OPTO-MECHANICAL SYSTEM AND METHOD HAVING CHAOS INDUCED STOCHASTIC RESONANCE AND OPTO-MECHANICALLY MEDIATED CHAOS TRANSFER

Abstract

An a system and method for chaos transfer between multiple detuned signals in a resonator mediated by chaotic mechanical oscillation induced stochastic resonance where at least one signal is strong and where at least one signal is weak and where the strong and weak signal follow the same route, from periodic oscillations to quasi-periodic and finally to chaotic oscillations, as the strong signal power is increased.

Claims

1. A method for chaos transfer between multiple signals comprising: transmitting multiple detuned signals in an optical micro cavity resonator with optomechanically induced oscillation where at least one signal is stronger than and detuned with respect to at least one other signal; and increasing the power of the at least one signal whereby as the power is increased the at least one signal and the at least one other signal follow the same route, from periodic oscillations to quasi-periodic and finally to chaotic oscillations.

2. The method for chaos transfer as recited in claim 1, where the at least one signal is an optical field pump exciting mechanical oscillations in the resonator and the at least one other signal is an optical field probe and where chaos transfer from the pump to the probe is mediated by the mechanical motion of the resonator.

3. The method for chaos transfer as recited in claim 2, where the at least one signal is a pump laser and the at least one other signal is a probe laser

4. The method for chaos transfer as recited in claim 3, where optomechanically induced chaos modulate the at least one other signal at a frequency of the mechanical oscillation.

5. The method for chaos transfer as recited in claim 4, where transmitting multiple signals in an optical micro cavity resonator includes coupling the at least one signal and the at least one other signal into and out of the micro cavity resonator with one or more of a waveguide, an optical fiber and free-space, and separating the at least one signal from the at least one other signal with a wavelength division multiplexer.

6. The method for chaos transfer as recited in claim 5, comprising: detecting the pump and probe signals for a maximal Lyapunov exponent and controlling increasing power of the at least one signal responsive to the maximal Lyapunov exponent detected.

7. The method for chaos transfer as recited in claim 6, comprising: detecting the at least one other signal with a photo detector

8. A system demonstrating chaos transfer between multiple signals comprising: a first signal generator configured to transmit a first signal through an optical micro cavity resonator; a second signal generator configured to transmit a second signal through the optical micro cavity resonator where the second signal generator is configured to transmit said second signal that is weaker than the first signal and second signal is detuned with respect to the first signal; and a photo detector and spectral analyzer configured to detect the transmitted light and calculating a maximal Lyapunov exponent of the first and second signals.

9. The system demonstrating chaos transfer as recited in claim 8, where the first signal generator is an optical field pump and the second signal generator is an optical field probe.

10. The system demonstrating chaos transfer as recited in claim 9, where the optical field pump is a pump laser and the optical field probe is a probe laser.

11. The system demonstrating chaos transfer as recited in claim 10, where the optical micro cavity resonator is configured to generate mechanical oscillations responsive to the first signal and modulate said second signal with the mechanical oscillations.

12. The system demonstrating chaos transfer as recited in claim 11, comprising: one or more of a waveguide, optical fiber and free space configured and positioned with respect to the optical micro cavity resonator to couple the first signal and the second signal into and out of the micro cavity resonator ; and a wavelength division multiplexer configured to separate the first signal from the second signal.

13. A method for chaos transfer between multiple signals comprising: transfering the optomechanically-induced chaos on an optical field in a microcavity resonator to weaker optical signal in the same microcavity resonator and said weaker optical signal is detuned each other in their optical frequencies and/or wavelengths by selectively tuning the signals such that their frequency is detuned from an optical resonance of the resonator by the frequency of the mechanical frequency which is excited by the optical field.

14. The method as recited in claim 13, comprising: controlling the mechanical oscillations with the optical field and hence optomechanically-inducing Kerr-like nonlinearity, chaos and backaction noise, such that stochastic resonance is observed, and such that the signal to noise ratio of the weaker probe field having a power below detection threshold; selectively increasing the power to the optical field suth that the weaker signal is detectable such that as the pump power increases the signal-to-noise ratio of the weaker signal increases up to its maximum value and then starts to decrease as the pump power continues to increase.

15. A method comprising: steering a waveguide-coupled microresonator or a microlaser to its exceptional point (EP); controlling the chirality of the light circulating in the microresonator thereby controlling the emission direction of the microlaser; and tuning the microresonator from an EP to another EP, such that the emission direction of the laser is be tuned from a unidirectional emission in the clockwise direction to a unidirectional emission in the counter-clockwise direction.

16. The method as recited in claim 15, comprising: steering the microresonator away from the EPs, thereby obtaining bidirectional.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0025] For a better understanding of the present technology as disclosed, reference may be made to the accompanying drawings in which:

[0026] FIG. 1a is a view of the microtoroid illustrating the mechanical motion induced by optical radiation force;

[0027] FIG. 1b is a typical transmission spectra obtained by scanning the wavelength of a tunable laser with a power well-below (red) and above (blue) the mechanical oscillation threshold;

[0028] FIG. 1c is A typical electrical spectrum analyzer (ESA) trace of the detected photocurrent below the mechanical oscillation threshold;

[0029] FIG. 2A through 2C are phase diagrams of the pump fields in periodic (left), quasi periodic (middle), and chaotic (right) regimes;

[0030] FIG. 2D through 2F is phase diagrams of the probe fields in periodic (left), quasi periodic (middle), and chaotic (right) regimes;

[0031] FIG. 2G is a Bifurcation diagram of the pump fields;

[0032] FIG. 2H is a Bifurcation diagram of the probe fields;

[0033] FIG. 3a is Maximal Lyapunov exponents for the pump (blue) and the probe (red) fields as a function of the pump power;

[0034] FIG. 3b is an illustration of the spectral response of a Bandwidth broadening of the probe as a function of the pump power;

[0035] FIG. 3c is a typical spectra obtained for the probe at different pump powers;

[0036] FIG. 3d is a typical spectra obtained for the probe at different pump powers;

[0037] FIG. 3e is a typical spectra obtained for the probe at different pump powers;

[0038] FIG. 4a is Signal-to-noise ratio (SNR) of the probe as a function of the pump power;

[0039] FIG. 4b is An illustration conceptualizing chaos-mediated stochastic resonance in an opto-mechanical resonator;

[0040] FIG. 4c is an illustration of increasing the pump power first increases the SNR to its maximum and then reduces itMean <?>;

[0041] FIG. 4d is an illustration of increasing the pump power first increases the SNR to its maximum and then reduces it, scaled standard deviation R of interspike intervals ?;

[0042] FIG. 5 is a schematic diagram illustration a configuration of the technology under test;

[0043] FIG. 6a is demonstration of power spectra for the pump and probe fields at various pump powers corresponding to periodic;

[0044] FIG. 6b is a demonstration of power spectra for the pump and probe fields at various pump powers corresponding to quasi-periodic;

[0045] FIG. 6c is a demonstration of power spectra for the pump and probe fields at various pump powers corresponding to chaotic regime;

[0046] FIG. 6d is a demonstration of power spectra for the pump and probe fields at various pump powers corresponding to periodic;

[0047] FIG. 6e is a demonstration of power spectra for the pump and probe fields at various pump powers corresponding to quasi-periodic;

[0048] FIG. 6f is a demonstration of power spectra for the pump and probe fields at various pump powers corresponding to chaotic regime;

[0049] FIG. 7a through 7f are a demonstration of opto-mechanically-induced period-doubling in the pump and probe fields;

[0050] FIG. 7g through 7l are a numerical simulation of opto-mechanically-induced period-doubling in the pump and probe fields;

[0051] FIG. 7m is an illustration of a mechanical transverse mode in a micro-toroid;

[0052] FIG. 7n is an illustration of a mechanical longitudinal mode in a micro-toroid;

[0053] FIG. 8a is an illustration of periodic mechanical motion of the microtoroid resonator when the pump and probe fields are both in the chaotic regime;

[0054] FIG. 8b is an illustration of periodic mechanical motion of the microtoroid resonator when Filtering by the mechanical resonator: the mechanical resonator works as a low-pass filter;

[0055] FIG. 9a illustrates the Maximum of the Lyapunov exponent for the pump (red spectra) and probe (blue spectra) fields showing effect of the pump-cavity detuning;

[0056] FIG. 9b is illustrates the Maximum of the Lyapunov exponent for the pump (red spectra) and probe (blue spectra) fields showing effect of the probe-cavity detuning on the maximum Lyapunov exponents of the pump and probe fields;

[0057] FIG. 9c is an illustration of Maximum of the Lyapunov exponent for the pump (red spectra) and probe (blue spectra) fields showing the effect of the damping rate of the pump;

[0058] FIG. 9d is an illustration of Maximum of the Lyapunov exponent for the pump (red spectra) and probe (blue spectra) fields showing the effect of damping rate of the probe on the maximum Lyapunov exponents of the pump and probe fields;

[0059] FIG. 10 is an illustration of a signal-to-noise ratio (SNR) for the pump and probe signals.

[0060] FIGS. 11a through 11c are an output spectra shos that the spectral location of the resonance peak do not change with increasing pump power;

[0061] FIGS. 11d through 11f are an output spectra obtained in the numerical simulations of stochastic resonance show that the spectral location of the resonance peak stays the same for increasing pump power;

[0062] FIGS. 11g through 11i are an output spectra obtained in the numerical simulations of coherence resonance which show that the spectral location of the resonance peaks change with increasing pump power;

[0063] FIGS. 12a through 12b is a mean interspike interval and its variation calculated from the output signal in the probe mode;

[0064] FIGS. 12c through 12d is a mean interspike interval and its variation obtained in the numerical simulation of stochastic resonance with input weak probe; and

[0065] FIGS. 12e through 12f is a mean interspike interval and its variation obtained in the numerical simulation of coherence resonance in our system without input weak probe.

[0066] FIGS. 13a-13c illustrate the experimental configuration used in the technology and the effect of scatterers.

[0067] FIGS. 14a-14h illustrate the experimental observation of scatterer-induced asymmetric backscattering.

[0068] FIGS. 15a and 15B illustrate Controlling directionality and intrinsic chirality of whispering-gallery-modes.

[0069] FIGS. 16a-16e illustrate Scatterer-induced mirror-symmetry breaking at an EP.

[0070] FIG. 17 illustrate Schematic of the setup with the definitions of the parameters and signal propagation directions.

[0071] FIGS. 18a and 18b illustrate the eigenmode evolution of the non-Hermitian system as a function of the effective size factor d and the relative phase angle ? between the scatterers

[0072] FIGS. 19a and 19b illustrate experimentally obtained mode spectra as the relative phase angle ? between the scatterers was varied

[0073] FIGS. 20a and 20b illustrate experimentally obtained evolution of eigenfrequencies as the relative size of the scatterers was varied at different relative phase angles ?

[0074] FIG. 21 illustrate experimentally obtained evolution of the splitting quality factor as a function of ? for fixed relative size factor

[0075] FIGS. 22a-22d illustrate weights of CW and CCW components in the eigenmodes as the relative phase difference ? between the two nanoscatterers is varied

[0076] FIGS. 23a and 23b compare the chirality as determined from the eigenvalue calculations for the lasing cavity with the chirality as determined from the transmission calculations.

[0077] FIG. 24 Comparison of the chirality definitions for ?.sub.TMA, ?.sub.lasing and ?.sub.transmission

[0078] FIGS. 25a-25d Asymmetric backscattering intensities |B.sub.CW/CCW|.sup.2 from a CW to a CCW wave [left panel: (A) and (C)] and from a CCW to a CW mode [right panel: (B) and (D)].

[0079] FIGS. 26a-26f Directionality with a biased input (CW) as a function of the relative phase difference between two scatterers (A).

[0080] While the technology as disclosed is susceptible to various modifications and alternative forms, specific implementations thereof are shown by way of example in the drawings and will herein be described in detail. It should be understood, however, that the drawings and detailed description presented herein are not intended to limit the disclosure to the particular implementations as disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the scope of the present technology as disclosed and as defined by the appended claims.

DESCRIPTION

[0081] According to the implementation(s) of the present technology as disclosed, various views are illustrated in FIG. 1-12 and like reference numerals are being used consistently throughout to refer to like and corresponding parts of the technology for all of the various views and figures of the drawing. Also, please note that the first digit(s) of the reference number for a given item or part of the technology should correspond to the Fig. number in which the item or part is first identified.

[0082] One implementation of the present technology as disclosed comprising an opto-mechanical system having opto-mechanically induced chaos and stochastic resonance teaches a novel system and method for opto-mechanically mediated chaos transfer between two optical fields such that they follow the same route to chaos. The opto-mechanical system can be utilized for encoding chaos on a weak signal for chaotic encoding that can be used in secure communication. Chaos induced stochastic resonance in opto-mechanical systems are also applicable for use in improving signal detection.

[0083] The technology as disclosed and claimed demonstrates generating and transferring optical chaos in an opto-mechanical resonator. The technology demonstrates opto-mechanically-mediated transfer of chaos from a strong optical field (pump) that excites mechanical oscillations, to a very weak optical field (probe) in the same resonator. The technology demonstrates that the probe and the pump fields follow the same route, from periodic oscillations to quasi-periodic and finally to chaotic oscillations, as the pump power is increased. The chaos transfer from the pump to the probe is mediated by the mechanical motion of the resonator, because there is no direct talk between these two largely-detuned optical fields. Moreover, the technology demonstrates stochastic resonance in an opto-mechanical system. The required stochastic process is provided by the chaotic dynamics and the opto-mechanical backaction.

[0084] The details of the technology as disclosed and various implementations can be better understood by referring to the figures of the drawing. Referring to FIGS. 1a through 1c, a basic configuration of the technology was tested, which included a fiber-taper-coupled WGM microtoroid resonator (FIG. 1a.). FIG. 1a is an illustration of a whispering-gallery mode microtoroid opto-mechanical microresonator illustrating the mechanical motion induced by optical radiation force. FIG. 1b illustrates a typical transmission spectra obtained by scanning the wavelength of a tunable laser with a power well below and above the mechanical oscillation threshold. At high powers, thermally induced linewidth broadening and the fluctuation due to the mechanical oscillations kick in. A close up view of the fluctuations in the transmission, obtained at a specific wavelength of the laser, reveals a sinusoidal oscillaton at a frequency ?.sub.m of the mechanical oscillation.

