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Quantum Mechanics

Radiation-pressure-driven micro-mechanical oscillator

Profile image of Tobias KippenbergTobias Kippenberg

Abstract

As Q factor is boosted in microscale optical resonant systems there will be a natural tendency for these systems to experience a radiationpressure induced instability. The instability is manifested as a regenerative oscillation (at radio frequencies) of the mechanical modes of the microcavity. The first observation of this radiation-pressure-induced instability is reported here. Embodied within a microscale, chip-based device reported here this mechanism can benefit both research into macroscale quantum mechanical phenomena [1] and improve the understanding of the mechanism within the context of LIGO [2]. It also suggests that new technologies are possible which will leverage the phenomenon within photonics.

Figures (5)
Fig. 1. Panel A illustrates the “below threshold” behavior where the optical pump wave at frequency Wis not strong enough to induce mechanical oscillations of the micro-toroid. Panel B illustrates the “above threshold” case for the n=3 vibrational mode. Mechanical oscillation at frequency ©) creates optical stokes (q@—Q.) and anti-stoked sidebands (q@+Q) in the transmitted pump wave. Inset of panel B shows the exaggerated cross-section of the third order mode and variation of the toriod radius as a result of these oscillations. in energy from the original photons by the RF frequency of the vibrations. Beating of the original pump wave with the up/down-shifted, optical sidebands therefore results in oscillations in optical power transmission at the mechanical frequencies of the structure. A rendering of this process is provided in Fig. 1. We observed these oscillations when the optical pump is coupled to sufficiently high quality factor optical modes (Q ~10’ or 10 ns photon lifetime at infra-red wavelengths). Spectral analysis of the detected, transmitted optical power using a high-resolution electrical spectrum analyzer (see Fig. 2) revealed extremely narrow peaks (linewidths of less than 10 Hz limited by the equipment resolution) at a frequency typically in the range of 10-100 MHz as well as at harmonics of this fundamental frequency. As can be seen in Fig. 2, two, distinct fundamental oscillation frequencies (and their harmonics [8]) were observed: a low frequency mode (~ 2-20 MHz) and a high frequency mode (~ 40-100 MHz).
Fig. 1. Panel A illustrates the “below threshold” behavior where the optical pump wave at frequency Wis not strong enough to induce mechanical oscillations of the micro-toroid. Panel B illustrates the “above threshold” case for the n=3 vibrational mode. Mechanical oscillation at frequency ©) creates optical stokes (q@—Q.) and anti-stoked sidebands (q@+Q) in the transmitted pump wave. Inset of panel B shows the exaggerated cross-section of the third order mode and variation of the toriod radius as a result of these oscillations. in energy from the original photons by the RF frequency of the vibrations. Beating of the original pump wave with the up/down-shifted, optical sidebands therefore results in oscillations in optical power transmission at the mechanical frequencies of the structure. A rendering of this process is provided in Fig. 1. We observed these oscillations when the optical pump is coupled to sufficiently high quality factor optical modes (Q ~10’ or 10 ns photon lifetime at infra-red wavelengths). Spectral analysis of the detected, transmitted optical power using a high-resolution electrical spectrum analyzer (see Fig. 2) revealed extremely narrow peaks (linewidths of less than 10 Hz limited by the equipment resolution) at a frequency typically in the range of 10-100 MHz as well as at harmonics of this fundamental frequency. As can be seen in Fig. 2, two, distinct fundamental oscillation frequencies (and their harmonics [8]) were observed: a low frequency mode (~ 2-20 MHz) and a high frequency mode (~ 40-100 MHz).
