Surface Scattering Loss of a Spherical Microcavity
The ultra-high quality factor(typically above 107)of the whispering-gallery microresonators such as microsphere,microtoroid,microbubble is mainly due to the total-internal-reflection and ultra-smooth surface.Usually,higher quality factor corresponding to narrower linewidth and longer photon life-time,which strongly increase the light-matter interaction and sensing performance,makes those microresonators the perfect platform for the nonlinear optics and optical sensing research.To further optimize the quality factor,analysis and suppressing the main loss channels is needed.The loss in a whispering-gallery microresonator is mainly depends on absorption loss,scattering loss and radiation loss.Among them,the radiation loss of the resonator is more obvious for small cavity or lower refractive contrast,while the absorption loss and scattering loss from the bulk cavity material have less dependence with the cavity size.Since the fast development of fabrication techniques,the bulk absorption and scattering loss can be strongly suppressed,while the surface scattering loss trends to become dominant for large cavities.However,the previous estimations of surface scattering have a range covers several orders of magnitude,which also become unacceptable for meeting the requirement of further improvement.In this article,the surface scattering loss of a spherical microcavity with the tiny random surface roughness(far below the optical wavelength)is analyzed by applying the equivalent current method.The surface function of the microcavity is first transfer to the equivalent polarization current which excite the secondary modes.By combining the simplified total radiation power of the excited modes with the estimation of the surface field component,a more precise estimation of surface scattering loss is finally provided and further verified via numerical calculation.In detail,this equivalent current can be expressed as the superposition of two tangential fields and one radial field on surface of the microsphere.Among them,the transverse electric(TE)modes can only be excited by the tangential current,while the transvers magnetic(TM)modes can be excited by both the tangential and radial currents.Here,for simplicity,the equivalent current expansion for four TE modes is calculated because they only have the tangential part,and the scattering power can thus be calculated by adding together all the radiation power of the excited TE and TM radiation modes,and the total scattering power from the modes with same angular momentum is proportional to the angular momentum.Note that,for the TE radiation modes,the current expansion coefficient is proportional to the field overlapping of the original TE mode,and the scattering field is similar to the radiation field of the original TE mode.The total scattering cross-section of the random surface is similar to a collection of dipole scatterers NV2p/λ4=R2σ2 B2/λ4,where N is the number of scatters,Vp is the Root Mean Square(RMS)volume of the scatterers,λ is the resonance wavelength,R is the cavity radius,σ2 and B is the variance and the correlation length of the surface function.Thus,the scattering quality factor shows a dependence of λ3 R/σ2 B2,This result agrees with the estimation given by GORODETSKY M L,and its two-dimension version λ2 R/σ2 B agrees with the perturbation theory developed by WIERSIG J.To verify this result,by applying the equivalent parameter,numerical simulation of the scattering power for the random sampling of surface point-like scatterers is demonstrated,and the surface current and the total scattering power agree well with the theory.This method can directly apply to TM modes by further consider the TM radiation mode excited by the radial part of the equivalent current.If the resonance modes have other loss channels such as a pillar in the real case,the low quality factor resonance modes should also be considered,and the scattering quality factor is further decreased.For longer correlation length,the equivalent current could introduce stronger coupling between modes,and also influence the scattering power.
Whispering gallery modeSpherical microcavityScattering lossQuality factorEquivalent current
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