On Frequency Combs in Monolithic Resonators

Nanophotonics 2016; 5 (2):363–391

Review Article Open Access

A. A. Savchenkov*, A. B. Matsko, and L. Maleki On Frequency Combs in Monolithic Resonators

DOI: 10.1515/nanoph-2016-0031 and 337 µm wavelengths) were measured to within a few Received November 12, 2015; accepted March 4, 2016 parts in 107 by mixing the laser emission with high order Abstract: Optical frequency combs have become indis- harmonics of a MW signal in a silicon diode [1]. The fre- pensable in astronomical measurements, biological fin- quencies of lasers stabilized to the 3.39 µm transition of gerprinting, optical metrology, and radio frequency pho- CH4 and the 9.33 µm and 10.18 µm transitions of CO2 were tonic signal generation. Recently demonstrated micror- measured with an IR frequency synthesis chain extending ing resonator-based Kerr frequency combs point the way upwards from the Cs frequency standard [2]. The absolute towards chip scale optical frequency comb generator re- frequency of a He–Ne laser (633 nm) stabilized with iodine taining major properties of the lab scale devices. This optical transition was measured with a total uncertainty 10 technique is promising for integrated miniature radio- of 1.6 parts in 10 by comparing with a known frequency, frequency and microwave sources, atomic clocks, opti- synthesized by summing the emission frequency of three cal references and femtosecond pulse generators. Here we different metrology lasers in a He–Ne plasma [3]. More re- 40 present Kerr frequency comb development in a historical cently, the frequency of a 657 nm Ca optical frequency perspective emphasizing its similarities and differences standard was determined against the Cs frequency stan- with other physical phenomena. We elucidate fundamen- dard by a phase-coherent chain with a total relative uncer- 13 tal principles and describe practical implementations of tainty of 2.5 parts in 10 [4]. Kerr comb oscillators, highlighting associated solved and Aside from this heroic efforts the demand for opti- unsolved problems. cal standards was fulfilled by locking environmentally- isolated lasers to optical atomic transitions. The accuracy Keywords: frequency comb, monolithic microresonator, of these standards was undetermined since the optical fre- whispering gallery mode resonator, photonics, nonlinear quency was not scaled with the primary MW standards. optics, four-wave mixing. Since the number of atiomic lines were limited, it was PACS: 42.65.Ky, 42.65.Yj, 42.65.Sf, 42.65.Tg, 42.70.Mp, impossible to create a source at an arbitrary desirable 42.79.Nv wavelength. Passive optical cavities used for laser stabilization and frequency stability transfer had inferior accuracy and relatively poor long term stability. A straight- 1 Introduction forward multistage multiplication of RF frequency to the optical frequency domain was impractical because of the Optical and microwave worlds were practically separated unavoidable phase noise increase, in accordance with ex- 2 before the advent of optical frequency combs. A few at- pression (νopt /νRF) , where νopt and νRF are values of tempts were made to use well developed radio frequency the optical and RF frequencies, respectively. These prob- (RF) and microwave (MW) atomic clocks for stabilizing and lems were solved in the optical domain by using self- measuring frequency of optical sources. These attempts stabilized pulsed lasers that replaced entire laboratories were hindered by absence of a simple technique to bridge full of equipment and resulted in a qualitative leap in tech- the RF and optics frequencies. The high accuracy goal of nology and science development. the measurements called for complex experiments involv- Initially, mode locked lasers were not considered as ing nonlinear frequency multiplication and mixing. The suitable metrological tools because of their limited stabil- experimental technique slowly evolved from far-infrared ity and absence of a clear correlation between their car- (IR) to visible wavelength. For instance, absolute frequen- rier optical frequency and the pulse repetition rate. Valida- cies of the far-IR transitions of the CN gas laser (311 µm tion of the concept of two-point stabilization of all compo- nentsof the spectra of a mode locked laser, for instace, by *Corresponding Author: A. A. Savchenkov: OEwaves Inc., 465 North locking the pulse repetition rate to an RF clock signal and Halstead Street, Ste. 140, Pasadena, CA 91107, U.S.A., self-stabilizing optical carrier via f −2f method, resulted in E-mail: [email protected] a revolution in the field [5]. Optical frequency spectrum of A. B. Matsko, L. Maleki: OEwaves Inc., 465 North Halstead Street, Ste. 140, Pasadena, CA 91107, U.S.A.

© 2016 A. A. Savchenkov et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 364 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators such a source was called a "frequency comb" for its appear- An atomic reference element or a stable optical cavity ance. The frequency comb became the ultimate single-step is needed for stabilization of the comb repetition rate. link between oscillators operating in RF and optical do- Packaged stabilized frequency comb generators, that in- mains. clude all the mentioned parts as well as stabilization elec- The optical tool of frequency metrology quickly found tronic loops, are currently well developed and commer- multiple applications beyond its initial utility becoming cially available. They are not broadly affordable because an indispensable asset of an optical laboratory. It is an ir- of the cost associated with the complexity. The devices replaceable part of modern optical clocks and the core of also are not mature enough to be utilized outside a lab- RF photonic oscillators with the record spectral purity. In oratory because of their sensitivity to environmental per- fact, the RF and MW signals produced by the frequency turbations such as temperature variations and mechani- comb rectified on a fast photodiode [6] allows achieving RF cal vibrations. Significant advances in the development of signals having spectral purity unattainable by any other transportable optical comb devices are made by several mean. commercial companies including Menlo Systems. The optical frequency comb is also utilized in search A crucial challenge to make the optical comb devices for remote exoplanets requiring a precise optical fre- broadly available is to improve them to the level of quartz quency reference [7]. Periodic Doppler shift of the stellar oscillators in terms of size, price, and robustness. These spectral lines carries information on mass of a potential parameters have to be improved by orders of magnitude to exoplanet and on its distance from the host star. Detection make the optical comb units comparable with the bulkiest of an Earth-mass planet at 1 a.u. distance from a Solar-like and the most expensive RF oscillators. Miniaturized, oon- star requires absolute precision of velocity measurement chip integrated source of an octave spanning stabilized co- on the order of 5 cm/s. Such measurements are carried out herent frequency spectrum will become an enabling tech- with solar spectrometers. Frequency comb-assisted cali- nology opening new doors in science and technology, sim- bration of the spectrometers increases the precision of ve- ilarly to the developments of semiconductor transistors locity measurement from 60 cm/s to 1 cm/s which is suffi- that started a technological revolution after the era of vac- cient for accurate detection of the planet. uum tubes. It is hardly possible, though, to achieve the Frequency combs are useful in optical communica- desirable improvement directly with conventional tech- tions. A conventional wavelength division multiplexer uti- nologies. A break-through scientific advance is needed. lizes multiple lasers to generate multiple optical channels. Microresonator-based frequency comb generators demon- Multiplexers quite often have a few hundred channels of strated nearly a decade ago [11] hold the promise to be- ∼10 GHz bandwidth. A chip-scale microresonator-based come such a break-through. comb produces an entire grid of equally spaced optical ref- Unlike conventional pulsed lasers, the Kerr frequency erences needed to sustain the channels [8, 9]. comb generator is a hyper-parametric oscillator. It is based Coherent frequency comb-based LIDAR (LIght Detec- on a cavity with nonzero cubic (Kerr) optical nonlinear- tion And Ranging system) combines advantages of both ity. The cavity is pumped with continuous wave (CW) light. time-of-flight and interferometric ranging techniques [10]. Multiple phase locked frequency harmonics are paramet- Combination of the two techniques ultimately allows for rically generated in the system when the pump power ex- 2 × 10−13 ranging precision measurements with coopera- ceeds a certain threshold determined by the cavity param- tive targets. The technique involves two frequency comb eters including quality (Q-) factor, nonlinearity, group ve- sources with slightly different repetition rates. One source locity dispersion, and mode volume. The phase of the har- is used for ranging. Its emission is bounced from the coop- monics of the comb oscillator operating in the coherent erative target and compared with the second source which regime depends on the phase of the optical pump. In con- is used as a local oscillator. The pulse-of-flight yields a pre- trast, the phase of the harmonics of mode locked lasers is cision of 3 µm, the measurements of relative phase of the determined by the lasing process. As the result, stabiliza- optical carrier provides a precision of 5 nm at long time tion of the Kerr frequency comb ultimately does not require scale. f − 2f self-referencing. The core piece of the conventional frequency comb is At this point it is unclear what is the ultimate scheme a pulsed (femtosecond) laser with a broad, ultimately an of a Kerr frequency comb generator that can be created us- octave-wide, frequency spectrum with a controlled repe- ing a CW pumped nonlinear microresonator. The proper- tition rate. A nonlinear optical mixer is required for sta- ties of the comb depend on the host material and the mor- bilization of the carrier-envelope offset (f-2f referencing) phology and size of the resonator; on the method of res- of the mode locked laser producing the frequency comb. onator integration and degree of its isolation from the en- A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 365 vironment; on noise of the pump laser; and on other pa- 2 Kerr comb fundamentals and rameters. It is currently not known what combination of these parameters is optimal. One reason is that the phys- realizations ical properties of Kerr frequency combs generated under realistic conditions in real resonators, including their ex- To produce a Kerr frequency comb one needs a CW laser citation, stabilization, coherence, and dynamics, are not and a nonlinear resonator with cubic optical nonlinearity. fully understood. However, there are plenty of first class The laser light is used for optical pumping of one of the res- research efforts that are advancing the field. In this re- onator modes. With the power of the light large enough, view we briefly highlight them and discuss possible di- the onset of the oscillation process occurs. It reveals it- rections for future development of Kerr frequency comb self by manifestation of optical frequency harmonics in the oscillators. Because of specificity of our own work, we light leaving the resonator. While properties of the gen- will pay more attention to Kerr frequency comb oscillators erated harmonics depend on the pumping light, e.g. the based on crystalline whispering gallery mode resonators phase of the harmonics depends on the pump phase in (WGMRs) (Fig. 1). the case of coherent (not chaotic) oscillation, there are independent variables in the system. For instance, relative phases of the harmonics experience a fundamental diffusion process [12, 13]. In what follows we briefly discuss the salient properties of the nonlinear process starting from the resonant four wave mixing in which polarization of the nonlinear medium is created by external field and con- cluding by true oscillation in which the polarization is created spontaneously from fluctuations of electromagnetic vacuum.

