Second harmonic generation in a two dimensional nonlinear quasi-crystal is demonstrated for the first time. Temperature and wavelength tuning of the crystal reveal the uniformity of the pattern while angle tuning reveals the dense nature of the crystal's Fourier spectrum. These results compare well with theoretical predictions showing the excellent uniformity of the crystal and suggest that more complicated ``nonlinear holograms'' should be possible.
Quasi-crystals are intriguing as they exhibit rotational symmetry and long range ordering but lack translational symmetry. 2-dimensional metal-dielectric patterns are interesting to make use of surface plasmon polariton (SPP) mediated local field enhancement and for near dispersionless SPP modes. In plasmonic crystals, the orientation and periodicity of the pattern dictate the polarization response and the discrete plasmon resonances while the interfaces define the plasmon dispersion. However, unique properties of plasmonic quasicrystals lead to polarization independence, designable k-space an…
Metasurface, a kind of two-dimensional structured medium, represents a novel platform to manipulate the propagation of light at subwavelength scale. In linear optical regime, many interesting topics such as planar metalens, metasurface optical holography and so on have been widely investigated. Recently, metasurfaces go into nonlinear optical regime. While it is recognized that the local symmetry of the meta-atoms plays vital roles, its relationship with global symmetry of the nonlinear metasurfaces remains elusive. According to the Penrose tiling and the newly proposed hexagonal quasicrystall…
Stable vortices with topological charge of 3 and 4 are examined numerically and analytically in photonic quasicrystals created by interference of 5 as well as 8 beams, in the cases of cubic as well as saturable nonlinearities. These structures are experimentally realizable, including a prototypical example of a stable charge 4 vortex. Direct numerical simulations corroborate the analytical and numerical linear stability analysis predictions.
We demonstrate theoretically and numerically that a warm fluid model of a plasma supports space-time quasicrystalline structures. These structures are highly nonlinear, two-phase, ion acoustic waves that are excited autoresonantly when the plasma is driven by two small amplitude chirped-frequency ponderomotive drives. The waves exhibit density excursions that substantially exceed the equilibrium plasma density. Remarkably, these extremely nonlinear waves persist even when the small amplitude drives are turned off. We derive the weakly nonlinear analytical theory by applying Whitham's averaged …
We prescribe a novel approach for continuously tailoring the local phase of the nonlinear polarizability which can lead to an arbitrary phase profile for harmonic generations. The introduced phase of the nonlinear polarizability is inherently a geometric Berry phase arising from the spin rotation coupling of light in the nonlinear regime. This approach provides new routes for controlling the optical nonlinear processes.
We propose novel multi-phase-matched process that starts with generation of a pair of symmetric second-harmonic waves. Each of them interacts again with the fundamental wave to produce two constructively interfering third harmonic waves collinear to the fundamental input wave.
We study the second-harmonic generation in left-handed metamaterials with a quadratic nonlinear response. We demonstrate a novel type of the exact phase matching between the backward propagating wave of the fundamental frequency and the forward propagating wave of the second harmonics. We show that this novel parametric process can convert a surface of the left-handed metamaterial into an effective mirror totally reflecting the second harmonics generated by an incident wave. We derive and analyze theoretically the coupled-mode equations for a semi-infinite nonlinear metamaterial. We also study…
The resonance effects on the optical second harmonic generation from 140 nm silver nanoparticles is studied experimentally by hyper-Rayleigh scattering and numerically by finite element method calculations. We find that the interferences between the broad dipolar and narrow octupolar surface plasmon resonances leads to nonlinear Fano profiles that can be externally controlled by the incident polarization angle. These profiles are responsible for the nonlinear plasmon-induced transparency in the second harmonic generation.
