Patent Publication Number: US-8111722-B1

Title: Low-noise RF oscillation and optical comb generation based on nonlinear optical resonator

Description:
PRIORITY CLAIMS AND RELATED PATENT APPLICATIONS 
     This patent document claims the priority of U.S. Provisional Application No. 61/033,409 entitled “LOW-NOISE RF OSCILLATORS BASED ON PHOTONIC FEEDBACK LOOP AND NONLINEAR OPTICAL RESONATOR” and filed on Mar. 3, 2008, U.S. Provisional Application No. 61/041,071 entitled “TUNABLE FREQUENCY COMB AND MICROWAVE PHOTONIC OSCILLATOR BASED ON FOUR WAVE MIXING IN MEDIA POSSESSING CUBIC NONLINEARITIES” and filed on Mar. 31, 2008, and U.S. Provisional Application No. 61/083,852 entitled “INJECTION LOCKING OF HYPERPARAMETRIC OPTICAL COMBS FOR GENERATION STABLE MICROWAVE AND RF SIGNALS” and filed on Jul. 25, 2008. The entire disclosures of the above patent applications are incorporated by reference as part of the disclosure of this document. 
    
    
     BACKGROUND 
     This application relates to signal oscillators based on photonic devices. 
     RF and microwave oscillators for generating signals in the RF and microwave frequencies may be constructed as “hybrid” devices by using both electronic and optical components to form opto-electronic oscillators (“OEOs”). See, e.g., U.S. Pat. Nos. 5,723,856, 5,777,778, 5,929,430, and 6,567,436. Such an OEO includes an electrically controllable optical modulator and at least one active opto-electronic feedback loop that comprises an optical part and an electrical part interconnected by a photodetector. The opto-electronic feedback loop receives the modulated optical output from the modulator and converted the modulated optical output into an electrical signal which is applied to control the modulator. The feedback loop produces a desired long delay in the optical part of the loop to suppress phase noise and feeds the converted electrical signal in phase to the modulator to generate the optical modulation and generate and sustain an electrical oscillation in RF or microwave frequencies when the total loop gain of the active opto-electronic loop and any other additional feedback loops exceeds the total loss. The generated oscillating signals are tunable in frequency and can have narrow spectral linewidths and low phase noise in comparison with the signals produced by other RF and microwaves oscillators. 
     SUMMARY 
     This document provides techniques and devices based on optical resonators made of nonlinear optical materials and nonlinear wave mixing to generate RF or microwave oscillations and optical comb signals. 
     In one aspect, this document provides a method for generating an RF or microwave oscillation. This method includes coupling a laser beam from a laser into a whispering-gallery-mode resonator formed of a nonlinear optical material with a third order nonlinearity at a power level above a pump threshold power level to cause an optical hyperparametric oscillation based on a nonlinear mixing in the resonator; and coupling light out of the resonator into a photodetector to produce an RF or microwave signal at an RF or microwave frequency with low noise. 
     In another aspect, a device is provided for generating an RF or microwave oscillation. This device includes a whispering-gallery-mode resonator formed of a nonlinear optical material exhibiting a third order nonlinearity; a laser to produce a laser beam that interacts with the nonlinear optical material to cause an optical hyperparametric oscillation due to a nonlinear mixing in the resonator; a first optical coupler that directly couples to the laser at one end of the first optical coupler to receive the laser beam and directly couples to the resonator to couple the laser beam into the resonator; a second optical coupler that directly couples to the resonator to output light out of the resonator; and a photodetector directly coupled to the second optical coupler to receive the output light and to produce an optical hyperparametric oscillation signal at an RF or microwave frequency with low noise. 
     In another aspect, a method is provided for generating an optical comb signal carrying different optical frequencies. This method includes coupling an optical pump beam from a laser into a whispering-gallery-mode resonator formed of a nonlinear optical material with a third order nonlinearity at a power level above a pump threshold power level to cause an optical hyperparametric oscillation based on a nonlinear mixing in the resonator to produce optical sidebands; and coupling light out of the resonator to generate an optical comb signal having the optical sidebands and a band at a frequency of the optical pump. 
     In yet another aspect, a device is provided to include an optical resonator formed of a nonlinear optical material exhibiting a third order nonlinearity and structured to circulate light in one or more resonator modes; a laser to produce a laser beam that interacts with the nonlinear optical material to cause an optical hyperparametric oscillation due to a nonlinear mixing in the resonator; an optical coupler that couples to the resonator to couple the laser beam from the laser along a first optical path into the resonator and couples light inside the resonator out as a resonator output beam along a second optical path; an optical reflector in the second optical path to reflect at least a portion of the resonator output beam back to the optical coupler which couples the reflected portion into the resonator in a way that the optical coupler couples light of the reflected portion inside the resonator out as a feedback beam to the laser; and a mechanism to control a phase of the feedback beam to establish a phase matching for locking the laser to the resonator based on injection locking. 
     These and other aspects and implementations are described in detail in the drawings, the description and the claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWING 
         FIGS. 1 ,  2 ,  3 ,  4 A,  4 B,  5 A and  5 B show examples of WGM resonators and optical coupling designs. 
         FIGS. 6A ,  6 B and  6 C show examples of RF or microwave oscillators based on nonlinear WGM resonators. 
         FIGS. 7 and 8  illustrate operations of nonlinear WGM resonators under optical pumping. 
         FIG. 9  shows a Pound-Drever-Hall (PDH) laser feedback locking scheme for locking a laser to a nonlinear WGM resonator. 
         FIGS. 10-15  show measurements of sample nonlinear WGM resonators for generating optical comb signals. 
         FIG. 16  shows an example for locking a laser to a resonator by using an external reflector. 
     
