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objects, mostly because of the Rayleigh wing scattering [ 4 ]. A way to overcome all these challenges is to exploit optomechanical cavities having both optical and mechanical high quality factors. Whispering Gallery Mode Resonators( WGMR) are great candidates since they highly confine both light and sound. In essence, WGMR are high-finesse dual photonic-phononic cavities( also known as phoxonic cavities) and have shown to be an excellent enhancement platform to study light-matter interactions, such as stimulated nonlinear optical processes and frequency generation [ 5 – 7 ]. On a more general note, the applications of WGMR are vast and deep, including microlasers [ 8 ], sensors [ 9, 10 ], parity-time systems [ 11 ], nonlinear frequency generation [ 12 ] and quantum optics [ 13 ]. Compared to traditional MEMS( MicroElectroMechanical Systems), the WGMR readout can reach quantum limited levels of detection or shot noise [ 14 ]. WGMR offer several advantages, such as scalability, stability, chemical robustness, real time detection. They can also be doped and / or coated with functional materials such as graphene [ 15 ]. In WGMR systems not only the coupled light will applies a force in terms of radiation pressure and excites optomechanical oscillations( OMO) [ 16 – 18 ], but also traditional acoustic fields will induce mechanical oscillations [ 19 ]. Indeed, optomechanical based sensors differ from traditional optical sensors [ 20 ] in their interaction method. Traditionally, optical sensing is based on evanescent detection or evanescent fluorescent excitation, whereas optomechanical and PA sensing relies on OMO.
Finally, in general terms, WGMR can be split into two families depending on their geometry. Resonators with an almost planar geometry( like, e. g. embedded integrated microrings) are typically referred to as 2D resonator, whereas resonators with a significant extention in all directions are referred to as 3D resonators. In this review, we will focus on 3D WGMR.
2 Whispering gallery mode resonators
According to the Oxford dictionary, a Whispering Gallery is a gallery or dome with acoustic properties such that a faint sound may be heard round its entire circumference. Lord Rayleigh studied and identified the Whispering Gallery modes( WGMs) in Saint Paul’ s cathedral in London [ 21 ] for the first time. In particular, this curvedguiding effect forced the acoustic wave to run along the gallery walls, producing a sticking effect to the wall, and was also energy efficient, allowing the wave to travel for the entire walkway length.
With the advent of laser technology and the possibility of refining extremely pure glasses, it was possible to translate this phenomenon into optics: using curved surfaces, an optical wave can be totally internally reflected while traveling around a circular medium. The fabrication materials for optical WGMR are mostly semiconductors, since the advanced fabrication techniques allow to produce with precision sub-micrometric structures, or glasses, since they minimize absorption losses, resulting in the best optical performances. All WGMR show cylindrical symmetry around
Figure 1. Series of panels showing examples of WGM resonators. Moving from left to right one has( a) a microsphere,( b) a microbubble,( c) a microbottle [ 26 ],( d) a microtoroid( adapted with permission from [ 27 ] Ó Optical Society of America).
one axis, have curved surfaces and high refractive index contrast with the surrounding medium. While running circular paths around the symmetry axis, light is subject to total internal reflections and the resulting guiding effect leads to the formation of the optical WGMs in close proximity to the interfaces(“ wall sticking”). In this regard, WGMs can be easily and effectively visualised as rings of light localised at the boundaries. The physical process leading to the WGMs formation is the interference of the guided wave with itself on the these closed paths [ 22 – 25 ]. This formation is analogous to the one happening in interferometers( e. g. Gires-Tournis, bowtie), where the guiding is instead provided by a set of mirrors. Consequently, WGMs form a set of discrete confined electromagnetic modes and the systems sustaining them are called Whispering Galley Mode Resonators( WGMR). Since the wave must run several round trips without significant attenuation for the interference process to happen, WGMR must be fabricated with low-loss materials and must have a little footprint. There is a wide variety of 3D structures that can support optical WGMs( see Fig. 1), such as microspheres, microdisks or microtoroids, and even hollow structures such as microbubbles and microbottles are possible.
More formally, the WGMs are solutions of the vectorial Helmholtz equations after taking into account the appropriate boundary conditions on both the electric and the magnetic fields. In particular, this mathematical solution defines both the resonance wavelengths of the WGMs and their spatial distribution: they are mostly localized within the surfaces( physical boundary), but the field extends in a very small fraction in the surrounding environment as an evanescent tail. In practical terms, the evanescent tails plays an important role since it allows to couple the WGMs to an external waveguide injecting and / or extracting light from the WGMR. The WGMR-waveguide coupling systems can be different, as discussed in Section 2.1, but for being effective the waveguide mode and the WGM need to