from Fig. 1(b); (c) curve-fitting results for the second-order mode around 1509.3 and 1519.6 nm; (d) extracted splitting Q (Qβ, dotted line with squares), intrinsic Q (Qi, cross), and scattering Q (Qss, solid line with circles) for the second-order (red, right) and fifth-order (green, left) radial modes, respectively. The vertical axis is shown using the logarithmic scale.

Fig. 2.
Fig. 2.

(a) Transmission scans of the second-order radial mode around 1509.3 nm at various input power levels; (b) measured, thermally induced wavelength shift Δλth (square) versus the relative input power along with a fit (solid line) for normalized linear absorption γla (linear absorption loss normalized to the cold-cavity loss) based on the model in [3]. The inset shows the measured, normalized nonlinear absorption γnla (shown by square, which is the nonlinear absorption loss normalized to the cold-cavity loss) versus the relative cavity energy along with a linear fit (solid line). The input power is the power sent from the testing laser, and the circulating power inside the resonator is approximated to be around 50 mW at 1 mW input power in our characterization setup.

Fig. 3.
Fig. 3.

(a) Schematic of the volume current method for scattering loss calculation; (b) simulated scattering Q (solid red line with circles) and splitting Q (solid green line with squares) for α=0.5π in Eq. (8). The red and green dotted lines correspond to the scattering and splitting Q, respectively, for the case with α=0 and no phase variation in Eq. (8). The vertical axis is shown using the logarithmic scale. (c) The autocorrelation function of the sidewall roughness with parameters used in (b).

Equations (8)

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Δr(x)Δr(x)=σ2exp(|xx|2Lc2),
Δr(ϕ)=12πnF(kn)eiknRϕ,
F(kn)=02πΔr(ϕ)einϕdϕ,
F(kn)F*(km)=2π3/2(σ2Lc)Rexp[(knLc2)2]δ(nm),
Az,m(r,θ,ϕ)=C0Ez,m(R,0)nF*(km+n)(ieiϕ)nJn[k0Rsin(θ)],
Aϕ,m(r,θ,ϕ)=12C0Eϕ,m(R,0)×nF*(kn+m)i(ieiϕ)n{Jn1[k0Rsin(θ)]Jn+1[k0Rsin(θ)]},
C0=iωhδn2R4πc2(eik0rr),
F(kn)=2π3/2(σ2Lc)Rexp[(knLc2)2]×[cos(α)+sin(α)Nn(0,1)]exp[2πiUn(0,1)](n>0),

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