Annotation of the results obtained in 2019
The scientific team led by V.E. Lobanov continued in 2019 a research on the grant of the Russian Science Foundation “New methods for generating broadband coherent optical frequency combs in microresonators”. 1) In the course of the project, a theory was developed that describes the effect of coherent (resonant) pumping on the formation of azimuthally modulated traveling / rotating modes in polariton ring microresonators in the presence of losses. The constructed theory was applied in the analysis of the excitation of rotating soliton structures by two-frequency coherent pumping, which is realized using structured laser beams, each of which can have a nontrivial phase distribution, i.e. carry a vortex characterized by a topological charge. It was found that simultaneous pumping at two frequencies leads to the formation of rotating structures, whose angular frequency of rotation is determined simultaneously by the frequencies and topological charges of the pump beams. The direction of rotation is also determined by the topological charges of the pump beams. The rotation frequency and effective energy corresponding to the stationary rotating modes were obtained analytically. A wide class of nonlinear rotating modes with various symmetries and nontrivial phase distributions was discovered. A connection between the mode symmetry and the topological charges of the pump beams was established. Nonlinear resonance curves were obtained for various combinations of the topological charges of the pump beams and it was demonstrated that an increase in the pump amplitude can lead to the bistability due to the nonlinear slope of the resonance curves, and up to five different solutions can coexist at the same pump frequency. The results were published in the journal Optics Letters. 2) An original experimental setup was created to study the effect of the self-injection locking by integrated (on-chip) microresonators. The spectrum of a multi-frequency laser diode with a central wavelength of 1535 nm, a total spectrum width of about 1 THz and a mode width of the order of 1 MHz was narrowed to a single line with a width of about 100 kHz. Using the obtained single-frequency source, optical comb generation in the same microresonator was first demonstrated. The maximum width of the comb spectrum was observed when using microresonators with a 1 THz free spectral range (FSR) and amounted to about 400 nm. The results were published in the journal Nature Communications. 3) Various types of nonlinear resonances in microresonators with quadratic nonlinearity were investigated and described, and their relationship with the existence of the soliton frequency combs was shown. A theory that describes solitons in quadratic microresonators as gap solitons was developed. Two types of solitons were found for different ratios of the signs of the group velocity dispersion coefficients at the first and second harmonics. For the case when the group velocity dispersion coefficients have opposite signs, solutions were found in the form of bright quasisolitons propagating at weakly modulated background of low intensity. For the case when the dispersion signs are the same, the existence of exponentially localized solitons was shown. It was proved that the transition between these types is associated with the closure of the forbidden gap in the spectrum of quasilinear waves. A methodology for matching the parameters of a microresonator to ensure efficient generation of frequency combs due to quadratic-nonlinear processes was also developed. The technique includes the selection of the operating point (laser frequency) and optimization of the shape for it. The matching frequency (190.8 THz) and the optimal parameters of a microresonator made from lithium niobate with five percent doping with magnesium oxide were calculated (large radius 501.89 μm and small 250 μm). An experimental setup based on microresonator made of lithium niobate (LiNbO3) and lithium tantalate (LiTaO3) was modernized by using a powerful multi-frequency Fabry-Perot laser diode with an output optical power of more than 100 mW. Microresonator made of lithium niobate with a thickness of 100 μm with a quality factor of 1*10^8, which allowed to reduce the half-wave voltage to 110 mV. A narrowing of the line of the laser diode was demonstrated due to the self-injection locking effect to the lithium niobate microresonator with the possibility of simultaneous fast electro-optical tuning using the Pockels effect. The linewidth of the laser diode locked to the lithium niobate microcavity was 4.7 kHz, while fast tuning of the resonance frequency was available up to 200 MHz (with 6.5 V). A study was also made of the regimes of multi-frequency self-injection locking of a laser diode to the lithium niobate microresonator of and the possibility of simultaneous generation of 4, 8, 12 locked lines was shown. The results were published in the journal Optics Express. 