We demonstrate the simultaneous generation of second and third harmonic signals from a telecom wavelength pump in a gallium phosphide (GaP) microdisk. Using analysis of the power scaling of both the second and third harmonic outputs and calculations of nonlinear cavity mode coupling factors, we study contributions to the third harmonic signal from direct and cascaded sum frequency generation processes. We find that despite the relatively high material absorption in gallium phosphide at the third harmonic wavelength, both of these processes can be significant, with relative magnitudes that depe…
We investigate the generation of ultraviolet (UV) second-harmonic radiation on the boundary of a UV transparent crystal, which is derived from the automatic partial phase matching of the incident wave and the total internal reflection. By adhering to another UV non-transparency crystal with larger second-order nonlinear coefficient χ^{(2)}, an nonlinear interface with large disparity in χ^{(2)} is formed and the enhancement of UV second-harmonic radiation is observed experimentally. The intensity of enhanced second harmonic wave generated at the nonlinear interface was up to 11.6 times at the …
Third-harmonic generation can be realized via both $χ^{(3)}$ and cascaded $χ^{(2)}$ nonlinear processes in a triply-resonant microcavity. It is still unknown how these processes interfere with each other and the optimization of the conversion efficiency still remains as a question. In this work, the interplay between the direct third-harmonic generation and the cascaded process combining of the second-harmonic generation and the sum-frequency generation are investigated. It is found that the interference effect between these two processes can be used to improve the conversion efficiency. By op…
We demonstrate that a superoscillating in time signal may be obtained as a nonlinear response on a harmonic low-frequency input. Using the realization of a superoscillating function proposed by (Huang et al. 2007 J. Opt. A: Pure Appl. Opt. 9 S285-8) as an example, we synthesize the response function of such a nonlinear transformer and investigated its robustness with respect to the frequency and amplitude variations of the input signal.
The nonlinear properties of a metamaterial sample composed of double-layer metallic patterns and voltage controllable diodes are experimentally investigated. Second harmonics and spectrum modulations are clearly observed in a wide band of microwave frequencies, showing that this kind of metamaterial is not only tunable by low DC bias voltage, but also behaves strong nonlinear property under a small power incidence. These properties are difficult to be found in normal, naturally occurring materials.
We present perturbation theory for analysis of generic third-order nonlinear processes in graphene integrated photonic structures. Optical response of graphene is treated as the nonlinear boundary condition in Maxwell equations. The derived models are applied for analysis of third harmonic generation in a graphene coated dielectric micro-fibre. The efficiency of up to few percent is predicted when using sub-picosecond pump pulses with energies of the order of $0.1$nJ in a sub-millimeter long fibre, when operating near the resonance of the graphene nonlinear conductivity $\hbarω=(2/3)E_F$.
The second harmonic generation is one of the most important applications of nonlinear effect, which has attracted great interests in nonlinear optics and microwave in the past decades. To the best of our knowledge, however, generating the second harmonics of surface plasmon polaritons (SPPs) has not been reported. Here, we propose to generate the second harmonics of spoof SPPs with high efficiency at microwave frequencies using subwavelength-scale nonlinear active device integrated on specially designed plasmonic waveguides, which are composed of two ultrathin corrugated metallic strips printe…
We study the properties of a tunable nonlinear metamaterial operating at microwave frequencies. We fabricate the nonlinear metamaterial composed of double split-ring resonators and wires where a varactor diode is introduced into each resonator so that the magnetic resonance can be tuned dynamically by varying the input power. We show that at higher powers the transmission of the metamaterial becomes power dependent, and we demonstrate experimentally power-dependent transmission properties and selective generation of higher harmonics.
