Third-harmonic Wave Research Articles

Bianisotropic Metasurface for Spatial Separation of Multiple Harmonic Waves

An approach to tuning harmonic responses of surface impedance is introduced to spatially separate fundamental, third-harmonic, and fifth-harmonic waves by focusing on the physical implementation perspective. The concept of the bianisotropic metasurface is adopted to refract the wave toward different angles by assigning identical phase gradients at fundamental and third-harmonic frequencies. On the other hand, high-sheet impedance is realized by maintaining the in-phase condition at the fifth harmonic to bypass the incident wave without refraction. The target properties of each unit cell are implemented using I-shaped patterns based on a three-stacked-layer topology. Different sizes and intervals are applied for the I-shaped patterns, and only their horizontal lengths are tuned to achieve the desired sheet impedances. The feasibility of the proposed approach is validated for the fundamental frequency of 10 GHz and two of odd harmonics through far-and near-field measurements. The results confirm that the fabricated metasurface separates the superimposed harmonic waves toward 44 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> , 10 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> , and 0 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> with the measured refraction efficiencies of 65.6%, 67.2%, and 74% at 10, 30, and 50 GHz, respectively.

High Efficiency Third-Harmonic Generation in a Medium with Quadratic Susceptibility Due to Cubic-like Nonlinearity Caused by Cascaded Second-Harmonic Generation

Third-harmonic generation (THG) is of interest for its various applications. We propose using the cascaded second-harmonic generation (SHG) to implement the frequency conversion process, which is similar to that occurring in a medium with cubic susceptibility. Physically, the process is based on consecutive generation of the second-harmonic and the sum frequency in the same crystal with quadratic susceptibility at large phase mismatching between the fundamental wave and the second-harmonic wave. In this case, at phase matching between the fundamental wave and the third-harmonic wave, THG occurs with high efficiency. To demonstrate such a possibility theoretically, we apply the multi-scale method to a set of Schrödinger equations, describing a three-wave interaction with the frequencies ω,2ω and 3ω in a medium with quadratic susceptibility, to derive modified equations describing the frequency tripling process. These equations are solved without using the fundamental wave energy non-depletion approximation. A THG efficiency equal to 94.5% is predicted theoretically. The analytical solution is confirmed by computer simulation results. We study how various factors, such as the incident pulse intensity, phase mismatching between interacting waves, group velocity mismatching of the pulses, and second-order dispersion of the wave packets influence the THG process.

Modeling of dual frequency combs and bistable solitons in third-harmonic generation

Phase-matching of the third-harmonic generation process can be used to extend the emission of radiation from Kerr microresonators into new spectral regions far from the pump wavelength. Here, we present a theoretical mean-field model for optical frequency combs in a dissipative and nonlinear χ(3)-based cavity system with parametric coupling between fundamental and third-harmonic waves. We investigate temporally dispersive dual-comb generation of phase-matched combs with broad bandwidth and anomalous dispersion of the fundamental field, individuating conditions for accessing a multistable regime that simultaneously supports two types of coupled bright cavity solitons. These bistable cavity solitons coexist for the same pump power and frequency detuning, while featuring dissimilar amplitudes of their individual field components. Third-harmonic generation frequency combs grant telecom pump laser sources a simultaneous and direct access to both the near-infrared and the visible regions, which may prove advantageous for the development of optical clocks and sensing applications.

Significant enhancement of third harmonic generation in graphene layers placed on a random grating supporting plasmonic Anderson localized states

We first theoretically investigate the third-harmonic generation (THG) in a metasurface in which trilayer graphene is deposited over a periodic structure of GaP-CaF2 unit cells. The results of simulation confirm that the output intensity of third-harmonic wave enhances by 229 times around the resonance frequency compared to a bare trilayer graphene surrounded by air. This is due to increase of nonlinear response because of the field enhancement in the graphene layer owing to the excitation of graphene plasmons. Second, to obtain more enhancement for THG, we propose and simulate a metasurface in which the trilayer graphene is deposited on a disordered GaP-CaF2 grating. For this random structure, significant THG occurs at multiple resonance frequencies induced by Anderson localization effect. Our simulation results demonstrate that the maximum output intensity of THG wave increases by 125 times compared to the periodic structure and 28,625 times compared to the bare trilayer graphene. The effects of disorder level are investigated on the output intensity of the THG signal. The number of peaks and the maximum value of THG wave increase with the disorder level. Finally, we study dependence of THG output intensity on the Fermi energy level of graphene. The emission peaks of the THG wave are blue-shifted as Fermi energy increases. Our theoretical prediction of significant enhancement of THG in the graphene based disordered structures needs some experimental evidences in the future to be employed in different applications such as imaging of biological tissues.