Fig. 1. A picture of a calcium fluorite broadband whispering gallery mode resonator. The resonator, having 6 mm in diameter, is properly shaped and polished to support light circulating around its a b circumference for forty microseconds. b

As the result of the artificial separation in the con- c sideration of the comb oscillators by the methods of their implementation (e.g. fiber loop oscillators, WGMR oscillators, on-chip oscillators), as well as by the name of the pro- d c cess (e.g. modulation instability lasing, hyper-parametric or parametric oscillation, Kerr comb generation), the over- d all picture of the development of the Kerr frequency comb a oscillators is difficult to grasp. In this paper we have at- tempted to unify the field. To improve understanding of the microresonator frequency comb oscillators we identify their similarities and differences with other, old and novel, physical phenomena and technologies. In what follows we Fig. 2. Nondegenerate four wave mixing results in formation of a show that i) physically the Kerr comb oscillators are far fourth photon "d" out of three photons "a", "b" and "c". The process from being unique and ii) the approaches reported so far is governed by χ(3) nonlinearity and obeys to energy and momen- on Kerr frequency comb generation differ from each other tum conservation laws. The FWM process is observed in a 6.9 mm calcium fluorite resonator pumped with three independent lasers only in the approach for their technical implementation. parked at different modes. The resonator has 109 loaded quality factor. 366 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

Kerr frequency comb generation is based on four wave -10 mixing (FWM). The concept of FWM processes interaction -20 of three electromagnetic fields in a nonlinear medium to -30 generate a fourth field (Fig. 2). Any two fields interacting -40 with the medium produce polarization not only at the ini- -50 tial frequency of the fields, but also at the sum and differ- Power, dBm dBm Power, -60 ence frequencies. The third field beats with the polariza- -70 tion frequency harmonics and scatters off the medium giv- -80 ing rise to the fourth field. These phenomena lead to the 1500 1550 1600 1650 classical picture of the FWM. Because any of the polariza- Wavelength, nm tion harmonics becomes a source of a new field, the FWM process can result in production of more than one field if -20 three fields interact in the material. Phase matching of this -40 nonlinear process defines the direction of the emission of dBm Power, the generated fields. -60 Four-wave mixing in extended three dimensional me- -80 dia is well known and is commonly modeled using plane 1545.2 1546.2 1547.2 1548.2 1549.2 waves. High-intensity pulsed lasers are usually required Wavelength, nm to observe the process. The efficiency of the FWM can increase in one-dimensional media, as in the case of prop- Fig. 3. An illustration of the competition between hyper-parametric agation through an optical fiber. Total internal reflection oscillation and frequency comb generation and stimulated Raman confining the light in the fiber results in maintaining high scattering (SRS). The eflciency of the processes depends onthe optical intensity along the fiber. The combination of the cavity spectrum, its local dispersion and bandwidth. Phase mismatch increases the comb threshold significantly in the case pre- long interaction length and high intensity light not only sented in the top image. The threshold value of SRS remains un- leads to increase of the efficiency of the optical processes changed since it does not require phase matching. As a result SRS initially studied in the bulk, but also produces numerous dominates. The harmonics around the pump frequency occur due novel FWM-related phenomena occurring due to modifica- to the four wave mixing between the pump light and the Raman tion of the phase matching conditions. frequency comb. The second image (below) shows the opposite situation. To observe this behavior we used a chip-scale 30 mW dis- Decreasing the dimensionality of the nonlinear tributed feedback diode laser to pump a calcium fluorite whispering medium to zero, as in monolithic optical microresonators, gallery mode resonator characterized with 10 GHz free spectral results in further increase of the FWM efficiency, allowing range and 1010 Q-factor. its observation at extremely low light levels, approaching countable numbers of photons per resonator mode. Zero 2.1 Early Studies dimensionality here means that the system has a discrete spectrum, as in an atom. The resonator can be considered Four wave mixing (including both self [14] and cross [15] as a lumped element with no spatial continuation. The- phase modulation), two-photon absorption [16], stimu- large field concentration leads to realization of 1012cm2 lated Raman scattering [17], third-harmonic generation and higher continuous wave optical intensities within the [18], and anti-Stokes frequency mixing [19] were among nonlinear material with low power diode lasers, resulting the first nonlinear effects extensively studied after thein- in observation of extreme optical phenomena on a chip. ception of laser. All these processes constitute the foun- The tight confinement of the field also allows true control dation for the Kerr frequency comb generation and asso- of the nonlinear phenomena: it is possible to severely sup- ciated phenomena, and are observed in optical microres- press one nonlinear process and enhance another one by onators. selecting the right morphology of the nonlinear microres- The third order nonlinear susceptibility χ(3) is respon- onator. The last feature is important for studying four-wave sible for the four-wave mixing processes [20]. In general, mixing, because this process can have lower threshold in χ(3) is a fourth rank tensor with 81 elements [21], and a properly shaped microresonator compared with stimu- each of these elements consists of a sum of 48 terms. The lated Brilloin scattering and stimulated Raman scattering number of independent terms is reduced through mate- (SRS) which usually compete with FWM in fibers (Fig. 3). rial symmetries. Cubic nonlinearity, χ(3), though, may have nonzero elements for any symmetry, unlike the quadratic A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 367 one, χ(2). Explicit expressions for the terms have been pub- fused amorphous materials, such as fused silica [38–42]. lished [22]. So far in the Kerr comb studies only one χ(3) ele- Although these materials feature small optical attenua- ment is considered either because the resonators are made tion, the highest quality factor of WGMs [40] has remained of amorphous materials or because the wave mixing oc- limited by Rayleigh scattering of residual volumetric in- curs among the waves of the same polarization collinear homogeneities and surface roughness [43] and contami- with the crystalline axis in crystalline structures. The prob- nation, e.g., with water molecules. This is generally the lem of FWM in crystalline resonant structures where sev- case, even though a resonator formed by surface tension eral elements of the nonlinearity tensor are important is forces has nearly a defect-free surface characterized by still unsolved. molecular-scale inhomogeneities. Crystalline WGM res- Coupling between the four waves may only occur onators are much less prone to Rayleigh scattering as when energy and momentum are both conserved [23]. The well as water absorption and, hence, have much higher four wave mixing leading to Kerr frequency comb gener- Q-factor [44–46]. ation also calls for photon number conservation. The en- Ultra–high Q WGM microresonators are commonly ergy conservation along with the photon number conser- used for enhancement of efficiency of nonlinear optical vation determines the frequencies of the generated har- processes such as three– [46, 47] and four–wave mixing monics. The momentum conservation defines the geome- [11, 48–51]. Strong spatial confinement of light as well as try of the process and its dependence on the dielectric sus- a long interaction time of light and the host medium of ceptibility of the material. the resonator results in significant decrease of the thresh- Since the energy transfer is a coherent phase olds of lasing [52, 53], stimulated Raman scattering [54– dependent-process, the energy exchange among the elec- 57], stimulated Brillouin scattering [58, 59], resonant opto- tromagnetic waves depends on their relative phase. If the mechanical scattering [60, 61], and other nonlinear pro- phase changes within the nonlinear medium, it could cesses. We focus here on phenomena related to resonant happen that the initially generated frequency harmon- hyper-parametric processes resulting in Kerr frequency ics disappear completely due to change of the direction of comb generation, leaving the other phenomena to be dis- their power exchange with the pump waves. Therefore, the cussed elsewhere. geometry of the nonlinear interaction has to be optimized. This may be achieved by either having a very short overlap length, or by choosing a small wave vector mismatch and 2.3 Oscillations and wave mixing in fiber proper interaction length. Phase matching becomes not resonators trivial in the case of low-dimensional structures and also in the case of very broad optical spectra involved in the Before discussing generation of Kerr frequency combs in interaction. In the first case the optical modes of the struc- microresonators it is important to review similar studies ture have to have nonzero geometrical overlap and also performed in optical fiber systems. Those studies took frequencies, defined by energy and photon number con- place earlier than microresonator ones, but physics of servation. The nonzero geometrical overlap means that the observed phenomena is rather similar to the physics the volume overlap integral of the spatial amplitudes of of Kerr frequency combs and other microresonator-based the four fields participating in the nonlinear process has phenomena. to be nonzero to observe generation of the frequency harmonics. In the second case the different optical harmonics experience different nonlinearity and the interaction effi- 2.3.1 Resonant oscillation ciency depends on their frequency [22]. The FWM process and the phase matching requirements are well described The oscillations occurring in coherently driven passive op- in textbooks on Nonlinear Optics, e.g. [24, 25]. tical fiber resonators were studied from the perspective of temporal cavity solitons [62, 63], belonging to a general class of dissipative solitons [64], modulation instability 2.2 Nonlinear Optics with Whispering lasers [65–69] as well as parametric oscillators [70–74]. Gallery Mode Resonators The microresonator Kerr frequency comb and the hyper-parametric oscillation in the fiber loops have nearly High-Q whispering gallery modes (morphology- identical properties. They are described by the same equa- dependent resonances) [26–35] were extensively studied tions [62, 75–77]. They are sustained due to continuous in liquid droplets [36, 37], as well as in solidified droplets of wave pumping and are phase locked to the pump (this 368 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators is true for the case of coherent, not chaotic, oscillations) monics by seeding modulated light to a fiber link [97], a [12, 78]. As a result, they are different from mode locked sub-micron silicon-on-insulator rib waveguide [98], and a lasers [79] and do not require two point locking for their coupled-resonator optical waveguides [99] have also been stabilization [80–82]. Fiber loop can support a large num- observed. Similar effect have been studied in microres- ber of non-interacting pulses that can be individually ad- onators [100, 101]. dressed [83–85] while microresonators also support mul- The basic differences between fiber and microres- tiple non-interacting pulses [86]. It is possible to control onator structures important for the wave mixing processes mode locking by modulation of the pump in both fiber cav- include size, shape (morphology), and finesse. The larger ities [87] and microcavities [88]. size is responsible for the smaller nonlinearity as well The differences between these processes are a few. as denser frequency spectrum of the basic (fundamental) One of them is the possibility of interaction of cavity soli- mode family of the fiber resonators, compared with mi- tons in a fiber cavity mediated by phonons [89]. Opti- croresonators. Microresonators usually support multiple cal pulses traveling in a fiber cavity excite, through elec- spatial mode families while fiber resonators only a single trostriction, transverse acoustic waves. The acoustic wave one due to differences in their geometry. The morphology excited by one optical pulse can influence another optical of the WGM resonator allows engineering the spectrum of pulse via time dependent change of the refractive index the modes and, hence, the resonator dispersion; this is not (the pulses are pulled into time dependent higher index re- possible with fiber resonators. On the other hand, the dis- gions associated with the acoustic wave) [90]. This type of persion of a fiber can be modified by engineering its core interaction seems to be suppressed in the microresonators and cladding refractive index profile. because of significant frequency difference between the The finesse of a microresonator can exceed a million mechanical modes of the resonator and round trip time of while finesse of a fiber resonator usually is less than ahun- light in the cavity. dred. The reason is that the microresonator is a seamless The difference associated with the size and finesse of structure while the fiber resonator needs splicing fibers the fiber loops and microresonators has multiple practi- in an out of a ring. Additionally, commercially available cal consequences. For instance, large values for the nor- fiber splitters used for in- and out-coupling with fiber rings malized GVD term, defined as a ratio of the difference be- are not better than 99/1. The finesse in microresonators is tween values of two adjacent free spectral ranges and a high in a very broad range of optical wavelengths deter- half width at the half maximum of the corresponding opti- mined by the resonator host material. cal modes (parameter D introduced in [12]) can be demon- Unlike fiber-based devices, microresonator devices do strated in high-Q microresonators. This value impacts the not suffer from Brillouin scattering (it is possible to observe phase matching of the hyper-parametric process and is re- Brillouin scattering in specially made resonators [58, 59]) sponsible for the number of stable multi-soliton solutions and are much less prone to Rayleigh scattering (the scatter- (supermodes) existing in the system [86]. ing can be suppressed by overloading the resonator mode Another difference is related to the ratio of the circula- [43]). Microresonators have no moving parts and coatings, tion time to the pulse duration time for mode locked pro- and are environmentally stable. They are polarization se- cesses observed in the fiber loops and microcavities. This lective. Those features make microresonators attractive for ratio is usually large in fiber loops but can be small (nearly a myriad of applications including optical frequency comb one) in microresonators. This is true for modulation insta- generation. bility (Turing rolls [91]) in anomalous GVD case and for It is possible to go further and find many commonal- generation of high order dark solitons in normal GVD case ities between microresonator processes and various non- [92] observed in microresonators (these regimes are dis- linear processes unrelated to optics, such as dipolar ex- cussed in what follows). citations in one-dimensional condensates [102], spatial dissipative solitons [103, 104], a long Josephson junction in periodic field [105], easy-axis ferromagnets in rotating 2.3.2 Resonant wave mixing magnetic field perpendicular to the easy axis [106, 107], and plasmas driven with radio frequency radiation [108]. Forced FWM processes in optical fibers and microres- All these processes can be described using similar math- onators also have a lot of in common. For example, phase ematical techniques and, hence, have somewhat similar matching of the FWM process in fibers can be achieved dynamics. Such an interconnection among various fields via different mode families of the fibers [93–95] and, simi- adds value to the Kerr frequency comb studies as they can larly, in microresonators [96]. Generation of frequency har- A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 369 improve the understanding of dynamics of completely dif- [25] ferent physical systems. n = n0 + n2I, (5) where I denotes the time-averaged spatially-normalized intensity of the optical field within the resonator, and n0 3 Theoretical Models and n2 are the linear and nonlinear refractive indices of the resonator host material. In this section we introduce two basic theoretical models According to (4) the power-dependent frequency shift utilized in studies of Kerr frequency comb generation. One of the optically pumped mode depends on the number of them is based on analysis of a set of ordinary differen- of photons in the mode (⟨N^ (0)⟩, ⟨... ⟩ stays for state av- tial equation (ODE) and the other – on forced nonlinear eraging) as ∆ω0 = −g⟨N^ (0)⟩. On the other hand, in ac- Schrödinger equation (NLSE). These models are equiva- cordance with (5), ∆ω0 = −ω0n2I/n0, and, by definition, 2 lent and are selected depending on practical problems that ~ω0⟨N^ (0)⟩ = Vñ0I/c (I = cn0|E0| /2π here is the maximal need to be analyzed. ODE are more convenient for analy- intensity of the electromagnetic wave within the mode), so sis of nonlinear processes in cavities with irregular mode we get [12] ~ω c n spectra, while NLSE is preferable for analysis of temporal g = ω 0 2 , (6) 0 Vñ n pulses generated in the cavities with smooth mode spec- 0 0 tra. The major practical difference between ODE and NLSE where V˜ is the mode volume. is that periodic boundary conditions, needed to describe The intensity of light has a certain spatial distribution the ring resonator systems, are implicit in ODE but not for within the resonator mode, not taken into account while NLSE. deriving Eq. (6). We present complex amplitude of the electric field of the running wave as a product of spatial and temporal parts, E(r)E0(t), where the dimensionless spatial 3.1 Unitary Evolution of a Nonlinear Mode distribution E(r) is normalized to unity max|E(r)| = 1, and define the mode volume as Let us consider an isolated (non-interacting with any ∫︁ 2 other) mode of a nonlinear resonator having eigenfre- V˜ = |E(r)| dV. (7) quency ω0 and characterized with annihilation (creation) V a a† operator ^ (^ ). The unitary time evolution of the mode is Because 2 described with Hamiltonian H^ = H^ 0 + V^, where n V˜ ~ω ⟨N^ (0)⟩ = 0 |E |2, (8) 0 2π 0 H^ = ω a^†a^, (1) 0 ~ 0 the amplitude of the electric field inside the resonator is V^ g a†a†aa = −~( /2)^ ^ ^^, (2) √︃ 2π~ω E = 0 A, (9) g is the coupling constant [109]. It is not necessary to use 0 n2V˜ 0 quantum operators to describe four wave mixing in the resonator modes. We use them here because it simplifies the where A is the slow amplitude of the field normalized such derivation of the equations of motion and allows straight- that |A|2 = ⟨N^ (0)⟩. forward verification of the correctness of the intermediate To correct Eq. (6) and take the spatial distribution of results. the electromagnetic field in the resonator into account we The Hamilton equation for the mode evolution note that the nonlinear part of the energy of the light in the nonlinear medium is defined as † a^˙ + iω0a^ = iga^ aâ^ (3) ∫︁ 3 U = χ(3)|E (t)|4 |E(r)|4dV (10) 2 0 has a solution V [︁ (︁ † )︁ ]︁ a^(t) = exp −i ω0 − ga^ (0)a^(0) t a^(0), (4) where we assumed that the medium is isotropic and homogeneous, so that the nonlinearity is polarization- showing that the photon number in the mode is conserved independent and (see [25]) (naturally) N^ (t) = N^ (0) = a^†(0)a^(0) and the frequency of 2 the mode decreases with the photon number increase. (3) n0c χ = n2. (11) The explicit expression for coupling constant g can be 12π2 derived using definition of the nonlinear refractive index 370 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