    
    
     DETAILED DESCRIPTION 
     This application describes implementations of a high frequency photonic microwave oscillator (e.g., in the X-W bands) based on the nonlinear process of four wave mixing (FWM) in crystalline whispering gallery mode resonators such as calcium fluoride or another material possessing cubic nonlinearity) that can be packaged in small packages. In FWM, the large field intensity in the high finesse WGM transforms two pump photons into two sideband photons, i.e., a signal photo and an idler photon. The sum of frequencies of the generated photons is equal to twice the frequency of the pumping light because of the energy conservation law. By supersaturating the oscillator and using multiple optical harmonics escaping the resonator (optical comb) the described oscillators can reduce the phase noise and increase spectral purity of the microwave signals generated on a fast photodiode. 
     The optical resonators may be configured as optical whispering-gallery-mode (“WGM”) resonators which support a special set of resonator modes known as whispering gallery (“WG”) modes. These WG modes represent optical fields confined in an interior region close to the surface of the resonator due to the total internal reflection at the boundary. For example, a dielectric sphere may be used to form a WGM resonator where WGM modes represent optical fields confined in an interior region close to the surface of the sphere around its equator due to the total internal reflection at the sphere boundary. Quartz microspheres with diameters on the order of 10˜10 2  microns have been used to form compact optical resonators with Q values greater than 10 9 . Such hi-Q WGM resonators may be used to produce oscillation signals with high spectral purity and low noise. Optical energy, once coupled into a whispering gallery mode, can circulate at or near the sphere equator over a long photon life time. 
     WGM resonators made of crystals described in this application can be optically superior to WGM resonators made of fused silica. WGM resonators made of crystalline CaF 2  can produce a Q factor at or greater than 10 10 . Such a high Q value allows for various applications, including generation of kilohertz optical resonances and low-threshold optical hyperparametric oscillations due to the Kerr nonlinear effect. The following sections first describe the exemplary geometries for crystal WGM resonators and then describe the properties of WGM resonators made of different materials. 
       FIGS. 1 ,  2 , and  3  illustrate three exemplary WGM resonators.  FIG. 1  shows a spherical WGM resonator  100  which is a solid dielectric sphere. The sphere  100  has an equator in the plane  102  which is symmetric around the z axis  101 . The circumference of the plane  102  is a circle and the plane  102  is a circular cross section. A WG mode exists around the equator within the spherical exterior surface and circulates within the resonator  100 . The spherical curvature of the exterior surface around the equator plane  102  provides spatial confinement along both the z direction and its perpendicular direction to support the WG modes. The eccentricity of the sphere  100  generally is low. 
       FIG. 2  shows an exemplary spheroidal microresonator  200 . This resonator  200  may be formed by revolving an ellipse (with axial lengths a and b) around the symmetric axis along the short elliptical axis  101  ( z ). Therefore, similar to the spherical resonator in  FIG. 1 , the plane  102  in  FIG. 2  also has a circular circumference and is a circular cross section. Different from the design in  FIG. 1 , the plane  102  in  FIG. 2  is a circular cross section of the non-spherical spheroid and around the short ellipsoid axis of the spheroid. The eccentricity of resonator  100  is (1−b 2 /a 2 ) 1/2  and is generally high, e.g., greater than 10 −1 . Hence, the exterior surface is the resonator  200  is not part of a sphere and provides more spatial confinement on the modes along the z direction than a spherical exterior. More specifically, the geometry of the cavity in the plane in which Z lies such as the zy or zx plane is elliptical. The equator plane  102  at the center of the resonator  200  is perpendicular to the axis  101  ( z ) and the WG modes circulate near the circumference of the plane  102  within the resonator  200 . 
       FIG. 3  shows another exemplary WGM resonator  300  which has a non-spherical exterior where the exterior profile is a general conic shape which can be mathematically represented by a quadratic equation of the Cartesian coordinates. Similar to the geometries in  FIGS. 1 and 2 , the exterior surface provides curvatures in both the direction in the plane  102  and the direction of z perpendicular to the plane  102  to confine and support the WG modes. Such a non-spherical, non-elliptical surface may be, among others, a parabola or hyperbola. Note that the plane  102  in  FIG. 3  is a circular cross section and a WG mode circulates around the circle in the equator. 
     The above three exemplary geometries in  FIGS. 1 ,  2 , and  3  share a common geometrical feature that they are all axially or cylindrically symmetric around the axis  101  ( z ) around which the WG modes circulate in the plane  102 . The curved exterior surface is smooth around the plane  102  and provides two-dimensional confinement around the plane  102  to support the WG modes. 
     Notably, the spatial extent of the WG modes in each resonator along the z direction  101  is limited above and below the plane  102  and hence it may not be necessary to have the entirety of the sphere  100 , the spheroid  200 , or the conical shape  300 . Instead, only a portion of the entire shape around the plane  102  that is sufficiently large to support the whispering gallery modes may be used to form the WGM resonator. For example, rings, disks and other geometries formed from a proper section of a sphere may be used as a spherical WGM resonator. 
       FIGS. 4A and 4B  show a disk-shaped WGM resonator  400  and a ring-shaped WGM resonator  420 , respectively. In  FIG. 4A , the solid disk  400  has a top surface  401 A above the center plane  102  and a bottom surface  401 B below the plane  102  with a distance H. The value of the distance H is sufficiently large to support the WG modes. Beyond this sufficient distance above the center plane  102 , the resonator may have sharp edges as illustrated in  FIGS. 3 ,  4 A, and  4 B. The exterior curved surface  402  can be selected from any of the shapes shown in  FIGS. 1 ,  2 , and  3  to achieve desired WG modes and spectral properties. The ring resonator  420  in  FIG. 4B  may be formed by removing a center portion  410  from the solid disk  400  in  FIG. 4A . Since the WG modes are present near the exterior part of the ring  420  near the exterior surface  402 , the thickness h of the ring may be set to be sufficiently large to support the WG modes. 