Third-harmonic-assisted four-wave mixing in a chip-based microresonator frequency comb generation.

Microcombs generated in photonic integrated circuits can provide broadband and coherent optical frequency combs with a high repetition rate from microwave to terahertz. Coherent microcombs formed in normal group velocity dispersion microresonators usually have a flat-top temporal profile, called platicon. Here, we propose a novel scheme to generate platicon in Si3N4 microresonator with the assistance of third-harmonic generation. The nonlinear coupling between the fundamental and the third-harmonic waves that draws support from third-order sum/difference frequency generation provides a new mechanism to achieve the phase matching of four-wave mixing in normal dispersion microresonators. We show that single or multiple platicons can be obtained by changing the third-harmonic nonlinear coupling strength and phase matching condition for third-order sum/difference frequency generation. Our work provides a promising solution to facilitate coherent and visible microcomb generation in a pure χ(3) microresonator, which is potential for self-referencing combs and optical clock stabilization.

Early fatigue damage evaluation based on nonlinear Lamb wave third-harmonic phase velocity matching

In this paper, third-harmonic phase velocity matching nonlinear Lamb waves are used for the early fatigue damage detection of aluminum alloys. An analysis is conducted regarding the relationship of third harmonic and third-order nonlinear parameters with fatigue accumulation. According to the analytical results, the amplitude of the third harmonic increases when the fatigue life falls below about 80%. However, when the fatigue life exceeds 80%, the third harmonic starts to decrease, and the third-order nonlinear parameters increase progressively before showing a downward trend. It can be found out that micro-cracks emerge with the accumulation of fatigue, thus reducing the amplitude of the third harmonic and causing the third-order nonlinear parameters to increase gradually and then decline. Furthermore, it is proposed to take the third harmonic amplitude and the third order nonlinear parameters as the indicators of early fatigue damage.

High relative-intensity blue light of supercontinuum generation in photonic crystal fibers

A distinct blue peak at 430 nm of the supercontinuum spectrum is generated in 4.6 m photonic crystal fiber with a high air-filling ratio, which is pumped by a 200 ps pulse laser at 1 µm. The amplitude of the blue peak is higher than that of the remaining pump pulse, and the relative intensity blue light of supercontinuum generation is 114%. We have systematically analyzed the formation mechanism and influence factors of the distinct blue peak with different fiber lengths and input pulses. The experimental results show that the wavelength position of the blue peak in the supercontinuum spectrum is almost unchanged, and both the fiber length and pulse width have a great influence on the intensity of the blue peak. In addition, we have discussed the fact that the dynamic changes of the blue peak may be affected by the ∼ 2200 n m peak through cross-phase modulation (XPM). It is possible that the high relative intensity of the blue peak in high-order modes may be originated from the XPM effect and the third-harmonic wave of the fundamental mode.

Investigation of Material Nonlinearity Measurements Using the Third-Harmonic Generation

