Concept of miniature optical pressure sensor based on coupled WGMs in a dielectric microsphere

We present the physical concept and sample engineering design of a new miniature pressure sensor based on the whispering gallery modes (WGMs) optically excited in a dielectric microsphere placed near a flexible reflective membrane which acts as an ambient pressure sensing element. WGMs excitation is carried out by free-space coupling of optical radiation to a microsphere. The distinctive feature of proposed sensor design is double excitation of optical eigenmodes by forward and backward propagating radiation reflected from a membrane that causes WGMs interference in particle volume. The optical intensity of resulting resonant field established in the microsphere carries information about the exact position of the pressure-loaded reflecting membrane. The sensitivity of the proposed sensor strongly depends on the quality factor of the excited resonant mode, as well as geometrical and mechanical parameters of the flexible membrane. Important advantages of the proposed sensor are miniature design (linear sensor dimensions depends only on the membrane diameter) and the absence of a mechanical contact of pressure-sensitive element with WGM resonator. Introduction A classical ambient pressure sensor is essentially a transducer of the pressure applied to its sensing element into an electrical, acoustic, microwave, or optical signal measured with a corresponding device. Pressure sensors are intensively studied up to now because of their extensive applications in mechanical, electrical, and biomedical engineering [Boyd2001]. To date, various concepts of optical pressure sensors are proposed and developed, which have obvious advantages such as miniaturization, immunity to electromagnetic interference, high sensitivity and signal transmission rate [Udd2011]. Most of the optomechanical pressure sensors are based on optical fibers [Zhu2005], interferometers (such as Mach-Zehnder [Luff1998] and Michelson [Chen1999]), or MEMS structures [Bao2005]. All of them have their disadvantages and advantages, an overview of which can be found in [Lee2003, Fraden2010, Udd2011]. A distinct family comprise the optical sensors which utilize optical high-quality resonant eigenmodes commonly referred to as the Whispering Gallery Modes (WGMs) excited within lossless dielectric bodies with high degree of geometrical symmetry [Foreman2015, Zheng2018]. Among them the pressure sensors based on ring [Zhao2012, Zhang2021], disk [Ma2017], racetrack [De2007], "bottle" [Gu2017, Sumetsky2019] and spherical [Ali2019] WGM micro-resonators are reported. WGM is an optical resonance localized close to the external boundary of the resonator and usually having a high-quality factor, i.e. a narrow linewidth [Ward2011, Wang2019]. The working principle of all optical WGM sensors is usually to measure the transmittance of a special optical feeder (optical fiber, or strip-line), used to excite and readout the WGM resonator signal while scanning the excitation frequency with a tunable laser. Once a WGM is excited, within WGM bandwidth the fiber transmittance drops dramatically, which is an indicator of tuning to the resonance. Any mechanical or thermal action applied to the WGM resonator changes its characteristics and the operating eigenfrequency which then is detected by a spectrometer. By the magnitude of WGM spectral displacement one can judge the level of external mechanical load on the resonator. WGM-based optical sensors are in principle immune to electromagnetic noise and thus can have a lower level of mechanical noise compared to widely used MEMS analogues. However, the bottleneck of these circuits is the requirement of mechanical contact between the loadsensitive sensor (membrane) and the WGM resonator that limits the scope of their applications, as well as the durability of the design. At the same time, WGM excitation in bulk microstructures can be carried out not only by evanescent electromagnetic fields using tapered fiber or prism couplers, but also by illuminating the microresonator with free-propagating optical radiation (direct free-space coupling). Despite the lower excitation efficiency [Cai2020], the free-space coupling has undeniable advantages because it does not require precise positioning of an optical coupler converting propagating optical wave into evanescent fields and the WGM resonator. Importantly, the efficiency of WGM excitation in a microparticle by direct radiation can be significantly increased using various techniques, such as side illumination by structured focused beams [Zemlyanov2000], or using the Mie scattering when placing the resonator near a reflecting substrate [Bobbert1986]. Recently, C. Liu et al. [Liu2000] showed by the numerical simulations that the WGMs excitation in a dielectric sphere located near a flat dielectric substrate with contrasting refractive index is accompanied by a substantial broadening and frequency shifting of the resonances, as well as WGM intensity