[0085] FIG. 1c illustrates a typical electrical system analyzer (ESA) trace of the detected photocurrent below the mechanical oscillation threshold. The inset shows the spectrum above the threshold. The traces represent the demonstrated data, and the curves are the best fitting. Referring to FIGS. 2A through H, Opto-mechanically-mediated chaos generation and transfer between optical fields A-C and D-F. Phase diagrams of the pump Figs (A-C) and the probe Figs (D-F) fields in periodic (left), quasi periodic (middle), and chaotic (right) regimes. The phase diagrams were obtained by plotting the first time derivative of the measured output power of the pump Figs (A-C) and the probe Figs (D-F) fields as a function of the respective output powers. Figs G, H, Bifurcation diagrams of the pump Fig (G) and the probe Fig (H) fields as function of the input pump power. The pump and probe enter the chaotic regime via the same bifurcation route. The ratios of the bifurcation intervals for the pump a.sub.1/a.sub.2 and probe ?.sub.1/?.sub.2 are both 4.5556. The ratio between the width of a tine and the width of one of its two subtines is b.sub.1/b.sub.2=2.6412 for the pump and {tilde over (b)}.sub.1/{tilde over (b)}.sub.2=2.8687 for the probe.

[0086] When the power of the pump field is increased, it is observed that the transmitted pump light transited from a fixed state to a region of periodic oscillations, and finally to the chaotic regime through period-doubling bifurcation cascades (see FIGS. 2A-2C). The periodic regime, with only a few sharp peaks, and the quasi-periodic regime, with infinite discrete sharp peaks, in the output spectrum of the pump field. Finally, the whole baseline of the output spectrum of the pump field increased, implying that the system entered the chaotic regime. All these results coincide very well with previous studies.

[0087] These phenomena observed for the pump field originate from the nonlinear opto-mechanical coupling between the optical pump field and the mechanical mode of the resonator. Intuitively, one may attribute this observed dynamic to the chaotic mechanical motion of the resonator. However, the reconstructed mechanical motion of the resonator, using the experimental data in the theoretical model, showed that the optical signal was chaotic even if the mechanical motion of the resonator was periodic. Thus, it can be concluded that the reason for the chaotic behaviour in the optical field in our experiments is the strong nonlinear optical Kerr response induced by the nonlinear coupling between the optical and mechanical modes.

[0088] Simultaneously monitoring the probe field reveals that as the pump power is increased, the probe, also, experienced periodic, quasi periodic, and finally chaotic regimes. More importantly, the pump and probe entered the chaotic regime via the same bifurcation route (FIG. 2), that is both optical fields experienced the same number of period-doubling cascades, and the doubling points occurred at the same values of the pump power. These features are clearly seen in the phase-space plots (FIG. 2A-2C and 2D-2F) and in the bifurcation diagrams (FIG. 2G and 2H). The demonstrated data fits very well with bifurcation, in which each periodic region is smaller than the previous region by the factor a.sub.1/a.sub.2=4.5556 for the pump and ?.sub.1/?.sub.2=4.5556 for the probe, and these factors are close to the first universal Feigenbaum constant 4.6692. The ratio between the width of a tine and the width of one of its two sub-tines for the pump is b.sub.1/b.sub.2=2.6412, and that for the probe {tilde over (b)}.sub.1/{tilde over (b)}.sub.2=2.8687, which are both close to the second universal Feigenbaum constant 2.5029 (two mathematical constants, which both express ratios in a bifurcated non-linear system).

[0089] In order to effectively demonstrate the present technology, the probe field is sufficiently weak such that it could not induce any mechanical oscillations of its own, and the large frequency-detuning between the pump field (in the 1550 nm band) and the probe field (in the 980 nm band) assured that there was no direct crosstalk between the optical fields. Thus the observed close relation between the route-to-chaos for the pump and probe fields can only be attributed to the fact that the periodic mechanical motion of the microresonator mediates the coupling between the optical modes via opto-mechanically-induced Kerr-like nonlinearity (the induced refractive index change is directly proportional to the square of the field instead of varying in linearity with it), and enables the probe to follow the pump field.

[0090] To demonstrate the technology, light from an external cavity laser in the 1550 nm band is first amplified by an erbium-doped fiber amplifier (EDFA) and then coupled into a microtoroid to act as the pump for the excitation of the mechanical modes. Optical transmission spectrum, is obtained by scanning the wavelength of the pump laser, which shows a typical Lorentzian lineshape (follows a fourier transform line broadening function) for low powers of the pump field (FIG. 1b). The quality factor of this optical mode was 10.sup.7. As the pump power is increased, the spectrum changed from a Lorentzian lineshape to a distorted asymmetric lineshape due to thermal nonlinearity. This helps to keep the pump laser detuned with respect to the resonant line of the microcavity. As a result, radiation-pressure-induced mechanical oscillations take place as reflected by the oscillations imprinted on the optical transmission spectra (FIG. 1b). This then leads to the modulation of the transmitted light at the frequency of the mechanical motion (FIG. 1b, inset). The Rf power versus frequency traces, obtained using an electrical spectrum analyzer (ESA), reveals a Lorentzian spectrum located at ?.sub.m?26.1 MHz with a linewidth of ?200 KHz, implying a mechanical quality factor of Q.sub.m?131, when the pump power is below the threshold of mechanical oscillation (FIG. 1c). For powers above the threshold, the linewidth narrowing is clearly observed (FIG. 1c, inset)

[0091] In order to demonstrate the effect of the mechanical motion induced by the strong pump field on a weak light field (probe light) within the same resonator, an external cavity laser with emission in the 980 nm band can be used. The power of the probe laser is chosen such that it does not induce any thermal or mechanical effect on the resonator, i.e., its power is well below the threshold of mechanical oscillations. The transmission spectra of the pump and the probe fields are separately monitored by photodiodes connected to an oscilloscope and an ESA. The probe resonance mode had a quality factor of 6?10.sup.6.

[0092] Referring to FIG. 5, a more detailed schematic diagram is provided of one implementation of the technology being demonstrated, which includes a pump and probe configuration. The pump (1550 nm band) and the probe (980 nm band) fields are coupled into and out of a microtoroid resonator via the same tapered fiber in the same direction. An Erbium-doped fibre amplifier EDFA is utilized for signal amplification. A PC is and a Polarization controller are utilized for control. A wavelength division multiplexer (WDM), a Photodetector for signal detection, and an Electrical spectrum analyzer (ESA) are utilized.

[0093] An optical pump field, provided by a tunable External Cavity Laser Diode (ECLD) in the 1550 nm band, is first amplified using an erbium-doped fiber amplifier (EDFA), and then coupled into a fiber, using a 2-to-1 fiber coupler, together with a probe field provided by a tunable ECLD in the 980 nm band. A section of the fiber is tapered, to enable efficient coupling of the pump and probe fields into and out of a microtoroid resonator. The pump and probe fields in the transmitted signals are separated from each other using a wavelength division multiplexer (WDM) and then sent to two separate photodetectors (PDs). The electrical signals from the PDs are then fed to an oscilloscope, in order to monitor the time-domain behavior, and also to an electrical spectrum analyzer (ESA) to obtain the power spectra.

[0094] It can be concluded that the intracavity pump and probe fields do not directly couple to each other, and that the probe and pump fields couple to the same mechanical mode of the microcavity with different coupling strengths. The technology demonstrates that in such a situation, the mechanical mode mediates an indirect coupling between the fields. The dynamical equation for the intracavity pump mode coupled to the mechanical mode of the cavity can be written as

{dot over (a)}.sub.pump?[?.sub.pump?i(?.sub.pump?g.sub.pumpX)]a.sub.pump+i??.sub.pump(t), (S1)

[0095] where a.sub.pump is the complex amplitude of the intracavity pump field, ?.sub.pump is the damping rate of the cavity pump mode, ?.sub.pump(t) represents the amplitude of the input pump field, ? is the pump-resonator coupling rate, ?.sub.pump is the frequency detuning between the input pump field and the cavity resonance, X is the position of the mechanical mode coupled to a.sub.pump, and g.sub.pump is the strength of the optomechanical coupling between the optical pump field and the mechanical mode. This equation can be solved in the frequency-domain by using the Fourier transform as

[00001]$\begin{array}{cc}{a}_{\mathrm{pump}}?\left(?\right)=\frac{-{\mathrm{ig}}_{\mathrm{pump}}}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}}.Math.{?}_{-?}^{+?}.Math.X?\left(?-{?}_1\right).Math.a_{\mathrm{pump}}?\left({?}_1\right).Math.d.Math..Math.{?}_1+\frac{i.Math..Math.{\mathrm{?.Math.}}_{\mathrm{pump}}?\left(?\right)}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}},& \left(\mathrm{S2}\right)\end{array}$

[0096] where a.sub.pump(?) X(?), and ?.sub.pump(?) are the Fourier transforms of the time-domain signals a.sub.pump(t), X(t), and ?.sub.pump(t). Since the dynamics of the mechanical motion X(t) is slow compared to that of the optical mode, the convolution term can be replaced in the above equation by the product a.sub.pump (?)X(?), under the slowly-varying envelope approximation, which then leads to

[00002]$\begin{array}{cc}\left[1-\frac{-{\mathrm{ig}}_{\mathrm{pump}}}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}}.Math.X?\left(?\right)\right].Math.a_{\mathrm{pump}}?\left(?\right)=\frac{-i.Math..Math.{\mathrm{?.Math.}}_{\mathrm{pump}}?\left(?\right)}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}}.& \left(\mathrm{S3}\right)\end{array}$

[0097] X(?) is in general so small that we have g.sub.pump.sup.2|X(?)|.sup.2?(???.sub.pump).sup.2+?.sub.pump.sup.2. Then using the identity 1/(1?x)?1+x, for x?1, we can re-write Eq. (S3) as

[00003]$\begin{array}{cc}{a}_{\mathrm{pump}}?\left(?\right)=\left[1+\frac{-{\mathrm{ig}}_{\mathrm{pump}}}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}}.Math.X?\left(?\right)\right].Math.\frac{-i.Math..Math.{\mathrm{?.Math.}}_{\mathrm{pump}}?\left(?\right)}{i?\left(?-{?}_{\mathrm{pump}}\right)+{?}_{\mathrm{pump}}}.& \left(\mathrm{S4}\right)\end{array}$

[0098] By multiplying the above equation with its conjugate and dropping the linear term of X(?), which is zero on average, we can obtain the relation between the spectrum S.sub.pump(?)=|a.sub.pump(?)|.sup.2 of the optical mode a.sub.pump and the spectrum of the mechanical motion S.sub.X(?)=|X(?)|.sup.2 as

[00004]$\begin{array}{cc}{S}_{\mathrm{pump}}?\left(?\right)=\frac{{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{pump}}^2}{{?}_{\mathrm{pump}}^2}.Math.{?}_{\mathrm{pump}}?\left(?\right)?\left[1+\frac{g_{\mathrm{pump}}^2}{{?}_{\mathrm{pump}}^2}.Math.{?}_{\mathrm{pump}}?\left(?\right).Math.S_X?\left(?\right)\right],& \left(\mathrm{S5}\right)\\ \mathrm{Where}& \\ {?}_{\mathrm{pump}}?\left(?\right)=\frac{{?}_{\mathrm{pump}}^2}{{?}_{\mathrm{pump}}^2+{\left(?-{?}_{\mathrm{pump}}\right)}^2}& \left(\mathrm{S6}\right)\end{array}$

[0099] is a susceptibility coefficient. By further introducing the normalized spectrum

[00005]$\begin{array}{cc}{\tilde{S}}_{\mathrm{pump}}?\left(?\right)=S_{\mathrm{pump}}?\left(?\right)-\frac{{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{pump}}^2}{{?}_{\mathrm{pump}}^2}.Math.{?}_{\mathrm{pump}}?\left(?\right),& \left(\mathrm{S7}\right)\end{array}$

[0100] the above equation can be written as

[00006]$\begin{array}{cc}{\tilde{S}}_{\mathrm{pump}}?\left(?\right)=\frac{{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{pump}}^2.Math.g_{\mathrm{pump}}^2}{{?}_{\mathrm{pump}}^4}.Math.{?}_{\mathrm{pump}}^2?\left(?\right).Math.S_X?\left(?\right),& \left(\mathrm{S8}\right)\end{array}$

[0101] A similar equation can be obtained by analyzing the spectrum of the optical mode a.sub.probe coupled to the probe field as

[00007]$\begin{array}{cc}{\tilde{S}}_{\mathrm{probe}}?\left(?\right)=\frac{{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{probe}}^2.Math.g_{\mathrm{probe}}^2}{{?}_{\mathrm{probe}}^4}.Math.{?}_{\mathrm{probe}}^2?\left(?\right).Math.S_X?\left(?\right),& \left(\mathrm{S9}\right)\\ \mathrm{Where}& \\ {?}_{\mathrm{probe}}?\left(?\right)=\frac{{?}_{\mathrm{probe}}^2}{{?}_{\mathrm{probe}}^2+{\left(?-{?}_{\mathrm{probe}}\right)}^2},& \left(\mathrm{S10}\right)\end{array}$

[0102] ?.sub.probe is the damping rate of the cavity mode coupled to the probe field, ?.sub.probe(t) represents the amplitude of the input probe field, ?.sub.probe is the detuning between the input probe field and the cavity resonance, and g.sub.probe is the coupling strength between the optical mode a.sub.probe and the mechanical mode.

[0103] From Eqs. (S8) and (S9), the relation between the normalized spectra {tilde over (S)}.sub.pump(?) and {tilde over (S)}.sub.probe (?) is obtain as

[00008]$\begin{array}{cc}{\tilde{S}}_{\mathrm{probe}}?\left(?\right)=G.Math.\frac{{?}_{\mathrm{probe}}^2?\left(?\right)}{{?}_{\mathrm{pump}}^2?\left(?\right)}.Math.{\tilde{S}}_{\mathrm{pump}}?\left(?\right),& \left(\mathrm{S11}\right)\\ \mathrm{Where}& \\ G=\frac{{\mathrm{.Math.}}_{\mathrm{probe}}^2.Math.g_{\mathrm{probe}}^2.Math.{?}_{\mathrm{pump}}^4}{{\mathrm{.Math.}}_{\mathrm{pump}}^2.Math.g_{\mathrm{pump}}^2.Math.{?}_{\mathrm{probe}}^4}.& \left(\mathrm{S12}\right)\end{array}$

[0104] If we assume that the detunings and damping rates of the optical modes are close to each other, i.e., ?.sub.pump??.sub.probe and ?.sub.pump??.sub.probe, we have ?.sub.probe.sup.2(?)/?.sub.pump.sup.2(?)?1, leading to

{tilde over (S)}.sub.probe(?)?G {tilde over (S)}.sub.pump(?). S(13)

[0105] This implies that the spectra of the pump and probe fields are correlated with each other. The correlation factor G is mainly determined by the opto-mechanical coupling strengths of the pump and the probe fields as well as the intensities of these fields.