Substituting (9) into (10), defining characteristic mode volume for the interaction as (it worth noting that the definition of the mode volume in form (12) is general for any nonlinear processes based on cubic nonlinearity [38, 43]) ∫︁ V = |E(r)|4dV, (12) V and noting that the interaction energy is equivalent to the interaction Hamiltonian, −U → V^, if the classical field am- Fig. 4. plitudes are substituted with the normal ordered quantum (a) A nonlinear ring resonator with a single linear coupler (e.g. a coupling prism). (b) A nonlinear ring resonator with two cou- A → a^ A* → a^† |A|4 → a^† 2 a^ 2 operators ( , , ( ) ( ) ), we find plers. that the corrected expression for the coupling coefficient becomes ~ω0c V n2 √︀ √︀ g ω E2 out(t) = 1 − T2 E in(t) + i T2 E1 0(t). (20) = 0 n . (13) 2 V˜n0 V˜ 0 τ Ln c T T T The correction term V/V˜ can be as small as 1/2 for WGMs Here 0 = / is the round trip time, , 1, and 2 are [43]. We also have to modify Eq. (5) as the power transmission coefficients of corresponding couplers (usually wavelength-dependent), Ei are the classical V amplitudes of the electric fields. The field Ein(t) belongs to n = n0 + n2 I, (14) V˜ a single spatial propagating mode, e.g. a Gaussian beam. to keep the consistency of the results. The condition of the spatial mode matching is required to ensure energy conservation. For example, from Eqs. (15) 2 2 2 and (16) we derive |E0(t)| + |Eout(t)| = |Ein(t)| + |E0(t − 2 3.2 Input-Output Formalism τ0)| . Since Eqs. (15,16) and (17-20) are linear, we can expand It is necessary to introduce a coupling between the exter- the classical formalism to the quantum one. Alternatively, nal world and the resonator modes to create a complete the equations can be derived using quantum approach tak- picture of the resonant four-wave mixing. We assume that ing into account interaction of the bound resonator mode the coupling is engineered in such a way that all the pump with modes of electromagnetic vacuum [109], but the re- light can be attenuated in the resonator mode, indepen- sult is essentially the same. dently on the linear refractive index of the resonator host material (for a discussion and review see [34, 110]). Prac- tically speaking, the best coupling efficiency (99.97%) so 3.3 Simplifications and Approximations far was achieved with tapered fibers [111]. The best coupling with prism evanescent field couplers was observed To derive a complete set of equations describing the res- to be 99% [110]. The input-output surfaces of the prism had onant self phase modulation process we have to link the anti-reflection coating. The limitations had purely techni- input-output equations with the Hamilton equations. It cal nature and can be overcome. can be done if we assume that (i) the cavity has high finesse We consider two input-output configurations shown (T ≪ 1); (ii) the intracavity field of the externally pumped in (Fig. 4) assuming the complete spatial matching of the mode can be presented as a product of slow and fast parts mode and the coupling elements. The input-output rela- E0 = E˜0 exp(−iωt), where ω is the carrier frequency of the tion [112, 113] for configuration in (Fig. 4a) is external nearly resonant pump; and (iii) the slow ampli- √ √ tude E˜0 does not change significantly during the round E (t) = i TEin(t) + 1 − TE (t − τ ), (15) τ E˜ t τ ≃ E˜ t τ E˜˙ t 0 √ √ 0 0 trip time 0. In this case 0( − 0√) 0( ) − 0 0( ), ≃ ≃ Eout(t) = 1 − TEin(t) + i TE0(t − τ0); (16) 1 − exp(iωτ0) i(ω0 − ω)τ0 and 1 − T 1 − T/2, so that Eqs. (15) and (16) can be rewritten as and input/output relation for configuration (Fig. 4b) is [︂ ]︂ √ ˙ T T iωt √︀ √︀ E˜0 + + i(ω0 − ω) E˜0 = i Ein e , (21) E1 0(t) = i T1 E1 in(t) + 1 − T1 E2 0(t − τ0), (17) 2τ0 τ0 √︀ √︀ √ E1 out(t) = 1 − T1 E1 in(t) + i T1 E2 0(t − τ0), (18) Eout = Ein + i TE0. (22) √︀ √︀ E2 0(t) = i T2 E2 in(t) + 1 − T2 E1 0(t), (19) A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 371