decrease as the particle approaches the substrate. In this case, the transverse magnetic (TM) resonant modes always demonstrate a red frequency shift, while the transverse electric (TE) resonances could be shifted toward the red and blue regions of the spectrum as the optical contrast of the substrate increases. Luk'yanchuk et al. [Lukyanchuk2000] numerically investigated the light scattering patterns of a spherical particle located near a dielectric substrate and found a strong dependence of the scattering amplitude and phase diagram on the gap between the sphere and the substrate. Using the charges and magnet images method in [Xifre-Perez2012] an experimental proof of the photonic interaction between a dielectric silica (Si) nanoresonator and its image behind a flat gold mirror is presented. It is shown that the scattering cross section of the silica resonant cavity is enhanced when placed near the metal mirror. Similar results are reported in the experimental work by A. Vasista et al. on the fluorescence of the Nile blue dye by WGM excitation in a 3 μm SiO2 microsphere [Vasista2018]. The sphere was placed on a gold reflecting substrate which the fluorophore was applied on. As shown, due to the contact with a metal mirror the splitting of the azimuthal resonant modes and the multiple WGM intensity enhancement took place. Interestingly, the impact of the reflecting substrate also manifests itself in nonlinear optical interactions, e.g., at the third harmonic generation in a Si nanodisk [Yao2020]. In this paper, we propose a new conceptual design of a miniature pressure sensor based on the effect of WGM excitation in a dielectric microsphere with dimensions of the order of an illuminating wavelength (mesowavelength particle). The distinctive features of the proposed sensor design are as follows: (a) a method of WGM excitation is the free-space illumination by an optical radiation tuned to the particular WGM resonance, (b) the placement of a pressure-sensitive element is contactless avoiding the mechanical impact on WGM resonator, and (c) pressure acquisition is achieved through the WGM intensity modulation (rather than its spectral shift) using the interference of WGMs excited by forward and backward propagating optical radiation upon reflection from a loaded flexible metal membrane. Specifically, due to the presence of optical reflection there is a double excitation of WGM in a spherical resonator, first by direct and then by reflected optical radiation. The optical intensity of the resulting WGM field is mediated by the interference between forward and reflected WGMs and depends on the relative position of the loaded flexible mirror. Using the finite elements method (FEM), we simulate such a sensor operation comprised of a 2 μm titanium oxide sphere and 100 nm thick gold membrane and show high sensitivity of the proposed concept to the ambient pressure. Working principle of a coupled WGMs (cWGM) pressure sensor The modus operandi of the prototype pressure sensor is shown in Fig. 1(a) and is based on the WGMs coupling being excited in a dielectric spherical microparticle (or microcylinder) by freely propagating optical radiation when reflected from a flexible mirror (reflecting membrane). WGMs interference can be constructive or destructive depending on the phase difference of the excited eigenmodes, which will lead to a change in the amplitude of the resulting optical field of the particle. In its turn, the phase difference of direct and reflected waves depends on the deflection magnitude of the flexible mirror arising under the action of ambient pressure excess acting from outside the mirror. Fig. 1. (a) Schematics illustrating the physical principle of the proposed cWGM pressure sensor, (b) the model sensing working characteristic. If some kind of quantum emitters (QEs) like quantum dots [Huber2020], spasers [Noginov2009], or nanoparticles with fluorescent substance [Lu2021, Sarkar2021, Grudinkin2015] are deposited inside the particle in the volume occupied by the resonance mode, then their emission intensity received at a photodetector will also change depending on the deflection magnitude of the sensitive mirror membrane. It is possible to choose such deflection range that the functional relation between intensity of QEs emission and mirror deflection will be unambiguous, which will allow one to reconstruct the value of measured overpressure (Fig. 1b). Computer simulations of cWGM excitation in a dielectric microsphere Consider the following geometry for the excitation in a microsphere the interfering (hereinafter, coupled) "whispering gallery" resonant modes. A dielectric microsphere with the radius R amounting to several optical wavelengths λ and with refractive index n is placed near a metal reflecting plate playing the role of a blind mirror and is illuminated from the back side by a plane electromagnetic wave with the amplitude E0 (Fig. 2a). Note that a dielectric with high refractive index in the visible and near-infrared range can also be