[0106] The relation between the spectra of the pump and probe signals shows that the opto-mechanical coupling strengths g.sub.pump and g.sub.probe of the pump and probe field to the excited mechanical mode determine how closely the probe field will follow the pump field. Clearly, these coupling strengths do not change the shape of the spectrum, and this is the reason why the probe signal follows the pump signal in the frequency domain and enters the chaotic regime via the same bifurcation route, despite the fact that they are far detuned from each other (FIG. 2G, 2H).

[0107] When demonstrating the technology, the mechanical motion is excited by the strong pump field, and the probe is chosen to have such a low power that it could not induce any mechanical oscillations. The large pump and probe detuning ensured that there is no direct coupling between them. The fact that both the pump and the probe are within the same resonator that sustains the mechanical oscillation naturally implies that both the pump and the probe are affected by the same mechanical oscillation with varying strengths, depending on how strongly they are coupled to the mechanical mode. The pump and probe spectra (FIG. 6) obtained by experimentation under these conditions agree well with the prediction given in Eq. (S13), in the sense that the spectra of the pump and the probe fields become correlated if they couple to the same mechanical mode. The slight differences in phase diagrams obtained in the demonstration (FIG. 2A-2C, 2D-2F) imply that different coupling strengths of the pump and probe to the same mechanical mode, due to the difference in their spatial overlaps with the mechanical mode, affect the trajectories and thus the phase diagrams.

[0108] One implementation of the technology as disclosed and claimed is configured to control chaos and stochastic noise. The technology is configured to control chaos and stochastic noise by increasing the pump power (1550 nm band) on the detected pump and the probe signals (980 nm band), on the degree of sensitivity to initial conditions and chaos in the probe. This is accomplished by calculating the maximal Lyapunov exponent (MLE) from the detected pump and probe signals. Lyapunov exponents quantify the sensitivity of a system to initial conditions and give a measure of predictability. They are a measure of the rate of convergence or divergence of nearby trajectories in phase space.

[0109] The behavior of the MLE is a good indicator of the degree of convergence or divergence of the whole system. A positive MLE implies divergence and sensitivity to initial conditions, and that the orbits are on a chaotic attractor. If, on the other hand, the MLE is negative, then trajectories converge to a common fixed point. A zero exponent implies that the orbits maintain their relative positions and they are on a stable attractor. The technology demonstrates that with increasing pump power the degree of chaos and sensitivity to initial conditions, as indicated by the positive MLE, first increase and then decreased after reaching its maximum, both for the pump and the probe fields (FIG. 3a). With further increase of the pump, the MLE becomes negative, indicating a reverse period-doubling route out of chaos into periodic dynamics. In addition to the pump power, the pump-cavity detuning and the damping rate of the pump affect the MLE for both the pump and the probe fields.

[0110] Referring to FIGS. 3b through 3e, Maximal Lyapunov exponents for the pump (blue) and the probe fields as a function of the pump power is illustrated. The Lyapunov exponents describe the sensitivity of the transmitted pump and the probe signals to the input pump power. Circles and diamonds are the exponents calculated from measured data. The blue and red curves are drawn as eye guidelines. FIG. 3b illustrate bandwidth broadening of the probe as a function of the pump power. Circles is the demonstrated data and the red curve is the fitting curve. The inset shows the cross-correlation between the pump and probe fields as a function of the pump powerTypical spectra obtained for the probe at different pump powers. Power increases from c to e, clearly showing the bandwidth broadening. The corresponding Lyapunov exponents and bandwidths are labelled in a and b.

[0111] The bandwidth D of the probe signal increases with increasing pump power (FIG. 3b-e), and the relation between the bandwidth D of the probe signal and the pump power P.sub.pump follows the power function D=?P.sub.pump.sup.1/2, with ?=1.65?10.sup.8 Hz/mW.sup.1/2 (FIG. 3b). This is contrary to the expectation that the less (more) chaotic the signal is, the smaller (larger) its bandwidth is. This can be attributed to the presence of both the deterministic noise from chaos and the stochastic noise from the opto-mechanical backaction. According to Newton's third law, for every action there is always an equal and opposite reaction. With similar inevitability, this time in quantum physics, for every measurement there is always a perturbation of the object being measured. This phenomenon, known as quantum back-action, could now be put to practical use because it can alter the frequency, position, and damping rate of a resonator. For example in an opto-mechanical system, radiation pressure caused by circulating photons create optomechanical oscillations and opto-mechanical dynamics, o[tpmechanical oscillations then back-action on the light (photons) and change their charateristics, inducing noise, shifting their frequency. The system is chaotic for the range of pump power where the maximal Lyapunov exponent is positive (FIG. 3a). For smaller or larger power levels, the system is not in the chaotic regime. Thus, chaos-induced noise is present only for a certain range of pump power.

[0112] The effect of opto-mechanical backaction, on the other hand, is always present in the power range shown in FIG. 3, and its effect increases with increasing pump power, where the higher the pump power, the larger the stochastic noise due to backaction (FIG. 3e has more backaction noise than FIG. 3d, which, in turn, has more than FIG. 3c).

[0113] In FIG. 3c and FIG. 3e (corresponding to zero or negative maximum Lyapunov exponent), the bandwidth is almost completely determined by the opto-mechanical backaction, with very small or no contribution from chaos. In FIG. 3d, the system is in the chaotic regime, and thus both chaos and the backaction contribute to noise, leading to a larger probe bandwidth in FIG. 3d than in FIG. 3c. At the pump power of FIG. 3e, on the other hand, the system is no more in chaotic. However, backaction noise reaches such high levels that it surpasses the combined effect of chaos and backaction noises of FIG. 3d. As a result, FIG. 3e has a larger bandwidth. Thus, for the present technology, the pump and the probe became less chaotic when the pump power was increased beyond a critical value; however, at the same time their bandwidths increased, implying more noise contribution from the optomechanical backaction. Therefore, the correlation between the pump and probe fields decreased with increasing pump power (FIG. 3b inset).

[0114] The technology as disclosed and claimed demonstrates stochastic resonance mediated by opto-mechanically-induced-chaos. Referring to FIGS. 4a through 4d, Opto-mechanically induced chaos-mediated stochastic resonance in an opto-mechanical resonator is illustrated. Referring to FIG. 4a, signal-to-noise ratio (SNR) of the probe as a function of the pump power is illustrated. The solid curve is the best fit to the demonstrated data (open circles). Referring to FIG. 4b, an illustration conceptualizing chaos-mediated stochastic resonance in an opto-mechanical resonator is provided. The mechanical motion mediates the pump-probe coupling and enables the pump field to control chaos, the strength of the opto-mechanical back-action, and the probe bandwidth. Hence, the pump controls the system's noise, where increasing the pump power first increases the SNR to its maximum and then reduces it. FIG. 4c, illustrates a Mean <?>, and FIG. 4d, a scaled standard deviation R of interspike intervals ?, obtained from experimental data for the probe (open circles) as a function of pump power, exhibiting the theoretically-expected characteristics for a system with stochastic resonance. The data points labelled as c, d and e correspond to the same points indicated in FIG. 3.

[0115] The technology as disclosed and claimed herein demonstrates that below a critical value, increasing the pump power increases the signal-to-noise ratio (SNR) of both the probe and the pump fields; however, beyond this value, the SNR decreased despite increasing pump power (FIG. 4a). When the pump is turned off (P.sub.pump?0 mW), the SNR of the probe signal is ?10 dB. The maximum value of the SNR is obtained for the pump power of P.sub.pump?15 mW. The relation between the pump power and the SNR of the probe is given by the expression (?/P.sub.pump)exp(??1?{square root over (P.sub.pump)}), with ?=0.825 mW and ?=7.4764 mW.sup.1/2. Combining the relation between the bandwidth and the pump power with the relation between the SNR and pump power, it is determined that the relation between the SNR and the bandwidth of the probe signal scales as SNR ? aD.sup.?2 exp (?b/D). This expression implies that SNR is not a monotonous function of the bandwidth D (i.e., noise), and that it is possible to increase the SNR by increasing the noise. This effect is referred to as stochastic resonance, which is a phenomenon in which the response of a nonlinear system to a weak input signal is optimized by the presence of a particular level of stochastic noise, i.e., the noise-enhanced response of an input signal. FIG. 4b provides a conceptual illustration of the mechanism leading to chaos-mediated stochastic resonance in our opto-mechanical system.

[0116] An observed noise benefit (FIG. 4a) can be described as stochastic resonance if the input (weak signal) and output signals are well-defined. When the technology is demonstrated, the input is given by the weak probe field (in the 980 nm band), and the output is the signal detected in the probe mode at the end of the fiber taper. In the rotated frame, and with the elimination of mechanical degrees of freedom, the optical system is described by a weak periodic input (i.e., the weak probe field) modulated by the frequency of the mechanical mode. The noise required for stochastic resonance can be either external or internal (due to the system internal dynamics). When demonstrating the present technology it is provided by both the opto-mechanical backaction and chaotic dynamics, which are both controlled by the external pump field.

[0117] At low pump powers, corresponding to periodic or less-chaotic regimes (i.e., negative or zero Lyapunov exponent), the contribution of the backaction noise is small, and chaos is not strong enough to help amplify the signal. Therefore, the SNR is low. At much higher pump power levels, the system evolves out of chaos. At the same time, the noise contribution to the probe from the opto-mechanical backaction increases with increasing pump power and becomes comparable to the probe signal. Consequently, the SNR of the probe decreases. Between these two SNR minima, the noise attains the optimal level to amplify the signal coherently with the help of intermode interference due to the chaotic map; and thus an SNR maximum occurs. Indeed, resonant jumps between different attractors of a system due to chaos-mediated noise as a route to stochastic resonance and to improve SNR.

[0118] The mean (?) (FIG. 4c) and scaled standard deviation R=?{square root over (?.sup.2??.sup.2)}/? (FIG. 4d) of the interspike intervals ? of the signals detected during the demonstration of the technology exhibit the theoretically-expected dependence on the noise (i.e., pump power) for a system with stochastic resonance. While (?) is not affected by the pump power and retains its value of 0.24 ?s (the resonance revival frequency of 26 MHz determined by the frequency of the mechanical mode), R attains a maximum at an optimal pump power (i.e., R is a concave function of noise). On the other hand, for a system with coherence resonance, increasing noise leads to a decrease in (?), and R is a convex function of the noise. It is known that in a system with coherence resonance the positions of the resonant peaks in the output spectra shift with increasing pump power, implying that the resonances are induced solely by noise. The resonant peak in our experimentally-obtained output power spectra, however, was located at the frequency of the mechanical mode, which modulated the input probe field, and its position did not change with increasing pump power (i.e., noise level), providing another signature of stochastic resonance. Thus, it can be concluded that the observed SNR enhancement is due to the chaos-mediated stochastic resonance, and hence the present technology constitutes the first observation of opto-mechanically-induced chaos-mediated stochastic resonance, which is a counterintuitive process where additional noise can be helpful.

[0119] The technology as disclosed and claimed demonstrates a bifurcation process and the route to chaos of the probe fields follow the route to chaos of the pump. When under test, the technology demonstrated a mechanical mode with a frequency of around 26 MHz, and the evolution of this mode as a function of the power of the input pump field.

[0120] Referring to FIG. 7, opto-mechanically-induced period-doubling in the pump and probe fields is illustrated. FIG. 7a illustrates test data for the technology under test, and FIG. 7b illustrates the results of numerical simulations showing first and second period-doubling processes for the pump (Lower spectra) and probe (Upper spectra) fields. The technology in one of various implementations as disclosed demonstrates a mechanical mode with a frequency of around 26 MHz, and demonstrates the evolution of this mode as a function of the power of the input pump field. As shown in FIG. 7a, both the pump and probe fields experience a period-doubling bifurcation as the input power of the pump field is increased. When the input pump power is low, the spectra of the pump and probe fields shows a peak at around 26 MHz. When the input pump power is increased above a critical value, a second peak appears just at half frequency of the main peak, i.e., ?13 MHz which corresponds to a period-doubling process. At higher powers, successive period-doubling events occur, leading to peaks located at frequencies of ?-th of the main peak. For example, the second period-doubling bifurcation leads to frequency peaks at 6.5 MHz for both the pump and the probe fields.

[0121] In FIG. 7b, the results of numerical simulations obtained is illustrated by solving the following set of equations

{dot over (a)}.sub.pump=?[?.sub.pump?i(?.sub.pump?g.sub.pumpX)]a.sub.pump+i??.sub.pump(t), (S14)

{dot over (a)}.sub.probe=?[?.sub.probe?i(?.sub.probe?g.sub.probeX)]a.sub.probe+i??.sub.probe(t), (S15)

{dot over (X)}=??.sub.mX+?.sub.mP, (S16) 2

{dot over (P)}=??.sub.mP??.sub.mX+g.sub.pump|a.sub.pump|.sup.2, (S17)

[0122] which describe the evolution of the pump and probe cavity modes and the mechanical mode. In a simulation, a single mechanical eigenmode with frequency 26 MHz can be considered, similar to what is demonstrated by the technology under test. Here, ?.sub.m and ?.sub.m are the frequency and damping rate of the mechanical mode. The probe signal is chosen to be very weak, so that it does not induce mechanical or thermal oscillations. Consequently, the mechanical mode was induced only by the pump field as described by the expression in Eq. (S17). The model explains the observations of the technology. It is clearly seen that the probe field follows the pump field during the bifurcation process.

