Problems In Nonlinear Optics Research Articles

Subwavelength Photorefractive Grating in a Thin‐Film Lithium Niobate Microcavity

AbstractSubwavelength gratings play a fundamental and pivotal role in numerous science and applications for wave manipulation, exhibiting distinctive features such as filtering, phase manipulation, and anti‐reflection. However, conventional fabrication methods for ultrasmall periodic structures are constrained by the fundamental optical diffraction limit, making it challenging to produce subwavelength gratings for optics. Here, a novel technique is demonstrated to build a reconfigurable subwavelength photorefractive grating (SPG) in a thin‐film lithium niobate on the platform of an optical microcavity. Such SPGs are optically induced through the photorefractive effect and the subwavelength features originate from the spatial phase modulations of the pump's standing wave. The resulting SPGs lead to the mode splitting of two counter‐propagating modes inside the microcavity, exhibiting an Electromagnetically Induced Transparency (EIT)‐like transmission spectrum. Moreover, the unique subwavelength characteristic of SPGs enables first‐order quasi‐phase‐matching for backward second‐harmonic generation, a long‐standing problem in nonlinear optics. Also, free‐space‐to‐chip vertical nonlinear frequency conversion can be achieved in a similar manner. These results provide a flexible approach toward fabricating subwavelength gratings, which holds significant potential in various applications such as nonlinear frequency conversion, optical communication, sensing, and quantum technologies.

Phase-encoding of loosely bound soliton molecules

Dissipative soliton molecules (DSMs) are of great interest for studying the complexity of nonlinear optical problems as they can map with the matter molecules for making interdisciplinary analogies. In contrast to strongly bound DSMs that have a short time separation between the bound solitons, the complex dynamics and underlying binding mechanism of loosely bound soliton molecules (LBSMs) with orders of magnitude longer time separation remain open questions. To this end, here, we explore real-time spectroscopy using a dispersive temporal interferometer (DTI) to visualize the dynamics of LBSMs in a mode-locked fiber laser and unveil their underlying phase-evolving mechanism. The DTI enables fringe-resolved spectroscopy in real time of the LBSM’s evolution by creating duplicates of the LBSM that results in a much closer time separation between the individual solitons of the LBSM. The real-time evolution of the LBSM’s phase exhibits a diverging sliding landscape, which is theoretically and experimentally proved to be closely associated with gain dynamics. Based on the understanding of its phase dynamics, we finally demonstrate programmable phase-encoding modulation of the LBSM through gain control. These efforts not only shed light on understanding the mechanism of long-range interactions in LBSMs but also provide an alternative approach for all-optical information processing.

Optimal Dirichlet boundary control by Fourier neural operators applied to nonlinear optics

We present an approach for solving optimal boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods in combination with collocation techniques. Thereby, Cl-regularity in time of the global discrete solution is ensured. The simulation data is used to train solution operators that effectively leverage the higher regularity of the data. The solution operator is represented by Fourier Neural Operators and can be used as the forward solver in the optimal Dirichlet boundary control problem.The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. The effectiveness of the approach is demonstrated on the problem of generating Terahertz radiation in periodically poled Lithium Niobate. The neural network is used as the solver in the optimal control setting to optimize the parametrization of the optical input pulse and maximize the yield of 0.3THz-frequency radiation.We exploit the periodic layering of the crystal to design the neural networks. The networks are trained to learn the propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. Our results indicate that the proposed method can achieve a speedup by a factor of 360 compared to classical methods. A comparison of our results to experimental data shows the potential to revolutionize the way we approach optimization problems in nonlinear optics.

Symmetry-conserving treatment of the random-phase problem in nonlinear coherent optics modeling in solids

Symmetry-conserving treatment of the random-phase problem in nonlinear coherent optics modeling in solids

Two-colour dissipative solitons and breathers in microresonator second-harmonic generation

Frequency conversion of dissipative solitons associated with the generation of broadband optical frequency combs having a tooth spacing of hundreds of giga-hertz is a topical challenge holding the key to practical applications in precision spectroscopy and data processing. The work in this direction is underpinned by fundamental problems in nonlinear and quantum optics. Here, we present the dissipative two-colour bright-bright and dark-dark solitons in a quasi-phase-matched microresonator pumped for the second-harmonic generation in the near-infrared spectral range. We also found the breather states associated with the pulse front motion and collisions. The soliton regime is found to be typical in slightly phase-mismatched resonators, while the phase-matched ones reveal broader but incoherent spectra and higher-order harmonic generation. Soliton and breather effects reported here exist for the negative tilt of the resonance line, which is possible only via the dominant contribution of second-order nonlinearity.

