**Scientific results:**
The main fundamental and applied results have been achieved in three directions of the project that are united by the objective of developing methods for creating strongly localized field structures using principles of coherent combination of laser beams and researching quantum and classic non-linear effects in systems with strong field polarization, both in the case of relatively low radiation intensity in waveguide systems and in the case of extremely high intensity that is sufficient to support a cascade of birth of electron–positron pairs in vacuum.

In the area of the coherent combination of beams and the synthesis of localized radiation structures:

- We have proposed a number of new approaches that allow to increase the efficiency of coherent combination in the scheme of a tiled aperture. In particular, we developed a highly efficient method for the coherent combination of a set of optical emitters located in one plane and having a aperture occupation ratio (for instance, radiation from an array of densely-packed single-mode optical fibers). The method allows to achieved an efficiency of coherent combination of over 98%, which is unattainable in the known schemes for this configuration of optical sources. We experimentally tested a method for the coherent combination of continuous radiation in a one-dimensional array of fiber waveguides. The coherent combination of fiber laser channels in a scheme with a tiled aperture has been studied for broadband and ultra-short pulses in the anti-phase distribution mode. In a scheme based on 8 independent fiber channels and based on multi-core fiber with a square 5×5 array of cores, we demonstrated coherent combination of ultra-short pulses at a wavelength of 1.03 μm while preserving the spectral and temporal parameters of the signal with an efficiency of over 80%, a high stability and a high quality of the combined beam M2=1.3. In a numerical modeling we demonstrated the possibility of increasing the efficiency to 90%.
- A new method has been proposed for transforming broadband and femtosecond radiation formed by an array of emitters in a tiled aperture scheme into a single rectangular-shaped beam. The method relies on anti-phase distribution of emitters and a simple optical scheme that can be implemented only using reflective elements and is therefore suitable for high-power radiation. The method can be scaled to a large number of emitters.
- A laser bench has been developed for researching coherent beam combination in various configurations. A multi-channel femtosecond laser master oscillator system has been developed that has adjustable parameters of radiation in the channels. The system allows to generate chirped pulses with a central wavelength of 1.03 μm and a spectrum width of 10 nm that supports pulse compression to ~200 fs. In the system we implemented the control of parameters of the output radiation: its frequency, the shape of the spectrum and the temporal profile of the pulse intensity (simultaneously in all the channels) as well as the phase of the signal (independently in each channel). Our researchers have developed an optoelectronic scheme for detecting and stabilizing of the phase of several powerful laser beams with respect to a single reference beam. The scheme supports the phase at a level of l/100 in the frequency range of up to ~1 kHz. A scheme has been developed for controlling the spatial shape of focused radiation on the basis of spatial phase modulators of light. A fiber probe was used to demonstrate the possibility of synthesizing field structures.
- A new method has been proposed for fast and unambiguous recovery of the temporal shape of ultra-short pulses relying on optical gating with a spectral resolution and spectral interferometry in a non-symmetrical configuration. The new method is free of disadvantages inherent to the known methods, such as the widely used standard method of optical gating with spectral resolution (SHG–FROG).
- A soliton pulse source with adjustable frequency in the sub-THz range has been developed — reconfigurable soliton crystals. The source can be used for increasing the mean power of pulsed laser systems.
- We have conducted a theoretical research of quantum features of the properties of radiation that occur when accounting for the quantum nature of light in the case of coherent combination of laser beams. A standard quantum noise limit has been found for radiation formed during the coherent combination of beams in a system with feedback. We determined the fundamental limitations for required the accuracy of phase synchronization between a large number of input laser beams in coherent beam combination schemes, imposed by quantum effects in phase stabilization systems with feedback. It was demonstrated that, under certain conditions, coherent beam combination can be more preferable than laser amplification up to the same power from the standpoint of the quantum noise characteristics of the radiation. In an experiment we have demonstrated the generation of radiation that is compressed along the amplitude quadrature at a level of –3 dB via combining independently compressed channels in orthogonal modes of polarization-preserving fiber, which, to some extent, models a coherent combination system for fiber channels.