[0123] As shown in FIGS. 7a-7l the technology demonstrates the existence of a second mechanical mode with frequency 5 MHz. This mode is excited when the pump power was increased to observe the second period-doubling process. Generally, one may think that this low-frequency mechanical mode would affect the bifurcation process of the 26 MHz mechanical mode, because these two mechanical modes are in the same micro-resonator and thus may couple to each other. However, the technology as disclosed does not demonstrate such a characteristic. Numerical simulations using COMSOL demonstrate that the mechanical modes at 26 MHz and 5 MHz are, respectively, transverse and longitudinal modes (FIG. 7c, 7d). Thus, they are orthogonal, which implies that there is minimal or no interaction between them.

[0124] Referring to FIGS. 7m and 7n a COMSOL simulation of the mechanical modes in a microtoroid is illustrated. The mechanical mode with frequency a, 26 MHz is a transverse mode whereas the one with frequency b, 5 MHz is a longitudinal mode. Both of these mechanical modes are observed, with the 5 MHz mode being excited only when the pump power is significantly high that the mode at 26 MHz experiences the second period doubling (FIG. 7a through 7l). The orthogonality of these mechanical modes implies that there is no direct coupling between them.

[0125] In order to understand how the co-existence of the pump and probe fields in the same opto-mechanical resonator affect their interaction with the system and with each other, consider the following Hamiltonian

[00009]$\begin{array}{cc}H={?}_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}+{\mathrm{.Math.}}_{\mathrm{probe}}?\left(a_{\mathrm{probe}}^{}+a_{\mathrm{probe}}\right)+g_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}.Math.X+\frac{{?}_m}{2}.Math.\left(X^2+P^2\right)+{?}_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}+{\mathrm{?.Math.}}_{\mathrm{pump}}?\left(a_{\mathrm{pump}}^{}+a_{\mathrm{pump}}\right)+g_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}.Math.X,& \left(\mathrm{S18}\right)\end{array}$

[0126] where the first (fourth) and second (fifth) terms are related to the free evolution of the probe a.sub.probe (pump a.sub.pump) field, and the third (sixth) term explains the interaction of the probe (the pump) field with the mechanical mode X. The last term corresponds to the free evolution of the mechanical mode.

[0127] First, consider only the probe field by eliminating the fourth, fifth and sixth terms. In this case, resulting at the Hamiltonian

[00010]$\begin{array}{cc}H={?}_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}+{\mathrm{?.Math.}}_{\mathrm{probe}}?\left(a_{\mathrm{probe}}^{}+a_{\mathrm{probe}}\right)+g_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}.Math.X+\frac{{?}_m}{2}.Math.\left(X^2+P^2\right).& \left(\mathrm{S19}\right)\end{array}$

[0128] By introducing the translational transformation

[00011]$\begin{array}{cc}\stackrel{.Math.}{X}=X+\frac{g_{\mathrm{probe}}}{{?}_m}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}},\stackrel{.Math.}{P}=P,& \left(\mathrm{S20}\right)\end{array}$

[0129] the Hamiltonian H can be re-expressed as

[00012]$\begin{array}{cc}H={?}_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{probe}}?\left(a_{\mathrm{probe}}^{}+a_{\mathrm{probe}}\right)-\frac{g_{\mathrm{probe}}^2}{2.Math..Math.{?}_m}.Math.{\left(a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}\right)}^2+\frac{{?}_m}{2}.Math.\left(\stackrel{.Math.}{X^2}+{\stackrel{.Math.}{P}}^2\right),& \left(\mathrm{S21}\right)\end{array}$

[0130] where we see that the nonlinear interaction between the probe field and the mechanical motion leads to an effective Kerr-like nonlinearity in the optical mode a.sub.probe, with its coefficient given as

[00013]$\begin{array}{cc}{?}_{\mathrm{probe}}=\frac{g_{\mathrm{probe}}^2}{2.Math.{?}_m},& \left(\mathrm{S22}\right)\end{array}$

[0131] where ?.sub.m is the frequency of the mechanical mode. Equation (S22) implies that the opto-mechanically-induced Kerr-like nonlinearity is dependent on (i) the opto-mechanical coupling between the optical and mechanical modes and (ii) the frequency of the mechanical mode.

[0132] Following a similar procedure, we can derive the coefficient of nonlinearity for the case when only the pump field is present. In such a case, resulting in

[00014]$\begin{array}{cc}H={?}_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}?\left(a_{\mathrm{pump}}^{}+a_{\mathrm{pump}}\right)+g_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}.Math.X+\frac{{?}_m}{2}.Math.\left(X^2+P^2\right).& \left(\mathrm{S23}\right)\end{array}$

[0133] By introducing the transformation

[00015]$\begin{array}{cc}\stackrel{.Math.}{X}=X+\frac{g_{\mathrm{pump}}}{{?}_m}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}},\stackrel{.Math.}{P}=P,& \left(\mathrm{S24}\right)\end{array}$

[0134] the Hamiltonian rewritten as

[00016]$\begin{array}{cc}H={?}_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}?\left(a_{\mathrm{pump}}^{}+a_{\mathrm{pump}}\right)-\frac{g_{\mathrm{pump}}^2}{2.Math..Math.{?}_m}.Math.{\left(a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}\right)}^2+\frac{{?}_m}{2}.Math.\left({\stackrel{.Math.}{X}}^2+{\stackrel{.Math.}{P}}^2\right).& \left(\mathrm{S25}\right)\end{array}$

[0135] Thus, the coefficient of the effective Kerr-like nonlinearity in the optical mode a.sub.pump becomes

[00017]$\begin{array}{cc}{?}_{\mathrm{pump}}=\frac{g_{\mathrm{pump}}^2}{2.Math..Math.{?}_m},& \left(\mathrm{S26}\right)\end{array}$

[0136] where ?.sub.m is the frequency of the mechanical mode and g.sub.pump is the strength of the coupling between the pump and mechanical modes.

[0137] Now let us consider the case where both the pump and probe fields exist within the same resonator and they are coupled to the same mechanical mode. In this case, by applying the transformation

[00018]$\begin{array}{cc}\tilde{X}=X+\frac{g_{\mathrm{probe}}}{{?}_m}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{pump}},+\frac{g_{\mathrm{pump}}}{{?}_m}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}},\tilde{P}=P,& \left(\mathrm{S27}\right)\end{array}$

[0138] re-express the Hamiltonian given in Eq. (S18) as

[00019]$\begin{array}{cc}H={?}_{\mathrm{probe}}.Math.a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{probe}}?\left(a_{\mathrm{probe}}^{}+a_{\mathrm{probe}}\right)-\frac{g_{\mathrm{probe}}^2}{2.Math..Math.{?}_m}.Math.{\left(a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}\right)}^2+\frac{{?}_m}{2}.Math.\left({\tilde{X}}^2+\widetilde{P^2}\right)+{?}_{\mathrm{pump}}.Math.a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}?\left(a_{\mathrm{pump}}^{}+a_{\mathrm{pump}}\right)-\frac{g_{\mathrm{pump}}^2}{2.Math..Math.{?}_m}.Math.{\left(a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}\right)}^2-\frac{g_{\mathrm{pump}}.Math.g_{\mathrm{probe}}}{{?}_m}.Math.\left(a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}\right).Math.\left(a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}\right).& \left(\mathrm{S28}\right)\end{array}$

[0139] Here the third and seventh terms are the coefficients of the Kerr-like nonlinearity derived earlier for the cases when only the probe or the pump fields exist in the opto-mechanical resonator. The last term, on the other hand, is new and implies an effective interaction between the pump and probe fields, if they both exist in the opto-mechanical resonator.

[0140] The dynamical equations of this system can be written as

{dot over (a)}.sub.pump=?[?.sub.pump?i(?.sub.pump?g.sub.pumpX)]a.sub.pump+i??.sub.pump, (S29)

{dot over (a)}.sub.probe=?[?.sub.probe?i(?.sub.probe?g.sub.probeX)]a.sub.probe+i??.sub.probe. (S30)

[0141] In the long-time limit (i.e., steady-state), we have {dot over (a)}.sub.pump, {dot over (a)}.sub.probe?0, which leads to

[00020]$\begin{array}{cc}{a}_{\mathrm{probe}}=\frac{i.Math..Math.?.Math..Math.{\mathrm{.Math.}}_{\mathrm{probe}}}{{?}_{\mathrm{probe}}-i?\left({?}_{\mathrm{probe}}-g_{\mathrm{probe}}.Math.X\right)}?\frac{i.Math..Math.{\mathrm{?.Math.}}_{\mathrm{probe}}}{{?}_{\mathrm{probe}}-i.Math..Math.{?}_{\mathrm{probe}}}+\frac{{\mathrm{?.Math.}}_{\mathrm{probe}}.Math.g_{\mathrm{probe}}}{{\left({?}_{\mathrm{probe}}-i.Math..Math.{?}_{\mathrm{probe}}\right)}^2}.Math.X,.Math.& \left(\mathrm{S31}\right)\\ a_{\mathrm{pump}}=\frac{i.Math..Math.?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}}{{?}_{\mathrm{pump}}-i?\left({?}_{\mathrm{pump}}-g_{\mathrm{pump}}.Math.X\right)}?\frac{i.Math..Math.?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}}{{?}_{\mathrm{pump}}-i.Math..Math.{?}_{\mathrm{pump}}}+\frac{?.Math..Math.{\mathrm{.Math.}}_{\mathrm{pump}}.Math.g_{\mathrm{pump}}}{{\left({?}_{\mathrm{pump}}-i.Math..Math.{?}_{\mathrm{pump}}\right)}^2}.Math.X.& \left(\mathrm{S32}\right)\end{array}$

[0142] If we further eliminate the degrees of freedom of the mechanical mode X from the above equations, then, under the conditions that ?.sub.pump=?.sub.probe, ?.sub.pump=?.sub.probe, and g.sub.pump=g.sub.probe, we have

a.sub.pump=(?.sub.pump/?.sub.probe)?.sub.probe. (S33)

[0143] By substituting this equation into the last term in Eq. (S28), we see that the last term of the Hamiltonian becomes

[00021]$\begin{array}{cc}\frac{g_{\mathrm{pump}}.Math.g_{\mathrm{probe}}}{{?}_m}.Math.\left(a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}\right).Math.\left(a_{\mathrm{pump}}^{}.Math.a_{\mathrm{pump}}\right).fwdarw.\frac{g_{\mathrm{pump}}.Math.g_{\mathrm{probe}}.Math.{\mathrm{.Math.}}_{\mathrm{pump}}^2}{{?}_m.Math.{\mathrm{.Math.}}_{\mathrm{probe}}^2}.Math.{\left(a_{\mathrm{probe}}^{}.Math.a_{\mathrm{probe}}\right)}^2,& \left(\mathrm{S34}\right)\end{array}$

[0144] from which we define the coefficient of nonlinearity as

[00022]$\begin{array}{cc}{\widetilde{?}}_{\mathrm{probe}}=\frac{g_{\mathrm{probe}}^2.Math.{\mathrm{.Math.}}_{\mathrm{pump}}^2}{{?}_m.Math.{\mathrm{.Math.}}_{\mathrm{probe}}^2}.& \left(\mathrm{S35}\right)\end{array}$

[0145] It is clear that even a very weak probe field can experience a strong Kerr nonlinearity, and hence a nonlinear dynamics, if the intensity of the pump is sufficiently strong. Thus, the system intrinsically enables an opto-mechanically-induced Kerr-like nonlinearity, which helps the optical pump and probe fields interact with each other. It is clear that the strength of the interaction can be made very high by increasing the ratio of the intensity of the input pump field ?.sub.pump.sup.2 to that of the input probe field ?.sub.probe.sup.2. With the configuration of the technology as tested, the pump field is at least three-orders of magnitude larger than the probe field. Thus the nonlinear coefficient {tilde over (?)}.sub.probe given in Eq. (S35) is increased by at least three-orders of magnitude, compared to the nonlinear coefficient ?.sub.probe given in Eq. (S22).

[0146] The trajectory of the mechanical motion can be estimated from the demonstration data. The mechanical mode excited in the microtoroid during the demonstration has a frequency of ?.sub.m=26.1 MHz and a damping rate of ?.sub.m=0.2 MHz, implying a quality factor of Q.sub.m?130 These values are used in the nonlinear opto-mechanical equations to reconstruct the mechanical motion. It is seen that the opto-mechanical resonator experiences a periodic motion (FIG. S5a) even when the detected optical pump field showed chaotic behavior. To explain this, we start from the following equation for the mechanical resonator

{dot over (X)}=??.sub.mX+?.sub.mP, (S36)

{dot over (P)}=??.sub.mP??.sub.mX+g.sub.pumpI(t), (S37)

[0147] where P is the momentum of the mechanical mode and I(t)=|a.sub.pump(t)|.sup.2 is the intensity of the pump with the field amplitude a.sub.pump. By introducing the complex amplitude

[0148] b=(X+iP)/?{square root over (2)}, Eqs. (S36) and (S37) can be rewritten as

{dot over (b)}=?(?.sub.m?i?.sub.m)b+g.sub.pumpl(t). (S38)

[0149] The above equation can be solved in the frequency domain as

[00023]$\begin{array}{cc}b?\left(?\right)=\frac{g_{\mathrm{pump}}}{i?\left(?-{?}_m\right)+{?}_m}.Math.I?\left(?\right),& \left(\mathrm{S39}\right)\end{array}$

[0150] from which we obtain

[00024]$\begin{array}{cc}{S}_b?\left(?\right)={.Math.b?\left(?\right).Math.}^2=\frac{g_{\mathrm{pump}}^2}{{\left(?-{?}_m\right)}^2+{?}_m^2}.Math.{.Math.I?\left(?\right).Math.}^2=\frac{g_{\mathrm{pump}}^2}{{?}_m^2}.Math.{?}_{b.Math..Math.I}?\left(?\right).Math.S_I?\left(?\right),& \left(\mathrm{S40}\right)\\ \mathrm{Where}& \\ {?}_{\mathrm{bI}}?\left(?\right)=\frac{{?}_m^2}{{\left(?-{?}_m\right)}^2+{?}_m^2}& \left(\mathrm{S41}\right)\end{array}$

[0151] is the susceptibility coefficient induced by the mechanical resonator and S.sub.I(?)=|I(?)|.sup.2 is the spectrum of I(t). As shown in FIG. 8a, the mechanical resonator works similar to a low-pass filter, which filters out the high-frequency components of I(t). In fact, the susceptibility coefficient X.sub.bI(?) modifies the shape of S.sub.I (?) and shrinks the spectrum S.sub.b(?) to the low-frequency regime. By such a filtering process, the mechanical motion of the resonator does not experience the high-frequency components typical of chaotic behavior, but instead remains in the periodic-oscillation regime, as shown in the reconstructed motion of the mechanical mode in FIG. 8b. FIG. 8 illustrates a reconstructed mechanical motion of the microtoroid resonator. FIG. 8a, illustrates periodic mechanical motion of the microtoroid when the pump and probe fields are both in the chaotic regime. FIG. 8b, illustrates filtering by the mechanical resonator where the mechanical resonator works as a low-pass filter which filters out the high-frequency components in the mechanical modes.