To rewrite the set of equations in a more conventional medium to sustain itself, unlike conventional lasing. The form we substitute T/(2τ0) = γ0c, where γ0c is the half comb can be produced in a lossy or lossless open system if width at the half maximum (HWHM) of the mode, deter- the CW pump is strong enough. The averaged energy of the mined by the coupling efficiency. Field amplitude Ein be- pump light should be conserved for the case of the lossless longs to the field propagating in the vacuum and mode nonlinear system. In other words, the optical power that matched with the corresponding resonator mode. Expec- enters the resonator should exit it unchanged. This energy tation value for the field is conservation law does not restrict the possibility of power √︂ redistribution among the harmonics. 2πPin iϕin Ein = e , (23) As with any nonlinear process, Kerr comb generation cn0A requires phase matching of the generated frequency har- P ϕ where in is the power of the external pump, in is the monics. Phase matching can be treated as the momentum A phase of the external pump, and is the effective cross- conservation law with respect to the wave vector of the section area of the pumped mode. With this substitutions pump and generated harmonics, and can be described as we obtain an overlap integral among the modes for the case of reso- [︀ ]︀ nant interaction [115]. A˙ + γ0c + i(ω0 − ω) A = F0, (24) The analysis of the generalized multimode hyper- Eout 2γc0 = 1 − A. (25) parametric process is rather complex. We can take ad- Ein F0 vantage of the single mode formalism described in the where √︂ previous sections and present the multimode interaction 2γ0c Pin iϕin F0 = i e , (26) Hamiltonian as ~ω0 ~g Eq. (25) results from Eq. (22), and the field amplitude A is V^ = − (ê†(t))2ê(t)2, (29) 2 normalized to the photon number within the mode, |A|2 = g ⟨N^ ⟩. Eq. (24) is the standard equation describing forced os- where is the coupling parameter obtained under as- cillations of a linear oscillator. This equation is written in sumption of complete space overlap of the resonator ω the rotating wave approximation. modes (13), 0 is the value of the optical frequency of the We have to merge (3) and (24) to describe the nonlinear center of the optical spectrum involved in the interaction c n behavior of the externally pumped mode. It is also worth process, is the speed of light in vacuum, 2 is the cubic ω to take unavoidable intrinsic linear loss of the resonator nonlinearity of the material at the carrier frequency 0, n into account. Using (17-20) and modeling the distributed and 0 is the linear index of refraction of the resonator host ω e internal loss as the second port of the device by substitu- material at frequency 0. The operator ^ is given by the sum of annihilation operators of the electromagnetic field tions T1/(2τ0) = γ0c and T2/(2τ0) = γ0, we obtain for the intracavity field for all the interacting resonator modes that we took into consideration: [︁ 2]︁ ∑︁ A˙ + γ0c + γ0 + i(ω0 − ω) − ig|A| A = F0, (27) ê = a^j . (30) j Eout 2γc = 1 − 0 A. (28) Ein F0 Equations describing the evolution of the field in the resonator modes are generated using Hamiltonian (29) and Eq. (27) takes into account nonlinear frequency shift the input-output formalism. For the case of a single exter- (self-phase modulation) occurring within a single mode nally pumped mode we have of the resonator. The derivation of this equation can be done in a simple and straightforward way if the internal i −iωt a^˙ j = −(γ0 + iωj)a^j + [V^ , a^j] + F0e δj0,j , (31) losses per round trip, linear and nonlinear phase shift ~ are accounted in the relation coupling E0(t) and E0(t − τ) where δj0,j is the Kronecker’s delta; j0 is the number of the (Eqs. (15) and (16)) [114]. externally pumped mode, γ0 = γ0c + γ0i is the half width at the half maximum for the optical modes, assumed to be the same for the all modes involved; and γ0c and γ0i stand 3.4 Kerr Comb Generation as a Multi-mode for coupling and intrinsic loss. The expression for the ex- Hyper-Parametric Process ternal optical pumping is given by Eq. (26). We neglect the quantum effects and do not take into account correspond- Generation of a Kerr frequency comb is a multi-mode non- ing Langevin noise terms. Introduction of these terms is linear process. It does not need an optical amplifying straightforward, though. 372 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