We describea theoretical analysis of the nonlinear dynamics of third-harmonic generation ($ω\to3ω$) via Kerr ($\chithree$) nonlinearities in a resonant cavity with resonances at both $ω$ and $3ω$. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of $\sim V/Q^2$ where $V$ is the modal volume and $Q$ is the lifetime, and can even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles.…
Optical microcavities are often used to realize enhanced nonlinear optical interactions for highly efficient second-harmonic generation. With increased pump power, the efficiency of nonlinear frequency conversion can be increased further, while some other unwanted nonlinear effects will also emerge, leading to complicated dynamics or instability. Here, we study the interplay between cascaded second- and third-order nonlinear processes and investigate their impact on the second-harmonic generation in microcavities. It is found that the non-degenerate optical parametric oscillation (OPO) appears…
Nonlinear signal even from a single molecule becomes visible at hot spots of plasmonic nanoparticles. In these structures, Fano resonances can control the nonlinear response in two ways. \textit{(i)} A linear Fano resonance can enhance the hot spot field, resulting enhanced nonlinear signal. \textit{(ii)} A nonlinear Fano resonance can enhance the nonlinear signal without enhancing the hot spot. In this study, we compare the enhancement of second harmonic signal at the steady-state obtained via these two methods. Since we are interested in the steady-state signal, we adapt a linear enhancement…
We propose an extensive discussion on the homogenization and scattering analysis of second-order nonlinear metasurfaces. Our developments are based on the generalized sheet transition conditions (GSTCs) which are used to model the electromagnetic responses of nonlinear metasurfaces. The GSTCs are solved both in the frequency domain, assuming an undepleted pump regime, and in the time-domain, assuming dispersionless material properties but a possible depleted pump regime. Based on these two modeling approaches, we derive the general second-harmonic reflectionless and transmissionless conditions…
Using harmonic emission from circularly polarized undulator is a procedure that is normally employed on synchrotron beamlines in order to extend the covered spectral range. A similar capability is likewise beneficial for next generation free-electron lasers. In this paper, we perform a first quantitative experimental analysis of the angular distribution of free-electron laser harmonic emission from helical undulators. Experimental results are compared to those obtained by means of a theoretical model based on the paraxial solution of Maxwell's equations.
Inclusion of the instantaneous Kerr nonlinearity in the FDTD framework leads to implicit equations that have to be solved iteratively. In principle, explicit integration can be achieved with the use of anharmonic oscillator equations, but it tends to be unstable and inappropriate for studying strong-field phenomena like laser filamentation. In this paper, we show that nonlinear susceptibility can be provided instead by a harmonic oscillator driven by a nonlinear force, chosen in a way to reproduce the polarization obtained from the solution of the quantum mechanical two level equations. The re…
We present experimental results on the observation of a bulk second-order nonlinear susceptibility derived from both free-space and integrated measurements in silicon nitride. Phase-matching is achieved through dispersion engineering of the waveguide cross-section, independently revealing multiple components of the nonlinear susceptibility, namely X(2)yyy and X(2)xxy. Additionally, we show how the generated second-harmonic signal may be actively tuned through the application of bias voltages across silicon nitride. The nonlinear material properties measured here are anticipated to allow for th…
The recent measurement of negative higher-order Kerr effect (HOKE) terms in gases has given rise to a controversial debate, fed by its impact on short laser pulse propagation. By comparing the experimentally measured yield of the third and fifth harmonics, with both an analytical and a full comprehensive numerical propagation model, we confirm the absolute and relative values of the reported HOKE indices.