4) A numerical study of the process of excitation of whispering gallery modes in microresonators was carried out using a hemispherical coupling element of the same material as the resonator. The possibility of the excitation of modes of various orders, including the fundamental mode, in such a system was demonstrated. A hemispherical silicon coupling element was made and the level of coupling with whispering gallery modes with an efficiency of over 35% at a wavelength of 1.55 μm was demonstrated. Thermooptical oscillations were observed at pump powers above 100 mW. A setup was developed to realize the effect of the self-injection locking at a wavelength of 2.25 μm in WGM resonators made of crystalline silicon using a hemispherical coupling element, which allows measuring the resonator parameters by the interferometric method. The width of the locking range exceeded 50 MHz; the coupling rate reached 15%. The Q-factor of the resonator at a wavelength of 2.25 μm was estimated by the interferometric method as exceeding 10^8, while the Q factor at a wavelength of 1.55 μm was measured more accurately and amounted to 2.0 * 10 ^ 8. 5) The generation of solitons in the self-injection locking regime in various spectral ranges was shown numerically. The multi-soliton, single-soliton regimes and the regime of soliton crystals generation were demonstrated. It has been shown analytically and numerically that the operating point of a self-injection locked laser always lies within the region of existence of solitons. An analytical model was developed that describes the effect of thermal nonlinearity on nonlinear processes in the self-injection locking regime, and the compensation of thermal shifts due to the locking to the resonator mode was shown in the stationary regime. The generation of platicons in the normal dispersion regime due to the self-injection locking effect was demonstrated. The generation thresholds for the backscattering coefficient were determined. The possibility of observation of the platicon drift and breather regimes was shown. The results were published in the journal Physical Review A. 6) The process of stabilization of semiconductor laser diodes by the crystalline microresonator in various spectral ranges from 780 nm to 1650 nm was investigated. Several fundamentally different modes were revealed: single-frequency generation with an ultra-narrow line, multi-frequency generation with a large distance (of the order of several nm) between the lines, Kerr soliton generation, and also a multi-frequency generation mode in which every first or every second line of the laser diode is stabilized and these lines can participate in four-wave interaction in a microcavity. A mode of supposedly passive mode locking of the laser source at wavelengths of 780 nm and 1550 nm was also observed. 7) The theory of splitting of nonlinear resonance due to the Rayleigh scattering and the accompanying interaction of the forward and backward waves was developed. It was shown that in the case of anomalous dispersion at low microresonator finesse value, solitons have complex dynamics, but at typical experimental parameters, the dynamics is close to the case realized in the absence of the backward wave. The effect of drift compensation was observed at low finesse values. For a sufficiently large backscattering coefficient, the presence of a backward wave leads to the suppression of soliton generation. In the case of normal dispersion, the interaction with the backward wave can lead to the modulational instability for the forward wave and to the nonlinear generation on the second branch of the nonlinear resonance curve.

 

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Annotation of the results obtained in 2017
The research team lead by Professor M.L. Gorodetsky in the Russian Quantum Center in 2017 conducted research in terms of the grant of the Russian Science Foundation "New methods for generating broadband coherent optical frequency combs in microresonators." Classic frequency combs marked with the 2005 Nobel Prize (T. Hansch and J. Hall) revolutionized precision optical measurements, but remain the part of well-endowed physical labs. Microresonator-based coherent Kerr combs combine ultracompactness and energy efficiency and open up wide opportunities for this technology to be transformed inti practical applications. The scientific group of Professor Gorodetsky holds a leading position in the world in this field. 1) Despite the compactness of the microresonators, in which the combs are generated, they require narrow-linewidth pumping lasers, which are usually much more bulky devices. However, back in the 1990s, at the Faculty of Physics of Moscow State University, it was proposed to use the same optical microresonators with whispering-gallery modes to narrow actively the diode laser generation line using whispering-gallery mode microresonators, using a wave resonantly backscattered in the microresonator. Now, such compact lasers with a line width of less than 1 kHz are already commercially available (OEwaves, USA). The combination of the self-injection locking of the laser line with the subsequent generation of Kerr's combs opens prospects for creating of integrated devices of a new generation. Within the framework of the studies planned for 2017, theoretical, numerical and experimental studies of the effect of the narrowing of the laser generation band due to the self-injection locking to an external microresonator were carried out. In radiophysics it is known that if a resonant load with a Q-factor higher than that of an oscillator is coupled to a single-loop oscillator, the resulting generation frequency will be more stable and will mainly be determined by the frequency of the external resonator, rather than the circuit of the generator. The stabilization coefficient can be many orders of magnitude. In the case under consideration, the main contour was a laser cavity having a Q-factor of the order of 10^3-10^4, and a stabilizing contour is a high-quality crystalline microresonator with a Q-factor greater