The set (31) should be supplied with an equation de- and also cannot predict stability of the pulses at times scribing the light leaving the resonator (similar to Eq. 28). longer than the ring down time. This problem is solved Assuming the pump power does not depend on time, the if the resonant multimode hyper-parametric process oc- relative amplitude of the output field is given by curring in a nonlinear resonator pumped with monochro- matic light is described using the Lugiato-Lefever equa- êout √︀ √︀ 2γ0c ê = 1 − 2γ0c τ0 − 1 − 2γ0i τ0 , (32) tion [103] (LL, a driven damped nonlinear Schrödinger êin F0 equation with periodic boundary conditions). Coexistence where it is assumed that the field is periodic ê(t−τ0) ≈ ê(t). of pulses and CW background for homogeneous nonlin- This approach, based on a set of ordinary differential ear Schrödinger equation [124, 125] is mathematically not equations (ODE), was introduced in [12] for three modes the same as coexistence of pulses on background for the comprising the pump and two sidebands and generalized damped driven nonlinear Schrödinger equation. The LL in [115]. Numerical simulations of the multimode Kerr fre- equation was introduced initially for spatial transversal quency comb using the set (31) were initially performed dissipative structures. The importance of periodic bound- by Y. Chembo et al. [115–117] and then followed by other ary conditions for correct description of Kerr frequency groups [86, 118–123]. The ODE set is suitable for descrip- combs was highlighted in [77]. This equation is valid at tion of generation of a comparably narrowband frequency times longer than the ring down time. Solutions of this comb for which parameter g is a constant independent on equation have been studied independently for the Kerr the interacting modes. frequency comb, and existence of steady mode locked regimes has been confirmed [62, 127–130]. It gives one grounds to expect that short optical pulses can be formed 3.5 Lugiato-Lefever (Forced Damped directly in the nonlinear microresonator [75–77, 86, 131]. Nonlinear Schrödinger) Equation Optical pulses can emerge by themselves from the nonlinear microresonator pumped with CW light, without any A high finesse optical ring cavity has a long ringdown time. pulse seeding. The pulses are located on top of the CW It means that the field within the cavity changes much background. The power loss is compensated by the exter- slower with respect to its round trip time, and the dynam- nal CW pump and mediated by the nonlinear interaction ics of the intracavity field can be considered as lossless on of the pulses with the background [129]. the time intervals much longer than the round trip time The LL equation for the optical field inside the nonlin- and much shorter than the ringdown time. Within this ap- ear ring resonator characterized with constant GVD reads proximation the externally pumped nonlinear optical cav- [76, 77] ity can be considered as a straight lossless nonlinear opti- ∂A i ∂2A β cal waveguide. The field evolution in such a waveguide can + 2Σ 2 = (33) ∂τ 2τ0 ∂t be studied using the standard 1D Nonlinear Schrödinger 2 [︀ ]︀ ig|A| A − γ0c + γ0 + i(ωj0 − ω) A + F0, Equation used in fiber optics. Since a solution of a lossless nonlinear Schrödinger equation with anomalous group where A(τ, t) is the slowly varying envelope of the elec- velocity dispersion, showing coexistence of short optical tric field, τ is the slow time, t is the retarded time, β2Σ = 2 pulse and continuous wave background [124, 125], has (2ωj0 − ωj0+1 − ωj0−1)(τ0/ωFSR) is the GVD parameter, been derived, it is possible to expect that the lossless opti- ωFSR ≃ (ωj0+1 − ωj0−1)/2 is the free spectral range of the cal cavity can support stable optical pulses that have dura- resonator (expressed here with angular frequency units). tion much shorter than the round trip time. The strength of By definition, time scale τ is much longer than the round this consideration is that it shows, within the boundaries trip time τ0. Eq. (32) can be used to find the amplitude of of the model, the phenomena observed in optical fibers light exiting the resonator. can be directly mapped to the phenomena that can take The ODE mode decomposition approach (31) is equiv- place in nonlinear cavities. Interestingly, even if such a fast alent to the properly formatted LL equation (33). The simi- phenomenon happens within the microcavity, its observa- larity of the approaches was shown in [75]. A proper op- tion using the light leaking from the resonator will be ob- timization of numerical algorithm makes the numerical structed due to the long ringdown time. A different type of evaluation time equivalent for the LL and ODE approaches probing must be developed to trace the fast phenomena, [123]. The possibility of the direct pulse generation in the like rogue wave generation [126]. resonator was demonstrated by the numerical solution of The approach discussed above is deficient since it the ODE set [119]. Analytical solutions of the LL equation does not show a route to the intracavity pulse formation describing the pulses associated with a soliton Kerr comb A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 373 were provided in [13, 86, 131] and technical as well as fun- equivalent straight waveguide. The FWM was realized us- damental quantum noise associated with the repetition ing 1550 nm pump and 1564 nm signal. Both the pump and rate of the pulses was analyzed in [13, 132, 133]. the signal were resonant with corresponding modes of the Each of the theoretical approaches has its own ad- ring. Continuous wave FWM based on bichromatic pump vantages. Hamilton ODEs (31) allow solving the problem was demonstrated in crystalline WGM resonators [100]. of nonlinear interaction of light confined in modes ofa Multiple frequency harmonics were observed in this ex- nonlinear microresonator with an irregular spectrum. This periment. Microresonator FWM with triple-frequency reso- class of problems pertains to the case when one or sev- nant pump was recently demonstrated and studied [142]. It eral modes are shifted from their unperturbed position was shown that increasing the complexity of the pumping because of interaction with other modes of the cavity. A mechanism also adds flexibility in controlling the nonlin- model of such an interaction can be easily incorporated ear process. It is worth noting that four-wave mixing pro- into the ODE set. The ODE approach allows solving prob- cess can combine three pump photons into one resulting lem of nonlinear interaction of selected arbitrary (limited) in third harmonic generation [50] (not described by the number of resonator modes. Hamiltonian formalism presented above). As in FWM with The LL approach allows studying mode locked bichromatic pump, this process does not have a threshold. regimes of Kerr frequency combs. It enables numerical The thresholdless frequency combs created with mod- tracing of the pulse evolution in the resonator, and makes ulation of the pump light are somewhat similar to the fre- possible simplified methods of analytical calculations of quency combs produced in resonant electro-optical mod- the properties of the pulse. The approach is especially con- ulators simultaneously fed with the pump light and an RF venient for studying generation of extremely short pulses signal [143–147]. The repetition rate of these combs is de- [134]. A generalized LL equation that takes into account termined by the modulation frequency. Unlike the pulse higher order dispersion was introduced in [75, 135]. Raman duration in a mode-locked laser, the pulse duration gen- scattering as well as self-steepening terms were added to erated in this system is not limited by the bandwidth of this equation in [136–138]. the laser gain, because the system is passive. The pulse The derivation of the LL as well as the set of Hamil- duration, however, is impacted by GVD of the cavity. The ton ODEs is limited by the mean-field approximation. It is frequency noise associated with the repetition rate can assumed that the overall nonlinear frequency shift of the be even higher than the frequency noise of the RF source cavity modes is much smaller than the cavity free spec- since the amplitude noise of the pump laser can be down- tral range. The more general case of Kerr frequency comb converted to the RF frequency domain. generation involving large optical pumping is analyzed in The resonant FWM can be used for quantum frequency [139] and existence of extremely broad frequency combs conversion [148, 149]. In this process a quantum state of (super cavity solitons) is predicted. light is transferred without destruction from one wavelength to another. High-Q nonlinear microresonator are attractive for this process because of their potentially low at- 4 Resonant Four Wave Mixing and tenuation. The FWM processes based on a bichromatic pump, Multi-Chromatic Pumping as described above, (note that modulated pump usually has more than two harmonics) were studied for the case Resonant Four Wave mixing (FWM) is arguably the sim- of thresholdless operation. Another class of oscillation is plest process occurring in nonlinear resonators. FWM was possible in this system. It starts oscillating when the pump observed and characterized in a WGM structure more than power is increased beyond a certain value. The major sig- 8 two decades ago. CS2 microdroplets with Q ∼ 10 (in- nature of this process is that the generated harmonic fre- ferred from the properties of the bulk liquid, the surface quencies are not always locked to the frequency of the imperfections are not taken into account [140]) were used pumps. The oscillation has many regimes and is able to for this purpose [55]. Droplets were excited as they fell support generation of intracavity pulses [150]. through a 514.5 nm CW argon-ion laser beam. The FWM The impact of pump modulation on comb formation process signal was observed between the pump and the was studied in several other works [88, 101, 151–154]. Stokes Raman line excited in the droplet. These include flat-topped dissipative solitonic pulse gen- A high-Q GaAs micro-ring resonator revealed 14% eration [152] and parametric seedling for stabilization of a FWM conversion efficiency at 10-mW peak pump power conventional Kerr frequency comb [88, 101]. The paramet- level [141]. This efficiency is 26 dB better compared with an ric seedling or injection locking of the frequency combs 374 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators can be used for improvement of their stability as well as herent optical pumping in a media possessing cubic non- for achieving low frequency phase noise. A relevant devel- linearity. Soliton mode locked Kerr frequency comb gen- opment took place in studies of fiber loop cavity solitons eration is a particular example of the hyper-parametric where it was shown that modulated pump can be used to oscillation corresponding to generation of stable tempo- control the position of light pulses [84]. To reduce the fluc- ral optical solitons within the medium. Generalized hyper- tuations of the repetition rate of the frequency comb it was parametric oscillation can have fixed instantaneous or suggested to take advantage of the injection locking prop- long term phase difference among the harmonics but those erties of the Kerr frequency comb oscillator and create an harmonics not necessarily constitute an optical pulse in opto-electronic oscillator that involves an active Kerr comb the time domain. We distinguish the variety of types of oscillator as a part of the optical loop of the opto-electronic hyper-parametric oscillation, described in this section, circuit [155]. from the soliton mode locked Kerr frequency comb generation described in the next section, since generation of short optical pulses out of a CW pump resonator that does 5 Hyper-parametric oscillation not have any conventional mode locker is a remarkable phenomenon. In what follows we will mention hard and soft exci- Hyper-parametric oscillation is one of the first nonlinear tation regimes. The soft oscillation onset is reached when processes observed in microresonators characterized with pump photons are not present in the resonator modes ini- cubic nonlinearity (Fig. 5). Historically, the processes that tially. The growth of the oscillation sidebands occurs adi- involve only a few pronounced generated frequency har- abatically in this case. The hard onset occurs with a dis- monics versus many harmonics have been dabbed the os- continuous jump of the intensity of oscillation sidebands cillation versus frequency comb generation. This is not to a certain finite level. The hard excitation occurs only right, in our opinion, since there is no pronounced phys- when pump photons are initially present in the resonator ical boundary between the oscillation and the frequency mode(s). The steady state solution corresponding to the comb generation. hard excitation cannot be reached adiabatically. In other Similarly to the frequency combs, the oscillation has words, slow modifications of any parameter of the res- both hard and soft excitation regimes. It can be coherent onator or the pump cannot bring the system to a stable so- and chaotic. It can be excited in resonators with normal lution requiring hard exitation. Here slow stands for time and anomalous GVD. Frequently, the regimes of the hyper- scale longer than typical time of the comb growth, which is parametric oscillation and the frequency comb generation longer compared to lifetime of the light confined in the res- are observed in the same resonators when the pump power onator. The hard excitation can also be achieved by nona- or loading is adiabatically modified. There is not interrup- diabatic change of the oscillator parameters. For instance, tions between the processes. change of the pump frequency faster than the resonator The oscillation has a lot in common with modulation ring down time. instability lasers [67–69], type-I and type-II Kerr frequency combs [156], mode-crossing based frequency comb [120], and Turing rolls (patterns) [91, 157]. The frequency combs 5.1 Anomalous GVD acssociated with these processes have a relatively small number of frequency harmonics and a triangular (in log 5.1.1 Sequential growth of the frequency comb scale) power spectrum. However, many frequency comb generators that have a larger number of harmonics as well A common feature of a class of hyper parametric as a diffrent envelope shape behave similarly to the os- oscillation-based processes, occurring in resonators char- cillators. We will talk about hyper-parametric oscillations acterized with anomalous GVD, is the dependence of dy- in this section. The "hyper-parametric" term [158] was se- namics of each higher order harmonic on the lower order lected to distinguish the four-photon process from the ones. It means that if one evaluates the temporal dynam- three photon parametric oscillation. Still, the term "para- ics of growth of the hyper-parametric oscillation-based metric" is frequently used to describe oscillation in optical frequency comb, one will observe that the higher order fiber as well as photonic waveguide systems [159]. harmonics emerge from the noise later than the lower or- For the sake of clarity we define the hyper-parametric der ones; and that the growth time constant is larger for oscillation as a nonlinear process comprising generation the higher order harmonics, so they reach the steady state of equally separated frequency harmonics out of CW co- A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 375

(a)

-1

10 3

-2 4

-3

10 amplitude

-4 Normalized

-12 -8 -4 0 4 8 12

Mode number

0 1 (b)

