RF-Photonic Frequency Stability Gear Box
Andrey B. Matsko, Anatoliy A. Savchenkov, Vladimir S. Ilchenko, David Seidel, Lute Maleki
aa r X i v : . [ phy s i c s . op ti c s ] N ov RF-Photonic Frequency Stability Gear Box
A. B. Matsko ∗ , A. A. Savchenkov, V. S. Ilchenko, D. Seidel, and L. Maleki OEwaves Inc., 465 N. Halstead St. Ste. 140, Pasadena, CA 91107
An optical technique based on stability transfer among modes of a monolithic optical microres-onator is proposed for long therm frequency stabilization of a radiofrequency (RF) oscillator. Weshow that locking two resonator modes, characterized with dissimilar sensitivity in responding toan applied forcing function, to a master RF oscillator allows enhancing the long term stability of aslave RF oscillator locked to two resonator modes having nearly identical sensitivity. For instance,the stability of a 10 MHz master oscillator characterized with Allan deviation of 10 − at 10 s canbe increased and transferred to a slave oscillator with identical stability performance, so that theresultant Allan deviation of the slave oscillator becomes equal to 10 − at 10 s. The method doesnot require absolute frequency references to achieve such a performance. PACS numbers:
Stable oscillators are required in every branch of mod-ern science and technology, so the demand for creation ofnovel devices with higher performance and smaller size,weight, and power consumption is continuously growing.On the other hand, the stability of simple free runningelectronic RF systems is limited both technically and fun-damentally. The increasing complexity of stabilizationcircuitry as well as usage of external reference units, e.g.atomic cells, is typically required to improve the per-formance of conventional electronic oscillators. In thisLetter we disclose an approach for stabilization of an RFoscillator using an RF photonic transducer consisting ofan optical microresonator possessing at least two familiesof modes with different sensitivity to an applied forcingfunction.An optical resonator can be used as a transducer be-tween optical and radio frequencies [1]. Nearly threedecades ago it was realized that locking the free spec-tral range (FSR) of a Fabry-Perot resonator to a radiofrequency oscillator, and simultaneous locking a laser fre-quency to an optical mode of the resonator, results in thestabilization of the laser frequency [2], since the opticalfrequency ω and the FSR frequency ω F SR are related as ω = l ω F SR + ∆ ω, (1)where l is an integer number, and ∆ ω is the frequencyshift arising due to the dispersive phase shift correctionswithin the resonator. In other words, the modes of theresonator serve as a bridge between RF and the opticalfrequencies [3].The approach was devised for an accurate transduc-tion of a known RF frequency, precisely defined by anRF clock, to the optical frequency domain. Implementa-tion of the method was problematic since it required anaccurate knowledge of the resonator dispersion frequencydependent parameter ∆ ω , which results from the disper-sion associated with mirror coatings. It became clear thatthe FSR of the resonator cannot be used as a constantfrequency marker, and the resonator frequency bridgeswere replaced with optical frequency combs [4] allowingtransfer of frequency and phase information from the RFto the optical frequency domain. While optical resonators are not particularly suitablefor absolute frequency metrology, they are useful for oscil-lator stabilization. Locking to a reference cavity is widelyutilized for stabilizion of laser frequency [5, 6] and RF os-cillators [7]. It was demonstrated [8] that locking a modeof an optical monolithic microresonator to an optical fre-quency reference results in suppression of the long termfrequency drift of the FSR of the resonator, as well asstabilization of the RF hyper-parametric oscillator basedon the resonator.An absolute frequency reference is not a prerequisitefor the long term frequency stabilization of an oscillator.Application of optical dual-mode stabilization technique,initially developed for stabilization of RF quartz oscil-lators [9], results in excellent long term stabilization ofthe spectrum of an optical microresonator, and