     An optical coupler is generally used to couple optical energy into or out of the WGM resonator by evanescent coupling.  FIGS. 5A and 5B  show two exemplary optical couplers engaged to a WGM resonator. The optical coupler may be in direct contact with or separated by a gap from the exterior surface of the resonator to effectuate the desired critical coupling.  FIG. 5A  shows an angle-polished fiber tip as a coupler for the WGM resonator. A waveguide with an angled end facet, such as a planar waveguide or other waveguide, may also be used as the coupler.  FIG. 5B  shows a micro prism as a coupler for the WGM resonator. Other evanescent couplers may also be used, such as a coupler formed from a photonic bandgap material. 
     WGM resonators can be used to provide an effective way to confine photons in small volumes for long periods of time. As such, WGM resonators have a wide range of applications in both fundamental studies and practical devices. For example, WGM resonators can be used for storage of light with linear optics, as an alternative to atomic light storage, as well as in tunable optical delay lines, a substitute for atomic-based slow light experiments. WGM resonators can also be used for optical filtering and opto-electronic oscillators, among other applications. 
     Amongst many parameters that characterize a WGM resonator (such as efficiency of in and out coupling, mode volume, free spectral range, etc.) the quality factor (Q) is a basic one. The Q factor is related to the lifetime of light energy in the resonator mode (τ) as Q=2πυτ, where v is the linear frequency of the mode. The ring down time corresponding to a mode with Q=2×10 10  and wavelength λ=1.3 μm is 15 μs, thus making ultrahigh Q resonators potentially attractive as light storage devices. Furthermore, some crystals are transparent enough to allow extremely high-Q whispering gallery modes while having important nonlinear properties to allow continuous manipulation of the WGMs&#39; characteristics and further extend their usefulness. 
     In a dielectric resonator, the maximum quality factor cannot exceed Q max =2πn 0 /(λα), where no is the refractive index of the material, λ is the wavelength of the light in vacuum, and α is the absorption coefficient of the dielectric material. The smaller the absorption, the larger is Q max . Hence, to predict the narrowest possible linewidth γ=τ −1  of a WGM one has to know the value of optical attenuation in transparent dielectrics—within their transparency window—within which the losses are considered negligible for the vast majority of applications. This question about the residual fundamental absorption has remained unanswered for most materials because of a lack of measurement methods with adequate sensitivity. Fortunately, high-Q whispering gallery modes themselves represent a unique tool to measure very small optical attenuations in a variety of transparent materials. 
     Previous experiments with WGM resonators fabricated by thermal reflow methods applicable to amorphous materials resulted in Q factors less than 9×10 9 . The measurements were performed with fused silica microcavities, where surface-tension forces produced nearly perfect resonator surfaces, yielding a measured Q factor that approached the fundamental limit determined by the material absorption. It is expected that optical crystals would have less loss than fused silica because crystals theoretically have a perfect lattice without inclusions and inhomogeneities that are always present in amorphous materials. The window of transparency for many crystalline materials is much wider than that of fused silica. Therefore, with sufficiently high-purity material, much smaller attenuation in the middle of the transparency window can be expected—as both the Rayleigh scattering edge and multiphonon absorption edge are pushed further apart towards ultraviolet and infrared regions, respectively. Moreover, crystals may suffer less, or not at all, the extrinsic absorption effects caused by chemosorption of OH ions and water, a reported limiting factor for the Q of fused silica near the bottom of its transparency window at 1.55 μm. 
     Until recently, one remaining problem with the realization of crystalline WGM resonators was the absence of a fabrication process that would yield nanometer-scale smoothness of spheroidal surfaces for elimination of surface scattering. Very recently this problem was solved. Mechanical optical polishing techniques have been used for fabricating ultrahigh-Q crystalline WGM resonators with Q approaching 10 9 . In this document, high quality factors (Q=2×10 10 ) in WGM resonators fabricated with transparent crystals are further described. 
     Crystalline WGM resonators with kilohertz-range resonance bandwidths at the room temperature and high resonance contrast (50% and more) are promising for integration into high performance optical networks. Because of small modal volumes and extremely narrow single-photon resonances, a variety of low-threshold nonlinear effects can be observed in WGM resonators based on small broadband nonlinear susceptibilities. As an example, below we report the observation of thermo-optical instability in crystalline resonators, reported earlier for much smaller volume high-Q silica microspheres. 
     There is little consistent experimental data on small optical attenuation within transparency windows of optical crystals. For example, the high sensitivity measurement of the minimum absorption of specially prepared fused silica, α=0.2 dB/km at λ=1.55 μm, (Δα≧10 −7  cm −1 ) becomes possible only because of kilometers of optical fibers fabricated from the material. Unfortunately, this method is not applicable to crystalline materials. Fibers have also been grown out of crystals such as sapphire, but attenuation in those (few dB per meter) was determined by scattering of their surface. Calorimetry methods for measurement of light absorption in transparent dielectrics give an error on the order of Δα≧10 −7  cm −1 . Several transparent materials have been tested for their residual absorption with calorimetric methods, while others have been characterized by direct scattering experiments, both yielding values at the level of a few ppm/cm of linear attenuation, which corresponds to the Q limitation at the level of 10 10 . The question is if this is a fundamental limit or the measurement results were limited by the imperfection of crystals used. 
     Selection of material for highest-Q WGM resonators must be based on fundamental factors, such as the widest transparency window, high-purity grade, and environmental stability. Alkali halides may not be suitable due to their hygroscopic property and sensitivity to atmospheric humidity. Bulk losses in solid transparent materials can be approximated with the phenomenological dependence
 