In the domain of the highly-efficient focusing of electromagnetic field and the modeling of quantum electrodynamic cascades in strongly focused fields:

- Using numerical modeling, we have researched the development of an electron-positron cascade in a field of laser beams focused in the form of electric or magnetic dipole that allow to achieve the highest intensity of electric and magnetic fields at a specified total power of laser radiation. We have studied the process of the departure of particles from the focal region where pairs are formed most actively. We have conducted numerical assessments of the threshold power at which the departure of particles is compensated by their birth, while if the threshold is exceeded, a vacuum breakdown occurs similar to in the same manner as avalanche breakdown in gases, in which the concentration of particles rises exponentially fast. In the case of electric-dipole focusing of 12 beams, the threshold power amounts to about 10 PW, while in the case of 2 beams it is 28 PW. Using numerical modeling, we have researched plasma structures formed as a result of the development of an election-positron cascade under the impact of laser beams that are focused, in the form of an electric dipole wave, on feed targets. The Laboratory has studied electron-positron plasma structures in systems with a total power ranging from 30 to 90 PW with from 3 to 12 beams. At the linear stage the initially homogeneous distribution becomes modulated with a number of maximums that is equal to the number of beams, which can be explained by the fact that the maximum growth of the cascade occurs in points of maximum amplitude. We demonstrated that, despite the rich diversity of configurations of the system with various numbers of beams, the nonlinear modes observed in the process of interaction correspond to nonlinear modes implemented in the ideal dipole wave: the layer formation mode and the electron-positron plasma pinching mode. This suggests that the described modes are implementable in experimental settings in systems with a relatively small number of beams. Also, using numerical modeling, we found that each mode is matched by specific dependencies of the directional diagram, of the generated gamma radiation on the power and the number of beams of laser radiation focused in the form of a dipole wave.
- We have demonstrated the stability of the researched modes of the interaction of ultra-powerful radiation with plasma during the development of a quantum electrodynamic cascade with respect to small perturbations of fields, which emphasizes their feasibility in principle in real experiments. It was demonstrated that qualitatively the structure of the fields and the dynamics of the development of a cascade is preserved even at maximum values of defocusing and dephasing. We found out that the main influence on the dynamics of the system is imposed by the defocusing of individual beams, as in the considered range of parameters defocusing has a significantly lower effect on the power due to the long duation of the pulses. At the same time, a decrease in the amplitude of the field is observed in the center, which leads to a decrease in the pace of the growth of the cascade and the number of arising particles, especially high-energy ones. For a wave with a power of 15 PW at a pulse duration of 30 fs and a target density of 10
^{22} cm^{-2},^{ }the efficiency of conversion to gamma photons falls from 42% (the ideal wave) to 23%. At powers of over 20 PW, the mode of the pinching of electron-positron plasma is also preserved when defocusing and dephasing of the laser beams is introduced, which will potentially allow to experimentally research this interaction mode as well.
- Our researchers have performed a numerical optimization on the parameters of the seed target in the problem of the generation of quantum electrodynamic cascades in a strongly focused field of ultra-short pulses. To conduct efficient numerical modeling, we solved problems of the optimization of the method of particles in cells to account for the rapidly growing number of particles, which allowed to achieve the high precision of the conducted computations. We determined the ranges of the densities of seed targets that allow the development of the cascade depending on their diameter, we also determined the influence of the duration of laser pulses and their configurations on these ranges of the densities. The efficiency of the conversion of laser energy to energy of photons and electron-positron pairs was also calculated, as well as the number of electron-positron pairs depending on the parameters of the target and the laser pulses. Depending on these parameters, we determined the energy and angular distributions of the generated gamma photons and electron-positron pairs.
- Using numerical modeling, within the developed finite-element model that describes the process of focusing of converging cylindrical and spherical (dipole) pulse waves, we have demonstrated that introducing third-, fifth- and seventh-order nonlinear absorption can reduce the characteristic spatial scale of established field structures in comparison with focusing in vacuum. It was observed that for fifth- and seventh-order nonlinear absorption it is also possible to increase the scale of localization of the field at relatively low values of the nonlinear absorption coefficient. As part of the reviewed problem of the dynamics of the establishment of the stationary mode, we have demonstrated that for the case of third-order nonlinear absorption in the process of focusing, the characteristic scale of field localization decreases monotonously until it reaches a constant value, whereas for the case of fifth- and seventh-order absorption the dynamics of the characteristic size of the structures in the process of focusing of pulsed radiation is more complex and can include several consecutive regions of monotonicity until it arrives at an established value.