an oscil-lator locked to this resonator [10–12]. In this Letter weintroduce another application of the optical dual modetechnique, and propose to use it for long term stabiliza-tion of an RF oscillator. The approch involves an opticalmicroresonator that enhances the efficiency of the elec-tronic feedback used to stabilize the oscillator.The microresonator-based RF photonic transduceruses two families of optical modes having significantlydifferent susceptibility to external conditions such as me-chanical pressure, voltage, temperature, etc. We selecttwo arbitrary modes, each of which belongs to one thetwo mode families, and lock the frequency difference be-tween those modes to a master RF oscillator. The lockingcan be implemented in various ways. For instance, let usassume that light emitted by a continuous wave (cw) laseris modulated with the RF signal. The carrier of the mod-ulated light is locked to the center of one of the opticalmodes using, e.g., the Pound-Drever-Hall technique [5].An external parameter(s) is (are) adjusted via an elec-tronic feedback so that the second selected optical modehas a frequency equal to the frequency of the modulationsideband. In this way the stability of the microresonatorspectrum becomes strongly correlated with the stabilityof the master RF oscillator, i.e. the spectrum of the res-onator becomes stable.Let us estimate in an explicit way the degree of thestability realized by this kind of locking. We assumethat the laser is tunable so its frequency ( ω ) follows thecorresponding optical mode. Locking of frequencies ofthe modulation sideband and the other selected resonatormode ( ω ) is realized via feedback to uncorrelated envi-ronmental (or applied) parameters q j changing the fre-quency of the both resonator modes. The drift of theoptical frequencies can be expressed as∆ ω i = X j α i,j ∆ q j + δω i , (2)where i = 0 , α i,j are the scaling parameters, ∆ q j arethe residual drifts of the corresponding environmental orapplied parameters. Here we also took into account thatthe frequencies of carrier and modulation sideband de-pend on the imperfection of the electronic locking, char-acterized via unknown detunings between the frequenciesof the optical harmonics and the corresponding resonatormodes ( δω i ). By definition, the frequency difference be-tween the carrier and the modulation sideband is givenby ∆ ω − ∆ ω = δω RF , (3)where δω RF is the residual frequency drift of the masteroscillator. The basic goal of the locking procedure is tominimize either h (∆ ω ) i or h (∆ ω ) i quadratic devia-tion (3).This problem has an explicit solution if only one envi-ronmental (or applied) parameter, e.g. the temperatureof the resonator T ( q ≡ T ), is important. This is themost practical case since the drift of the ambient temper-ature usually results in frequency drift of the oscillatorlocked to the microresonator. We infer from Eq. (3) thatthe residual temperature drift is given by the locking cir-cuit ∆ T = ( δω RF − δω + δω ) / ( α ,T − α ,T ), and thestability of the optical harmonics is∆ ω i = α i,T α ,T − α ,T ( δω RF − δω + δω ) + δω i . (4)Ultimately, if α ,T and α ,T are significantly different,the stability of the optical harmonics is given by the sta-bility of the RF master oscillator, h (∆ ω i ) i ∼ h ( δω RF ) i .Therefore, the technique allows transferring the stabilityof an RF oscillator to the optical domain in such a waythat the relative stability of the laser locked to the res-onator becomes much larger compared with the relativestability of the RF master oscillator .It is useful to estimate the efficiency of the method.Let us consider a freely suspended MgF WGM microres-onator [10] and assume that ω ( ω ) is the frequency ofits ordinarily (extraordinarily) polarized mode, and δω and δω are negligible, so that α ,T = − n o ∂n o ∂T − R ∂R∂T , (5) α ,T = − n e ∂n e ∂T − R ∂R∂T , (6) where n e and n o are extraordinary and ordinary refrac-tive indices of the material, and R is the radius of the res-onator (see Fig. 1). The requirement of free suspension isneeded to exclude from the consideration the thermally-dependent strain of the resonator. Taking into account( ∂n o /∂t ) /n o = 0 . ∂n e /∂t ) /n e = 0 .