[0152] Lyapunov exponents quantify sensitivity of a system to initial conditions and give a measure of predictability. They are a measure of the rate of convergence or divergence of nearby trajectories. A positive exponent implies divergence and that the orbits are on a chaotic attractor. A negative exponent implies convergence to a common fixed point. Zero exponent implies that the orbits maintain their relative positions and they are on a stable attractor. The present technology as disclosed shows how the pump power affects the maximum Lyapunov exponent of the pump and probe fields. In FIG. 9, numerical results are presented regarding the effect of the frequency detuning between the cavity resonance and the pump, frequency detuning between the cavity resonance and the probe, and the damping rates of the pump and probe on the maximum Lyapunov exponent. As seen in FIG. 9a, Lyapunov exponents of the pump and probe fields vary with increasing frequency detuning between the pump and the cavity resonance. As the frequency detuning of the pump increases, Lyapunov exponent increases from negative to positive values, attaining its maximum value at a detuning value of ?.sub.pump?0.9 ?.sub.m. With further increase of detuning, it decreases and returns back to negative values. Thus, with increasing detuning of the pump from the cavity resonance, the system evolves first to chaotic regime and then gets out of chaos into a periodic dynamics.

[0153] This is similar to the behavior observed for the varying pump field. Interestingly, both the pump and probe fields follow the same dependence on the pump-cavity detuning. When examining the effect of probe-cavity detuning (FIG. 9b), It can be determined that varying probe-cavity detuning affects only the maximum Lyapunov exponent of the probe, and the pump Lyapunov exponent is not affected. The reason for this is that in the demonstration of the technology and in these simulations, the power of the probe field is kept sufficiently weak that it does not affect the pump field. A similar trend is seen in the case of varying the damping rates of the pump and probe modes, that is varying the damping rate of the pump affects Lyapunov exponents of both the pump and probe (FIG. 9c) but varying the damping rate of the probe affects only the Lyapunov exponent of the probe (FIG. 9d). FIG. 9c shows that with increasing damping rate the maximum Lyapunov exponent decreases from a positive value down to negative values. This can be explained as follows. Increasing damping rate, decreases the quality factor of the resonator which in turn reduces the intracavity field intensity. As a result optomechanical oscillation is gradually suppressed and the degree of the chaos induced by optomechanical interaction decreases.

[0154] FIG. 9 illustrates the maximum of the Lyapunov exponent for the pump (Upper spectra) and probe (Lower spectra) fields. FIG. 9a illustrates the effect of the pump-cavity detuning, 9b, the effect of probe-cavity detuning, 9c, the effect of the damping rate of the pump, and for FIG. 9d the effect of damping rate of the probe on the maximum Lyapunov exponents of the pump and probe fields.

[0155] In order to further illustrate the stochastic resonance phenomenon, first, focus on the dynamics of the optical mode coupled to the probe field a.sub.probe. The total Hamiltonian of the optical modes a.sub.pump, a.sub.probe, and the mechanical mode can be written as in Eq. (S18). By introducing the translation transformation in Eq. (S27) and getting rid of the degrees of freedom of the mechanical mode and the optical mode coupled to the pump field a.sub.pump, the Hamiltonian in Eq. (S18) can be re-expressed as

H=?.sub.probea.sub.probe.sup.a.sub.probe+??.sub.probe(a.sub.probe.sup.+a.sub.probe)?{tilde over (?)}.sub.probe(a.sub.probe.sup.a.sub.probe).sup.2, (S42)

[0156] where {tilde over (?)}.sub.probe is given in Eq. (S35). We can see that the nonlinear opto-mechanical coupling leads to an effective fourth-order nonlinear term in the optical mode a.sub.probe. Introducing the normalized position and momentum operators

[00025]$\begin{array}{cc}{x}_{\mathrm{probe}}=\frac{1}{\sqrt{2}}.Math.\left(a_{\mathrm{probe}}^{}+a_{\mathrm{probe}}\right),p_{\mathrm{probe}}=\frac{i}{\sqrt{2}}.Math.\left(a_{\mathrm{probe}}^{}-a_{\mathrm{probe}}\right),& \left(\mathrm{S43}\right)\end{array}$

[0157] we write the following dynamical equation by dropping some non-resonant terms and introducing the noise terms:

{dot over (x)}.sub.probe=??.sub.probex.sub.probe+?.sub.probep.sub.probe, (S44)

{dot over (p)}.sub.probe=??.sub.probex.sub.probe??.sub.probep.sub.probe+{tilde over (?)}.sub.probex.sup.3+??.sub.probe(t)+?(t), (S45)

[0158] where ?(t) is a noise term with a correlation time negligibly small when compared to the characteristic time scale of the optical modes and mechanical mode of the optomechanical resonator:

?(t)?(t=2D?(t?t), (S46)

[0159] with D denoting the strength of the noise. Subsequently, we arrive at the second-order oscillation equation

{umlaut over (x)}.sub.probe+2?.sub.probe{dot over (x)}.sub.probe=?(?.sub.probe.sup.2+?.sub.probe.sup.2)x.sub.probe+{tilde over (?)}.sub.probe?.sub.probex.sub.probe.sup.3+??.sub.probe?.sub.probe(t)+?.sub.probe?(t). (S47)

[0160] Under the condition that ?.sub.probe??.sub.probe in the overdamped limit, the above second-order oscillation equation can be reduced to

[00026]$\begin{array}{cc}{\stackrel{.}{x}}_{\mathrm{probe}}=-\frac{{?}_{\mathrm{probe}}^2}{2.Math.{?}_{\mathrm{probe}}}.Math.x_{\mathrm{probe}}+{\widetilde{?}}_{\mathrm{probe}}.Math.\frac{{?}_{\mathrm{probe}}}{2.Math.{?}_{\mathrm{probe}}}.Math.x_{\mathrm{probe}}+?.Math.\frac{{?}_{\mathrm{probe}}}{2.Math.{?}_{\mathrm{probe}}}.Math.{\mathrm{.Math.}}_{\mathrm{probe}}?\left(t\right)+\frac{{?}_{\mathrm{probe}}}{2.Math.{?}_{\mathrm{probe}}}.Math.??\left(t\right).& \left(\mathrm{S48}\right)\end{array}$

[0161] If introducing the normalized time unit ?=(2?.sub.probe/?.sub.probe)t, arriving at

[00027]$\begin{array}{cc}\frac{d}{d.Math..Math.?}.Math.x_{\mathrm{probe}}=-{?}_{\mathrm{probe}}.Math.x_{\mathrm{probe}}+{\widetilde{?}}_{\mathrm{probe}}.Math.x_{\mathrm{probe}}^3+?.Math..Math.{\mathrm{.Math.}}_{\mathrm{probe}}?\left(?\right)+??\left(?\right).& \left(\mathrm{S49}\right)\end{array}$

[0162] which is a typical equation leading to the stochastic resonance phenomenon. The signal-to-noise ratio (SNR) for such a system is given by

[00028]$\begin{array}{cc}S.Math..Math.N.Math..Math.R=\frac{{?}_{\mathrm{probe}}^2.Math.{?}_m^2.Math.{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{probe}}^2}{8.Math.\sqrt{2}.Math.D^2.Math.g_{\mathrm{probe}}^4}.Math.\exp ?\left(-\frac{{?}_{\mathrm{probe}}^2.Math.{?}_m}{8.Math.g_{\mathrm{probe}}^2.Math.D}\right).& \left(\mathrm{S50}\right)\end{array}$

[0163] Since the strength of the noise D is related to the pump power P.sub.pump by D=?P.sub.pump.sup.1/2, the relation between the SNR and the pump power can be re-written as

[00029]$\begin{array}{cc}S.Math..Math.N.Math..Math.R=\frac{{?}_{\mathrm{probe}}^2.Math.{?}_m^2.Math.{?}^2.Math.{\mathrm{.Math.}}_{\mathrm{probe}}^2}{8.Math.\sqrt{2}.Math.{?}^2.Math.P_{\mathrm{pump}}.Math.g_{\mathrm{probe}}^4}.Math.\exp ?\left(-\frac{{?}_{\mathrm{probe}}^4.Math.{?}_m}{8.Math..Math.g_{\mathrm{probe}}^2.Math.?.Math..Math.P_{\mathrm{pump}}}\right),& \left(\mathrm{S51}\right)\end{array}$

[0164] which implies that the SNR is not a monotonous function of the pump power P.sub.pump and hence it is possible to increase the SNR by increasing the pump power (i.e., subsequently by increasing the bandwidth D and hence the noise). Following the same procedure one can derive SNR for the pump in a straightforward way.

[0165] In FIG. 10, we give the SNR versus pump power for both the probe and pump fields measured in our experiments together with the best fit according to Eq. (S51) for the probe and the similar expression for the pump. Keeping ? and ? as free parameters, we found the best fits with ?=0.825 mW and ?=7.4764 in W.sup.1/2 for the probe and with ?=2.6388 mW and ?=6.47 mW.sup.1/2 for the pump.

[0166] FIG. 10 illustrates the Signal-to-noise ratio (SNR) for the pump and probe signals. The technology demonstrates a signal-to-noise ratio (SNR) of the probe (blue open circles) and pump (red diamonds) signals as a function of the pump power. Solid curves are the best fits to the experimental data.

[0167] As discussed above, stochastic resonance is a phenomenon in which the response of a nonlinear system to a weak input signal is optimized by the presence of a particular level of noise, i.e., the noise-enhanced response of a deterministic input signal. Coherence resonance is a related effect demonstrating the constructive role of noise, and is known as stochastic resonance without input signal. Coherence resonance helps to improve the temporal regularity of a bursting time series signal. The main difference between stochastic resonance and coherence resonance is whether a deterministic input signal is input to the system and whether the induced SNR enhancement is the consequence of the response of this deterministic input. With at least on implementation of the present technology, a weak probe signal, which is modulated by the mechanical mode of the optomechanical resonator at the frequency ?.sub.m=26 MHz, acts as a periodic input signal fed into the system. In order to confirm that the observed phenomenon in the technology as demonstrated is stochastic resonance rather than coherence resonance, numerical simulations are performed and compared the results with the present technology demonstration results. The dynamical equations used for numerical simulation are given by

{dot over (a)}.sub.pump=?[?.sub.pump?i(?.sub.pump?g.sub.pumpX)]a.sub.pump+i??.sub.pump(t)+D.sub.pump?.sub.pump(t), (S52)

{dot over (a)}.sub.probe=?[?.sub.probe?i(?.sub.probe?g.sub.probeX)]a.sub.probe+i??.sub.probe(t)+D.sub.probe?.sub.probe(t), (S53)

{dot over (X)}=??.sub.mX+?.sub.mP, (S54)

{dot over (P)}=??.sub.mP??.sub.mX+g.sub.pump|a.sub.pump|.sup.2+D.sub.m?.sub.m(t), (S55)

[0168] with parameters ?.sub.pump/?.sub.m=?.sub.probe/?.sub.m=1, ?.sub.pump/?.sub.pump=0.1, ?.sub.probe/?.sub.probe=0.1,

[0169] ?.sub.m/?.sub.m=0.01, g.sub.pump/?.sub.pump=g.sub.probe/?.sub.probe=0.1, 78/?.sub.pump=?.sub.pump/?.sub.pump?1,

[0170] D.sub.pump/?.sub.pump=0.1, D.sub.probe/?.sub.probe=0.1, D.sub.m/?.sub.m=0.1. ?.sub.pump(t), ?.sub.probe(t), ?.sub.m(t) are white noises such that

E[?.sub.i(t)]=0,E[?.sub.i(t)?.sub.j(t)]=?.sub.ij?(t?t), (S56)

[0171] where E(.Math.) is average over the noise. In the case of stochastic resonance, ?.sub.probe/?.sub.probe=0.1, and in the case of coherence resonance ?.sub.probe/?.sub.probe=0 to simulate the system with a weak probe input and without the weak probe input, respectively.

[0172] FIG. 11 illustrates an output spectra obtained in the experiments and in the numerical simulations of stochastic resonance and coherence resonance at various pump powers. FIG. 11a, illustrates an Output spectra obtained in the demonstration testing show that the spectral location of the resonance peak do not change with increasing pump power. FIG. 11b, illustrates an output spectra obtained in the numerical simulations of stochastic resonance show that the spectral location of the resonance peak stays the same for increasing pump power, similar to what was observed in the demonstration testing. FIG. 11c, illustrates an output spectra obtained in the numerical simulations of coherence resonance which show that the spectral location of the resonance peaks change with increasing pump power. From left to the right, the input pump power is increased.

[0173] The output spectra obtained from the demonstration of the technology is compared (FIG. 11a) with the results of numerical simulations where the theoretical model introduced above is considered with and without weak probe input to simulate stochastic resonance (FIG. 1b) and coherence resonance (FIG. 11c).

[0174] It is seen that in the output spectra obtained from the technology demonstration (FIG. 11a) and the simulations with weak probe input (FIG. 11b), the position of the resonant peaks are not affected by increasing pump power. The spectral position of the resonant peak in the output spectra is fixed at the frequency of the periodic input signal. However, for the case, with no weak probe input, simulating coherence resonance, the positions of the resonant peaks in the output spectra shift with increasing pump power, implying that the resonances are induced by noise. Thus, the behavior of the resonances in the output spectra obtained in the demonstration testing agrees with what one would expect for stochastic resonance, and it is completely different that what one would expect for coherence resonance.