than 10^8. The coupling of the laser cavity with the microresonator occurs due to the resonant backward Rayleigh scattering of the laser radiation in the microresonator. An analytic theory describing the effect of the laser frequency self-injection locking due to backscattering in a microresonator was first developed and allowed to describe the main stabilization regimes and to obtain expressions for the stabilization coefficient and the width of the locking band. The obtained expressions were verified experimentally in an experimental setup consisting of a laser diode with a distributed optical coupling at a wavelength of 1.55 μm connected with a microresonator of magnesium fluoride with a Q-factor of 10^9. The obtained analytical estimates for the width of the locking band and the width of the generation line coincide with good accuracy with the experimental data. The results are published in the Optics Express journal. In the course of the project, the generation of soliton combs in a microresonator made from magnesium fluoride was experimentally demonstrated in the self-injection locking regime when the microresonator was pumped by a simple laser diode (power ~ 100-300 mW) having many longitudinal modes in the generation band ~ 10 nm. Due to the optical feedback using the microresonator and the resulting self-injection locking effect, the narrowing of the laser diode spectrum and the stabilization of the generation frequency at the frequency of the microresonator mode were shown. Due to the mechanism of longitudinal mode competition under optical feedback conditions, up to 50% of the laser diode power is converted into one narrow emission line ~ 1 kHz. In the course of the work, it was shown that the use of single mode but multifreuency diodes for generating frequency combs makes it possible to achieve stable singe-frequncy regie of oscillation. The high power of the laser at the resonator mode frequency makes it possible to generate efficiently a soliton comb using a compact laser diode. In this case, in contrast to the DFB diode, multifrequency diodes have at the same price an order of magnitude greater power and efficiency. As a result of the experiments, Kerr combs were first obtained by pumping by a compact laser diode with a wide spectrum at wavelengths of 1550 nm and 1650 nm. The width of the optical spectrum of the comb was 20-30 nm, while pure low-noise beatnotes were observed at a frequency equal to the intermode distance (or FSR) of the microresonator, with a width less than 1 kHz. The results were reported at international conferences CLEO / Europe (Munich, Germany), ICTON (Girona, Spain), PIERS (Singapore) and are preparing for publication. 2) The conducted studies showed that the main parameter determining the generation efficiency of the frequency comb, its shape and spectral width is the mode dispersion, i.e. the dependence of the frequency of the mode on its propagation constant (the azimuthal number in this case). This value is determined by the material (material dispersion) and the shape of the resonator (geometric dispersion). It was shown earlier that by controlling the cross-sectional shape of the microresonator, it is possible to reduce the number of families of the modes, and also to show a significant improvement and expansion of the comb spectrum. In the course of the research, the criteria for optimal dispersion dependence were formulated and the problem of determining the optimum shape of the resonator for achieving it was formulated. To solve this problem, a semi-analytic theory has been developed that makes it possible to calculate the dispersion of spheroidal resonators for subsequent numerical optimization. To study more complex forms, a program is written in the Comsol environment, using the finite element method for numerical calculation of the microresonator dispersion. This approach allows us to optimize the shape of the resonator. A search of the simplest forms of resonators was carried out and it was shown that the trapezoidal shape of the section, which can be reproduced on the diamond turning machine available in the group, makes it possible to achieve good results. 3) During the project, a theoretical model for the generation of Kerr frequency combs in the Raman scattering band was developed. The formalism of coupled-mode equations previously successfully used to model the generation and dynamics of Kerr frequency combs and dissipative solitons has been extended to take Raman scattering into account by introducing a term responsible for the interaction of the optical field with the local molecular oscillator and the addition of the corresponding dependence to the dielectric susceptibility. The proposed model, in addition to a more adequate description of the interaction of the Raman and Kerr effects, is extremely convenient for describing the process of generating an optical frequency comb in microresonators with a complex dispersion law or a small number of participating modes. On the basis of the proposed model, numerical simulation of the process of generation of optical frequency combs in the normal dispersion frequency range of a microresonator was carried out, taking into account the Raman scattering effect. The possibility of generating soliton-like pulses in the normal dispersion mode, platicons, previously discovered by performers, was first studied and the dynamics of such structures in the Raman scattering band was investigated for the first time. It was shown that Raman scattering has a strong