25 ppm/K,and ( ∂R/∂T ) /R = 9 ppm/K we find h (∆ ω ) i / ∼ h ( δω RF ) i / if the frequency of the light correspondsto 1.5 µ m wavelength. The long term drift of the laser be-comes only an order of magnitude larger compared withthe long term drift of the master RF oscillator. FIG. 1: Illustration of the thermal shift of the spectrum of aMgF WGM microresonator. If the temperature of the res-onator changes by ∆ T , the ordinarily polarized modes shift by α ,T ∆ T in frequency, while extraordinarily polarized modesshift by α ,T ∆ T . To stabilize the frequency of the resonatorspectrum, we propose to lock the frequency difference be-tween one mode belonging to the family of ordinarily polarizedmodes, and the other mode belonging to the family of ordi-narily polarized modes, to an RF master oscillator havingfrequency ω RF . This kind of locking not only stabilizes theentire optical spectrum, but also enhances the relative stabil-ity of the frequency difference between any two optical modesbelonging to the same mode family (e.g. ω RF , as shown inthe picture) beyond the stability of the master oscillator. The approach is similar to the technique described in[2], since the resonator operates as a transducer of thestability of an RF master oscillator to the optical fre-quency domain. However, the transduction efficiencyof the approach proposed here is orders of magnitudehigher. In fact in the approach described in [2] theresonator FSR was locked to an RF oscillator. In thiscase the frequency drift of the optical modes exceeds( ω /ω RF ) δω RF , which is a rather large value comparedwith 27 δω RF .The proposed locking technique is efficient for suppres-sion of the frequency drift because of a single parameter q i . If two independent drifting parameters are present(e.g. T and q ), and temperature T is the parameter usedin the feedback loop, in accordance with Eqs. (2) and (4),we find∆ ω = α ,T α ,T − α ,T δω RF + α ,T α ,q − α ,T α ,q α ,T − α ,T ∆ q, (7)where for the sake of simplicity we have neglected δω and δω . The technique helps to suppress the drift of param-eter q only if it influences the resonator modes involvedin the locking process such that α ,T α ,q ≈ α ,T α ,q .Therefore, to cancel the drift of q one needs to createanother locking loop. For instance, if one manages toreach ∆ T = ξ ( δω RF − δω + δω ) / ( α ,T − α ,T ) and∆ T = (1 − ξ )( δω RF − δω + δω ) / ( α ,q − α ,q ) usingtwo different electronic feedback loops, where 1 > ξ > α ,i = α ,i We have shown that it is possible to transfer the ab-solute stability of a master RF oscillator to a mode ofan optical microresonator. Developing this idea further,we use the properties of an optical resonator to suggestthe relative drift of the FSR and the optical frequenciesbelonging to the same mode family, are the same [1, 2]∆ ω F SR ω F SR = ∆ ω ω . (8)Therefore, using the stabilization procedure describedabove we are able to stabilize the resonator FSR suchthat ∆ ω F SR ω F SR = α ,T α ,T − α ,T δω RF ω RF ω RF ω . (9)In other words, the long term stability of the FSR exceedsthe stability of the master RF oscillator if the proposedstabilization procedure is used. In the particular case when the ordinarily and extraor-dinarily polarized resonator modes are used for locking,a stronger condition compared with (9) can be derived.For any mode pair having the same polarization and fre-quencies ω and ω the relative drift of the frequency difference ω − ω = ω RF is the same as the relativedrift of the optical frequency. Selecting ω RF = ω RF weconclude that the optical resonator stabilized by lockingto a master RF oscillator has two optical modes with rel-ative long term frequency stability much better than thestability of the master oscillator:∆ ω RF ω RF = α ,T α ,T − α ,T ω RF ω δω RF ω RF . (10)Those two stable modes can be used for stabilization ofeither a slave RF oscillator or the same master oscillator.In the case of slave oscillator one just need to lock it tothe corresponding pair of optical modes. In the case ofthe stabilization of the master RF oscillator, the opticalresonator should be inserted into the electronic feedbackloop. Proper time constants and gains of the loops shouldbe selected to achieve the optimal level of stabilization.Let us consider a 10 MHz master oscillator character-ized with Allan deviation of 10 − at 10 s. In accordancewith Eq. (10) the optical resonator made out MgF al-lows achieving relative stability for two ordinary opticalmodes separated by 10 MHz at the level characterizedby Allan deviation of 1 . × − at 10 s. This longterm stability can be transferred to a slave RF oscilla-tor locked to the optical modes with an opto-electronicfeedback loop.To conclude, we have shown that an optical resonatorcan be used as a frequency stability transducer for RFoscillators. Creation of the suggested opto-electronic cir-cuits will allow ultra-stable compact RF oscillators with-outthe need for absolute frequency references. We be-lieve this technique can significantly improve applicationswhere small and efficient highly stable oscillators are re-quired. [1] Z. Bay, G. G. Luther, and J. A. White, ”Measurementof an Optical Frequency and the Speed of Light,” Phys.Rev. Lett. , 189192 (1972).[2] R. G. DeVoe and R. G. Brewer, ”Laser frequency divi-sion and stabilization,” Phys. Rev. A(R) , 2827-2829(1984).[3] J. Ye and T. W. 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