Theory Of Second-harmonic Generation Research Articles

Resonant Second-Harmonic Generation as a Probe of Quantum Geometry.

Nonlinear responses are actively studied as probes of topology and band geometric properties of solids. Here, we show that second harmonic generation serves as a probe of the Berry curvature, quantum metric, and quantum geometric connection. We generalize the theory of second harmonic generation to include Fermi surface effects in metallic systems, and finite scattering timescale. In doped materials the Fermi surface and Fermi sea cause all second harmonic terms to exhibit resonances, and we identify two novel contributions to the second harmonic signal: a double resonance due to the Fermi surface and a higher-order pole due to the Fermi sea. We discuss experimental observation in the monolayer of time reversal symmetric Weyl semimetal WTe_{2} and the parity-time reversal symmetric topological antiferromagnet CuMnAs.

铌酸锂微纳波导中的二阶非线性光学效应研究进展

<p indent="0mm">Lithium niobate (LiNbO₃, LN) crystal has a large second-order nonlinear coefficient (<italic>d</italic>₃₃=25.2 pm/V@1064 nm), wide transparent window (0.35–5 μm), and stable periodic microdomain structure preparation, which is a good platform for the study and application of second-order nonlinear optical effects. Lithium niobate thin film (LNTF) is regarded as a promising integrated photonics platform due to its excellent linear and nonlinear optical properties. As the basic unit of an integrated optical system, lithium niobate micro/nano waveguide has been studied as a transmission and control device, and it shows excellent second-order nonlinear optical characteristics. The second-order nonlinearity plays an important role in modern optics, including second-harmonic generation (SHG), sum- and difference-frequency generation, parametric down conversion, and parametric oscillation. Efficient and compact wavelength converters based on second-order nonlinearity are key components for a wide range of applications, including entangled photon sources, optical parametric oscillators, and optical parametric amplifiers. The second-order nonlinear polarization intensity is proportional to the square of the electric field intensity. Micro/nano optical waveguides can improve the light intensity in the device through the local characteristics of the spatial light field, to improve the conversion efficiency of the nonlinear processes. And significantly enhanced electric field strength makes the normalized nonlinear conversion efficiency of LNTF waveguides exceed that of the reverse proton<bold> </bold>exchange lithium niobate waveguides (frequency conversion efficiency 150% W^–1 cm^–2@1550 nm) by one order of magnitude. The nonlinear conversion efficiency of LNTF micro/nano waveguides can be further improved through mode phase matching (MPM) or quasi-phase matching (QPM) to optimize the spatial overlap between the eigenmodes involved in the nonlinear process. The detailed theory, fabrication and application were demonstrated in the manuscript as follows: (1) The theory of second harmonic generation. As an example of second-order nonlinear process, the factors affecting the conversion efficiency of second harmonic generation were analyzed, and can be summarized as nonlinear coefficient, mode special overlap, mode area and phase-matching condition. (2) The fabrication of periodically poled lithium niobate waveguides and periodically poled technics. Based on the micro fabrication processes such as electron beam lithography, dry etching and chemico-mechanical polishing, the propagation loss could be reduced to <sc>0.027 dB/cm.</sc> With the combination of low loss waveguides and the electrode polarization process, periodically poled lithium niobate microstructures and the detailed fabrication processes were discussed in the manuscript. (3) Efficient second harmonic generation based on MPM and QPM. It was reported that the normalized efficiency can reach 1000% W^–1 cm^–2, and we described broadband and tunable SHG in detail. (4) Spontaneous parametric down conversion. With the development of quantum information technology and its application in the fields of secure communication and precision measurement, quantum photon sources with high speed, high brightness and high purity have attracted more and more attention. Due to the strong optical confinement, the micro/nano waveguides reduce the pump power and improve the quality of generated photon pairs, which is helpful to realize the integrated optical path of quantum information processing. In summary, this paper reviews the recent development of second-order nonlinear optical effects in LNTF micro/nano waveguides. Firstly, the theory of second harmonic conversion is introduced, and then the fabrication and periodically poled techniques of LNTF micro/nano waveguides are discussed. As an example, the efficient SHG and parametric down conversion of LNTF waveguides were detailedly demonstrated. The processes of mode phase matching and quasi-phase matching technics, tunable SHG and wideband frequency doubling effect were emphatically investigated.