influence on the dynamics of their propagation. If the value of the Raman amplification exceeds a certain critical value, then narrow platicons become unstable, and wide platicons are transformed into complex chaotic complex structures without a clear threshold of loss of stability. In addition to the practical important study of the possibility of expanding the spectral coverage of optical frequency combs, this result represents an important area of fundamental research related to the physics of complex nonlinear structures. The results are published in the Optics Express journal. In experimental studies, Raman combs at 1630 nm with a width of 10 nm were observed in microresonator manufactured from BaF2 pumped by a laser at 1550 nm. Also, lines near the pumping frequency of 1550 nm were observed. The optical comb was not in a coherent state, there were a lot of beat signals at different frequencies of 0-15 GHz. The obtained experimental data are in good agreement with the results of numerical simulation. The results were reported at the PECS-2017 conference in Svetlogorsk. 4) In the course of the project, it was theoretically predicted that under conditions typical for recently published experiments on the generation of frequency combs with a spectral width exceeding the optical octave, the spectral distance between the comb teeth is switched to the self-trapping mode, i.e. becomes independent of the pump frequency and, accordingly, the effect of pump noise on this important characteristic of the comb becomes minimal in comparison with the regime when self-trapping is absent. The reason for self-trapping is the creation of dispersion traps for a soliton pulse, which forms a comb in the spectral representation. The transition between the two modes is controlled by the relationship between the length of the resonator and the length of the attenuation of the dispersion radiation. This effect was illustrated using numerical simulation. 5) The development of the theory of generation of frequency combs in resonators has begun, where two spatial dimensions play an important role. As an example of such a resonator, it is proposed to use bottle-shaped microresonators. At this stage, the analytical theory developed within the framework of the Lugiato-Lefever model and the properties of the one-dimensional limit, which are necessary for the transition to two-dimensional geometry, were studied. Calculations of bistability and instability conditions were carried out, which will be further generalized to the two-dimensional case.

 

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Annotation of the results obtained in 2018
The research team under the guidance of Professor Mikhail Gorodetsky in the Russian Quantum Center in 2018 continued research on the grant of the Russian Science Foundation “New methods for generating broadband coherent optical frequency combs in microresonators”. 1) In the course of the project, a model that describes the generation of frequency combs in the regime of the self-injection locking of the pump laser to a nonlinear resonator was developed, and numerical simulation of the generation of solitons and platicons was initiated. An original theoretical model which makes it possible to describe the phenomenon of the self-injection locking of the multi-frequency laser to a high-quality optical microresonator was also developed. The possibility of converting a multi-frequency laser emission spectrum to an effectively single-frequency with full power conservation as well as to multi-frequency with a small number of lines using an optical microresonator is shown numerically. The threshold value of the feedback level was found, at which the effective transformation of the multi-frequency spectrum occurs. It was shown that, in the case of a multi-frequency laser, the Bogatov effect is pronounced, leading to the characteristic spectrum profile of a self-injection locked laser. An original experiment scheme was developed, which allows to study in detail the process of the self-injection locking of a multi-frequency laser and various regimes of its operation. The transformation of the spectrum of a multi-frequency laser into a single-frequency one, as well as simultaneous generation at several frequencies, was demonstrated. The obtained experimental data are in good agreement with the results of numerical simulation. Also, an experimental setup was designed to study the possibility of simultaneously self-injection locking of the large number of lines in the laser spectrum under the condition of the equality of the intermode distances of the microresonator and the laser and the first experiments were carried out. Experimental setup was created for generating frequency combs in the region of the normal group velocity dispersion (at a pump wavelength of 780 nm). In the course of the experiments, the platicon-like spectra of the Kerr frequency combs were observed. The results of the researches were published in the journals Optics Express and Nature Photonics. 