State-of-the-art experiments employ strong ultrafast optical fields to study the nonlinear response of electrons in solids on an attosecond time-scale. Notably, a recent experiment retrieved a 3rd order nonlinear susceptibility by comparing the nonlinear response induced by a strong laser field to a linear response induced by the otherwise identical weak field. In parallel, experiments have demonstrated high harmonic generation (HHG) in solids, a highly nonlinear process that until recently had only been observed in gases. The highly nonlinear nature of HHG has the potential to extract even hi…
Scattering has usually be considered as detrimental for optical focusing or imaging. Recently, more and more research has shown that strongly scattering materials can be utilized to focus coherent light by controlling or shaping the incident light. Here, purposeful focusing of second-harmonic waves, which are generated and scattered from nonlinear turbid media via feedback-based wavefront shaping is presented. This work shows a flexible manipulation of both disordered linear and nonlinear scattering signals, indicating more controllable degrees of freedom for the description of turbid media. T…
Optical second-harmonic generation (SHG) is studied for the confined geometry of a circular cylindrical waveguide or optical fiber. A model situation of high symmetry is considered where the material with nonlinear susceptibility is isotropic and distributed in radially symmetric manner about the axis. In addition it is assumed that the material of high second-order nonlinearity consists of a thin circular layer, with periodic variation in the axial direction - similar to a usual quasi-phasematched configuration. One can optimize the efficiency of SHG by choosing the period of the array such t…
Organic cavity polaritons are bosonic quasi-particles that arise from the strong interaction between organic molecular excitons and photons within microcavities. The spectral dispersion of third harmonic generation near resonance with the cavity polariton states is studied experimentally via angle-resolved reflected third harmonic generation measurements with several pump wavelengths. In addition, a three-step nonlinear optical transfer matrix model is used to simulate the third harmonic generation using the sum-over-states dispersive nonlinear coefficients, which include hybrid exciton-photon…
Since the first observation of second harmonic generation (SHG), there have been extensive studies on this nonlinear phenomenon not only to clarify its physical origin but also to realize unconventional functionalities. Nonetheless, a widely accepted model of SHG with rigorous experimental verification that describes the contributions of different underlying microscopic mechanisms is still under debate. Here, we examine second harmonic far-field radiation patterns over a wide angle from metallic structures with different resonances, to reveal the structure-dependent contributions from distinct…
Plasmonic waveguides provide an integrated platform to develop efficient nanoscale ultrafast photonic devices. Theoretical models that describe nonlinear optical phenomena in plasmonic waveguides, usually, only incorporate bulk nonlinearities, while nonlinearities that arise from metallic constituents remained unexplored. In this work, we present a method that enables a generalized treatment of the nonlinearities present in plasmonic waveguides and use it to calculate second-harmonic generation from free electrons through a hydrodynamic nonlocal description. As a general application of our met…
Vectorial nonlinear optics refers to the investigation of optical processes whose nonlinear polarization (NP) undergoes spin-orbit coupling (SOC) interactions where, in general, the driving light field or the new field generated by the interaction containing the SOC property. To contribute to fundamental knowledge in this domain, we examine the type-II second harmonic generation (SHG) induced by vectorial laser modes. First, we provide a general theory to analyze the vectorial SHG process. Second, by using two typical vector modes as examples, we show how the SOC of the pump field dictates non…
A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying semi-infinite regions, and (ii) two- and three-dimensional (2D and 3D) quasiperiodic (QP) patterns, which are built as superpositions of plane-wave modes with incommensurate spatial periodicities. These topics are selected for the present article, dedicated to the 70th birthday of Professor M. I. Tribelsky, due to the impact made on them by papers published by P…
We study the families of nonlinear modes described by the nonlinear Schrödinger equation with the PT-symmetric harmonic potential $x^2-2iαx$. The found nonlinear modes display a number of interesting features. In particular, we have observed that the modes, bifurcating from the different eigenstates of the underlying linear problem, can actually belong to the same family of nonlinear modes. We also show that by proper adjustment of the coefficient $α$ it is possible to enhance stability of small-amplitude and strongly nonlinear modes comparing to the well-studied case of the real harmonic pote…
Beams that experience third-harmonic generation (THG) also experience Kerr effects. With Kerr effects, beams do not take simple Gaussian shapes, but exhibit nonlinear dynamics. These nonlinear dynamics have an effect on the THG accumulated by focusing and then diverging beams. We formulate a self-consistent and complete set of nonlinear Schroedinger equations for a pair of coupled beams -- a fundamental and its third-harmonic. Numerical simulations show that the Kerr nonlinearities allow some third-harmonic to propagate to the far-field even for zero or negative phase-mismatch. This is because…