Analytical theory of second harmonic generation from a nanowire with noncentrosymmetric geometry

We analytically investigate the effect of a non-centrosymmetric geometry in the optical second harmonic (SH) generation from a particle made of a centrosymmetric material, in the interior of which quadratic optical processes are suppressed. We consider a cylindrical particle with a cross-section that is slightly deformed away from a circle and with a radius much smaller than the wavelength. We calculate the induced linear and nonlinear fields perturbatively in terms of the deformation parameter and obtain the nonlinear dipolar and quadrupolar hyperpolarizabilities, whose spectra we evaluate for metallic and dielectric materials. We show that for very small deformations the dipolar contribution to the response competes with the quadrupolar term, and may even be dominant. We explore the spectra of the hyperpolarizability and identify the contributions to its structure for metallic and dielectric particles. We also discuss the nature of SH radiation at various frequencies and find that it may be dominated by the dipolar or the quadrupolar term, or that both may compete yielding non-symmetric radiation patterns. Our calculation may be employed to assess, calibrate and test numerical SH calculations.

Second harmonic generation and spontaneous parametric down-conversion in Mie nanoresonators

The recent active research in the field of nonlinear Mie nanoresonators has led to significant increase of harmonic generation efficiency from subwavelength structures. The progress in the second order nonlinear processes became evident after switching to nanoresonators made of materials that possess a bulk nonlinearity such as semiconductor and perovskite-type materials. Despite of that, the theory of second-harmonic generation from spherical Mie nanoparticles with a bulk nonlinearity has not been developed yet. In this work, we present a theoretical approach based on Mie theory for describing the second order nonlinear processes in all-dielectric nanoresonators. We focus on the second harmonic generation (SHG) process as well as on the inverse one, the process of spontaneous parametric down-conversion (SPDC). We demonstrate the SHG-SPDC correspondence in terms of Mie resonances and identify the selection rules governing these second order nonlinear processes. We show how one can control the spatial correlations of photons in SPDC from a single Mie-nanoparticle by properly choosing the parameters of the system, in particular, allowing for Kerker-type unidirectional emission of photons.

Theory of SHG in a medium with combined nonlinear response

Second-harmonic generation (SHG) is used in many practical applications such as substance diagnostics and imaging of various processes as well as for frequency conversion. It is well known that frequency doubling may also be used for the generation of a tripled frequency wave due to mixing of optical radiation at basic frequency with optical radiation at doubled frequency. In this case, the relationship between phases of interacting waves with basic frequency and doubled frequency plays an important role. Therefore, the solution of the corresponding problem for wave phase evolutions is an important issue. In this paper, we provide such a solution for the SHG of high-intensity femtosecond pulses, taking into account the influence of a cubic nonlinear response on frequency doubling by using an original approach together with the long pulse duration approximation and plane wave approximation. The main feature of our approach is the absence of using the basic wave energy non-depletion approximation and using the problem invariants. It should be stressed that the frequency conversion process under consideration possesses a multi-stability property: there are two modes of SHG at certain values of intensity at basic frequency. We derive the wave intensity and phase evolution for both interacting waves for each of the modes. The derived formulas are verified by computer simulation based on using corresponding nonlinear Schrodinger equations. It is very important that we demonstrate the robustness of the derived formulas for the inhomogeneous shape of the pulse. They can be applied to computing of pulse shapes and beam profiles, even if reverse energy conversion occurs. As an example, we consider SHG for an incident Gaussian pulse with basic frequency.

On the theory of second-harmonic generation in 2D nonlinear photonic crystals with arbitrary domain structures

Theoretical analysis is given to second-harmonic generation in 2D nonlinear photonic crystals of rectangular symmetry with rectangular domains. Formulas for the nonlinear response of the crystal with an arbitrary domain structure are derived within the approximation of a given field. A possibility of calculating time profiles of second-harmonic pulses down to the femtosecond range is demonstrated.

On theory of second harmonic generation in 2D nonlinear photonic crystals with arbitrary structures

On theory of second harmonic generation in 2D nonlinear photonic crystals with arbitrary structures

Theory of second-harmonic generation in a chirped 2D nonlinear optical superlattice under nonlinear Raman-Nath diffraction

We analyze second-harmonic generation (SHG) in a two-dimensional nonlinear optical superlattice (NLOS) with its modulation period being chirped in the propagation direction and constant in the transverse direction. This results in efficient multiple SHG via nonlinear Raman–Nath diffraction. We obtain exact analytical expressions for a SH amplitude generated in chirped 2D NLOSs and for its quasi-phase-matching bandwidth. The results of analytical calculations are in excellent agreement with the numerical ones. We show that the process is robust to angular deviations of NLOS and it can be applied to enable tunable and broadband frequency conversion.