[0175] Next, the mean interspike intervals are compared and its scaled standard deviation calculated from the output signal measured in our experiments with the results of numerical simulations of the technology in the one or more implementations disclosed when a weak probe field is used as an input (case of stochastic resonance) and when there is no input probe field (case of coherence resonance). The interspike interval is defined as the mean time between two adjacent spikes in the time-domain output signals,

[00030]$\begin{array}{cc}???=\underset{N.fwdarw.?}{\lim }.Math.\frac{1}{N}.Math.\underset{i=1}{\overset{N}{.Math.}}.Math.{?}_i,& \left(S.Math.\mathrm{.57}\right)\end{array}$

[0176] where ?.sub.i is the time between the i-th and (i+1)-th spikes. The variation R of the interspike intervals which is defined as the scaled standard devistion of the mean interspike interval is given as

[00031]$\begin{array}{cc}R=\frac{\sqrt{?{?}^2?-{???}^2}}{???}.& \left(S.Math.\mathrm{.58}\right)\end{array}$

[0177] FIG. 12 illustrates a mean interspike interval and its variation for the probe mode. FIG. 12a, illustrates a mean interspike interval and its variation calculated from the output signal in the probe mode obtained in the experiments. FIG. 12b, illustrates a mean interspike interval and its variation obtained in the numerical simulation of stochastic resonance in our system (with input weak probe). FIG. 12c illustrates a mean interspike interval and its variation obtained in the numerical simulation of coherence resonance in our system (without input weak probe). Experimental results agree well with the simulation results of stochastic resonance, and demonstrate a completely different dynamics than the coherence resonance. This imply that the observed phenomenon in the experiments is stochastic resonance.

[0178] In FIG. 12, illustrates the results of the demonstration test for the technology (FIG. 12a) and the numerical simulations for stochastic resonance (FIG. 12b) and for coherence resonance (FIG. 12c). The pump power dependence of ? and R obtained for our experimental data and that obtained for the numerical simulation of stochastic resonance agree well, that is in both the experiments and numerical simulations we see that pump power does not affect ? much, and R reaches a maximum at an optimal pump power (i.e., R is a concave). From the results of the simulations of coherence resonance, we see that (i) the mean interspike interval ? drops gradually with increasing pump power, and (ii) R is a concave function, exhibiting a minimum at an optimal pump power. The very good agreement between what is observed in the technology demonstration testing and the results of the numerical simulations of stochastic resonance in the theoretical model describing the present technology strongly supports that observed in the experiments is stochastic resonance rather than coherence resonance.

[0179] The various implementations of chaos induced stochastic resonance in opto-mechanical systems as shown above illustrate a novel system and method for opto-mechanically mediated chaos transfer between two optical fields such that they follow the same route to chaos. A user of the present technology as disclosed may choose any of the above implementations, or an equivalent thereof, depending upon the desired application. In this regard, it is recognized that various forms of the subject of chaos induced stochastic resonance in opto-mechanical system could be utilized without departing from the scope of the present invention.

[0180] **Chirality lies at the heart of the most fascinating and fundamental phenomena in modern physics like the quantum Hall effect, Majorana fermions and the surface conductance in topological insulators as well as in p-wave superconductors. In all of these cases chiral edge states exist, which propagate along the surface of a sample in a specific direction. The chirality (or handedness) is secured by specific mechanisms, which prevent the same edge state from propagating in the opposite direction. For example, in topological insulators the backscattering of edge-states is prevented by the strong spin-orbit coupling of the underlying material.

[0181] Transferring such concepts to the optical domain is a challenging endeavor that has recently attracted considerable attention. Quite similar to their electronic counterparts, the electromagnetic realizations of chiral states typically require either a mechanism that breaks time-reversal symmetry or one that gives rise to a spin-orbit coupling of light. Since such mechanisms are often not available or difficult to realize, alternative concepts have recently been proposed, which require, however, a careful arrangement of many optical resonators in structured arrays. Here we demonstrate explicitly that the above demanding requirements on the realization of chiral optical states propagating along the surface of a system can all be bypassed by using a single resonator with non-Hermitian scattering. The key insight in this respect is that a judiciously chosen non-Hermitian out-coupling of two near-degenerate resonator modes to the environment leads to an asymmetric backscattering between them and thus to an effective breaking of the time-reversal symmetry that supports chiral behaviour. More specifically, we show that a strong spatial chirality can be imposed on a pair of WGMs in a resonator in the sense of a switchable direction of rotation inside the resonator such that they can be tuned to propagate in either the clockwise (cw) or the counterclockwise (ccw) direction.

[0182] In our experiment we achieved this on-demand tunable modal chirality and directional emission using two scatterers placed in the evanescent field of a resonator. When varying the relative positions of the scatterers the modes in the resonator change their chirality periodically reaching maximal chirality and unidirectional emission at an exceptional point (EP) a feature which is caused by the non-Hermitian character of the system.

[0183] FIG. 13, illustrated the experimental configuration used in the technology and the effect of scatterers. (A) Illustration of a WGM resonator side-coupled to two waveguides, with the two scatterers enabling the dynamical tuning of the modes. cw and ccw are the clockwise and counterclockwise rotating intracavity fields. a.sub.cw(ccw) and b.sub.cw(ccw) are the field amplitudes propagating in the waveguides. ?: relative phase angle between the scatterers. (B) Varying the size and the relative phase angle of a second scatterer helps to dynamically change the frequency detuning (splitting) and the linewidths of the split modes revealing avoided crossings (top panel) and an EP (lower panel). (C) Effect of ? on the frequency splitting 2 g, difference ?.sub.diff and sum ?.sub.sum of the linewidths of split resonances when relative size of the scatterers were kept fixed (FIG. 19-21).

[0184] The setup consists of a silica microtoroid WGM resonator that allows for the in- and out-coupling of light through two single-mode waveguides (FIG. 13A and 17). The resonator had a quality factor Q?3.9?10.sup.7 at the resonant wavelength of 1535.8 nm. To probe the scattererinduced chirality of the WGMs, and to simulate scatterers we used two silica nanotips whose relative positions (i.e., relative phase angle ?) and sizes within the evanescent field of the WGMs were controlled by nanopositioners.

[0185] First, using only the waveguide with ports 1 and 2 (FIG. 13A), we determined the effect of the sizes and positions of the scatterers on the transmission spectra. With the first scatterer entering the mode volume, we observed frequency splitting in the transmission spectra due to scattererinduced modal coupling between the cw and ccw travelling modes. Subsequently, the relative position and the size of the second scatterer were tuned to bring the system to an EP (FIGS. 13, B and C, and 18-21) which is a non-Hermitian degeneracy identified by the coalescence of the complex frequency eigenvalues and the corresponding eigenstates. EP acts as a veritable source of non-trivial physics in a variety of systems. Depending on the amount of initial splitting introduced by the first scatterer and ?, tuning the relative scatterer size brought the resonance frequencies (real part of eigenvalues) closer to each other, and then either an avoided crossing or an EP was observed (FIG. 13B, 20 and 21). At the EP both the frequency splitting 2 g and the linewidth difference ?.sub.diff of the resonances approach zero, whereas the sum of their linewidths m.sub.sum remains finite (FIG. 13C, 20 and 21). An EP does not only lead to a perfect spectral overlap between resonances, but also forces the two corresponding modes to become identical. Correspondingly, a pair of two counter-propagating WGMs observed in closed Hermitian resonators turns into a pair of co-propagating modes with a chirality that increases the closer the system is steered to the EP (FIG. 22-24).

[0186] To investigate this modal chirality in detail we used both of the waveguides and monitored the transmission and reflection spectra at the output ports of the second waveguide for injection of light from two different sides of the first waveguide (FIG. 14). In the absence of the scatterers, when light was injected in the cw direction, a resonance peak was observed in the transmission and no signal was obtained in the reflection port [FIG. 14A(i)]. Similarly, when the light was injected in the ccw direction, the resonance peak was observed in the transmission port with no signal in the reflection port [FIG. 14B(i)]. When only one scatterer was introduced, two split resonance modes were observed in the transmission and reflection ports regardless of whether the signal was injected in the cw or ccw directions [FIG. 14, A(ii) and 2B(ii)], implying that the field inside the resonator is composed of modes travelling in both cw and ccw directions. When the second scatterer was introduced and its position and size were tuned to bring the system to an EP, we observed that the transmission curves for injections from two different sides were the same but the reflection curves were different [FIG. 14, A(iii) and B(iii)]: while the reflection shows a pronounced resonance peak for the ccw input, this peak vanishes for the cw input. The fact that the transmission curves for different input ports are the same follows from reciprocity, which is well-fulfilled in our system. On the other hand, the asymmetric backscattering (reflection) is the defining hallmark of the desired chiral modes, for which we provide here the first direct measurement in a microcavity (FIG. 25 and supplementary text 20).

[0187] FIG. 14. Experimental observation of scatterer-induced asymmetric backscattering. (A, B) When there is no scattering center in or on the resonator, light coupled into the resonator through the first waveguide in the cw (A(i)) [or ccw (B(i))] direction couples out into the second waveguide in the cw (A(i)) [or ccw (B(i))] direction: the resonant peak in the transmission and no signal in the reflection. A(ii), B(ii), When a first scatterer is placed in the mode field, resonant peaks are observed in both the transmission and the reflection regardless of whether the light is input in the cw (A(ii)) or in the ccw (B(ii)) directions. A(iii), B(iii), When a second scatterer is suitably placed in the mode field, for the cw input there is no signal in the reflection output port (A(iii)), whereas for the ccw input there is a resonant peak in the reflection, revealing asymmetric backscattering for the two input directions. Inset of B(iii) compares the two backscattering peaks in A(iii) and B(iii). Estimated chirality is ?0.86.

[0188] The crucial question to ask at this point is how the chiralityan intrinsic property of a mode that we aim to demonstrate-can be distinguished from the simple directionality (or sense of rotation) imposed on the light in the resonator just by the biased input. To differentiate between these two fundamentally different concepts based on the experimentally obtained transmission spectra, we determined the chirality and the directionality of the field within the WGM resonator using the following operational definitions: the directionality defined as D=(?{square root over (I.sub.bccw)}??{square root over (I.sub.bcw)})/(?{square root over (I.sub.bccw)}+?{square root over (I.sub.bcw)}) simply compares the difference of the absolute values of the light amplitudes measured in the ccw and cw directions, irrespective of the direction from which the light is injected (FIG. 13A and 14). We observed that varying the relative distance between the scatterers changed this directionality, but the initial direction, that is the direction in which the input light was injected, remained dominant (FIG. 15A). The intrinsic chirality of a resonator mode is a quantity that is entirely independent of any input direction and therefore not as straightforward to access experimentally. One can, however, get access to the chirality a through the intensities measured in the used four-port setup as a=(?{square root over (I.sub.14)}??{square root over (I.sub.23)})/(?{square root over (I.sub.14)}?{square root over (I.sub.23)}), where I.sub.jk denotes the intensity of light measured at the k-th port for the input at the j-th port (FIG. 13A and 17, supplementary text 19 and 20). Note that to obtain ? the reflection intensities obtained for injections from two different sides are compared. The chirality thus quantifies the asymmetric backscattering, similar to what is shown in FIG. 14A(iii) and 14B(iii). If the backscattering is equal for both injection sides (I.sub.14=I.sub.23) the chirality is zero, implying symmetric backscattering and orthogonal eigenstates. In the case where backscattering for injection from one side dominates, the chirality approaches 1 or ?1 depending on which side is dominant. The extreme values ?=?1 are, indeed, only possible when the eigenvalues and eigenvectors of the system coalesce, that is, when the system is at an EP. By changing the relative phase angle between the scatterers, we obtained quite significant values ???0.79 of chirality with both negative and positive signs (FIG. 15B). The strong chiralities observed in FIG. 15B are linked to the presence of two EPs, each of which can be reached by optimizing ? such that asymmetric scattering is maximized for one of the two injection directions (FIGS. 18, 22, 23 and 25, supplementary text 17 and 20).

[0189] FIG. 15. Controlling directionality and intrinsic chirality of whispering-gallery-modes. (A) Directionality and (B) chirality a of the WGMs of a silica microtoroid resonator as a function of ? between the two scatterers.

[0190] FIG. 16. Scatterer-induced mirror-symmetry breaking at an EP. In a WGM microlaser with mirror symmetry the intracavity laser modes rotate both in cw and ccw directions and thus the outcoupled light is bidirectional and chirality is zero. The scatterer-induced symmetry breaking allows tuning both the directionality and the chirality of laser modes. (A) Intensity of light out-coupled into a waveguide in the cw and ccw directions as a function of ?. Regions of bidirectional emission, and fully unidirectional emission are seen. (B) Chirality as a function of ?. Transitions from non-chiral states to unity (?1) chirality at EPs are clearly seen. Unity chirality regions correspond to unity unidirectional emission regions in (A). (C, D, E) Finite element simulations revealing the intracavity field patterns for the cases labeled as C, D and E in (A) and (B). Results shown in (C)-(E) were obtained for the same size factor but different ?: (C) 2.628 rad; (D) 2.631 rad; and (E) 2.626 rad. P.sub.1 and P.sub.2 denote the locations of scatterers.

[0191] Finally, we addressed the question how this controllably induced intrinsic chirality can find applications and lead to new physics in the sense that the intrinsic chirality of the modes is fully brought to bear. The answer is to look at lasing in such devices since the lasing modes are intrinsic modes of the system. Previous studies along this line were restricted to ultrasmall resonators on the wavelength scale, where modes were shown to exhibit a local chirality and no connection to asymmetric backscattering could be established. Here we address the challenging case of resonators with a diameter being multiple times the wavelength (>50?), for which we achieved a global and dynamically tunable chirality in a microcavity laser that we can directly link to the non-Hemitian scattering properties of the resonator. In our last set of experiments, we achieved a global and dynamically tunable chirality in a microcavity laser that we can directly link to the non-Hemitian scattering properties of the resonator. We used an Erbium (Er.sup.3+) doped silica microtoroid resonator coupled to only the first waveguide, which was used both to couple into the resonator the pump light to excite Er.sup.3+ions and to couple out the generated WGM laser light. With a pump light in the 1450 nm band, lasing from Er.sup.3+ions in the WGM resonator occurred in the 1550 nm band. Since the emission from Erbium ions couples into both the cw and ccw modes and the WGM resonators have a rotational symmetry, the outcoupled laser light typically does not have a pre-determined out-coupling direction in the waveguide. With a single fiber tip in the mode field, these initially frequency degenerate modes couple to each other creating split lasing modes. Using another fiber tip as a second scatterer, we investigated the chirality in the WGM microlaser by monitoring the laser field coupled to the waveguide in the cw and ccw directions. For this situation the parameters a and D from above can be conveniently adapted to determine the chirality of lasing modes based on the experimentally accessible quantities. Note that for the lasing modes chirality and directionality are equivalent as they both quantify the intrinsic dynamics of the laser system. We observed that by tuning the relative distance between the scatterers, the chirality of the lasing modes and with it the directional out-coupling to the fiber can be tuned in the same way as shown for the passive resonator (FIG. 15).