-10

4 -20

-30

-40

0 10 20 30 40 50 Relative amplitude of the sidebands, dB

Normalized time, t

Fig. 6. Sequential growth can occur with both phase- and mode locked frequency combs. A numerical simulation characterizing growth of the frequency comb based on a hyper-parametric oscillation with soft excitation [160]. (a) Steady state frequency spectrum of a hyper-parametric oscillation based Kerr frequency comb generated in a WGM resonator. (b) Time dependence of the normalized amplitudes of the first five harmonics of the optical frequency comb Fig. 5. An example of a hyper-parametric oscillation frequency spec- generated in the WGM resonator. The growth of the harmonics con- mm ≃ 9 trum observed in a ∅2.6 CaF2 Q 10 WGM resonator pumped firms cascaded FWM process [161]. with continuous wave 1549.7 nm 8 mW light. Simultaneous generation of photon pairs (c and d) out of two pump photons (a and b) constitutes degenerate four wave mixing. The process represents Initial observations of both narrow [48, 49] and broad an all-optical oscillation. The peak (e) occurs because of the ampli- frequency combs [11, 165] can be considered as a hyper- fied spontaneous emission of the optical amplifier utilized inthe parametric oscillation process if we adopt the criterion of experiment. smooth growth of the high order harmonics from the lower order ones to be a feature of the oscillator. Such process- approximately at the same time as the lower order ones eswere discovered initially resulting in "fractal" growth (Fig. 6). theory of uniform comb formation in resonator character- This particular type of hyper-parametric oscillation is ized with relatively small anomalous GVD [166]. It was as- characterized with soft excitation regime [119] as well as sumed that in the first stage a primary frequency comb high dynamic stability [91] and is analogous to the op- forms (hyper-parametric oscillation starts). At the second tical fiber-based processes. It was known for some time stage each primary harmonic form its own secondary, sec- that propagation of CW light in a nonlinear anomalous ondary, comb. These secondary combs were considered in- GVD medium becomes unstable at certain power [162– dependent with slightly different repetition rate. Finally, 164]. The (modulation) instability transforms the CW light at the third stage, all the secondary harmonics phase lock into a tightly packed train of optical pulses. The instabil- due to the cross-modulation effect. The locking occurs if ity occurs for both positive and negative GVD if the CW the primary spacing is nearly an integer multiple of the light is confined in a fiber ring [67–69]. It was notedin secondary spacing. If this is not the case, a chaotic [167] or [70] that the modulational instability laser is really a fiber partially chaotic [168] regimes can occur instead of phase optical parametric oscillator, in which pump, signal, and locking. It is possible to use pump power and the pump idler waves are resonated inside a fiber ring cavity. One laser frequency as levers to adjust the repetition rates of needs several Watts of power to excite modulation insta- the frequency combs and achieve phase locking. bility in a straight fiber. A fiber resonator with high enough To form a mode locked comb (mode locking was in- finesse becomes unstable at hundred milliwatts [69]. The ferred from the envelope shape while phase locking was generated spectrum is relatively narrow (it does not exceed confirmed by generation of a spectrally pure RF signal by a THz), which adds to the similarity between observed the comb on a photodiode) one could either increase the hyper-parametric oscillation in microcavities and the fiber pump power or increase the resonator Q-factor [169]. The systems. reported process of the comb transformation does not have a threshold and is similar in both cases. In the case of 376 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators the increasing Q-factor two parameters change: intracav- The model describing Kerr comb formation via pri- ity power as well as effective GVD of the resonator. Fig- mary and secondary comb excitation may be mimicked by ure 7 illustrates development of such a broad frequency the process related to linear interaction of different mode comb [169, 170]. In this experiment the detuning of the fre- families of the same resonator [120, 171–173]. Even a single quency of the pumping light from the corresponding op- mode resonator contains two orthogonally polarized mode tical resonance was kept as small as possible, though this families that can interact due to imperfectness of the res- parameter was not directly controlled. It was not measured onator shape as well as material inhomogeneity [174]. Mul- if the laser was parked at the blue or red slope of the res- timode monolithic cavities usually have many mode fami- onance. No comb was observed when the coupling prism lies interaction of which results in a significant deteriora- touched the MgF2 WGM resonator surface and the loaded tion of the resonator spectrum to the level of unpredictabil- Q-factor became too small to support the comb generation. ity, as was observed experimentally [175]. The loading was then decreased until hyper-parametric The process, similar to the excitation of the primary oscillation started and several spectral lines of the pri- frequency comb, is observed in the case of sufficiently mary comb separated by almost a THz appeared. A fur- strong interaction between two mode families having rel- ther decrease of the resonator loading resulted in the re- atively large difference between their FSRs [176]. Such an duction of the repetition rate of the frequency harmonics interaction results in a nearly periodic frequency depen- (GVD increases) and creation of higher density secondary dent pattern of GVD occurring due to the Vernier effect. As combs (as the power increases). After stabilization of the a result, certain modes have the lowest excitation thresh- resonator loading, adjustment of the semiconductor laser old. They support generation of the lowest threshold ("pri- current produced stable Kerr frequency combs with desir- mary") frequency comb when the pump power exceeds able repetition rates and spectral shapes. this threshold. Further increase of the power results in generation of the secondary frequency combs having slightly different repetition rates. These secondary combs can be

-20 (a) phase locked via mutual injection as was confirmed by ex-

-40 periments [177].

-60 In our opinion, the process of formation of the hyper- parametric oscillation-based frequency combs is more

1535 1540 1545 1550 likely to occur due to mode crossings then due to the dy-

-20 (b) namics of light confined in an ideal resonator characterized with cubic nonlinearity and small anomalous GVD. -40 While in our numerical simulation we observed forma-

-60 tion of the primary comb in the case of the ideal (with no mode interactions) resonator, we did not observe forma- 1535 1540 1545 1550

Optical power, dBm tion of the independent coherent secondary comb clusters

-20 (c) that synchronize due to mutual injection locking. Instead, -40 the primary comb tends to fill-up with secondary sideband

-60 and become incoherent if the power of the pump increases or the frequency detuning value decreases. Additionally, 1535 1540 1545 1550 the theoretical model does not explain forming complex

W avelength, nm comb envelope patters observed in experiments [176–178]. The first observed Kerr frequency comb, obtained in Fig. 7. Experimentally observed growth of the frequency comb gen- an on-chip microtoroidal fused silica cavity [11], was most erated in the MgF2 resonator (FSR=34.67 GHz) pumped with a constant power CW light. The loading of the resonator decreases (the likely impacted by the mode GVD modification due to the Q-factor increases) from (a) to (c). The comb with higher repetition mode crossings in the resonator. The frequency comb was rate (hyper-parametric oscillation) is generated in the overloaded definitely phase locked, because of the measured value of resonator. Increase of the loaded Q-factor results in filling the gaps the stability of the comb lines; however its envelope did between the modes and eventually results in locking of all the har- not resemble the envelope of the soliton comb. The same monics. The figure is adapted from [169]. consideration applies to the observations of phase locking of the comb harmonics [156, 166, 179]. The envelope shape of this type of frequency comb is significantly different from the shape of the envelope of a mode locked A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 377

(soliton) frequency comb. Introduction of mode crossings seems to be important to explain these processes and a further theoretical study is needed to understand this process completely.

A hyper-parametric oscillation can involve several Power, dBm mode families at the same time. One of examples of this 0 process is generation of a multi-octave frequency spec- -10 trum in a crystalline microresonator [96]. The intrinsic -20 GVD of the resonator is too high to allow such an oscilla- -30 tion within the same mode family and the nonlinear pro- -40 cess takes place via overlapping modes belonging to dif- -50 ferent families. Such an interaction leads to generation of -60 -70 bizarre and seemingly irregular spectra (Fig. 8). It is pos- 350 400 450 500 550 600 650 700 sible to show that each component of the spectra can be Wavelength, nm explained by nonlinear mixing among frequency harmonics of several primary hyper-parametric oscillations as well as stimulated Raman scattering component. Interestingly, such processes are usually observable in normal GVD resonators, as they have lower threshold if compared with threshold of generation of normal GVD frequency combs. Fig. 8. Hyper-parametric oscillation in a normal GVD microres- A visualisation of the spectral lines of the broad onator. A GVD distortion resulting from the linear mode crosstalk frequency comb performed with a diffraction grating is results in favorable conditions for hyper-parametric oscillations shown at the pictures at the top and at the bottom pan- within a very wide wavelength range. The experiment involves a 780 nm pump light coupled into a MgF2 microresonator. The pump els of (Fig. 8). Strong 780 nm laser pump is visible near produces sets of several tone pairs extending to the UV region. the left edge of each picture as a small brownish oval. Its The blue wing of the spectra is shown in the middle of the illus- intensity appears weak due to low camera sensitivity at in- tration. The step-like spectral feature at 475nm is a feature of the frared part of spectra. Bright spots of different color from optical spectrum analyzer. The wide line at 515 nm is a feature of the deep red to emerald belong to comb components at visi- pump DFB laser. The span of such a Kerr-comb exceeds two octaves. The top and bottom panels illustrate different realizations of the ble part of the blue wing of the Kerr comb. Note that some process. Tuning between the regimes is achieved by changing the comb components are produced at transverse modes since temperature of the resonator. the modes far field has two maxima along vertical direction. Comb components of wavelengths longer than deep red are invisible since the sensitivity of the camera used in in resonators impacted by mode crossing leading to local these measurements drops very fast into infrared. disruption of the resonator GVD. To reach the hard excitation regime experimentally one needs to abruptly change either the power or the fre- 5.1.2 Hard excitation of hyper parametric oscillations quency of the pump laser. This technique works for both Kerr frequency comb as well as the hyper parametric os- Hyper-parametric oscillation occurring in the resonators cillation excitation. One of the standard ways of achieving characterized with anomalous GVD usually has soft exci- the regime is scanning the pump laser frequency through tation and sequential growth, as discussed above. Hard the chaotic regime (which is usually available even at excitation is usually attributed to the generation of mode certain frequencies in a set of a few modes) fast enough locked frequency combs discussed in the next section. in comparison with the resonator ring down time [167]. However, the hard excitation regime is also possible in the Non-adiabatic pump power change also can be utilized case of a hyper-parametric oscillation that involves a few [181, 182]. This allows collecting some energy in the res- modes under conditions of anomalous GVD [180]. No short onator modes needed to reach the hard excitation regime. pulses are generated in this case. To observe these regimes It is also possible in some cases to switch between various experimentally one needs to consider a resonator having oscillation regimes avoiding the chaotic and unstable re- a few modes available for the four wave mixing. The other gions by adiabatic changing the parameters of the pump modes should be eliminated in some way. This is possible light [183]. 378 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