Coherent Harmonic Generation (CHG), and in particular Nonlinear Harmonic Generation (NHG), is of importance for both short wavelength Free-Electron Lasers (FELs), in relation with the achievement of shorter wavelengths with a fixed electron-beam energy, and high-average power FEL resonators, in relation with destructive effects of higher harmonics radiation on mirrors. In this paper we present a treatment of NHG from helical wigglers with particular emphasis on the second harmonic. Our study is based on an exact analytical solution of Maxwell's equations, derived with the help of a Green's fun…
Nonlinear frequency conversion in optics can originate the coherent light at the wavelength where it is hard or unlikely to achieve by directly lasing and is a fundamental topic in science and engineering covering both classical and quantum regions. The critical requirement for efficient nonlinear frequency conversion is phase-matching, a momentum conservation relation between the fundamental and harmonic light. It is dreaming of a technique for compensating the phase-mismatching in universal nonlinear materials and wavelength ranges, since the phase-matching was proposed in 1962. Here, an add…
We present a novel Rotational Anisotropy Nonlinear Harmonic Generation (RA-NHG) apparatus based primarily upon reflective optics. The data acquisition scheme used here allows for fast accumulation of RA-NHG traces, mitigating low frequency noise from laser drift, while permitting real-time adjustment of acquired signals with significantly more data points per unit angle rotation of the optics than other RA-NHG setups. We discuss the design and construction of the optical and electronic components of the device and present example data taken on a GaAs test sample at a variety of wavelengths. Th…
We present a method of performing high-speed rotational anisotropy nonlinear optical harmonic generation experiments at rotational frequencies of several hertz by projecting the harmonic light reflected at different angles from a sample onto a stationary position sensitive detector. The high rotational speed of the technique, $10^3$ to $10^4$ times larger than existing methods, permits precise measurements of the crystallographic and electronic symmetries of samples by averaging over low frequency laser power, beam pointing, and pulse width fluctuations. We demonstrate the sensitivity of our t…
Time-dependent nonlinear media, such as rapidly generated plasmas produced via laser ionization of gases, can increase the energy of individual laser photons and generate tunable high-order harmonic pulses. This phenomenon, known as photon acceleration, has traditionally required extreme-intensity laser pulses and macroscopic propagation lengths. Here, we report on a novel nonlinear material$-$an ultrathin semiconductor metasurface$-$that exhibits efficient photon acceleration at low intensities. We observe a signature nonlinear manifestation of photon acceleration: third-harmonic generation o…
Nonlinear metasurfaces offer new paradigm for boosting optical effect beyond limitations of conventional materials. In this work, we present an alternative way to produce pronounced third-harmonic generation (THG) based on nonlinear field resonances rather than linear field enhancement, which is a typical strategy for achieving strong nonlinear response. By designing and studying a nonlinear plasmonic-graphene metasurface at terahertz regime with hybrid guided modes and bound states in the continuum modes, it is found that a THG with a narrow bandwidth can be observed, thanks to the strong res…
We construct families of ordinary and gap solitons (GSs), including solitary vortices, in the two-dimensional (2D) system based on the nonlinear-SchrÄodinger/Gross-Pitaevskii equation with the 2D or quasi-1D (Q1D) periodic linear potential, combined with the periodic modulation of the cubic nonlinearity (also in the 2D or Q1D form), which is, generally, incommensurate with the linear potential, thus forming a \nonlinear quasicrystal". Stable vortices are built as complexes of four peaks with the separation between them equal to the double period of the linear potential. The system may be reali…
We study the influence of third-harmonic generation (THG) and negative frequency polarization terms in the self-phase modulation (SPM) of short and intense pulses in Kerr media. We find that THG induces additional symmetric lobes in the SPM process. The amplitude of these new sidebands are greatly enhanced by the contributions of the negative frequency Kerr (NFK) term and the shock operator. We compare our theoretical predictions based on the analytical nonlinear phase with simulations carried out by using the full unidirectional pulse propagation equation (UPPE).
We demonstrate supermode-based second harmonic generation in an integrated nonlinear interferometer made of linear and nonlinear directional couplers. We use a fully-fibered pump shaper to demonstrate second harmonic generation pumped by the symmetric or anti-symmetric fundamental spatial modes. The selection of the pumping mode and thus of a specific SHG spectral profile is achieved through the selection of the fundamental wavelength and via a robust phase setting scheme. We use two methods: either post-selecting or actively setting the pumping mode. Such a modal phase matching paves the way …
We study specific features of resonant mode conversion in nonlinear waveguides stimulated by the bi-harmonic longitudinal modulation of its parameters, which includes changes of the waveguide depth as well as its bending (in the one-dimensional case) or spiraling (in the two-dimensional case). We demonstrate the possibility of simultaneous excitation of higher-order modes of different parities and topologies with controllable energy weights. The output mode composition is highly sensitive to the variation in the input power and detuning from the resonant modulation frequency.