Highly efficient second-harmonic generation from a two-dimensional linear–nonlinear photonic crystal heterostructure

The theory of second-harmonic generation (SHG) in two-dimensional (2D) linear–nonlinear photonic crystal (PC) heterostructure has been developed by means of transmission matrices and a Green’s function approach based on lattice sums. Using this method, we can calculate the conversion efficiency of second-harmonic waves in various 2D PC heterostructures. Not only has great enhancement of SHG been observed when the frequencies of both fundamental frequency and second frequency waves are located at the photonic band edges, but some unique phenomena for the SHG based on resonant states in the heterostructure have also been found. The resonant SHG can appear at the corresponding frequency of the resonant state in the range of 160° of the incident angle around the normal direction, although the value of the SHG changes with the incident angle. The second-harmonic conversion efficiency appears to have a higher value in the heterostructure containing several nonlinear PC slabs because of the competition between phase matching and the decrease in transmittance. We believe that these findings are very beneficial for the design of SHG devices.

Complete pump depletion by autoresonant second harmonic generation in a nonuniform medium

In this paper, we develop for the first time to our knowledge an analytical theory of second harmonic generation (SHG) in a generic nonuniform &#x3C7;(2) medium. It is shown that by varying the properties of the medium gradually enough, the system can enter an autoresonant state in which the phases of the fundamental pump and of the generated second harmonic wave are locked. The effect of autoresonance allows efficient transfer of energy between the waves and, due to the continuous phase-locking in the system, all the energy of the pump could be converted to the second harmonic. Simple closed-form expressions for the waves amplitudes as a function of the longitudinal coordinate are derived, and an explicit criterion for the stability of the autoresonant state is obtained. Our analytical theory is compared to the numerical solution of the coupled mode equations, which are found to be in excellent agreement with each other. The analytical closed-form expressions that we derive could be very useful for practical design of SHG devices with increased performances, such as highly efficient, wideband frequency converters.

Theory of second harmonic generation of ultrashort pulses for collinear achromatic phase matching

Achromatic phase matching (APM), a favorable method in increasing phase matching bandwidth, is explored for second harmonic generation of ultrashort pulses based on a typical grating telescope system. A set of coupled equations incorporating angular dispersion is constructed in the space–time domain. An analytic solution with a pump of tilting pulsed Gaussian beam to the equations is given under the undepleted pump approximation. With the aid of matrix formalism, some properties of the conversion are demonstrated. Though a maximal phase matching bandwidth is obtained, angular dispersion makes the conversion active only around the geometrical focus. Theory shows that APM does not require an overall pulse-front matching in the conversion process.

Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres

The theory of second harmonic generation (SHG) in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials is developed by means of multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency and second harmonic, as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other distributions of the metallic spheres. It is shown that the linear and nonlinear optical response of all ensembles of metallic spheres is strongly influenced by the excitation of localized surface plasmon-polariton resonances. The physical origin for such a phenomenon has also been analyzed.

Phase effects at second-harmonic generation in metamaterials

We develop the theory of second-harmonic generation in metamaterials where fundamental radiation is considered in the constant-intensity approximation taking into account the phase changes of interacting waves. We investigate the second-harmonic wave intensity in metamaterials at different parameters of the problem under consideration. We consider a self-action effect of the light wave in metamaterials in view of the phase changes of all the interacting waves and compare this effect with an analogous effect in usual homogeneous quadratic media. We show that it is possible to change the phase velocity of the pump wave by varying such parameters as the pump intensity, nonlinear medium length, and phase mismatch between the interacting waves. The metamaterials under consideration provide a possibility to change the phase of the fundamental radiation of order of magnitude higher than those in the usual homogeneous quadratic media.

Study of the properties of the SHG with diode lasers

An experimental study of the second-harmonic generation (SHG) properties with nondiffraction-limited radiation is presented. This includes the dependency of the normalized conversion efficiency on the crystal length, the beam quality, and the focusing conditions. A Gauss–Schell-model-based theory of SHG is applied, which gives an improved simulation of the experimental results in comparison to the standard Boyd–Kleinman theory for nondiffraction-limited beams.