Rigorous solution to second harmonic generation considering transmission and reflection of light at air–crystal interface

Considering the transmission and reflection of TE-polarized pump light at the air–crystal interface, the second harmonic generation (SHG) in a lithium niobate (LN) crystal is investigated theoretically and systematically in this work. In previous studies, the theoretical analyses of reflection and transmission of incident wave in the process of nonlinear frequency conversion were not considered in LN crystal on account of the complicated calculations. First, we establish a physical picture describing that a beam of light in TE mode transports in the LN crystal considering transmission and reflection at the crystal surface and generates nonlinear second-order optical polarization in crystal. Then we analytically derive the reflection coefficient and transmission coefficient of pump light by using the dispersion relationships and electromagnetic boundary conditions. We construct the nonlinear coupled wave equations, derive and present the small signal approximation solution and the general large signal solution exactly. Under the transmission model and reflection model, we find that the conversion efficiency of the second-harmonic wave is obviously dependent on transmission coefficient and other general physical quantities such as the length of LN crystal and the amplitude of pump light. Our analytical theory and formulation can act as an accurate tool for the quantitative evaluation of the SHG energy conversion efficiency in an LN crystal under practical situations, and it can practically be used to treat other more complicated and general nonlinear optics problems.

Nonlinear Lorentz Model for Explicit Integration of Optical Nonlinearity in FDTD

Including optical nonlinearity in FDTD software in a stable, efficient, and rigorous way can be challenging. Traditional methods address this challenge by solving an implicit form of Maxwell’s equations iteratively. Reaching numerical convergence over the entire numerical space at each time step demands significant computational resources, which can be a limiting factor for the modeling of large-scale three-dimensional nonlinear optics problems (complex photonics devices, laser filamentation, ...). Recently, we proposed an explicit methodology based on a nonlinear generalization of the Lorentz dispersion model and developed example cases where it was used to account for both linear and nonlinear optical effects. An overview of this work is proposed here.

Численное моделирование когерентных и турбулентных структур излучения методом нелинейных интегральных отображений

The propagation of stable coherent entities of an electromagnetic field in nonlinear media with parameters varying in space can be described in the framework of iterations of nonlinear integral transformations. It is shown that for a set of geometries relevant to typical problems of nonlinear optics, numerical modeling by reducing to dynamical systems with discrete time and continuous spatial variables to iterates of local nonlinear Feigenbaum and Ikeda mappings and nonlocal diffusion-dispersion linear integral transforms is equivalent to partial differential equations of the Ginzburg-Landau type in a fairly wide range of parameters. Such nonlocal mappings , which are the products of matrix operators in the numerical implementation, turn out to be stable numerical-difference schemes, provide fast convergence and an adequate approximation of solutions. The realism of this approach allows one to take into account the effect of noise on nonlinear dynamics by superimposing a spatial noise specified in the form of a multimode random process at each iteration and selecting the stable wave configurations. The nonlinear wave formations described by this method include optical phase singularities, spatial solitons, and turbulent states with fast decay of correlations. The particular interest is in the periodic configurations of the electromagnetic field obtained by this numerical method that arise as a result of phase synchronization, such as optical lattices and self-organized vortex clusters.

Failure of the N‐wave interaction approximation without imposing periodic boundary conditions

AbstractThe N‐wave interaction (NWI) system appears as an amplitude system in the description of N nonlinearly interacting and linearly transported wave packets in dispersive wave systems such as the water wave problem or problems in nonlinear optics. The purpose of this paper is twofold. First we give a new simplified proof for the failure of the NWI approximation in case of resonances which are located at integer multiples of a basic wave number k0 in the original ‐spatially periodic dispersive wave system. Secondly, we give a first rigorous proof that an amplitude system fails in the description of an original system, without imposing periodic boundary conditions on the original system.

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion.

An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

Interaction solutions of the first BKP equation

In this paper, we study the interaction solutions of the first BKP equation by using the Hirota direct method and taking long-wave limit. In order to obtain the interaction solutions, the multi-soliton solutions are firstly derived using the Hirota direct method, then the interaction solutions are successfully constructed by properly choosing appropriate parameters and taking long-wave limit on the soliton solutions. These parameters have great influences on the propagation directions, shapes as well as energy. Moreover, the dynamic properties of these obtained solutions are illustrated vividly by some graphs. The results in this work could be used to solve nonlinear problems in nonlinear optics and engineering field.

The Cauchy problem for nonlinear quadratic interactions of the Schrödinger type in one dimensional space

In this work, we study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations with quadratic nonlinearities, which appears modeling problems in nonlinear optics. We obtain the local well-posedness for data in Sobolev spaces with low regularity. To obtain the local theory, we prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, we establish the global well-posedness in Hs(R)×Hs(R), for some negatives indexes s. The proof of our global result uses the I-method introduced by Colliander et al.

Graphics processing unit‐based solution of nonlinear Maxwell's equations for inhomogeneous dispersive media

SummaryA new approach is developed for fast solution of complex dynamic problems in nonlinear optics. The model consists of the nonlinear Maxwell's equations coupled with time‐dependent electron density equation. The approach is based on the Finite‐Difference Time‐Domain and the auxiliary differential equation methods for frequency‐dependent Drude media with a time‐dependent carrier density, changing due to Kerr, photoionization, avalanche, and recombination effects. The system of nonlinear Maxwell‐Ampere equations is solved by an iterative fixed‐point procedure. The proposed approach is shown to remain stable even for complex nonlinear media and strong gradient fields. Graphics‐processing‐units technique is implemented by using an efficient algorithm enabling solution of massively 3‐dimensional problems within reasonable computation time.