5.2 Normal GVD

It is widely believed that anomalous GVD is needed to achieve modulation instability in fibers. However, it was b shown that if the fiber is used to create a ring cavity, the c modulation instability occurs in the case of normal GVD as 0 well [67]. The reason for this is the availability of frequency -10 detuning of the pump laser frequency from the mode frequency as an additional knob to compensate for the non- -20 d linear cross/self modulation effects [68]. The analysis of a the instability involving three modes only: the pump and -30 the two sidebands. While this model is oversimplified, it -40 gives a lot of insight for generation of both narrow and dBm Power, broad frequency combs. -50 It was shown that the hyper-parametric oscillation can occur at any GVD value for the case of a microresonator -60 in only a few modes are involved. Similarly to the fibers, 1547.3 1547.8 1548.3 1548.8 formation of only two symmetric sidebands with respect Wavelength, nm to the carrier were initially considered [12, 180]. It was assumed that the hard oscillation onset always takes place in Fig. 9. Spectrum of a frequency comb generated due to interac- the case of normal GVD. A more detailed study has shown tion among different mode families in 2a CaF WGM resonator. This that the normal GVD hyper parametric oscillator also has frequency comb is not supported by the properties of an ideal res- soft excitation regime within some specific range of pa- onator having no mode crossing. Intracavity avoided mode crossing distorts local FSR of the resonator by as much as 100 MHz. The res- rameters [184]. onator is depicted at top left corner. Distortion of local dispersion The first observation of the low repetition rate Kerr compensates for practically any material and geometrical comb- frequency combs in CaF2 resonators was based on mode unfavorable dispersion offset and allows for comb formation within crossings and had the standard development mechanism a narrow range of modified dispersion. The asymmetry of the spec- for the combs of this kind (Fig. 9) [165]. Initially a primary trum is a signature of the mode interaction. comb emerged, and then a broad frequency comb with irregular envelope was generated. The repetition rate of the Kerr frequency comb generation is the natural formation primary comb was determined by the interaction between of high-contrast optical pulses within the microresonator, different families of the WGMs. The signatures of mode pumped solely with continuous wave light. Generic hyper- crossings can be observed at the optical spectrum of the parametric oscillation-mediated frequency combs corre- comb (Fig. 10). The particular resonator had 2.55 mm in di- spond to low contrast pulses [91] with duration (and sepa- ameter and was pumped with 1550 nm light, which cor- ration) comparable with the round trip time of the light in 2 responds to normal GVD β2 ≃ 2.3 ps /km [165]. Simi- the resonator. lar results were obtained with a bigger CaF2 resonator. A The mode locked frequency comb pulse formation is Kerr comb was observed in a CaF2 0.78 cm diameter WGM generally associated with the hard excitation regime [119]. resonator, pumped with 1320 nm light, and characterized The pulses appear as unity objects; they cannot be con- 2 with β2 ≃ 7.7 ps /km normal GVD [114]. As shown below, sidered from the perspective of a cascading FWM process. a truly mode locked frequency comb generated in a normal Each pulse represents a dynamic supermode of the nonlin- GVD resonator has a significantly different envelope. ear multi-mode structure (Fig. 11). We found that that soft excitation of mode locked Kerr frequency comb is possible when GVD of the resonator is comparably large and pump 6 Mode Locked Frequency Combs power is relatively small. Discussion of this particular case will be presented elsewhere. As we mentioned above, mode locked Kerr frequency comb generation is an example of a hyper-parametric process. The the major difference between conventional hyper- parametric oscillation-mediated process and mode locked A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 379

-20

-40

-60 b c 0 1500 1520 1540 1560 1580 1600

-30 -10

-40 Power, dBm Power,

-50 -20 -60 a -70 -30 1538 1539 1540 d

W avelength, nm -40

Fig. 10. Frequency comb generated in a CaF2 WGM resonator -50 pumped with 50 mW of power. Presence of the frequency harmonics Power, dBm occurring within a single comb line due to mode crossings is clearly -60 seen. The lines affected by the mode crossings have higher power compared with the other lines. The experiment is described in [165]. -70 1530 1540 1550 1560 Wavelength, nm 6.1 Importance of small cavities in mode locked lasers Fig. 11. Components of a coherent wide comb are strongly coupled to each other. They hardly constitute independent OPO pairs. In- It was recognized long ago that short optical cavities are stead, all components of the frequency comb structure behave as a single supermode. The comb emerges spontaneously and cannot be important for stable generation of optical pulses with high explained by a sequential generation process. repetition rates. The dense mode spectrum and nonlinearity of long fiber ring cavities result in various kinds of undesirable instabilities. For instance, an optical soliton and ii) the pulse duration is shorter than the inverse cavity becomes unstable as the soliton-laser cavity approaches FSR. The first regime is studied in [192–195]. The second the length of several soliton periods [185, 186]. Long cav- regime dealing with propagation of short pulses in WGM ity based harmonically mode-locked lasers suffer from the cavities, which is more relevant to generation of mode supermode noise [187]. locked Kerr frequency combs, was examined in [196–200]. Application of short cavities solve most of these prob- Time resolved measurements of picosecond optical pulses lems enabling generation of high repetition rate opti- propagating in dielectric spheres [197] and subpicosecond cal pulse trains. For example, 2 ps pulses at a 16.3 GHz terahertz pulse propagation in a dielectric cylinder [198] repetition rate were generated with a 0.25 cm-long ac- were reported, and microcavity internal fields created by tively mode-locked monolithic laser [188]; 420 GHz subpicosecond pulses were discussed theoretically [199]. Gen- harmonic synchronous mode locking was realized in a eral theoretical analysis of the propagation was presented laser cavity of total length approximately 174 µm [189]. and the existence of solitons in microresonators was pre- A significant supermode noise suppression was demon- dicted before the mode locked frequency combs emerged strated by inserting a small high-finesse Fabry-Perot res- [196]. Usage of active mode locking mechanism was sug- onator to the cavity of an actively mode-locked laser [190, gested to excite the solitons [200]. 191]. Even though the possibility of stable propagation of short optical pulses in a microresonator was predicted independently, the generation of Kerr comb solitons was a 6.2 Pulse propagation in WGM resonators new development since the solitons emerged on a DC background that compensated for the unavoidable attenuation Pulse propagation in WGM resonators has been studied of the pulses. This is rather different than the case of pas- more recently, but still before the development of the mode sive stable propagation of the pulses in the resonator. locked Kerr frequency comb oscillator. It is reasonable to distinguish between two regimes of optical pulse propagation in a microresonator: i) the pulse duration exceeds the inverse of the free spectral range (FSR) of the cavity, 380 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

6.3 Soliton Kerr combs plete understanding of the coherence properties of a Kerr frequency comb is also as yet out of reach. The first demonstration of the soliton mode locked regime While discussing practically important features of in a microresonator was reported in [86]. The pulses were Kerr optical frequency comb generators it is important to generated in a MgF2 resonator having 35.2 GHz FSR and compare them with other existing mode locked lasers. The 450 kHz loaded bandwidth and clearly measured using major advantage of the Kerr frequency comb oscillators frequency-resolved optical gating (FROG) technique. Ex- is related to its small size and low power consumption. istence of the soliton mode locked regime was indepen- The microresonator can be tightly integrated with a semi- dently confirmed for SiN [201] and fused silica [182] res- conductor laser chip to create a fully functioning turn-key onators. A fully coherent 2/3 octave frequency comb was device having less than a cubic centimeter volume and observed in a SiN microresonator [181, 202]. The optical consuming less than a Watt of power (where 90% of the spectra resembling soliton formation were observed as power consumption goes to the electronic drivers as well well [203, 204]. as temperature stabilization of the microresonator). One of While anomalous GVD is considered as optimal for major feature allowing achievement of this tight integra- soliton Kerr combs, normal GVD resonators can sup- tion is related a possibility of self-injection locking of the port generation of dark solitons without involvement laser to the microresonator generating the frequency comb of any mode locking mechanism or GVD interruptions. (Fig. 12). This phenomenon was predicted [121, 184, 205, 206] and demonstrated experimentally [92]. Both narrow (hyper- Power, dBm parametric oscillation like) [207, 208] and broad Kerr fre- -20 quency combs were demonstrated under conditions of -30 normal GVD [92, 205] and no visible interaction of the -40 resonator spectrum due to mode crossings. The existence -50 domain of the stable normal GVD mode locked combs -60 is rather narrow [121, 209] and a different type of stable -70 pulses can exist in this case provided a possibility of small -80 interruption of the mode spectrum due to the resonator -90 mode families anti-crossing [152, 153]. It was noticed re- 1533 1553 1573 cently, that the dark pulses are related to so-called switch- Wavelength, nm ing waves [210–212]. a b c 6.4 Unsolved problems d There are many unsolved problems related to Kerr frequency combs. The low efficiency of the nonlinear process resulting in the mode locked Kerr comb is one of them [137]. The efficiency, defined as the ratio of the pumppower Fig. 12. Self-injection locked Kerr frequency comb oscillator. Self- and averaged pulse train power, degrades with growth of injection locking of a DFB laser (a) to a whispering gallery mode the comb spectral width, and is inversely proportional to (d) is achieved through resonant intracavity backscattering. The the number of comb lines. A solution has to be developed emission of the laser is evanescently coupled into the whispering to circumvent this restriction to enable generation of oc- gallery cavity via a coupling prism (c) and a collimating lens (b). This tave spanning Kerr frequency combs with a relatively low locking technique allows for tight laser-mode frequency detuning control and achieving a very narrow laser line. An example of a repetition rate. Other problems include generation of a packaged all-optical self injection locked Kerr comb generator is coherent octave spanning Kerr frequency comb, realiza- shown at the picture above (top, right). Unlike thermal locking self- tion of self-stabilization of the comb, generation of fre- injection locking is supported at both at blue and red slopes of quency combs in visible and ultra violet parts of optical the mode. As a result, coherent combs with both CaF2 and MgF2 spectrum (mid-IR frequency combs were recently demon- resonators were achieved with the laser self-injection locked to the strated [213, 214]), as well as complete integration of the mode that generates the comb [96]. An example of the frequency spectrum of a coherent comb produced by the described technique comb generator with the pump laser on a chip. The com- is shown at the chart above (top, left). A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 381

The packaged microresonator comb [204] has less ist a large variety of the time-dependent solitons. Some of than 10 nm 3 dB spectral width, which is broad enough them change symmetrically with respect to the pulse cen- for some applications. The major unsolved problem is re- ter, some – not. The last case is observed with high order lated to generation of a coherent frequency comb covering pulses [206]. an octave in optical frequency domain and still consuming less than a Watt of power (currently existing broad comb generators consume more than a Watt of optical power 6.5.2 Akhmediev breathers which precludes their integration on a chip). The octave spanning is needed for the f-2f self locking of the frequency The observation of a KM soliton does not depend on the comb [5]. geometrical position of the observer with respect to the resonator rim. In contrast, the AB solution is localized in space and, hence, if a 1D resonant analogy of the AB soli- 6.5 Breathers ton exists, its observation should be dependent on the geometrical position of the observer. This is possible if a stand- The suggestion [75, 76] that the Kerr comb generation can ing wave is excited in the cavity. Similar conclusion is re- be analyzed using a driven damped nonlinear Schrödinger lated to the study of ultrashort rogue optical waves in the equation or the Lugiato-Lefever equation (LL) has made CW pumped microresonators [126]. Again, the LL formal- feasible application of numerous results of earlier stud- ism is suitable for description of these processes if the time ies of this equation, as well as the insight provided by scale is long enough in comparison with the round trip the study of physical systems described by this equation time. Therefore, the time scale of the rogue wave formation to the optical microcavity case. Among others, LL equa- should be much slower than the round trip time. tion has a number of dynamic solutions including the so Breathers have not been directly observed in microres- called Kuznetsov-Ma (KM) solitons [215–217] and Akhme- onators as yet. However, their existence was validated ex- diev breathers (AB) [218, 219]. These solutions correspond perimentally using long optical fiber cavities [225]. The to time dependent envelope pulses formed on a finite back- observed breathing period was approximately ten round ground plain wave and belong to a general class of time de- trip times, which corresponds to the numerical simula- pendent dissipative solitons that exist in various physical tions. The breathers should be studied deeper for the systems [220, 221]. case of microresonator-based Kerr comb generators and proper experiments should be devised to observe the phenomenon. 6.5.1 Kuznetsov-Ma solitons