With respect to harmonic generation measurements in isotropic materials, the amplitude of the third-harmonic wave generally depends on both β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (the square of the second-order nonlinear parameter β) and y (the third-order nonlinear parameter). Therefore, some care should be taken when measuring these parameters using the third-harmonic amplitude. The purpose of this paper is to investigate detailed theoretical and measurement techniques for the accurate determination of β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> which dominates the third-harmonic amplitude in most biological fluids and crystalline solids. The theory deals with harmonic generation in materials with cubic nonlinearity and defines the formula for measuring β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with corrections for diffraction and attenuation. These corrections are derived from the Westervelt equation and play an important role in the measurement of nonlinear parameters. The third-harmonic amplitude that varies with the distance in water is obtained for the measurement of β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . β is also measured from the second-harmonic amplitude for comparison. We also confirm the required input voltage to stably generate the third harmonic and discuss the effects of diffraction and attenuation correction on the β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> determination. The measured β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> in the propagation range of 0.05-0.2 m agrees well with the square of the measured β, revealing that the third-harmonic amplitude is closely related to β <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , not y. This paper covers comprehensive theories, experimentation, and analysis related to the measurement of third-harmonic generation in isotropic media and can be immediately applied to crystalline solids.

Enhanced third-harmonic generation via s- and p-polarized optical surface modes of 1D photonic crystal structure

Generation of third-harmonic radiation, visible to the naked eye as a collimated green beam, is obtained via femtosecond excitation of the optical surface modes (SMs) propagating along a one-dimensional (1D) photonic crystal (PC) for s- and p-polarizations separately. For both polarizations, the PC SMs exist at the fundamental and third-harmonic frequencies, which allows efficient nonlinear conversion at the phase-matching points. The pattern of the third-harmonic surface wave scattering is detected and modeled, and it is shown that this pattern reveals the mode structure of the PC. Applications of the studied 1D PC structure for the experimental testing of 2D nonlinear materials are discussed.

Broadband Quasi-Phase-Matched Harmonic Generation in an On-Chip Monocrystalline Lithium Niobate Microdisk Resonator.

We reveal a unique broadband natural quasi-phase-matching (QPM) mechanism underlying an observation of highly efficient second- and third-order harmonic generation at multiple wavelengths in an x-cut lithium niobate (LN) microdisk resonator. For light waves in the transverse-electric mode propagating along the circumference of the microdisk, the effective nonlinear optical coefficients naturally oscillate periodically to change both the sign and magnitude, facilitating QPM without the necessity of domain engineering in the micrometer-scale LN disk. The second-harmonic and cascaded third-harmonic waves are simultaneously generated with normalized conversion efficiencies as high as 9.9%/mW and 1.05%/mW^{2}, respectively, thanks to the utilization of the highest nonlinear coefficient d_{33} of LN. The high efficiency achieved with the microdisk of a diameter of ∼30 μm is beneficial for realizing high-density integration of nonlinear photonic devices such as wavelength convertors and entangled photon sources.

Phase-matching and Peak Nonlinearity Enhanced Third-Harmonic Generation in Graphene Plasmonic Coupler

Optical third-harmonic generation (THG) is important for various applications in nanomedicine, photocatalysis, and biosensing, but advances are held back due to small third-order nonlinear susceptibility ${\ensuremath{\chi}}^{(3)}$ and phase mismatch between the fundamental and third-harmonic waves. This work uses a graphene plasmonic coupler to achieve tunable, highly efficient THG through combining phase matching with resonant three-photon transitions. The study not only offers understanding and guidance for the problem at hand, but also points to further exploration of tunable, subwavelength coherent sources and ultracompact signal processing in the midinfrared and terahertz spectral regimes.

Ultrasonic harmonic generation from materials with up to cubic nonlinearity.

This letter considers the combined effects of quadratic and cubic nonlinearity on plane wave propagation in generally anisotropic elastic solids. Displacement solutions are derived that represent the fundamental, second-, and third-harmonic waves. In arriving at the solutions, the quadratic and cubic nonlinearity parameters for generally anisotropic materials are defined. The effects of quadratic and cubic nonlinearity are shown to influence the amplitude and phase of the fundamental wave. In addition, the phase of the third-harmonic depends on a simple combination of the quadratic and cubic nonlinearity parameters. Nonlinearity parameters are given explicitly for materials having isotropic and cubic symmetry. Lastly, acoustic nonlinearity surfaces are introduced, which illustrate the nonlinearity parameters as a function of various propagation directions in anisotropic materials.