Fig. 2. The measured, spectral content of pump-power (at 1550 nm) transmission as observed on an electrical spectrum analyzer (bandwidth set at 100kHz). Two families of frequencies are observed. Those at lower frequency are driven by oscillation of an n=1 vibrational mode and those at higher frequency by an n=3 mode. Harmonics of the fundamental mechanical frequency are caused by the nonlinear transfer characteristic of the waveguide resonator system (see footnote 8). The inset shows the numerically modelled cross-sectional plot (exaggerated for clarity) of the strain field for the first and third vibrational eigen-modes of a toroidal silica micro-cavity on a silicon post. The stress field is superimposed (color coded). As evident from the modelling, the mechanical oscillations cause a displacement of the toroidal periphery and thereby induce a shift in the whispering gallery mode resonant frequency. The eigen-frequencies of the mechanical modes of the toroidal structures were investigated numerically to confirm their mechanical origin. Close agreement of the measured RF oscillation frequencies with the results of the numerical modelling (less than 2% discrepancy) confirms that the first (n=1) and third (n=3) order flexural modes are responsible for creating the observed low- and high- frequency families of oscillations, respectively. The inset in Fig. 2 shows the numerically modelled cross-sectional view of the n=1 and n=3 modes, while Fig. 3 provides the comparison between the numerical and experimental mechanical eigen- frequencies versus the toroid “overhang length” L (see cross sectional view in Fig. 2 for the definition of L). The mechanical origin of these oscillations was also confirmed by lowering a metallic micro-probe into proximity with the plane of the silica disk connecting the toroid to the silicon pillar (i.e. not the toroid itself, where the optical mode lies). Since the probe is far removed from the toroidal whispering gallery, it affects only the mechanical and not the optical properties of the structure. The optical power oscillations were observed to fully quench upon probe contact.
Fig. 2. The measured, spectral content of pump-power (at 1550 nm) transmission as observed on an electrical spectrum analyzer (bandwidth set at 100kHz). Two families of frequencies are observed. Those at lower frequency are driven by oscillation of an n=1 vibrational mode and those at higher frequency by an n=3 mode. Harmonics of the fundamental mechanical frequency are caused by the nonlinear transfer characteristic of the waveguide resonator system (see footnote 8). The inset shows the numerically modelled cross-sectional plot (exaggerated for clarity) of the strain field for the first and third vibrational eigen-modes of a toroidal silica micro-cavity on a silicon post. The stress field is superimposed (color coded). As evident from the modelling, the mechanical oscillations cause a displacement of the toroidal periphery and thereby induce a shift in the whispering gallery mode resonant frequency. The eigen-frequencies of the mechanical modes of the toroidal structures were investigated numerically to confirm their mechanical origin. Close agreement of the measured RF oscillation frequencies with the results of the numerical modelling (less than 2% discrepancy) confirms that the first (n=1) and third (n=3) order flexural modes are responsible for creating the observed low- and high- frequency families of oscillations, respectively. The inset in Fig. 2 shows the numerically modelled cross-sectional view of the n=1 and n=3 modes, while Fig. 3 provides the comparison between the numerical and experimental mechanical eigen- frequencies versus the toroid “overhang length” L (see cross sectional view in Fig. 2 for the definition of L). The mechanical origin of these oscillations was also confirmed by lowering a metallic micro-probe into proximity with the plane of the silica disk connecting the toroid to the silicon pillar (i.e. not the toroid itself, where the optical mode lies). Since the probe is far removed from the toroidal whispering gallery, it affects only the mechanical and not the optical properties of the structure. The optical power oscillations were observed to fully quench upon probe contact.
Fig. 3. Numerical calculation (solid line) and measured (points) dispersion relation for the fundamental (n=1) and third order (n=3) flexural modes of a micro-toroid as a function of the free hanging length of the disk structure. The inset shows the agreement between the numerical and measured frequencies. This system can be modelled using a set of coupled differential equations: one governins the harmonic motion of the flexing toroid and a second governing the resonant optical field [2] The equation of motion for deformation (more precisely, the displacement of the whisperin; vallery) is that of a damped, harmonic oscillator driven by radiation pressure (generated by th circulating optical field at the periphery of the micro-toroid). The optical field in the cavity, o1 the other hand, couples to this mechanical motion through the dependence of the cavity resonant frequency on mechanical deformation. By solving this system of equations self Sonsistently a mechanical gain term can be defined which is a linear function of the circulatin; optical power and that offsets intrinsic sources of mechanical damping as given below. Thi leads to a threshold optical pump power for onset of mechanical oscillations as follows:
Fig. 3. Numerical calculation (solid line) and measured (points) dispersion relation for the fundamental (n=1) and third order (n=3) flexural modes of a micro-toroid as a function of the free hanging length of the disk structure. The inset shows the agreement between the numerical and measured frequencies. This system can be modelled using a set of coupled differential equations: one governins the harmonic motion of the flexing toroid and a second governing the resonant optical field [2] The equation of motion for deformation (more precisely, the displacement of the whisperin; vallery) is that of a damped, harmonic oscillator driven by radiation pressure (generated by th circulating optical field at the periphery of the micro-toroid). The optical field in the cavity, o1 the other hand, couples to this mechanical motion through the dependence of the cavity resonant frequency on mechanical deformation. By solving this system of equations self Sonsistently a mechanical gain term can be defined which is a linear function of the circulatin; optical power and that offsets intrinsic sources of mechanical damping as given below. Thi leads to a threshold optical pump power for onset of mechanical oscillations as follows:
Fig. 4. Measured amplitude response (points) of the mechanical vibrations of an n=1 mechanical mode as a function of driving-force frequency (modulation frequency of the pump power). Circles (green), triangles (red), and stars (blue) represent the data for 2 LW sa LW and 9 UW of average pump power. The inset shows the effect of the optical power on the linewidth of the mechanical oscillator inferred from the theoretical fits (such as the solid lines in the main Fig.). A linear fit shows a threshold of 11 ,yW for the mechanical oscillations and an intrinsic quality factor of 630 for the measured mechanical mode of the toroidal structure.
Fig. 4. Measured amplitude response (points) of the mechanical vibrations of an n=1 mechanical mode as a function of driving-force frequency (modulation frequency of the pump power). Circles (green), triangles (red), and stars (blue) represent the data for 2 LW sa LW and 9 UW of average pump power. The inset shows the effect of the optical power on the linewidth of the mechanical oscillator inferred from the theoretical fits (such as the solid lines in the main Fig.). A linear fit shows a threshold of 11 ,yW for the mechanical oscillations and an intrinsic quality factor of 630 for the measured mechanical mode of the toroidal structure.
Fig. 5. Measured mechanical oscillator displacement as a function of the optical pump power showing threshold behaviour. Oscillations initiate at about 20 LW of input power and start to saturate for higher values of pump power. This saturation is associated with the lower optical- mechanical coupling at displacements large enough to shift the resonant frequency of the optical mode by greater than its linewidth. Inset shows the dependence of measured threshold power on optical quality factor. Data taken from Ref. [14]. These data were obtained using different optical modes in a single micro-toroid (so that only optical Q would be varied). The slope of the fit in a log-log scale is 3+0.3, which agrees well with inverse cubic behaviour expected for radiation-pressure-induced regenerative oscillations (as opposed to inverse quadratic behaviour for photothermal effects).
Fig. 5. Measured mechanical oscillator displacement as a function of the optical pump power showing threshold behaviour. Oscillations initiate at about 20 LW of input power and start to saturate for higher values of pump power. This saturation is associated with the lower optical- mechanical coupling at displacements large enough to shift the resonant frequency of the optical mode by greater than its linewidth. Inset shows the dependence of measured threshold power on optical quality factor. Data taken from Ref. [14]. These data were obtained using different optical modes in a single micro-toroid (so that only optical Q would be varied). The slope of the fit in a log-log scale is 3+0.3, which agrees well with inverse cubic behaviour expected for radiation-pressure-induced regenerative oscillations (as opposed to inverse quadratic behaviour for photothermal effects).

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