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Surface optomechanics: calculating optically excited acoustical whispering gallery modes in microspheres

John Zehnpfennig, Gaurav Bahl, Matthew Tomes, and Tal Carmon

Open Access Open Access

Abstract

Stimulated Brillouin scattering recently allowed experimental excitation of surface acoustic resonances in micro-devices, enabling vibration at rates in the range of 50 MHz to 12 GHz. The experimental availability of such mechanical whispering gallery modes in photonic-MEMS raises questions on their structure and spectral distribution. Here we calculate the form and frequency of such vibrational surface whispering gallery modes, revealing diverse types of surface vibrations including longitudinal, transverse, and Rayleigh-type deformations. We parametrically investigate these various modes by changing their orders in the azimuthal, radial, and polar directions to reveal different vibrational structures including mechanical resonances that are localized near the interface with the environment where they can sense changes in the surroundings.

©2011 Optical Society of America

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Figures (5)
Fig. 1
Fig. 1 Illustration of the mechanical whispering gallery resonance in a sphere. Deformation of the outer surface describes the exaggerated mechanical deformation. The cuts reveal also the internal deformation as indicated by colors.

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Fig. 2
Fig. 2 Rayleigh, transverse, and longitudinal whispering gallery modes where energy is confined within a wavelength distance from the interface. The major deformation is either radial, polar, or azimuthal as indicated by arrows. Mϕ in this calculation is 20, indicating that 20 acoustical wavelengthes are resonating along the spehre circumference. Color represents absolute value of the deformation.

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Fig. 3
Fig. 3 High-order mechanical whispering gallery modes. Top: Increasing the mode order in the radial and polar directions for mechanical whispering gallery modes in a silica sphere. Color represents deformation. Mϕ = 20. Bottom: We depict several of the top modes and present them in 3D. The presented section is one acoustic wavelength in the azimuthal direction. The equator is seen to deform into a sine where the Rayleigh mode the deformation is in the radial direction and for the transverse mode the deformation is in the polar direction.

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Fig. 4
Fig. 4 High-order mechanical whispering gallery modes. The calculated speed of sound is shown as a function of the azimuthal mode order, Mϕ . The first three transverse orders are given for the Rayleigh-, transverse-, and longitudinal families. At large Mϕ , the speed of sound asymptotically converges to the relevant speed of sound in bulk media (see Table 1).

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Fig. 5
Fig. 5 Vibration frequencies for the various modes in a r = 100 micron silica sphere as a function of their azimuthal mode order. Left, with Mϕ typical to forward Brillouin excitation. Right, with Mϕ typical to backward Brillouin excitation. The shadowed regions estimate how high resonance frequencies can go for each of these modes via relying on high order transverse members of this mode family. The shadowed region is bounded in the Mϕ direction as estimation from momentum conservation consideration. We assume excitation with 1.5-micron telecom pump.

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Tables (1)

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Table 1 Analytically and Numerically Calculated Speeds of Sound a

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Equations (2)

Equations on this page are rendered with MathJax. Learn more.

ρ ˜ = A a ( t ) T a ( θ , r ) e i ( M φ a φ − ϖ a t ) , E ˜ p = A p ( t ) T p ( θ , r ) e i ( M φ p φ − ϖ p t ) , E ˜ S = A S ( t ) T S ( θ , r ) e i ( M φ S φ − ϖ S t ) .
∂ 2 ρ ˜ ∂ t 2 − V ∇ 2 ρ ˜ − b ∂ ∂ t ∇ 2 ρ ˜ = 1 2 ε 0 γ θ ∇ 2 〈 ( E ˜ p + E ˜ S ) 2 〉 .
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