Wave propagation in two‐dimensional left‐handed non‐linear transmission line metamaterials

Recent research on harmonic generations in one-dimensional (1D) left-handed non-linear transmission line (LH NLTL) metamaterials has shown that the fundamental wave and its higher harmonics can realise quasi-phase matching, which in turn ensures the high conversion efficiency of the harmonics. In this study, the authors first present the electromagnetic (EM) theory to describe the generation and propagation of both second-harmonic (SH) and third-harmonic (TH) waves in a 2D non-linear transmission line metamaterial. Then, with the aid of the microwave circuit simulations, they fabricate a 2D LH NLTL metamaterial and demonstrate that an EM wave propagation in such medium has three states: left-handed (LH), right-handed (RH), and non-propagation. Furthermore, the response of the medium to applied field intensity shows that it can realise self-tuning between LH and RH states for the TH waves.

Second and third harmonic waves excited by focused Gaussian beams.

Harmonic generation by tightly-focused Gaussian beams is finding important applications, primarily in nonlinear microscopy. It is often naively assumed that the nonlinear signal is generated predominantly in the focal region. However, the intensity of Gaussian-excited electromagnetic harmonic waves is sensitive to the excitation geometry and to the phase matching condition, and may depend on quite an extended region of the material away from the focal plane. Here we solve analytically the amplitude integral for second harmonic and third harmonic waves and study the generated harmonic intensities vs. focal-plane position within the material. We find that maximum intensity for positive wave-vector mismatch values, for both second harmonic and third harmonic waves, is achieved when the fundamental Gaussian is focused few Rayleigh lengths beyond the front surface. Harmonic-generation theory predicts strong intensity oscillations with thickness if the material is very thin. We reproduced these intensity oscillations in glass slabs pumped at 1550nm. From the oscillations of the 517nm third-harmonic waves with slab thickness we estimate the wave-vector mismatch in a Soda-lime glass as Δk(H)= -0.249μm(-1).

Noncollinear third-harmonic generation with large angular acceptance by noncritical phase matching in KDP crystal.

The scheme of prefocusing to focus sum frequency generation (SFG) without a lens is proposed and experimentally verified in this study. Noncollinear type-I noncritical phase-matching SFG to generate a third-harmonic wave with large angular acceptance is presented. The principle of broad angular acceptance and the advantages of this PM configuration are also described in detail. External angular bandwidth of 7.33° for noncollinear SFG was measured in a 2 mm long KH2PO4 (KDP) crystal, which is in reasonably good agreement with the theoretical calculation. The mechanism of broadband SFG and the prefocusing scheme make it possible for the realization of convergent third-harmonic generation without involving a lens, which provides a promising way to avoid damage to optical components during the focusing of high-energy UV light in high-power laser facilities.

Comparison of wave load effects on a TLP wind turbine by using computational fluid dynamics and potential flow theory approaches

Tension Leg Platform (TLP) is one of the concepts which shows promising results during initial studies to carry floating wind turbines. One of the concerns regarding tension leg platform wind turbines (TLPWTs) is the high natural frequencies of the structure that may be excited by nonlinear waves loads. Since Computational Fluid Dynamics (CFD) models are capable of capturing nonlinear wave loads, they can lead to better insight about this concern. In the current study, a CFD model based on immersed boundary method, in combination with a two-body structural model of TLPWT is developed to study wave induced responses of TLPWT in deep water. The results are compared with the results of a potential flow theory-finite element software, SIMO-RIFLEX (SR). First, the CFD based model is described and the potential flow theory based model is briefly introduced. Then, a grid sensitivity study is performed and free decay tests are simulated to determine the natural frequencies of different motion modes of the TLPWT. The responses of the TLPWT to regular waves are studied, and the effects of wave height are investigated. For the studied wave heights which vary from small to medium amplitude (wave height over wavelength less than 0.071), the results predicted by the CFD based model are generally in good agreement with the potential flow theory based model. The only considerable difference is the TLPWT mean surge motion which is predicted higher by the CFD model, possibly because of considering the nonlinear effects of the waves loads and applying these loads at the TLPWT instantaneous position in the CFD model. This difference does not considerably affect the important TLPWT design driving parameters such as tendons forces and tower base moment, since it only affects the mean dynamic position of TLPWT. In the current study, the incoming wave frequency is set such that third-harmonic wave frequency coincides with the first tower bending mode frequency. However, for the studied wave conditions a significant excitation of tower natural frequency is not observed. The high stiffness of tendons which results in linear pitch motion of TLPWT hull (less than 0.02 degrees) and tower (less than 0.25 degrees) can explain the limited excitement of the tower first bending mode. The good agreement between CFD and potential flow theory based results for small and medium amplitude waves gives confidence to the proposed CFD based model to be further used for hydrodynamic analysis of floating wind turbines in extreme ocean conditions.