Both KM- and AB-like soliton solutions can formally ex- 6.5.3 Chaotic solutions ist in a CW-pumped high finesse nonlinear ring microresonator [121]. KM solitons correspond to an optical pulse In addition to breathers the LL equation brings to life that propagates in the resonator, where the envelope of chaotic solutions. The dynamic chaos was demonstrated the pulse periodically changes at a time scale much longer experimentally by every group working in the field. The compared with the round trip time. The corresponding first observation of an octave spanning frequency comb spectral width of the frequency comb forming the pulse suggested the possibility of chaos, since the linewidth slowly varies in time, as well. Such a behavior is known to of each comb harmonic was measured to exceed both be an example of Fermi-Pasta-Ulam recurrence [219, 222– the pump laser linewidth and the optical mode band- 224] corresponding to counterintuitive periodic behavior width [226]. The first concize demonstration of the chaotic of a complex nonlinear system with many degrees of free- regime in a crystalline WGM resonator as well as a discus- dom. It fits well the assumptions of the mathematical sion of its importance to reach the hard excitation regimes model of the Kerr combs. Indeed, by derivation, the LL of the soliton combs was presented in [167]. equation is valid for the study of a processes that is slow compared with the round trip time of the resonator. Both bright [121] and dark [206] KM breathers exist. The bright breathers are observed for the case of anomalous GVD, while dark breathers are observed in the normal GVD resonators. Depending on initial conditions, there ex- 382 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

7 Dispersion Engineering interference pattern at the resonator surface [230]. Similar pattern formation was achieved with microtoroids [231]. Such an interaction of co-propagating modes can be used The resonant enhancement of frequency conversion pro- to correct local GVD of a resonator to achieve a desirable cesses is the most efficient when both the pumping light regime of the Kerr frequency comb generation. and the light generated due to the nonlinear interaction Importance of mode crossing for Kerr comb genera- are confined in corresponding WGM modes, and con- tion was discovered [120] and confirmed by multiple stud- versely, the nonlinear process can be suppressed com- ies [171–176, 232]. Some of them assert usefulness of mode pletely if there is no WGM mode to support the light that crossings (see, e.g., [120]), while others claim their detri- would be produced by the nonlinear interaction. For in- mental role for generation of the Kerr frequency comb (see, stance, stimulated Brillouin scattering is observed either e.g., [175]). In our opinion, both statements are correct. in low-Q WGM resonators or in WGM resonators specifi- The local dispersion modification due to mode cross- cally made to observe the effect. SBS is not observed in ing results in achievement of soft excitation regime of the high-Q resonators with sparse spectra where there is no oscillation characterized with excitation of the fundamen- WGM that could support the Stokes component of light. tal (harmonics separated by single FSR) frequency comb. Kerr frequency comb generation is less demanding on This regime can be achieved in resonators characterized phase matching. Still, to achieve generation of a bright with both normal and anomalous GVD. It is possible to soliton mode locked comb one needs to create a mode fam- generate frequency combs with harmonics separated by ily with anomalous GVD having nearly a flat wavelength single, double, triple, etc. FSR separation in the same res- dependence. Since a WGM spectrum can be very irregu- onator by a controllable modification of the resonator tem- lar, especially in larger resonators, finding the conditions perature or pump power. Thermal dependence of the local when the nonlinear process is resonantly phase and mode GVD in birefringent resonators like those made with MgF matched is one of the most challenging tasks related to the 2 allows controlling the local GVD value in real time and observation of the resonantly enhanced nonlinear interac- can be used for Kerr comb stabilization [233]. It is some- tions. Fortunately, the WGM spectrum can be engineered times enough to change the resonator temperature by a and a desirable pattern of WGM can be created by chang- few mK to suppress comb generation. The generation is re- ing the shape of the resonator and the refractive index stored if the temperature is restored. Local mode coupling distribution of the resonator host material. In the follow- helps to ignite mode locked regimes in single normal GVD ing sections we overview some known correction mecha- ring resonators [201]. It is also possible to use a set of cou- nisms. pled resonators to control the GVD using the mode interaction principle. In this way Kerr comb generation, repe- 7.1 Local correction tition rate selection, and mode locking are achieved with coupled SiN microrings controlled via an on-chip micro- heater [234]. The frequency combs generated in this way Most microresonators contain many mode families. These have relatively narrow optical spectra but are character- mode families are independent in ideal devices. However, ized with high stability and high relative coherence of the imperfection of the resonator host material, surface, and frequency harmonics. Therefore, artificial introduction of morphology lifts the orthogonality of the resonator modes mode crossings can be considered as a tool for local correc- by breaking the resonator symmetry. Presence of sub- tion of the resonator GVD and is helpful for Kerr frequency wavelength inhomogeneities results in resonant Rayleigh comb generation. scattering and strong coupling of clockwise and counter On the other hand, avoided mode crossings, occurring clockwise modes [43, 227], while large scale deviations re- due to linear mode coupling, prevent soliton formation sults in avoided crossings of modes belonging to differ- when affecting resonator modes in the vicinity of the pump ent mode families. The frequency pulling magnitude was laser frequency [175]. These crossings disrupt growth of measured to reach gigahertz scale in crystalline resonators the soliton frequency comb. They also introduce uncer- [228, 229]. tainty to the GVD value in a multimode resonator [120]. A clear visual demonstration of the mode cross- This "noise" prohibits practical creation of multimode mi- coupling of co-propagating modes was performed with a croresonators having a single mode family with very small silica microsphere immersed into a dye solution. Resonant GVD value, which is frequently required for generation of frequencies of several cavity modes running in the same octave spanning soliton Kerr frequency combs. direction were simultaneously excited and formed a static A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators Ë 383

An example of highly coherent Kerr frequency comb protrusions on a cylindrical preform [239, 240]. In general, generation demonstrated in CaF2 resonator with 9.9 GHz suggested methods of global GVD manipulation for Kerr FSR is shown in (Fig. 13) [235]. The spectral patterns of frequency comb generation include i) selection of differ- the combs hint that a cluster of modes is formed in the ent mode families for dispersion optimization [241–243], resonator. The resonator has nearly zero normal GVD and ii) modification of the resonator morphology [173, 203, 232, generation of this type of a stable frequency combs looks 244–252], and iii) development of composite resonators or unlikely there without admitting the mode crossing hy- resonators with spatially inhomogeneous material proper- pothesis. ties [134, 209, 246, 253–257]. An important dispersion correction mechanism was

discovered in SiN microrings. It was found that the wave- -20

(a) (b) length dependence of the Q-factors of the modes partici- -30 pating in the Kerr frequency comb formation acts as a fast -40

saturable absorber and can be used for mode locking and -50 generation of bright solitons in resonators characterized

-60 with normal GVD [201].

Optical power, dBm power, Optical -70

1559 1560 1561 1559 1560 1561 Wavelength dependent gain, e.g. Raman gain, can

W avelength, nm also initiate mode locked Kerr frequency combs [258]. Low threshold stimulated Raman scattering (SRS) was ob- Fig. 13. Frequency comb generated in a CaF2 WGM resonator characterized with 9.9 GHz FSR and pumped with CW light. Panels (a) served in microresonators before the emergency of the and (b) stand for different comb patterns observed in the resonator. Kerr frequency comb [56]. This process does not require The laser is self-injection locked to the resonator mode (this is the any phase matching of pump and signal and frequently first demonstration of usefulness of self-injection locking for the competes with Kerr frequency comb generation [12, 169, frequency comb generation). The experiment is described in [235]. 232, 259]. Cascaded mixture of multiple FWM with SRS frequently creates comb-like spectra around the carrier and SRS tone. However, if wave mixing is not supported by the modal structure of the resonator, it is possible to 7.2 Global correction generate a mode locked SRS frequency comb that does not produce significant sidebands around the pump fre- The total resonator GVD consists of the material and geo- quency (Fig. 14) [258]. Similarly to generation of multi- metric contributions. While it is hard to change the disper- octave hyper-parametric oscillation [96], the mode-locked sion of the material, it is possible to change the morphol- SRS is observed when the pumping is performed at nor- ogy of the system and achieve the desirable GVD. This was mal GVD where the generation of the conventional Kerr fre- realized with respect to fiber optics four decades ago. For quency comb has a higher threshold. instance, phase-matched nonlinear mixing in an optical fiber was achieved using the dispersion of the waveguide modes to compensate for the dispersion of the material 8 Conclusion [93–95]. More recently it was found that micro-structuring of waveguides can provide control over the GVD to improve We reviewed some aspects of hyper-parametric oscillation supercontinuum generation [236, 237]. It is reasonable to as well as Kerr frequency comb generation from the his- expect that similar techniques should work for the case of torical perspective and traced the path made by discover- microresonators. ies during the last decade. While this review is not com- Mode crossing alters the local GVD of the resonator plete by any means, it reflects the major developments in allowing generation of spectrally narrow Kerr frequency the field and shows the great potential for further studies combs. Further extension of the comb envelope requires in the area to address many remaining unanswered ques- a cross-talk-free mode family within a very broad range tions. A discussion of the applications of the frequency of wavelength as well as flat anomalous GVD. To perform combs is left beyond the scope of the paper. However, we global GVD modification different techniques can be ap- believe that the development of chip-scale devices utiliz- plied. It is possible to reduce the modal density and crossing the frequency combs to their full capacity will become coupling by changing the resonator shape, e.g. reducing the major driver for the research of the microresonator Kerr the mode volume by making oblate toroidal resonators frequency combs in the years to come. [42, 238] or by shaping the resonators as a low contrast 384 Ë A. A. Savchenkov et al., On Frequency Combs in Monolithic Resonators

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