Second harmonic generation from tetragonal centrosymmetric crystals

We present a phenomenological theory of second harmonic generation (SHG) from tetragonal centrosymmetric crystals. It includes the lowest order bulk response, which is of electric quadrupole and magnetic dipole symmetry, and the lowest order surface response, which is of electric dipole symmetry. Based on the symmetry properties of the nonlinear response tensor, complete expressions are obtained for the SHG expected in reflection from a crystal with a (001) surface orientation. We suggest experimental configurations that can be used to extract particular tensor elements. We find that dichroism is present in SHG from tetragonal centrosymmetric crystals, although the crystal is not chiral.

Efficiency of second harmonic generation for partially coherent light in anisotropic crystals

Abstract This paper analyzes the second harmonic generation (SHG) efficiency of light with partial temporal coherence due to depolarization effects in birefringent media. It discusses relations between SHG efficiency fading, light source spectrum, crystal birefringence, and phase matching conditions. The efficiency of SHG pumped by the partially coherent light beam that may depolarize light in nonlinear birefringent crystal is also analyzed. The basic theory of SHG with its modification for partially coherent light with depolarization and some numerical calculations of the SHG process are described.

Optimum formation of the response of aperiodic nonlinear crystals in the process of second harmonic generation

The theory of second harmonic generation in nonlinear optical crystals that is caused by the quasiphase-matched interaction of the waves with a random domain structure is developed. The theoretical analysis is performed using the approximation of the intensity fixed within the domain length. It is shown that the developed method allows one to accurately describe the energy exchange between the interacting waves at any values of the efficiency of second harmonic generation. Within the framework of the developed theory, the so-called simulated-annealing algorithm for the formation of the response of the medium is created, which is responsible for the spectral dependence of the efficiency of the second harmonic generation with any specified profiles under conditions of energy depletion of the fundamental radiation.

Nonlinear optical phenomena in nanowires

The theory of second harmonic generation (SHG) in nanowires with different dielectric constants from their environment is developed. A nonuniform distribution of the optical electric field created by image forces results in strong SHG even in materials with inversion symmetry. Integral intensity, spatial distribution, and the polarization of the SHG depend dramatically on the light polarization, as well as on the mutual orientation of the light wave vector and nanowire axis. In semiconducting or dielectric nanowires, the SHG is maximal when the light polarization is parallel to the nanowire axis. In metallic nanowires, the SHG is a function of the light frequency ω being maximal when ω or 2ω coincide with the surface plasmon frequency. At the first resonant frequency, the SHG is caused mostly by light polarized perpendicularly and at the second frequency—parallel to the nanowire axis.

Exact solution for second harmonic generation in XFELs

The generation of harmonic radiation via a non-linear mechanism, driven by electron bunching at the fundamental frequency, is an important option in the operation of high-gain Free-Electron Lasers (FELs). By utilizing harmonic generation at a large scale facility, the production of intense radiation at shorter wavelengths for the same electron beam energy is feasible. This paper describes a theory of second harmonic generation in planar undulators with particular attention to X-Ray FELs (XFELs). Our study is based on an exact analytical solution of Maxwell’s equations, derived with the help of a Green’s function method. Up-to-date theoretical understanding of second harmonic generation is limited to the estimation of the total radiation power, which is based on a comparison of the right hand side of the wave equation for the first harmonic with the right hand side of the equation for the second harmonic, the latter being incorrectly modified. The exact solution should be obtained by solving the wave equation itself. Our work yields correct parametric dependencies and specific predictions of additional properties such as polarization, angular distribution of the radiation intensity and total power. The most surprising prediction is the presence of a vertically polarized part of the second harmonic radiation, whereas current knowledge predicts a horizontally polarized field.

Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media

There has been developed the theory of second harmonic generation of the intensive laser fields in the existence of both quadratic and cubic polarization in medium in the constant-intensity approximation accounting for the reverse effect of the excited wave on the exciting one and simultaneously allowing us to take into account phase mismatch and the damping of all the interacting waves. It is shown that the changes of pump intensity through self-phase and cross-phase modulation processes effect optimum phase relationship between interacting waves, the change of the spatial beating period. The conditions of compensating undesirable phase shifts between interacting waves have been determined, the analytical expressions for calculation of optimum values of phase mismatch, length of noncentrosymmetrical medium and the spatial beating period are offered. It is shown that in the absence of linear phase mismatch with an increase of basic radiation intensity the spatial beating period is being reduced. The numerous analysis has been made of frequency doubling process efficiency for KDP and LiNbO3 crystals.