In the domain of quantum effects in systems with strong localization:

- We have experimentally demonstrated the generation of light with compression of quantum uncertainty (light compression) with the use of the Kerr effect during the propagation of ultra-short soliton pulses in a new scheme based on nonlinear fiber. A unique and highly reliable scheme for generating polarization-compressed light based on the use of fiber that preserves polarization and is highly stable without feedback systems, unlike the known fiber schemes.
- The Laboratory has conducted a numerical modeling of the process of quantum compression, found the optimal parameters of the scheme. We theoretically demonstrated the possibility of increasing the power of compressed light when large mode area fibers and multi-core fibers are used and found the optimal parameters of fiber and ultra-short pulses to achieve the maximum compression rate. A soliton pulse source has been developed with the frequency adjustable in the THz range. It can be used to increase the mean power of compressed light. We have studied, theoretically and in a numerical modeling, the non-classical properties of optical solitons and continuous laser signals in fibers with Kerr nonlinearity on the basis of quartz, tellurium, sulfide and selenide glasses with the purpose of producing quadrature-compressed light. In optimal cases we demonstrated the possibility of compressing noises to a level significantly stronger than 10 dB at wavelengths of both 1.55 µm and 2 µm.
- We have conducted a theoretical research of the possibility of forming Gaussian entangled state of light during the propagation of radiation along a one-dimensional lattice of connected one-dimensional waveguides that have cubic or quadratic nonlinearity. In the case of a system with quadratic nonlinearity, photon pairs in the entangled state are born in the process of spontaneous parametric scattering of the pumping radiation. We found the threshold value of the power of the pumping radiation that is caused by the geometry of the reviewed problem. As the threshold is exceeded, the growth of the mean number of photons changes from linear to exponential. The optimal ratio of the coefficient of connection of waveguides and the amplitude of the pumping radiation has been found, which corresponds to the maximum entanglement of the output state. In a system with cubic nonlinearity, we demonstrated that the propagation of the pumping radiation in the transverse direction significantly changes the evolution of the number of photons in the signal and idle mode. We demonstrated that, contrary to the case of quadratic nonlinearity, in which a pronounced entanglement forms between the pairs located symmetrically with respect to the center of the lattice, in the case of cubic nonlinearity there is an entanglement between the waveguides located at the boundary of the propagation of the pumping radiation.
- Our researchers have built a theory of photon emission by a non-equilibrium open quantum system in the in a subwave quasi-two-dimensional system. We found the optimal ratio of the parameter of diffraction connection of a subwave planar electrodynamic system to the environment and the broadening constant of the interzone junction of interzone transition that ensures the maximization of spontaneous photon radiation.
- The Laboratory has conducted a systemic theoretical research of volume and surface polaritons in Weyl semimetal with disturbed symmetry with respect to time reversal. An easy and efficient theoretical model has been proposed that describes an electron structure with two zones and two Dirac points with various chiralities. It was demonstrated that information on the electron structure of Weyl semimetals can be unambiguously extracted from measurements of the scattering, transmission, reflection and the polarization of electromagnetic waves. Plasmons with unusual properties were found that propagate in Weyl and Dirac systems in a quantizing magnetic field.
- We have demonstrated the possibility of implementing acoustically-induced transparency in stainless steel foil for photons from a synchrotron Mössbauer source in a mode that is important for possible applications, as it ensures the synchronization of the moments of the formation of photons and the phases of oscillation of foil. It was shown that the intensity of a single-photon wave packet that passed through foil gains regular amplitude modulation that decreases as the frequency of oscillation of foil increases and its optical thickness decreases. The mechanisms causing this modulation were uncovered. The experimental conditions have been determined that allow to observe the deceleration of photons with energies of 14.4 keV form a synchrotron Mössbauer source to 24 m/s in stainless steel foil enriched with the 57Fe nuclide at room temperature by acoustically-induced transparency. We theoretically demonstrated the possibility of transforming quasi-monochromatic high-power radiation with a photon energy of 14.4 keV emitted by a
^{57}Co radioactive Mössbauer source to ultra-short pulses with pre-defined spectro-temporal characteristics in an acoustically resonance-controlled nuclear absorber comprised of stainless steel foil enriched with the 57Fe nuclide.
- A universal test bench has been created for the experimental research of the properties of optical microresonators. We experimentally demonstrated the generation of optical frequency combs in quartz microspheres with a Q factor of ~2×10
^{7} in various modes, including chaotic frequency combs, Raman frequency combs, combs in the dissipative soliton mode in the telecommunication range, which was confirmed by the results of mathematical modeling. We conducted experimental research of linear and nonlinear properties of tellurium microspheres. A model has been built, whose results were verified experimentally, that describes the thermal shift of resonance modes of a whispering gallery when the energy of the pumping radiation is thermalized. We numerically researched the linear and nonlinear properties of solid and hollow tellurium microspheres, found the dependency of the dispersion and nonlinearity on the geometrical parameters and the radiation wavelength.
- Technologies for manufacturing high Q factor resonators relying on soft glasses have been created. In a As
_{2}S_{3}
microsphere we, for the first time, achieved wideband readjustment of single-mode Raman generation in the 1.610–1.663 μm range by adjusting the pumping radiation wavelength in the 1.522–1.574 μm range. The results are in good agreement with the results of numerical modeling within the Lugiato–Lefever equation. In As_{2}S_{3}
microspheres we demonstrated, for the first time, that it is possible to control Raman generation in the telecommunication range with the use of an external low-coherence source in the visible range. Namely, we researched the mode of the starting of single-mode Raman generation during pumping at a wavelength of 1530 nm with the use of an auxiliary laser diode at a wavelength of 650 nm for the thermo-optical control of the resonance frequencies of modes of a whispering gallery. The Laboratory implemented and theoretically explained various effects leading to the transformation of the radiation of continuous narrowband pumping radiation due to nonlinear and laser processes in tellurium and quartz microspheres. In tellurium microspheres doped with erbium ions our researcher have experimentally demonstrated multi-mode laser generation in the L range at a central wavelength of 1605 nm that is started when in-band pumping is switched on at various fixed wavelengths when the threshold power is significantly exceeded. We theoretically and experimentally studied stationary and dynamic dependencies of thermo-optical shifts of resonance (natural) frequency of microspheres on the power of the thermalized pumping radiation. Our studies demonstrated that the temperature sensitivity almost does not depend on the diameter of a microsphere, while the relaxation time is proportional to the diameter squared, which can be used for the creation of thermo-optical microsensors.