2) 2) An original theoretical theoretical model that describes the combined influence of Kerr and Brillouin scattering effects on the generation of optical frequency combs was developed. A technique that allows to speed up the numerical simulation of the processes under consideration was proposed. It was shown numerically that in the Brillouin scattering band with normal dispersion of a microresonator, efficient generation of the Brillouin scattering cascade lines is possible only with precise matching of the free spectral range of the microresonator with the Brillouin scattering frequency. In the absence of such an agreement, generation is possible only on the non-fundamental mode family of of the microresonator with lower efficiency. To verify the proposed model, an experiment was conducted to generate frequency combs in a BaF2 microresonator with a diameter of d = 3.9 mm (free spectral range of 16.689 GHz), the wavelength of a continuous pump laser was 1550 nm, and the width of the loaded resonance line was 500 kHz. In the forward and backward waves, the generation of cascade lines of stimulated Brillouin scattering, separated from each other by one free spectral range, was observed. A theoretical model that takes into account the effects of the interaction of elastic and inelastic scattering in optical microcavities, including the interaction of the forward and backward waves, Kerr and Raman effects was also developed. It was shown numerically that the linear coupling of the forward and backward waves can lead to modulation instability and, consequently, to the generation of frequency combs in the normal dispersion regime. It was also found that in the anomalous dispersion range, the interaction of waves propagating in the opposite direction can lead to a drift of solitons and, thereby, either compensate or enhance the effect of the drift, which appears due to the third-order dispersion, or from Raman scattering. 3) A modernized software package was created to calculate the optimal shape of the microresonator and verify the results of form optimization. Profiles of microresonators, optimal for the generation of wideband combs, were calculated. It was shown that for the manufacturing of microresonators by the method of diamond turning, additional polishing is required, which limits the ability of this method to create microresonators with the optimal shape. It was proposed to study the possibility of creating microresonators using high-power femtosecond lasers. 4) A system of coupled mode equations taking into account the second-order nonlinearity for simulating nonlinear generation in a microresonator was developed and analyzed. A model for calculating the phase matching conditions in a microresonator with quadratic nonlinearity was prepared. An assumption about the mechanism of frequency combs generation in quadratic microresonators due to the effect of effective cubic nonlinearity resulting from non-phase-matched (going without phase matching conditions) parametric processes was made. The development of effective numerical schemes for modeling nonlinear processes in quadratic microresonators, also taking into account the periodic modulation of nonlinearity, was initiated. For experimental studies, a technique for producing microresonators from quadratically nonlinear materials, such as lithium niobate (LiNbO3) and lithium tantalate (LiTaO3) was developed. Microresonators were made of lithium niobate with a quality factor of ~ 2*10^8, as well as lithium tantalate with a quality factor of ~ 7*1 ^7 with different profiles. A study of the optimal coupling tuning with such microresonators was performed. A design of an experimental setup for the excitation of whispering gallery modes in quadratically nonlinear optical crystals for different wavelengths (1550 nm, 780 nm) was developed and implemented. Coupling efficiency was about 40%. The effectiveness of the method of electro-optical adjustment using the Pockels effect was shown. 5) A technique was developed and the simultaneous generation of stable soliton states in different spatial modes of a single crystalline whispering gallery mode microresonator was experimentally demonstrated. The generation of two solitons propagating in one direction and two solitons propagating in opposite directions was shown. the simultaneous excitation of three solitons at different mode families, two of which propagated in one direction, and the third in the opposite direction was also demonstrated. It was shown that the resulting Kerr optical frequency combs are mutually coherent, have different repetition frequencies and are suitable for applications in double- or triple-comb setups. The possibility of applying the generated signals to spectroscopy problems was demonstrated. 6) An analytical model, which is a two-dimensional generalization of the Lugiato-Lefever model, correctly describing the process of generating two-dimensional frequency combs and the formation of two-dimensional dissipative soliton structures in optical microresonators was developed. Various regimes of field evolution were investigated and optimal conditions for the excitation of stable two-dimensional solitons with regular spectra in two-dimensional bottle-shaped microresonators taking into account the type of structure of the resonator modes in both the transverse and azimuth directions were found. A series of two-dimensional dissipative solitons localized both in the azimuth and transverse directions was found. The regions of stability in terms of the pump frequency of such solitons were found. It was shown that outside the stability domains, either their attenuation is observed, or chaotic oscillations of the field amplitude with a sequence of quasi-collapses, or the formation of stable breathers. It was shown that the structure of the excited solitons substantially depends on the position of the pump center in the transverse direction. The results were published in Optics Letters.

 

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