Phase-mismatched Second-harmonic Generation Research Articles

Carrier-envelope offset frequency dynamics of a 10-GHz modelocked laser based on cascaded quadratic nonlinearities.

Cascaded quadratic nonlinearities from phase-mismatched second-harmonic generation build the foundation for robust soliton modelocking in straight-cavity laser configurations by providing a tunable and self-defocusing nonlinearity. The frequency dependence of the loss-related part of the corresponding nonlinear response function causes a power-dependent self-frequency shift (SFS). In this paper, we develop a simple analytical model for the SFS-induced changes on the carrier-envelope offset frequency (fCEO) and experimentally investigate the static and dynamic fCEO dependence on pump power. We find good agreement with the measured dependence of fCEO on laser output power, showing a broad fCEO tuning capability from zero up to the pulse repetition rate. Moreover, we stabilize the relative intensity noise to the -157dBc/Hz level leading to a tenfold reduction in fCEO-linewidth.

Invited Article: Multiple-octave spanning high-energy mid-IR supercontinuum generation in bulk quadratic nonlinear crystals

Bright and broadband coherent mid-IR radiation is important for exciting and probing molecular vibrations. Using cascaded nonlinearities in conventional quadratic nonlinear crystals like lithium niobate, self-defocusing near-IR solitons have been demonstrated that led to very broadband supercontinuum generation in the visible, near-IR, and short-wavelength mid-IR. Here we conduct an experiment where a mid-IR crystal is pumped in the mid-IR. The crystal is cut for noncritical interaction, so the three-wave mixing of a single mid-IR femtosecond pump source leads to highly phase-mismatched second-harmonic generation. This self-acting cascaded process leads to the formation of a self-defocusing soliton at the mid-IR pump wavelength and after the self-compression point multiple octave-spanning supercontinua are observed. The results were recorded in a commercially available crystal LiInS2 pumped in the 3-4 μm range with 85 fs 50 μJ pulse energy, with the broadest supercontinuum covering 1.6-7.0 μm. We measured up 30 μJ energy in the supercontinuum, and the energy promises to scale favorably with an increased pump energy. Other mid-IR crystals can readily be used as well to cover other pump wavelengths and target other supercontinuum wavelength ranges.

Femtosecond mode locking based on adiabatic excitation of quadratic solitons

We demonstrate a new approach for pulse formation in mode-locked lasers, based on exciting intracavity solitons in a two-dimensionally patterned quasi-phase-matching (QPM) grating. Through an adiabatic following process enabled by an apodized QPM crystal, we transiently excite multicolor nonlinear states within the crystal, utilize their advantageous properties for pulse formation and stabilization, and then convert the energy back to the resonating laser pulse before the end of the crystal in order to suppress losses. This idea gives access to large nonlinearities that would otherwise be too lossy for use intracavity. In our case, the states accessed are self-defocusing Kerr-like nonlinearities based on phase-mismatched second-harmonic generation. The QPM device has an additional transverse gradient, for tuning the nonlinearity and to aid in laser self-starting. We demonstrate the technique in a semiconductor saturable absorber mirror mode-locked laser with Yb:CALGO as the gain medium, producing 100 fs pulses at 540 MHz repetition rate, with 760 mW of average output power. We present comprehensive theoretical and numerical modeling of the laser to understand the new mode-locking regime. Our approach offers a flexible and compact route to managing nonlinearities inside laser cavities while suppressing the losses that could otherwise prevent or deteriorate mode-locked operation, and is particularly interesting for highly compact bulk, fiber, and waveguide lasers with gigahertz repetition rates and operating wavelengths from the near- to mid-infrared spectral regions.

SESAM modelocked Yb:CaGdAlO_4 laser in the soliton modelocking regime with positive intracavity dispersion

We demonstrate femtosecond SESAM modelocking in the near-infrared by using cascaded quadratic nonlinearities (phase-mismatched second-harmonic generation, SHG), enabling soliton modelocking in the normal dispersion regime without any dispersion compensating elements. To obtain large and negative self-phase modulation (SPM) we use an intracavity LBO crystal, whose temperature and angles are optimized with respect to SPM, nonlinear losses, and self-starting characteristics. To support femtosecond pulses, we use the very promising Yb:CaGdAlO(4) (CALGO) gain material, operated in a bulk configuration. The LBO crystal provides sufficient negative SPM to compensate for its own GDD as well as the positive GDD and SPM from the gain crystal. The modelocked laser produces pulses of 114 fs at 1050 nm, with a repetition rate of 113 MHz (average output power 1.1 W). We perform a detailed theoretical study of this soliton modelocking regime with positive GDD, which clearly indicates the important design constraints in an intuitive and systematic way. In particular, due to its importance in avoiding multi-pulsed modelocking, we examine the nonlinear loss associated with the cascading process carefully and show how it can be suppressed in practice. With this modelocking regime, it should be possible to overcome the limits faced by current state of the art modelocked lasers in terms of dispersion compensation and nonlinearity management at high powers, suppression of Q-switching in compact GHz lasers, and enabling femtosecond soliton modelocking at very high repetition rates due to the high nonlinearities accessible via cascading combined with eliminating the need for intracavity dispersion compensation.