Boundary-value problems for elliptic functional-differential equations and their applications

Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics. Bibliography: 137 titles.

Emission of terahertz pulses from vanadium dioxide films undergoing metal–insulator phase transition

This paper describes research on the optics of functional materials, which can change their dielectric properties according to their function. Vanadium dioxide is a good example of such a material where the insulator-to-metal phase transition offers the possibility to control dielectric properties and to use them as a triggering element for photonic applications in the wide spectral range from optical to terahertz frequencies. We observed emission of terahertz (THz) radiation from VO2 films in insulating and conductive phase states under femtosecond pulse irradiation. We found that the efficiency of THz emission increases up to 30 times after the insulator-to-metal phase transition. This process occurs in thin films while it is fundamentally forbidden in the bulk material, and polarization analysis of the emitted radiation reveals the crucial importance of interface contributions. The properties of the THz radiation emitted by VO2 are determined by displacement photocurrents at the VO2–air and VO2–substrate interfaces induced by the incident laser light. In each phase state the contributions of the two boundaries are different. Properties of the effective dielectric susceptibility χ(2) tensor for the insulating phase were defined. In demonstrating the conversion of optical into THz radiation in VO2 films, we found that fundamental symmetry restrictions are not applicable to problems of nonlinear optics of thin films.

Using Lorentz’s theorem for studying the propagation wave packets in a nonlinear medium

We obtain integral relationships expressing the amplitude of wave-packet field on the output surface of a layer via the field amplitude on the input surface, the field on the side surface of the layer, and the integral of the nonlinear currents inside the layer at the previous times in the spectral and spatiotemporal forms. These relationships allow one to perform studies and develop numerical simulation algorithms of propagation and interaction of wave packets which have wide frequency and angular spectra, including calculations of excitation of evanescent waves in the case of sharp focusing or formation of supernarrow filaments. Using the obtained relationships, we propose an algorithm that employs a simple iteration of the first order and allows one to significantly speed up the calculations. The possibility of using the fast Fourier transform to develop algorithms for numerical simulation of the boundary-value problems of nonlinear optics is demonstrated.

Analysis of three-wave mixing in one-dimensional nonlinear multilayer structures with pump depletion

We present a general theoretical approach that combines the iterative technique with the nonlinear transfer matrix method to solve the general problems of three-wave mixing with pump depletion in nonlinear multilayer structures. The theoretical model is directly derived from nonlinear coupled wave equations, shows fast numerical convergence and maintains conservation of total energy perfectly for general structures. We have employed this approach to discuss second harmonic generation in photonic bandgap crystals with slow light effect and obtained a conversion efficiency of about 25% for both the forward and backward second harmonic waves. The detailed wave evolution process and energy conversion mechanism of sum frequency generation and difference frequency generation in perfect and imperfectly quasiphase-matching structures have been systematically investigated and their subtle and strong dependence on the initial conditions of incident waves has been revealed. The developed theoretical approach can be very useful for efficient and quantitative analysis of various nonlinear optical problems and help to design optimum nonlinear multilayer structures for specific high-efficiency nonlinear three-wave mixing processes.

Scalar time domain modeling and coupling of second harmonic generation process in GaAs discontinuous optical waveguide

We present in this work the scalar potential formulation of second harmonic generation process in chi((2)) nonlinear analysis. This approach is intrinsically well suited to the applications of the concept of circuit analysis and synthesis to nonlinear optical problems, and represents a novel alternative method in the analysis of nonlinear optical waveguide, by providing a good convergent numerical solution. The time domain modeling is applied to nonlinear GaAs asymmetrical waveguide with dielectric discontinuities in the hypothesis of quasi phase matching condition in order to evaluate the efficiency conversion of the second harmonic signal. The accuracy of the modeling is validated by the good agreement with the published experimental results. The effective dielectric constant method allows to extend the analysis also to 3D optical waveguides.

Problems of nonlinear optics and acousto-optics in the curriculum of “Quantum Electronics” specialization

Problems of nonlinear optics and acousto-optics in the curriculum of “Quantum Electronics” specialization

Hamiltonian formulation of coupled-mode theory in waveguiding structures

We present a Hamiltonian formulation of coupled mode theory for scenarios in which the coupled modes are associated with different "parent structures," such as two nearby waveguides. The relativistic nature of the photon leads to the complication that not any set of orthonormal modes can be used as a basis if the associated amplitudes are to satisfy canonical commutation relations. This difficulty is circumvented by the introduction of "dressed parent modes," which are in fact seen to be the "coupled modes" of the system. While an exact solution of the linear problem within the restricted basis of interest formally must be found before these modes can be constructed, in practice they can be constructed directly from the modes of the parent structures. The approach can be applied to periodic parent structures, such as photonic crystal waveguiding structures, as well as to simpler waveguides. We illustrate the accuracy of the various approximations employed by studying two sample systems in detail. We derive the linear coupled mode equations, and show how the approach can be immediately generalized from the linear regime to treat problems in nonlinear quantum optics.