[0192] As depicted in FIG. 16A, depending on the relative distance between the scatterers one can have a bidirectional laser or a unidirectional laser, which emits only in the cw or the ccw direction. For the bidirectional case, one can also tune the relative strengths of emissions in cw and ccw directions. As expected, the chirality is maximal (?1) for the relative phase angles where strong unidirectional emission is observed (FIG. 16B), and chirality is close to zero for the angles where bidirectional emission is seen. This confirms that by tuning the system to an EP the modes can be made chiral and hence the emission direction of lasing can be controlled: in one of the two EPs, emission is in the cw and in the other EP the emission is in the ccw direction. Thus, by transiting from one EP to another EP the direction of unidirectional emission is completely reversed: an effect demonstrated for the first time here. The fact that the maximum possible chirality values for the lasing system are reached here very robustly can be attributed to the fact that the non-linear interactions in a laser tend to reinforce a modal chirality already predetermined by the resonator geometry.

[0193] To relate this behavior to the internal field distribution in the cavity, we also performed numerical simulations which revealed that when the intracavity intensity distribution shows a standing-wave pattern with a balanced contribution of cw and ccw propagating components and a clear interference pattern, the emission is bidirectional, in the sense that laser light leaks into the second waveguide in both the cw and ccw directions (FIG. 16C). However, when the distribution does not show such a standard standing-wave pattern but an indiscernible interference pattern, the emission is very directional, such that the intracavity field couples to the waveguide only in the cw or the ccw direction depending on whether the system is at the first or the second EP (FIG. 16, D and E). We also confirmed that the presence or absence of an interference pattern in the field distribution is also linked with a bi- or uni-directional transmission, respectively, observed in FIG. 15 for the passive resonator (FIG. 26).

[0194] Summarizing, we have demonstrated chiral modes in whispering-gallery-mode microcavities and microlasers via geometry-induced non-Hermitian mode-couplings. The underlying physical mechanism that enables chirality and directional emission is the strong asymmetric backscattering in the vicinity of an EP which universally occurs in all open physical systems. We believe that our work will lead to new directions of research and to the development of WGM microcavities and microlasers with new functionalities. In addition to controlling the flow of light and laser emission in on-chip micro and nanostructures, our findings have important implications in cavity-QED for the interaction between atoms/molecules and the cavity light. They may also enable high performance sensors to detect nanoscale dielectric, plasmonic and biological particles and aerosols, and be useful for a variety of applications such as the generation of optical beams with a well-defined orbital angular momentum (OAM) (such as OAM microlasers, vortex lasers, etc.) and the topological protection in optical delay lines.

[0195] Two-Mode-Approximation (TMA) model and the eigenmode evolution. In this section we briefly review the two-mode approximation (TMA) model and the eigenmode evolution in whispering-gallery-mode (WGM) microcavities with nanoscatterer-induced broken spatial symmetry, as described briefly in the main text. This will help to understand the basic relationship between asymmetric backscattering of counter-propagating waves and the resulting co-propagation, non-orthogonality, and chirality of optical modes. We furthermore derive how the chirality of a lasing mode can be measured by weakly coupling two waveguides to the system. As a complementary schematic of the setup shown in FIG. 13. FIG. 17 presents the details of the involved parameters and the input/output signal directions for clarification.

The TMA model used in our analysis was first phenomenologically introduced for deformed microdisk cavities and was later rigorously derived for the microdisk with two scatterers. The main approach is to model the dynamics in the slowly-varying envelope approximation in the time domain with a Schr?dinger-like equation.

[00032]$\begin{array}{cc}1.Math.\frac{d}{\mathrm{dt}}.Math.?=H.Math..Math.?& \left(S.Math.\mathrm{.59}\right)\end{array}$

[0196] **Here ?, is the complex-valued two-dimensional vector consisting of the field amplitudes of the CCW propagating wave ?.sub.CCW. and the CW propagating wave ?.sub.CW. The former corresponds to the angular dependence in real space, and the latter to ; the positive integer m is the angular mode number. Since the microcavity is an open system, the corresponding effective Hamiltonian,

[00033]$\begin{array}{cc}H=\left(\begin{array}{cc}{?}_0& A\\ B& {?}_0\end{array}\right)& \left(S.Math.\mathrm{.60}\right)\end{array}$

is a 2?2 matrix, which is in general non-Hermitian.

[0197] FIG. 17. Schematic of the setup with the definitions of the parameters and signal propagation directions. a.sub.j, in (a.sub.j, out) denotes the input (output) signal amplitude from the j-th port. K.sub.0, K.sub.1are the cavity decay rate and the cavity-waveguide coupling coefficient, respectively. d.sub.1 (d.sub.2) denotes the effective scattering size factor of the first (second) nanoscatterer (corresponding to the spatial overlap between the scatterer and the optical mode), which is varied by changing the distance between the scatterer and the microresonator. The angle ? denotes the relative phase angle between the scatterers.

The real parts of the diagonal elements ?.sub.c are the frequencies and the imaginary parts are the decay rates of the resonant traveling waves. The complex-valued off-diagonal elements A and B are the backscattering coefficients, which describe the scattering from the CW (CCW) to the CCW (CW) travelling wave. In general, in the open system the backscattering is asymmetric, |A|?|B|, which is allowed because of the non-Hermiticity of the Hamiltonian. The complex eigenvalues of H are,

[00034]$\begin{array}{cc}{?}_{?}={?}_c?\sqrt{\mathrm{AB}}& \left(S.Math.\mathrm{.61}\right)\end{array}$

to which the following complex (not normalized) right eigenvectors are associated,

[00035]$\begin{array}{cc}{?}_{?}=\left(\begin{array}{c}\sqrt{A}\\ ??B\end{array}\right).& \left(S.Math.\mathrm{.62}\right)\end{array}$

As shown in the text, the asymmetric scattering is closely related with the evolution of the eigenmodes, especially in the vicinity of the exceptional points (EP), where either of the backscattering coefficients A or B is zero and both the eigenvalues (S.61) and the eigenvectors (S.62) coalesce. To verify this interesting feature, we specifically checked the eigenmode evolution in our system both theoretically and experimentally. For the particular case of the WGM microtoroid perturbed by two scatterers the matrix elements of H are determined as follows,

[00036]$\begin{array}{cc}\begin{array}{c}.Math.{?}_c=.Math.{?}_0+V_1+U_1+V_2+U_2\\ =.Math.{?}_c-\frac{{?}_0+2.Math..Math.{?}_1}{2}.Math.1+V_1+U_1+V_2+U_2\end{array}& \left(S.Math.\mathrm{.63}\right)\\ .Math.A=\left(V_1-U_1\right)+\left(V_2-U_2\right).Math..Math.\text{?}& \left(S.Math.\mathrm{.64}\right)\\ .Math.B=\left(V_1-U_1\right)+\left(V_2-U_2\right).Math..Math.\text{?}& \left(S.Math.\mathrm{.65}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

where ?.sub.c denotes the intrinsic cavity resonant frequency, and ?.sub.0 and ?.sub.1 are the cavity decay rate and the cavity-waveguide coupling coefficient. The quantities 2V.sub.j and 2U.sub.j are given by the complex frequency shifts for positive- and negative-parity modes introduced by j-th particle (j==1,2) alone. These quantities can be calculated for the single-particle-microdisk system either fully numerically [using, e.g., the finite-difference time-domain method (FDTD), the boundary element method (BEM)], or analytically using the Green's function approach for point scatterers with U.sub.j=0. Here we used the finite element method (FEM). In our simplified model U.sub.i is set to zero since |U.sub.1|?|V.sub.1|. FIG. 18 presents the evolution of the eigenfrequencies of our system (obtained with FEM simulations) as the phase difference angle ? and the effective size factor d are tuned. The EPs can be clearly observed where the eigenfrequencies coalesce, as pointed out in both FIG. 18A and 18B.

[0198] FIG. 18. The eigenmode evolution of the non-Hermitian system as a function of the effective size factor d and the relative phase angle ? between the scatterers. (A) Real part of the eigenmodes ?.sub.?. (B) Imaginary part of the eigenmodes ?.sub.?. Two exceptional points are clearly seen. EP: Exceptional Point.

[0199] Experimental observation of an EP by tuning the size and position of two scatterers. In our experiments with a silica microtoroid WGM resonator, we chose a mode for which there was no observable frequency splitting in the transmission spectra before the introduction of the scatterers. We probed the scatterer-induced chiral dynamics of the WGMs, using two silica nanotips whose relative positions (i.e., relative phase angle ?) and sizes within the evanescent field of the WGMs were controlled by nanopositioners (FIG. 13). The size ratio of the scatterers was tuned by enlarging the volume of one of the nanotips within the resonator mode field while keeping the volume of the other nanotip fixed.

[0200] FIG. 19. Experimentally obtained mode spectra as the relative phase angle ? between the scatterers was varied. ? increased continuously from (i) to (viii). Mode coalescence is clearly seen in (v). Modes bifurcated again when ? was increased further (vi-viii). The evolution of the eigenmodes of the system was obtained by coupling two waveguides to the system (FIG. 13& 17) and monitoring the transmission spectra (FIG. 19) as the wavelength of a tunable laser was scanned. The two eigenmodes coalesced clearly as the phase difference angle ? between the 1.sup.st and the 2.sup.nd nanoscatterer was varied to the vicinity of the EP but bifurcated again as ? was further increased. We also checked the evolution of the eigenfrequencies when the effective size of the 2.sup.nd scatterer was varied at different phase difference angles ?.

[0201] FIG. 20. Experimentally obtained evolution of eigenfrequencies as the relative size of the scatterers was varied at different relative phase angles ?. (A) Difference of the real parts of the eigenfrequencies (frequency splitting or frequency detuning). (B) Imaginary parts (linewidths) of the eigenfrequencies.

[0202] FIG. 21. Experimentally obtained evolution of the splitting quality factor as a function of ? for fixed relative size factor. In FIG. 13B of the main text, we presented the evolution of the frequency splitting 2 g, linewidth difference ?.sub.diff and the sum y.sub.sum of the linewidths of split modes as a function of the relative phase angle ?. In FIGS. 20 & 21 we provide more experimental results to further clarify how the relative phase angle ? and the relative size factor of the scatterers affect the spectra of the split resonance modes and help to drive the system to the vicinity of an EP. FIG. 20 depicts the evolution of the amount of frequency splitting and the linewidths of the split resonances as a function of the size factor at different values ? implying that when the relative size factor is varied, the system can or cannot reach an EP depending on the relative phase angle ? between the scatterers: For some values of ?, the system experiences avoided crossing. The resolvability of the frequency splitting in a transmission spectrum was previously quantified by the splitting quality factor, which is defined as the ratio of the frequency splitting 2 g to the sum ?.sub.sum, of the linewidths of the split resonances. Experimental results shown in FIG. 21 clearly show that when the resonances coalesce at an EP, the splitting quality factor reaches its minimum.

[0203] Emission and chirality analysis for the lasing cavity. As a consequence of the non-Hermitian character of the Hamiltonian the eigenvectors (S.62) are in general not orthogonal. This happens whenever the backscattering is asymmetric,

[00037]$.Math.A.Math.?.Math.B.Math.,$

as

[00038]${?}_{+}^{*}.Math.{?}_{-}=.Math.A.Math.-.Math.B.Math..$

The asymmetric backscattering

[00039]$.Math.A.Math.?.Math.B.Math.$

also implies that both modes have a dominant component that increases the closer the system is steered to the EP (FIG. 22). This corresponds to a dominant propagation direction in real space. We quantify this imbalance by the chirality

[00040]$\begin{array}{cc}{?}_{\mathrm{TMA}}=\frac{.Math.A.Math.-.Math.B.Math.}{.Math.A.Math.+.Math.B.Math.}& \left(S.Math.\mathrm{.66}\right)\end{array}$

[0204] In contrast to the original definition of the chirality, this chirality parameter also provides information on the sense of rotation not just on its absolute magnitude. For a balanced contribution, |A|?|B|, the chirality is close to 0. In the case where the CCW (CW) component dominates, |A|>|B|, (|A|<|B|), the chirality approaches 1 (?1) and both modes become copropagating. It is possible to create a situation of full asymmetry in the backscattering, i.e. a.fwdarw.?1. In this case, either A or B vanishes, while the other component is nonzerol. Solving the Schr?dinger Eq. (S.59), we get the eigenfrequencies of the system Eq. (S.61). The corresponding eigenmodes Eq. (S.62) can be further expressed as

?.sub.?=?.sub.CCW??{square root over (B/A?.sub.CW)}?(S.67)

In the experiments, the chirality (S.66) of the eigenmodes of the system can be obtained by coupling waveguides to the system (as shown in FIG. 17) and by inducing lasing (e.g., Raman lasing in silica resonators or lasing from Erbium ions in Erbium doped silica resonators) within the system. Using coupled mode theory and the assumption that there is no backscattering of light from the waveguide into the cavity one can relate the amplitudes in the waveguide to the coefficients A and B via

a.sub.cw,out=??{square root over (?.sub.1)}.sup.a?.sub.CW=??{square root over (?.sub.1)}.sup.a?{square root over (B)}(S.68)

a.sub.ccw,out=??{square root over (?.sub.1)}?.sub.CCW=??{square root over (?.sub.1)}.sup.a?{square root over (A)}(S.69)

Hence, the chirality of the lasing system can be obtained from the waveguide amplitudes as

[00041]$\begin{array}{cc}.Math.{?}_{\mathrm{testing}}=\frac{{.Math.\text{?}.Math.}^2-{.Math.\text{?}.Math.}^2}{{.Math.\text{?}.Math.}^2-{.Math.\text{?}.Math.}^2}& \left(S.Math.\mathrm{.70}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

where a.sub.ccwout can be either a.sub.1out or and can be either or or . The same formula can also be used in full numerical calculations to extract the chirality of the quasi-bound states of the system for comparison to the result of the two-mode approximation of Eq. (S.66).