Walk-off reduction, using an external optical plate and Bessel–Gaussian interaction

To reduce the walk-off angle of the extraordinary third-harmonic ultraviolet wave at 355 nm generated by type II KTiOPO4 and type I β-BaB2O4 optical crystals, and the Gaussian output beam of a Q-switched Nd:YAG laser, a simple theoretical model was developed based on a rotatable BK7 plate of variable thickness. By rotating the plate up to 35° along the beam direction, we reduced the walk-off angle up to ∼ 13%. The same phenomenon is predicted by the model, confirming the performance of the model. It is found that, due to the walk-off effect, the intensity profile of the third-harmonic generation beam is slightly degraded. To compensate for the observed phenomena and further reduce the walk-off, we used a combination of a convex lens and an axicon to transform the beam profile of the interacting fundamental and second-harmonic generation waves to the zero-order Bessel–Gaussian form. As a result, the walk-off is decreased to ∼48.81 mrad, providing ∼30% relative reduction. By using the same BK7 plate rotated up to 35° along the third-harmonic beam direction, the walk-off angle is further reduced to 38.9 mrad. Moreover, it is observed that the beam profile of the emerged Bessel–Gaussian third-harmonic generation beam remains unchanged with no degradation.

Third-harmonic generation at a silver/vacuum interface using surface plasmon polaritons

We present an analytical study of a third-harmonic generation process by surface plasmon polaritons at a Kerr-nonlinear-metal/dielectric interface. Using a planar silver/vacuum interface, we obtain a solution to the frequency-tripling process, showing a semibound third-harmonic surface wave. Unlike the fundamental surface-bound mode, the nonlinear signal radiates into the metal, while remaining bound at the dielectric side of the interface. The propagation of the tripled frequency wave into the metal is generally lossy, except within a narrow range of ultraviolet frequencies, where it is minimally attenuated. The latter is enabled by a transparency window that is unique to the permittivity of silver.

Third-harmonic pulse generation in one-dimensional photonic crystal structures

Enhanced third-harmonic generation in a one-dimensional photonic crystal doped with third-order nonlinear medium was numerically investigated using the multiple-scale method and the split-step Fourier transform. The optimal fundamental frequency for third-harmonic wave generation was determined from the transmission spectrum. The third-harmonic pulse intensities grow, depending on the structure thickness and the fundamental-frequency detuning parameter, which determines the band-edge phase matching condition. Furthermore, the total energy output of third-harmonic pulses, depending on the fundamental-frequency pulse width, may be more than 1000 times the energy produced by a phase-matched bulk medium. A narrow pulse with bandwidth less than the band-edge transmission peak enables high conversion efficiency. The maximum conversion efficiency of the forward component may be 12 to 13 orders of magnitude greater than that of the backward component.

A double-beam magnetron-injection gun for third-harmonic continuous wave 1-THz gyrotron

The concept of a continuous wave 1-kW/1-THz gyrotron operated at the third cyclotron harmonic of the transverse electric TE9,7 operating mode has been developed. To suppress the mode competition effects in a terahertz gyrotron, we propose a scheme with two generating helical electron beams (HEBs) formed in a double-beam triode magnetron-injection gun (MIG), where both emitters of the electron beams are located on a common cathode of the conventional MIG. An optimal geometry of the MIG electrodes is found. It is shown that in a proposed scheme two HEBs having close pitch factors and a moderate velocity spread can be formed. This makes them suitable for high-efficiency single-mode generation in the high frequency gyrotron at high harmonic.