α≅α UV   e   λ     UV     /λ +α R λ −4 +α IR   e   −λ     IR     /λ ,  (1)
 
where α UV , α R , and α IR  represent the blue wing (primary electronic), Rayleigh, and red wing (multiphonon) losses of the light, respectively; λ UV , and λ IR  stand for the edges of the material transparency window. This expression does not take into account resonant absorption due to possible crystal impurities. Unfortunately, coefficients in Eq. (1) are not always known.
 
     One example of nonlinear materials for fabrication of high-Q WGM resonators with optical nonlinear behaviors is calcium fluoride (CaF 2 ). This material is useful in various applications because of its use in ultraviolet lithography applications at 193 and 157 nm. Ultrapure crystals of this material suitable for wide aperture optics have been grown, and are commercially available. According to recently reported measurements on scattering in CaF 2  α=3×10 −5  cm −1  at 193 nm, extremely small scattering can be projected in the near-infrared band corresponding to the limitation of Q at the level of 10 13 . 
     Lattice absorption at this wavelength can be predicted from the position of the middle infrared multiphonon edge, and yields even smaller Q limitations. Because of residual doping and nonstoichiometry, both scattering and absorption are present and reduce the Q in actual resonators. An additional source for Q limitation may be the scattering produced by the residual surface inhomogeneities resulting from the polishing techniques. At the limit of conventional optical polishing quality (average roughness σ=2 nm), the estimates based on the waveguide model for WGM surface scattering yield Q≅10 11 . 
     We studied WGM resonators fabricated with calcium fluoride and a few other crystalline materials made of LiNbO 3 , LiTaO 3  and Al 2 O 3 , and measured their quality factors. CaF 2  resonators were fabricated by core-drilling of cylindrical preforms and subsequent polishing of the rim of the preforms into spheroidal geometry. The fabricated resonators had a diameter of 4-7 millimeters and a thickness of 0.5-1 mm. The fabricated Calcium fluoride resonators had a Q factor of about 2×10 10 . 
     Measurement of the Q was done using the prism coupling method. The intrinsic Q was measured from the bandwidth of the observed resonances in the undercoupled regime. Because of different refraction indices in resonators, we used BK7 glass prisms (n=1.52) for silica (n=1.44) and calcium fluoride (n=1.43), diamond (n=2.36) for lithium niobate (n=2.10, 2.20), and lithium niobate prism (n=2.10) for sapphire (n=1.75). We used extended cavity diode lasers at 760 nm, distributed feedback semiconductor lasers at 1550 nm, and solid-state YAG lasers at 1319 nm as the light source. 
     A high-Q nonlinear WGM resonators can be used for achieving low-threshold optical hyperparametric oscillations. The oscillations result from the resonantly enhanced four-wave mixing occurring due to the Kerr nonlinearity of the material. Because of the narrow bandwidth of the resonator modes as well as the high efficiency of the resonant frequency conversion, the oscillations produce stable narrow-band beat-note of the pump, signal, and idler waves. A theoretical model for this process is described. 
     Realization of efficient nonlinear optical interactions at low light levels has been one of the main goals of non-linear optics since its inception. Optical resonators contribute significantly to achieving this goal, because confining light in a small volume for a long period of time leads to increased nonlinear optical interactions. Optical whispering gallery mode (WGM) resonators are particularly well suited for the purpose. Features of high quality factors (Q) and small mode volumes have already led to the observation of low-threshold lasing as well as efficient nonlinear wave mixing in WGM resonators made of amorphous materials. 
     Optical hyperparametric oscillations, dubbed as modulation instability in fiber optics, usually are hindered by small nonlinearity of the materials, so high-power light pulses are required for their observation. Though the nonlinearity of CaF 2  is even smaller than that of fused silica, we were able to observe with low-power continuous wave pump light a strong nonlinear interaction among resonator modes resulting from the high Q (Q&gt;5×10 9 ) of the resonator. New fields are generated due to this interaction. 
     The frequency of the microwave signal produced by mixing the pump and the generated side-bands on a fast photodiode is stable and does not experience a frequency shift that could occur due to the self- and cross-phase modulation effects. Conversely in, e.g., coherent atomic media, the oscillations frequency shifts to compensate for the frequency mismatch due to the cross-phase modulation effect (ac Stark shift). In our system the oscillation frequency is given by the mode structure and, therefore, can be tuned by changing the resonator dimensions. Different from resonators fabricated with amorphous materials and liquids, high-Q crystalline resonators allow for a better discrimination of the third-order nonlinear processes and the observation of pure hyperparametric oscillation signals. As a result, the hyperoscillator is promising for applications as an all-optical secondary frequency reference. 
     The hyperparametric oscillations could be masked with stimulated Raman scattering (SRS) and other non-linear effects. For example, an observation of secondary lines in the vicinity of the optical pumping line in the SRS experiments with WGM silica microresonators was interpreted as four-wave mixing between the pump and two Raman waves generated in the resonator, rather than as the four-photon parametric process based on electronic Kerr nonlinearity of the medium. An interplay among various stimulated nonlinear processes has also been observed and studied in droplet spherical microcavities. 
     The polarization selection rules together with WGM&#39;s geometrical selection rules allow for the observation of nonlinear processes occurring solely due to the electronic nonlinearity of the crystals in crystalline WGM resonators. Let us consider a fluorite WGM resonator possessing cylindrical symmetry with symmetry axis. The linear index of refraction in a cubic crystal is uniform and isotropic, therefore the usual description of the modes is valid for the resonator. The TE and TM families of WGMs have polarization directions parallel and orthogonal to the symmetry axis, respectively. If an optical pumping light is sent into a TE mode, the Raman signal cannot be generated in the same mode family because in a cubic crystal such as CaF 2  there is only one, triply degenerate, Raman-active vibration with symmetry F 2g . Finally, in the ultrahigh Q crystalline resonators, due to the material as well as geometrical dispersion, the value of the free spectral range (FSR) at the Raman detuning frequency differs from the FSR at the carrier frequency by an amount exceeding the mode spectral width. Hence, frequency mixing between the Raman signal and the carrier is strongly suppressed. Any field generation in the TE mode family is due to the electronic nonlinearity only, and Raman scattering occurs in the TM modes. 
     Consider three cavity modes: one nearly resonant with the pump laser and the other two nearly resonant with the generated optical sidebands. Our analysis begins with the following equations for the slow amplitudes of the intracavity fields
 
 {dot over (A)}=−Γ   0   A+ig[|A|   2 +2| B   + | 2 +2| B   − | 2   ]A+ 2 igA*B   +   B   −   +F   0 ,
 
 {dot over (B)}   + =−Γ +   B   +   +ig[ 2| A|   2   +|B   + | 2 +2| B   − | 2   ]B   +   +igB*   −   |A|   2 ,
 
 {dot over (B)}   − =−Γ −   B   −   +ig[ 2| A|   2 +2| B   + | 2   +|B   − | 2   ]B   −   +igB*   +   |A|   2 ,
 
where Γ o =i(ω o −ω)+γ o  and Γ ± =i(ω ± −{tilde over (ω)} ± )+γ ± , γ o , γ + , and γ −  as well as ω o , ω + , and ω −  are the decay rates and eigenfrequencies of the optical cavity modes respectively; ω is the carrier frequency of the external pump (A), {tilde over (ω)} ±  and {tilde over (ω)} ±  are the carrier frequencies of generated light (B +  and B − , respectively). These frequencies are determined by the oscillation process and cannot be controlled from the outside. However, there is a relation between them (energy conservation law): 2ω={tilde over (ω)} ± +{tilde over (ω)} − . Dimensionless slowly varying amplitudes A, B + , and B −  are normalized such that |A| 2 , |B + | 2 , and |B − | 2  describe photon number in the corresponding modes. The coupling constant can be found from the following expression
 
 g=hω   0   2   n   2   c/Vn   0   2  
 
where n 2  is an optical constant that characterizes the strength of the optical nonlinearity, n o  is the linear refractive index of the material, V is the mode volume, and c is the speed of light in the vacuum. Deriving this coupling constant we assume that the modes are nearly overlapped geometrically, which is true if the frequency difference between them is small. The force F o  stands for the external pumping of the system F o =(2γ o P o /ω o ) 1/2 , where P o  is the pump power of the mode applied from the outside.
 
     For the sake of simplicity we assume that the modes are identical, i.e., y + =y − =y o , which is justified by observation with actual resonators. Then, solving the set (1)-(3) in steady state we find the oscillation frequency for generated fields 
                 ω   -       ω   ~     ⁢           ⁢   …       ⁢           =           ω   ~     +     -   ω     =       1   2     ⁢     (       ω   +     -     ω   ⁢           ⁢   …       ⁢           )           ,         
i.e., the beat-note frequency depends solely on the frequency difference between the resonator modes and does not depend on the light power or the laser detuning from the pumping mode. As a consequence, the electronic frequency lock circuit changes the carrier frequency of the pump laser but does not change the frequency of the beat note of the pumping laser and the generated sidebands.
 
     The threshold optical power can be found from the steady state solution of the set of three equations for the slow amplitudes of the intracavity fields: 
                 P   th     ≃     1.54   ⁢           ⁢     π   2     ⁢           ⁢         n   0   2     ⁢   V         n   2     ⁢   λ   ⁢           ⁢     Q   2             ,         
where the numerical factor 1.54 comes from the influence of the self-phase modulation effects on the oscillation threshold. The theoretical value for threshold in our experiment is P th ≈0.3 mW, where n o =1.44 is the refractive index of the material, n 2 =3.2×10 −16  cm 2 /W is the nonlinearity coefficient for calcium fluoride, V=10 −4  cm 3  is the mode volume, Q=6×10 9 , and λ=1.32 μm.
 