[0205] FIG. 22. Weights of CW and CCW components in the eigenmodes as the relative phase difference ? between the two nanoscatterers is varied, away from EP and in the vicinity of EP, with two different size factors of the 2.sup.nd nanoscatterer, according to Eq.(S.67). Evolution of the eigenfrequencies and CW (CCW) weights in the eigenmodes as ? is varied for (A) and (B) V.sub.1=1.5?0.1 i, V.sub.2=1.0997?0.065 i, and (C) and (D) V.sub.1=1.5?0.1 i, V.sub.2=1.4999?0.104 i. Note that for the size factor used in (A) and (B) eigenmodes cannot reach the EP whereas for the size factor used in (C) and (D) the eigenmodes can reach the EP and a strong asymmetric distribution of the CW/CCW weights appears in the vicinity of EP. Insets are the zoom-in plots in the vicinity of EP. In (C) and (D), two EPs are clearly seen.

[0206] Chirality analysis and comparison between the lasing and the transmission models. In this section we extend the TMA to describe the transmission of light through waveguide-cavity systems as illustrated in FIG. 17, which is also the setup for the results and the analysis shown in FIG. 15 of the main text. We allow for incoming waves from the upper left with amplitude a.sub.1,in and from the upper right with amplitude a.sub.2,in, such that it is possible to couple into the WGMs in either the CW or the CCW directions. Based on coupled mode theory we add a coupling term to Eq. (S.59) and arrive at

[00042]$\begin{array}{cc}.Math.1.Math.\frac{d}{\mathrm{dt}}.Math.?=H.Math..Math.?+L.Math.\sqrt{{?}_1}.Math.\left(\text{?}\right)& \left(S.Math.\mathrm{.71}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

with ?.sub.1 denoting the waveguide-resonator coupling coefficient. The losses due to coupling of the cavity to the waveguides are included in the diagonal elements ?.sub.2 of the Hamiltonian (S.60). Assuming that there is no backscattering of light between the microcavity and the waveguides (which is justified when the distance between cavity and waveguides is sufficiently large) we derive the outgoing amplitudes in the lower waveguide as

[00043]$\begin{array}{cc}{?}_{\mathrm{TMA}}=\frac{.Math.A.Math.-.Math.B.Math.}{.Math.A.Math.+.Math.B.Math.}& \left(S.Math.\mathrm{.66}\right)\\ a_{3,\mathrm{out}}=-\sqrt{K_1}.Math.{}^{*}?_{\mathrm{CW}}& \left(S.Math.\mathrm{.72}\right)\\ a_{4,\mathrm{out}}=-\sqrt{K_1}.Math.{}^{*}?_{\mathrm{CCW}}& \left(S.Math.\mathrm{.73}\right)\end{array}$

We can choose ?.sub.1 to be real as we are only interested in the absolute values of a.sub.3,out and a.sub.4,out. For a CW excitation with a.sub.1-in at a fixed frequency ?.sub.e we find from Eqs. (S.72)-(S.73)

[00044]$\begin{array}{cc}.Math.\text{?}=\frac{1.Math..Math.{?}_1?\left(\text{?}-{?}_e\right)}{{\left(\text{?}-{?}_e\right)}^2-\mathrm{AB}}.Math.\text{?}& \left(S.Math.\mathrm{.74}\right)\\ .Math.\text{?}=\frac{-1.Math..Math.{?}_1.Math.A}{{\left({?}_c-{?}_e\right)}^2-\mathrm{AB}}.Math.\text{?}& \left(S.Math.\mathrm{.75}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

Analogously, for a CCW excitation via a.sub.2-in we find

[00045]$\begin{array}{cc}.Math.\text{?}=\frac{-1.Math..Math.{?}_1.Math.B}{{\left({?}_c-{?}_e\right)}^2-\mathrm{AB}}.Math.\text{?}& \left(S.Math.\mathrm{.76}\right)\\ .Math.\text{?}=\frac{-1.Math..Math.{?}_1?\left({?}_c-{?}_e\right)}{{\left({?}_c-{?}_e\right)}^2-\mathrm{AB}}.Math.\text{?}& \left(S.Math.\mathrm{.77}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

The asymmetric backscattering expresses itself here by the fact that the numerator of a.sub.4,out in Eq. (S.75) is proportional to A, whereas the numerator of a.sub.3,out in Eq.(S.76) is proportional to B. Assuming that the input amplitudes a.sub.1-in and a.sub.2-in are the same, we find the chirality as defined by Eq. (S.66) in terms of the transmission amplitudes to be

[00046]$\begin{array}{cc}.Math.{?}_{\mathrm{transmission}}=\frac{.Math.\text{?}.Math.-.Math.\text{?}.Math.}{.Math.\text{?}.Math.+.Math.\text{?}.Math.}& \left(S.Math.\mathrm{.78}\right)\\ \text{?}.Math.\text{indicates text missing or illegible when filed}& \end{array}$

where a.sub.4,out (a.sub.3,out) has been obtained by injecting light at port 1 (2). The crucial difference between the formulas for the chirality as measured in the lasing system [Eq. (S.70)] and the formula for the chirality |a|.sup.2 measured in a transmission experiment [Eq. (S.78)] is that in the former the intensities, |a|.sup.2 of the outgoing waveguide modes are used, whereas in the latter only the modulus of the amplitudes, |a|, appear.

[0207] In order to compare the two different chirality formulas, Eqs. (S.70) and (S.78), we have performed numerical calculations using a finite element method where we have solved the inhomogeneous Helmholtz equation. The calculations were restricted to the transverse magnetic (TM) polarization in two dimensions. The geometry of the system is shown in FIG. 17. The parameters for the waveguides and scatterers have been chosen such that the scatterers perturb the eigenvalues of the system much stronger than the waveguides coupled to the resonator. Therefore, the chirality is determined primarily through the scatterers, similar to the experiment. One of the scatterers had a fixed position, situated at an angle of ?/2 with respect to the waveguides. The second scatterer was situated on the opposite side of the disk and its position was given by the angle ? between the scatterers. The effective size factor, d.sub.2, of the second scatterer (which is the spatial overlap between the scatterer and the optical mode) was varied by changing the distance between the scatterer and the resonator. In the calculations the angle ? was varied between 2.91 and 3.06, and the size factor d.sub.2 was varied between 0.01 and 0.04. The waveguides, as well as the microresonator had an effective refractive index of n=1.444. The system was excited by injecting light into the waveguides at any of the ports 1-4 with frequency ?.sub.eachieved by placing a line source f (y) at the corresponding side of the system (marked by a black dashed line in FIG. 17), which excites only the fundamental mode f(y,?.sub.e) of the waveguide. Both, the spatial profile f(y,?.sub.e) of the fundamental mode, as well as the ropagation coefficient ?.sub.x were found through matching conditions at the dielectric waveguide interface. The computational domain was truncated by a reflectionless perfectly matched layer, which absorbs all scattered outgoing waves. The incoming and outgoing amplitudes a.sub.1-4(in,out) of the waveguide modes were extracted by projecting the solution of the inhomogeneous Helmholtz equation onto the individual (fundamental) waveguide modes.

[0208] In FIG. 23 we compare the chirality as determined from the eigenvalue calculations for the lasing cavity with the chirality as determined from the transmission calculations. The chirality is obtained under variation of the two positional parameters (d.sub.2, ?) of the second scatterer. We chose to vary two parameters in order to be able to exactly reach the exceptional points where the chirality features an absolute maximum, i.e. ?=?1. In the parameter range shown in FIG. 23 two pairs of EPs are depicted where each pair features two EPs of opposite chirality. The pattern of EP pairs is roughly repetitive when extending the scanned interval of angle ? as long as the scatterer does not come close to one of the attached waveguides. In the calculations we observe an excellent agreement between the two chirality definitions such that we can indeed assume that both methods yield a good estimate for the internal chirality of the whispering gallery modes induced by the presence of the two scatterers.

[0209] FIG. 23. Comparison of the chirality obtained (A) through a full numerical eigenvalue calculation by Eq. (S.70) and (B) through a full numerical transmission calculation by Eq. (S.78). The dependence of the chirality is plotted with respect to the position of the second scatterer given by both the angle between the scatterers, ?, as well as by the effective size factor, d.sub.2. Both formulas yield very similar values for the chirality validating Eqs. (S.70) and (S.78).

[0210] In a next step we explicitly compared the full numerical results to the results from the TMA model. For this, we calculated the parameters A, ?, and ?.sub.c through separate eigenvalue calculations for each of the scatterers, where no waveguides were attached to the system. The value for the coupling coefficient ?.sub.1 has been determined from transmission calculations from port 1 to port 3 with no scatterers present. In FIG. 24 the chirality definitions of Eqs. (S.66), (S.70) and (S.78) are compared to each other for the case that the distance of the 2.sup.nd nanotip is fixed at the same distance as the 1.sup.st nanotip, i.e. d.sub.2=0.02. Similar to FIG. 23 we again observe an excellent agreement between the numerical calculations. For the TMA model we find that it correctly predicts the angles at which the chirality becomes minimal/maximal, but the exact values differ. The reason for this is that the TMA model does not include other scattering processes as, for example, from the resonator to the waveguide.

[0211] FIG. 24. Comparison of the chirality definitions for ?.sub.TMA, ?.sub.lasing and ?.sub.transmission. In the calculations the second scatterer has an effective size factor d.sub.2=0.02 and the angle ? is varied.

[0212] FIG. 25. Asymmetric backscattering intensities |B.sub.CW/CCW|.sup.2 from a CW to a CCW wave [left panel: (A) and (C)] and from a CCW to a CW mode [right panel: (B) and (D)]. The results are obtained from a full numerical transmission calculation using a finite element method [upper panel: (A) and (B)], as well as from the TMA model [lower panel: (C) and (D)]. Both models yield the same frequencies at which the backscattering intensities peak, but the overall intensities differ from each other since additional scattering processes as from the waveguide to the resonator are not included in the TMA. In each panel the backscattering intensity is shown as functions of the injected frequency detuning ?.sub.e??.sub.a and the angular position ? of the second nanotip. Dashed lines mark the local minima of backscattering intensities, corresponding to the chirality maxima and minima. The asymmetric backscattering is shown by the shifted intensity patterns with respect to the angle ?.

[0213] The asymmetric backscattering which results in the intriguing chirality behavior in FIG. 24 can also be observed by looking at the normalized backscattering intensity |B.sub.CCW|.sup.2=|a.sub.CCW,out|.sup.2/|a.sub.CW,in|.sup.2 from the CW to CCW traveling mode and the similarly defined |B.sub.CW|.sup.2. From Eq. (S.70) it follows that an exceptional point (with an absolute chirality maximum) is reached when either of the backscattering intensities |B.sub.CW/CCW|.sup.2 is zero. Hence, a chirality maximum (minimum) can be found by minimizing the backscattering intensity |R.sub.CCW|.sup.2 (|R.sub.CW|.sup.2). This strategy has also been used in the experiment and the corresponding data is shown in FIG. 14 of the main text. The EPs corresponding to opposite chiralities occur at slightly different angles ?, which manifests itself by shifting the two backscattering intensity pattern |B.sub.CW/CCW|.sup.2 with respect to the angle ? as shown in FIG. 25. Here, the angles ? at which the backscattering |B.sub.CW/CCW|.sup.2 becomes minimal are indicated by dashed lines. In addition, both the results for the TMA model and the numerical transmission calculations are plotted. The frequencies at which the backscattering intensities |B.sub.CW/CCW|.sup.2 peak match very well between the two models; however, the predicted overall intensities differ due to the differences in the models.

[0214] Directionality analysis for the biased input case in the transmission model. As discussed in the main text, the intrinsic chirality is different from the directionality when light is injected into the resonator in a preferred direction such as in the CW or the CCW direction (i.e., we referred to this as the biased input). Our experiments described in the main text revealed that varying the relative distance (relative spatial phase) between the scatterers affects the amount of light coupled out of the resonator into the forward direction (i.e., in the direction of the input) and into the backward direction (i.e., in the opposite direction of the input); however, the amount of light coupled out of the resonator into the forward direction always remains higher than that in the backward direction.

[0215] FIG. 26. Directionality with a biased input (CW) as a function of the relative phase difference between two scatterers (A). Summary of the results obtained in the numerical simulation and the fitting curve using the theoretical model. (B-F), Results of finite element simulations at different relative phase angles ? but fixed size factor revealing the intracavity field patterns and output direction in the waveguides. ? values are: (B) 2.590 rad; (C) 2.617 rad; (D) 2.625 rad; (E) 2.631 rad; and (F) 2.653 rad. P.sub.1 and P.sub.2 denote the locations of the scatterers.

[0216] FIG. 26 depicts the results of finite element simulations with COMSOL validating our experimental observations presented in FIGS. 14&15 in the main text. It is seen that directionality is always negative taking values between its minimum and maximum values by changing the relative phase angle. Decreasing directionality implies the presence of scattering into the direction opposite to the direction of the injected light. Backward scattering, however, remains always weaker than forward scattering. Simulations reveal that when the intracavity field forms a standing-wave pattern with well-defined nodal lines, light couples out from the resonator in both the cw and ccw directions (FIG. 26B); however, when nodal lines are washed out and the field profile deviates from the standing-wave pattern light couples out from the resonator in the direction of the input (FIG. 26D). A relation between the visibility of the nodal lines (and the standing-wave pattern) and the ratio of the light coupled into cw and ccw directions is clearly seen (FIG. 26).

[0217] As is evident from the foregoing description, certain aspects of the present technology as disclosed are not limited by the particular details of the examples illustrated herein, and it is therefore contemplated that other modifications and applications, or equivalents thereof, will occur to those skilled in the art. It is accordingly intended that the claims shall cover all such modifications and applications that do not depart from the scope of the present technology as disclosed and claimed.

[0218] Other aspects, objects and advantages of the present technology as disclosed can be obtained from a study of the drawings, the disclosure and the appended claims.