In this work, the polar tetrahedron [PN$_2$O$_2$] was revealed as a new deep-ultraviolet (deep-UV) nonlinear optically active unit. Accordingly, a thermodynamically stable compound (PNO) consisting of the polar [PN$_2$O$_2$] units was predicted and suggested as a promising candidate of deep-UV nonlinear optical (NLO) material. Compared with other deep-UV materials known to date, PNO possesses the strongest second harmonic generation (SHG) coefficient (about 6 times that of KH$_2$PO$_4$ (KDP)). Moreover, its three-dimensional connectivity endows it with good mechanical and thermal properties. T…
Topology-driven nonlinear light-matter effects open up new paradigms for both topological photonics and nonlinear optics. Here, we propose to achieve high-efficiency second-harmonic generation in a second-order photonic topological insulator. Such system hosts highly localized topological corner states with large quality factors for both fundamental and second harmonic waves, which could be matched perfectly in frequency by simply tuning the structural parameters. Through the nonlinear interaction of the doubly resonant topological corner states, unprecedented frequency conversion efficiency i…
We present a method, based on noncollinear second harmonic generation, to evaluate the non-zero elements of the nonlinear optical susceptibility. At a fixed incidence angle, the generated signal is investigated by varying the polarization state of both fundamental beams. The resulting polarization charts allows to verify if Kleinman symmetry rules can be applied to a given material or to retrieve the absolute value of the nonlinear optical tensor terms, from a reference measurement. Experimental measurements obtained from Gallium Nitride layers are reported. The proposed method does not requir…
Speckle patterns generated in a disordered medium carry a lot of information despite the apparent complete randomness in the intensity pattern. When the medium possesses $χ^{(2)}$ nonlinearity, the speckle is sensitive to the phase of the incident fundamental light, as well as the light generated within. Here, we examine the speckle decorrelation in the fundamental and second-harmonic transmitted light as a function of varying power in the fundamental beam. At low incident powers, the speckle patterns produced by successive pulses exhibit strong correlations, that decrease with increasing powe…
Intense extreme-ultraviolet (XUV) pulses enable the investigation of XUV-induced nonlinear processes and are a prerequisite for the development of attosecond pump - attosecond probe experiments. While highly nonlinear processes in the XUV range have been studied at free-electron lasers (FELs), high-harmonic generation (HHG) has allowed the investigation of low-order nonlinear processes. Here we suggest a concept to optimize the HHG intensity, which surprisingly requires a scaling of the experimental parameters that differs substantially from optimizing the HHG pulse energy. As a result, we are…
Strong light-matter coupling within electromagnetic environments provides a promising path to modify and control chemical and physical processes. The origin of the enhancement of nonlinear optical processes such as second-harmonic and third-harmonic generation (SHG and THG) due to strong light-matter coupling is attributed to distinct physical effects which questions the relevance of strong coupling in these processes. In this work, we leverage a first-principles approach to investigate the origins of the experimentally observed enhancement of resonant SHG and THG under strong light-matter cou…
We present a method to measure quadratic Terahertz optical nonlinearities in Terahertz time-domain spectroscopy. We use a rotating linear polarizer (a polarizing chopper) to modulate the amplitude of the incident THz pulse train. We use a phase-sensitive lock-in detection at the fundamental and the second harmonic of the modulation frequency to separate the materials' responses that are linear and quadratic in Terahertz electric field. We demonstrate this method by measuring the quadratic Terahertz Kerr effect in the presence of the much stronger linear electro-optic effect in the (110) GaP cr…
We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions by perturbation theory, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of H…
In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time (PT)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with PT -symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral metho…