     The above equation suggests that the efficiency of the parametric process increases with a decrease of the mode volume. We used a relatively large WGM resonator because of the fabrication convenience. Reducing the size of the resonator could result in a dramatic reduction of the threshold for the oscillation. Since the mode volume may be roughly estimated as V=2πλR 2 , it is clear that reducing the radius R by an order of magnitude would result in 2 orders of magnitude reduction in the threshold of the parametric process. This could place WGM resonators in the same class as the oscillators based on atomic coherence. However, unlike the frequency difference between sidebands in the atomic oscillator, the frequency of the WGM oscillator could be free from power (ac Stark) shifts. 
     Analysis based on the Langevin equations describing quantum behavior of the system suggests that the phase diffusion of the beat-note is small, similar to the low phase diffusion of the hyperparametric process in atomic coherent media. Close to the oscillation threshold the phase diffusion coefficient is 
                 D   beat     ≃         γ   0   2     4     ⁢           ⁢       ℏω   0       P     B   out             ,         
where P Bout  is the output power in a sideband. The corresponding Allan deviation is σ beat /ω beat =(2D beat /tω 2   beat ) 1/2 . We could estimate the Allan deviation as follows:
 
σ beat /ω beat ≅10 −13   /√{square root over (r)} 
 
for γ 0 =3×10 5  rad/s, P Bout =1 mW, ω 0 =1.4×10 15  rad/s and ω beat =5×10 10  rad/s. Follow up studies of the stability of the oscillations in the general case will be published elsewhere.
 
     Our experiments show that a larger number of modes beyond the above three interacting modes could participate in the process. The number of participating modes is determined by the variation of the mode spacing in the resonator. Generally, modes of a resonator are not equidistant because of the second order dispersion of the material and the geometrical dispersion. We introduce D=(2ω o −ω + −ω − )/γ o  to take the second order dispersion of the resonator into account. If |D|≧1 the modes are not equidistant and, therefore, multiple harmonic generation is impossible. 
     Geometrical dispersion for the main mode sequence of a WGM resonator is D≅0.41c/(γ 0 Rn 0 m 5/3 ), for a resonator with radius R; ω + , ω 0 , and ω −  are assumed to be m+1, m, and m−1 modes of the resonator (ω m Rn ωm =mc, m&gt;&gt;1). For R=0.4 cm, γ 0 =2×10 5  rad/s, m=3×10 4  we obtain D=7×10 −4 , therefore the geometrical dispersion is relatively small in our case. However, the dispersion of the material is large enough. Using the Sellmeier dispersion equation, we find D≅0.1 at the pump laser wavelength. This implies that approximately three sideband pairs can be generated in the system (we see only two in the experiment). 
     Furthermore, the absence of the Raman signal in our experiments shows that effective Raman nonlinearity of the medium is lower than the value measured earlier. Theoretical estimates based on numbers from predict nearly equal pump power threshold values for both the Raman and the hyperparametric processes. Using the expression derived for SRS threshold P R ≅π2n 0   2 V/Gλ 2 Q 2 , where G≅2×10 −11  cm/W is the Raman gain coefficient for CaF 2 , we estimate P th /P R ≈1 for any resonator made of CaF 2 . However, as mentioned above, we did not observe any SRS signal in the experiment. 
     Therefore, because of the long interaction times of the pumping light with the material, even the small cubic nonlinearity of CaF 2  results in an efficient generation of narrow-band optical sidebands. This process can be used for the demonstration of a new kind of an all-optical frequency reference. Moreover, the oscillations are promising as a source of squeezed light because the sideband photon pairs generated in the hyperparametric processes are generally quantum correlated. 
     Photonic microwave oscillators can be built based on generation and subsequent demodulation of polychromatic light to produce a well defined and stable beat-note signal. Hyperparametric oscillators based on nonlinear WGM optical resonators can be used to generate ultrastable microwave signals. Such microwave oscillators have the advantage of a small size and low input power, and can generate microwave signals at any desired frequency, which is determined by the size of the resonator. 
     Hyperparametric optical oscillation is based on four-wave mixing among two pump, signal, and idler photons by transforming two pump photos in a pump beam into one signal photon and one idler photon. This mixing results in the growth of the signal and idler optical sidebands from vacuum fluctuations at the expense of the pumping wave. A high intracavity intensity in high finesse WGMs results in χ(3)based four-photon processes like hω+hω→h(ω+ωM)+h(ω−ωM), where ω is the carrier frequency of the external pumping, and ωM is determined by the free spectral range of the resonator ωM=ΩFSR. Cascading of the process and generating multiple equidistant signal and idler harmonics (optical comb) is also possible in this oscillator. Demodulation of the optical output of the oscillator by means of a fast photodiode results in the generation of high frequency microwave signals at frequency ωM. The spectral purity of the signal increases with increasing Q factor of the WGMs and the optical power of the generated signal and idler. The pumping threshold of the oscillation can be as small as microWatt levels for the resonators with ultrahigh Q-factors. 