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Implemented results of research:

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Technologies and products based on the scientific results of the Laboratory are at the development stage. We are developing a laser at a wavelength of 2.3 μm based on tellurium fibers for scientific and biomedical applications. Our researchers are developing a technology for generating optical combs in microresonators for telecommunication applications and optical computations.

**Education and career development:**

- Three Candidate of Sciences dissertations and two Doctor of Sciences dissertations have been prepared and defended.
- Two postgraduate students have completed one short-term internship and one long-term internship in the directions «Squeezing the quantum noise of short pulses of light» and «Quantum properties of optical solitons in nonlinear fibers» at the Max Planck Institute for the Science of Light under the supervision of Gerhard Leuchs. Two postgraduate students have completed the advanced training course «Quantum optical communication technologies» on the grounds of ITMO University. Four postgraduate students and one undergraduate have completed the online advanced training course «Optical frequency combs for classical and quantum communication systems»on the basis of the Institute of Telecommunications of Riga Technical University.
- Members of the academic team of the Laboratory have developed and implemented new education courses for students of Nizhniy Novgorod State University on the topic of the Laboratory’s work, namely: «Measuring ultra-short optical pulses» and «Nonlinear propagation of femtosecond pulses in fiber optical waveguides» (series of research and educational case studies, 10 hours, an extension to the program of the course «Research and educational experiment»), lectures «Fiber ultra-short pulse lasers», «Fiber quantum optics», «What is nonlinear optics?» (as an extension to the program of the course «Basics of scientific communication»); the lecture course «Nonlinear waves in optics» (22 academic hours), the online lecture course «Nonlinear and quantum effects in optical fibers and microresonators» (18 academic hours) for undergraduate and postgraduate students. New textbooks have been created: «Dispersion and nonlinear properties of spherical microresonators based on various glasses: a study guide»; «Collection of individual assignments in physics. Thermodynamics and molecular physics. Part 3». Employees of the Laboratory participated in the organization of seminars for young researchers with the scientific schools «Nonlinear waves–2020» and «Nonlinear waves–2022».

**Collaborations:**

- G. G. Devyatykh Institute of Chemistry of High-Purity Substances of the Russian Academy of Sciences: joint research and publications, participation in joint grants.
- Max Planck Institute for the Science of Light (Germany): joint work, organization of internships for postgraduate students, joint publications.
- Sant Longowal Institute of Engineering and Technology (India): academic visits of employees, a collaborative grant, a memorandum of cooperation, collaborative articles
- Sant Longowal Institute of Eng. & Technology; Punjabi University; University of Lucknow (India): academic visits of employees, a completed joint grant, joint articles.
- Institute of Telecommunications of the Riga Technical University (Latvia): academic visits of employees, joint work and publications, organization of remote internships for undergraduate and postgraduate students, development of lecture courses, signing of a Memorandum of Understanding, MoU.
- University of Queensland (Australia): joint research, collaborative publications.
- Texas A&M University (USA): academic visits, joint research, collaborative publications.
- University of Gothenburg (Sweden): joint research, collaborative articles.