     There are several problems hindering the direct applications of the hyperparametric oscillations. One of those problems is related to the fact that the optical signal escaping WGM resonator is mostly phase modulated. Therefore, a direct detection of the signal on the fast photodiode does not result in generation of a microwave. To go around this discrepancy, the nonlinear WGM resonator can be placed in an arm of a Mach-Zehnder interferometer with an additional delay line in another arm of the interferometer. The optical interference of the light from the two arms allows transforming phase modulated signal into an amplitude modulated signal which can be detected by an optical detector to produce a microwave signal. 
       FIG. 6A  shows an example of a hyperparametric microwave photonic oscillator in an optical interferometer configuration with a first optical path  1611  having the nonlinear WGM resonator  630  and a second optical path  612  with a long delay line. Light from a laser  601  is split into the two paths  611  and  612 . Two coupling prisms  631  and  632  or other optical couplers can be used to optically couple the resonator  630  to the first optical path  611 . The output light of the resonator  630  is collected into a single-mode fiber after the coupling prism  632  and is combined with the light from the optical delay line. The combined light is sent to a photodiode PD  650  which produces a beat signal as a narrow-band microwave signal with low noise. A signal amplifier  660  and a spectrum analyzer  660  can be used downstream from the photodiode  650 . 
       FIG. 6B  shows an example of a hyperparametric microwave photonic oscillator in which the oscillator is able to generate microwave signals without a delay in the above interferometer configuration in  FIG. 6A . This simplifies packaging the device. 
       FIG. 6C  shows an oscillator where a laser diode  601  is directly coupled to an optical coupling element CP 1  ( 631 , e.g., a coupling prism) that is optically coupled to the WDM nonlinear resonator  630  and a second optical coupling element CP 2  ( 632 , e.g., a coupling prism) is coupled to the resonator  630  to produce an optical output. The photodiode PD  650  is coupled to the CP 2  to convert the optical output received by the photodiode  650  into a low noise RF/microwave signal. 
     The above designs without the optical delay line are based on single sideband four wave mixing process occurring in the resonators. A single sideband signal does not require any interferometric technique to generate a microwave signal on the photodiode. 
     An example of the single-sideband signal is shown in  FIG. 7 , which shows experimentally observed spectrum of the hyperparametric oscillator. The oscillator has only one sideband separated with the carrier by resonator FSR (12 GHz), unlike to the usual hyperparametric oscillator having symmetric sidebands. The optical signal generates 12 GHz spectrally pure microwave signal on a fast photodiode. 
     The single-sideband oscillator is suitable for packaging of the device in a small package. The process occurs due to the presence of multiple frequencies degenerate optical modes in a WGM resonator. The modes interfere on the resonator surface. The interference results in specific spatial patterns on the resonator surface. Each sideband generated in the resonator has its own unique pattern. Selecting the right geometrical position of the output coupler on the surface of the resonator, it is possible to retrieve the carrier and only one generated sideband. 
       FIG. 8  illustrates that monochromatic light interacting simultaneously with several degenerate or nearly degenerate modes of a WGM resonator results in the interference pattern on the resonator surface. The pattern is stationary in time if the modes are completely degenerate. Selecting right position for the output coupler allows detection of the output light (e.g., at point A). At the point B, however, there is a null in the optical field so that no light is detected when the coupler is placed at B. 
     Therefore, a single sideband oscillator can be made by using a nonlinear WGM resonator with comparably high spectral density and an output evanescent field coupler that can be positioned in the proximity of the resonator surface. We have shown experimentally that by selecting the proper point on the resonator surface it is possible to observe optical hyperparametric oscillations with only one sideband generated. Such an oscillation can be demodulated directly on a fast photodiode. 
     The hyperparametric oscillator produces a high spectral purity for the microwave signal generated at the output of the photodetector. We have measured phase noise of the signals and found that it is shot noise limited and that the phase noise floor can reach at least −126 dBc/Hz level. To improve the spectral purity we can oversaturate the oscillator and generate an optical comb. Microwave signals generated by demodulation of the optical comb have better spectral purity compared with the single-sideband oscillator. Optical comb corresponds to mode locking in the system resulting in generation of short optical pulses. We have found that the phase noise of the microwave signal generated by the demodulation of the train of optical pulses with duration t and repetition rate T is given by shot noise with a power spectral density given by 
                 S   ϕ     ⁡     (   ω   )       ≈         2   ⁢     ℏω   0           P   ave     ⁢     ω   2         ⁢           ⁢       4   ⁢     π   2     ⁢   α   ⁢           ⁢     t   2         T   4               
where ω 0  is the frequency of the optical pump, P ave  is the averaged optical power of the generated pulse train, α is the round trip optical loss. Hence, the shorter is the pulse compared with the repetition rate the smaller is the phase noise. On the other hand we know that T/t is approximately the number of modes in the comb N. Hence, we expect that the comb will have much lower (N^2) phase noise compared with usual hyperparametric oscillator having one or two sidebands.
 
     Nonlinear WGM resonators with the third order nonlinearities, such as CaF2 WGM resonators, can be used to construct tunable optical comb generators. A CaF2 WGM resonator was used to generate optical combs with 25 m GHz frequency spacing (m is an integer number). The spacing (the number m) was changed controllably by selecting the proper detuning of the carrier frequency of the pump laser with respect to a selected WGM frequency. Demodulation of the optical comb by means of a fast photodiode can be used to generate high-frequency microwave signals at the comb repetition frequency or the comb spacing. The linewidth of generated 25 GHz signals was less than 40 Hz. 
     Such a comb generator includes a laser to produce the pump laser beam, a nonlinear WGM resonator and an optical coupling module to couple the pump laser beam into the nonlinear WGM resonator and to couple light out of the nonlinear WGM resonator. Tuning of the frequencies in the optical comb can be achieved by tuning the frequency of the pump laser beam and the comb spacing can be adjusted by locking the pump laser to the nonlinear WGM resonator and controlling the locking condition of the pump laser. 
       FIG. 9  shows an example of such a comb generator. Pump light from the laser, e.g., a 1550 nm tunable laser coupled to a fiber, is sent into a CaF2 WGM resonator using a coupling prism, and was retrieved out of the resonator using another coupling prism. The light escaping the prism is collimated and coupled into a single mode fiber. The coupling efficiency can be set, e.g., higher than 35%. The resonator may have a conical shape with the rounded and polished rim. The CaF2 WGM resonator used in our tests is 2.55 mm in diameter and 0.5 mm in thickness. The intrinsic Q factor was on the order of 2.5×10 9 . The proper shaping of the resonator can reduce the mode cross section area to less than a hundred of square microns. The resonator can be packaged into a thermally stabilized box, e.g., by using a thermoelectric cooler (TEC), to compensate for external thermal fluctuations. The optical output of the resonator was directed to an optical spectrum analyzer (OSA) to measure the optical spectral properties of the output and a photodetector and an RF spectrum analyzer (RFSA) to measure the RF or microwave spectral properties of the output of the photodetector. 
     In  FIG. 9 , the laser frequency is locked to a mode of the WGM resonator. As illustrated, a Pound-Drever-Hall laser feedback locking system is used where a part of the optical output of the WGM resonator is used as the optical feedback for laser locking. The level and the phase of the laser locking are set to be different for the oscillating and nonoscillating resonator. Increasing the power of the locked laser above the threshold of the oscillation causes the lock instability. This is expected since the symmetry of the resonance changes at the oscillation threshold. The lock parameters can be modified or adjusted while increasing the laser power to keep the laser locked. While the laser is locked to the WGM resonator, the detuning of the laser frequency from the resonance frequency of the WGM resonator can be changed to tune the comb by modifying the lock parameters. 
     When the WGM resonator is optically pumped at a low input level when the pumping power approaches the threshold of the hyperparametric oscillations, no optical comb is generated and a competition of stimulated Raman scattering (SRS) and the FWM processes is observed. The WGM resonator used in our tests had multiple mode families of high Q WGMs. We found that SRS has a lower threshold compared with the FWM oscillation process in the case of direct pumping of the modes that belong to the basic mode sequence. This is an unexpected result because the SRS process has a somewhat smaller threshold compared with the hyperparametric oscillation in the modes having identical parameters. The discrepancy is due to the fact that different mode families have different quality factors given by the field distribution in the mode, and positions of the couplers. The test setup was arranged in such a way that the basic sequence of the WGMs had lower Q factor (higher loading) compared with the higher order transverse modes. The SRS process starts in the higher-Q modes even though the modes have larger volume V. This happens because the SRS threshold power is inversely proportional to VQ 2 . 
     Pumping of the basic mode sequence with larger power of light typically leads to hyperparametric oscillation taking place along with the SRS.  FIG. 10  shows a measured frequency spectrum of the SRS at about 9.67 THz from the optical carrier and hyperparametric oscillations observed in the CaF2 resonator pumped to a mode belonging to the basic mode sequence. The structure of the lines is shown by inserts below the spectrum. The loaded quality factor Q was 10 9  and the pump power sent to the modes was 8 mW. Our tests indicated that hyperparametric and SRS processes start in the higher Q modes. The frequency separation between the modes participating in these processes is much less than the FSR of the resonator and the modes are apparently of transverse nature. This also explains the absence of FWM between the SRS light and the carrier. 
     The photon pairs generated by FWM are approximately 8 THz apart from the pump frequency as shown in  FIG. 10 . This is because the CaF2 has its zero-dispersion point in the vicinity of 1550 nm. This generation of photon pairs far away from the pump makes the WGM resonator-based hyperparametric oscillator well suited for quantum communication and quantum cryptography networks. The oscillator avoids large coupling losses occurring when the photon pairs are launched into communication fibers, in contrast with the traditional twin-photon sources based on the χ(2)down-conversion process. Moreover, a lossless separation of the narrow band photons with their carrier frequencies several terahertz apart can be readily obtained. 
     In the tests conducted, optical combs were generated when the pump power increased far above the oscillation threshold. Stable optical combs were generated when the frequency of the laser was locked to a high Q transverse WGM. In this way, we observed hyperparametric oscillation with a lower threshold compared with the SRS process. Even a significant increase of the optical pump power did not lead to the onset of the SRS process because of the fast growth of the optical comb lines. 
       FIG. 11  shows examples of hyperparametric oscillation observed in the resonator pumped with 10 mW of 1550 nm light. Spectra (a) and (b) correspond to different detuning of the pump from the WGM resonant frequency. The measured spectrum (a) shows the result of the photon summation process when the carrier and the first Stokes sideband, separated by 25 GHz, generate photons at 12.5 GHz frequency. The process is possible because of the high density of the WGMs and is forbidden in the single mode family resonators. 
     The growth of the combs has several peculiarities. In some cases, a significant asymmetry was present in the growth of the signal and idler sidebands as shown in  FIG. 11 . This asymmetry is not explained with the usual theory of hyper-parametric oscillation which predicts generation of symmetric sidebands. One possible explanation for this is the high modal density of the resonator. In the experiment the laser pumps not a single mode, but a nearly degenerate mode cluster. The transverse mode families have slightly different geometrical dispersion so the shape of the cluster changes with frequency and each mode family results in its own hyperparametric oscillation. The signal and idler modes of those oscillations are nearly degenerate so they can interfere, and interference results in sideband suppression on either side of the carrier. This results in the “single sideband” oscillations that were observed in our tests. The interfering combs should not be considered as independent because the generated sidebands have a distinct phase dependence, as is shown in generation of microwave signals by comb demodulation. 
       FIG. 12  shows (a) the optical comb generated by the CaF2 WGM resonator pumped at by a pump laser beam of 50 mW in power, and (b) the enlarged central part of the measurement in (a). The generated optical comb has two definite repetition frequencies equal to one and four FSRs of the resonator.  FIG. 13  shows the modification of the comb shown in  FIG. 12  when the level and the phase of the laser lock were changed.  FIG. 13(   b ) shows the enlarged central part of the measurement in  FIG. 13(   a ). 
     The interaction of the signal and the idler harmonics becomes more pronounced when the pump power is further increased beyond the pump threshold at which the single sideband oscillation is generated.  FIGS. 12 and 13  show observed combs with more than 30 THz frequency span. The envelopes of the combs are modulated and the reason for the modulation can be deduced from  FIG. 13(   b ). The comb is generated over a mode cluster that changes its shape with frequency. 
     The above described nonlinear WGM resonator-based optical comb generator can be tuned and the controllable tuning of the comb repetition frequency is achieved by changing the frequency of the pump laser. Keeping other experimental conditions unchanged (e.g., the temperature and optical coupling of the resonator), the level and the phase of the laser lock can be changed to cause a change in the comb frequency spacing. The measurements shown in  FIGS. 11-13  provide examples for the tuning. This tuning capability of nonlinear WGM resonator-based comb generators is useful in various applications. 
     Another feature of nonlinear WGM resonator-based comb generators is that the different modes of the optical comb are coherent. The demodulation of the Kerr (hyperparametric) frequency comb so generated can be directly detected by a fast photodiode to produce a high frequency RF or microwave signal at the comb repetition frequency. This is a consequence and an indication that the comb lines are coherent. The spectral purity of the signal increases with increasing Q factor of the WGMs, the optical power of the generated sidebands, and the spectral width of the comb. The output of the fast photodiode is an RF or microwave beat signal caused by coherent interference between different spectral components in the comb. To demonstrate the coherent properties of the comb, a comb with the primary frequency spacing of 25 GHz was directed into a fast 40-GHz photodiode with an optical band of 1480-1640 nm.  FIG. 14  shows the recorded the microwave beat signal output by the 40-GHz photodiode.  FIG. 14(   a ) shows the signal in the logarithmic scale and  FIG. 14(   b ) shows the same signal in the linear scale.  FIG. 14(   c ) shows the spectrum of the optical comb directed into the 40-GHz photodiode. The result of the linear fit of the microwave line indicates that the generated microwave beat signal has a linewidth less than 40 Hz, indicating high coherence of the beat signal. A microwave spectrum analyzer (Agilent 8564A) used in this experiment has a 10 Hz video bandwidth, no averaging, and the internal microwave attenuation is 10 dB (the actual microwave noise floor is an order of magnitude lower). No optical postfiltering of the optical signal was involved. 
       FIG. 14  also indicates that the microwave signal is inhomogeneously broadened to 40 Hz. The noise floor corresponds to the measurement bandwidth (approximately 4 Hz). The broadening comes from the thermorefractive jitter of the WGM resonance frequency with respect to the pump laser carrier frequency. The laser locking circuit based on 8-kHz modulation used in the test is not fast enough to compensate for this jitter. A faster lock (e.g., 10 MHz) may be used to allow measuring a narrower bandwidth of the microwave signal. 
     The comb used in the microwave generation in  FIG. 14(   c ) has an asymmetric shape. Unlike the nearly symmetric combs in  FIGS. 12 and 13 , this comb is shifted to the blue side of the carrier. To produce the comb in  FIG. 14(   c ), the laser was locked to one of the modes belonging to the basic mode sequence. We observed the two mode oscillation process as in  FIG. 10  for lower pump power that transformed into the equidistant comb as the pump power was increased. The SRS process was suppressed. 
     In a different test, an externally modulated light signal was sent to the nonlinear WGM resonator as the optical pump.  FIG. 15  shows measured chaotic oscillations measured in the optical output of the nonlinear WGM resonator. The resonator was pumped with laser light at 1550 nm that is modulated at 25 786 kHz and has a power of 50 mW. The generated spectrum is not noticeably broader than the spectrum produced with a cw pumped resonator and the modes are not equidistant. 
     Therefore, optical frequency combs can be generated by optically pumping a WGM crystalline resonator to provide tunable comb frequency spacing corresponding to the FSR of the resonator. The combs have large spectral widths(e.g., exceeding 30 THz) and good relative coherence of the modes. The properties of the generated combs depend on the selection of the optically pumped mode, and the level and the phase of the lock of the laser to the resonator. 
     The above described generation of optical combs using optical cubic nonlinearity in WGM resonators can use laser locking to stabilize the frequencies of the generated optical comb signals. As illustrated in  FIG. 9 , a Pound-Drever-Hall (PDH) laser feedback locking scheme can be used to lock the laser that produces the pump light to the nonlinear WGM resonator. The PDH locking is an example of laser locking techniques based on a feedback locking circuit that uses the light coupled of the resonator to produce an electrical control signal to lock the laser to the resonator. The level and the phase of the lock are different for the oscillating and non-oscillating resonators. Increasing the power of the locked laser above the threshold of the oscillation causes the lock instability. This locking of the laser can facilitate generation of spectrally pure microwave signals. Tests indicate that the unlocked comb signals tend to have border linewidths (e.g., about MHz) than linewidths generated by a comb generator with a locked laser, e.g., less than 40 Hz as shown in  FIG. 14 . 
     Alternative to the Pound-Drever-Hall (PDH) laser feedback locking, Rayleigh scattering inside a WGM resonator or a solid state ring resonator can be used to lock a laser to such a resonator in a form of self injection locking. This injection locking locks a laser to a nonlinear resonator producing a hyperparametric frequency comb by injecting light of the optical output of the nonlinear resonator under optical pumping by the laser light of the laser back into the laser under a proper phase matching condition. The optical phase of the feedback light from the nonlinear resonator to the laser is adjusted to meet the phase matching condition. 
     Two feedback mechanisms can be used to direct light from the nonlinear resonator to the laser for locking the laser. The first feedback mechanism uses the signal produced via Rayleigh scattering inside the nonlinear resonator. The light caused by the Rayleigh scattering traces the optical path of the original pump light from the laser to travel from the nonlinear resonator to the laser. 
     The second feedback mechanism uses a reflector, e.g., an additional partially transparent mirror, placed at the output optical path of the nonlinear resonator to generate a reflection back to the nonlinear resonator and then to the laser.  FIG. 16  shows an example of a device  1600  that locks a laser  1601  to a nonlinear resonator  1610 . The nonlinear resonator  1610  can be a ring resonator, a disk resonator, a spherical resonator or non-spherical resonator (e.g., a spheroid resonator). An optical coupler  1620 , which can be a coupling prism as shown, is used to provide optical input to the resonator  1610  and to provide optical output from the resonator  1610 . The laser  1601  produces and directs a laser beam  1661  to the coupling prism  1620  which couples the laser beam  1661  into the resonator  1610  as the beam  1662  circulating in the counter-clock wise direction inside the resonator  1610 . The light of the circulating beam  1662  is optically coupled out by the optical coupler  1620  as a resonator output beam  1663 . A reflector  1640  is placed after the coupling prism  1620  in the optical path of the resonator output beam  1663  to reflect at least a portion of the resonator output beam  1663  back to the coupling prism  1620 . Optical collimators  1602  and  1631  can be used to collimate the light. The reflector  1640  can be a partial reflector to transmit part of the resonator output beam  1663  as an output beam  1664  and to reflect part of the resonator output beam as a returned beam  1665 . The reflector  1640  may also be a full reflector that reflects all light of the beam  1663  back as the returned beam  1665 . The feedback beam  1665  is coupled into the resonator  1610  as a counter propagating beam  1666  which is coupled by the coupling prism  1620  as a feedback beam  1667  towards the laser  1601 . The feedback beam  1667  enters the laser  1601  and causes the laser to lock to the resonator  1610  via injecting locking. 
     The above laser locking based on optical feedback from the nonlinear resonator  1610  based on either the Rayleigh scattering inside the resonator  1610  or the external reflector  1640  can be established when the optical phase of the feedback beam  1667  from the resonator  1610  to the laser  1601  meets the phase matching condition for the injection locking. A phase control mechanism can be implemented in the optical path of the feedback beam  1667  in the Rayleigh scattering scheme or one or more beams  1661 ,  1662 ,  1663 ,  1665 ,  1666  and  1667  in the scheme using the external reflector  1640  to adjust and control the optical phase of the feedback beam  1667 . As illustrated, in one implementation of this phase control mechanism, the reflector  1540  may be a movable mirror that can be controlled to change its position along the optical path of the beam  1663  to adjust the optical phase of the feedback beam  1667 . The phase of the returning signal  1667  can also be adjusted either by a phase rotator  1603  placed between the laser  1601  and the coupler  1620  or a phase rotator  1663  placed between the coupler  1620  or collimator  1631  and the external reflector or mirror  1640 . A joint configuration of using both the Rayleigh scattering inside the resonator  1610  and the external reflector  1640  may also be used. The selection of the configuration depends on the operating conditions including the loading of the resonator  1610  with the coupler  1620  as well as the strength of the Rayleigh scattering in the resonator  1610 . Such a locking technique can be used allow avoiding technical difficulties associated with using the PDH locking and other locking designs. 
     While this document contains many specifics, these should not be construed as limitations on the scope of an invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments of the invention. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or a variation of a subcombination. 
     Only a few implementations are disclosed. Variations and enhancements of the described implementations and other implementations can be made based on what is described and illustrated in this document.