Few-cycle solitons and supercontinuum generation with cascaded quadratic nonlinearities in unpoled lithium niobate ridge waveguides

Formation and interaction of few-cycle solitons in a lithium niobate ridge waveguide are numerically investigated. The solitons are created through a cascaded phase-mismatched second-harmonic generation process, which induces a dominant self-defocusing Kerr-like nonlinearity on the pump pulse. The inherent material self-focusing Kerr nonlinearity is overcome over a wide wavelength range, and self-defocusing solitons are supported from 1100 to 1900 nm, covering the whole communication band. Single cycle self-compressed solitons and supercontinuum generation spanning 1.3 octaves are observed when pumped with femtosecond nanojoule pulses at 1550 nm. The waveguide is not periodically poled, as quasi-phase-matching would lead to detrimental nonlinear effects impeding few-cycle soliton formation.

Dispersive shock waves in phase-mismatched second-harmonic generation

We investigate wave-breaking and dispersive shock wave formation driven by a pulse undergoing second-harmonic generation in a quadratic medium. We show that the process is accessible in the regime of high phase-mismatch (cascading limit) and weak dispersion. Insight into the phenomenon is obtained by means of a suitable hydrodynamic reduction of the equations that govern the mixing.

Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation

This paper concerns the theoretical, numerical, and experimental study of the second-harmonic-generation (SHG) process under conditions of phase and group-velocity mismatch and aims to demonstrate the dimensionality transition of the SHG process caused by the change of the fundamental wave diameter. We show that SHG from a narrow fundamental beam leads to the spontaneous self-phase-matching process with, in addition, the appearance of angular dispersion for the off-axis frequency components generated. The angular dispersion sustains the formation of the short X pulse in the second harmonic (SH) and is recognized as three-dimensional (3D) dynamics. On the contrary, the large-diameter fundamental beam reduces the number of the degrees of freedom, does not allow the generation of the angular dispersion, and maintains the so-called one-dimensional (1D) SHG dynamics, where the self-phase-matching appears just for axial components and is accompanied by the shrinking of the SH temporal bandwidth, and sustains a long SH pulse formation. The transition from long SH pulse generation typical of the 1D dynamics to the short 3D X pulse is illustrated numerically and experimentally by changing the conditions from the self-defocusing to the self-focusing regime by simply tuning the phase mismatch. The numerical and experimental verification of the analytical results are also presented.

Complete spatial and temporal locking in phase-mismatched second-harmonic generation

We experimentally demonstrate simultaneous phase and group velocity locking of fundamental and generated second harmonic pulses in Lithium Niobate, under conditions of material phase mismatch. In phase-mismatched, pulsed second harmonic generation in addition to a reflected signal two forward-propagating pulses are also generated at the interface between a linear and a second order nonlinear material: the first pulse results from the solution of the homogeneous wave equation, and propagates at the group velocity expected from material dispersion; the second pulse is the solution of the inhomogeneous wave equation, is phase-locked and trapped by the pump pulse, and follows the pump trajectory. At normal incidence, the normal and phase locked pulses simply trail each other. At oblique incidence, the consequences can be quite dramatic. The homogeneous pulse refracts as predicted by material dispersion and Snell's law, yielding at least two spatially separate second harmonic spots at the medium's exit. We thus report the first experimental results showing that, at oblique incidence, fundamental and phase-locked second harmonic pulses travel with the same group velocity and follow the same trajectory. This is direct evidence that, at least up to first order, the effective dispersion of the phase-locked pulse is similar to the dispersion of the pump pulse.

Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses

We simulate and discuss novel spatio-temporal propagation effects that relate specifically to pulsed, phase-mismatched second-harmonic generation in a material characterized by simultaneously negative permittivity and permeability, having finite length. Using a generic Lorentz model for the dielectric permittivity and magnetic permeability, the fundamental and second-harmonic frequencies are tuned so that the corresponding indices of refraction are negative for the pump and positive for the second-harmonic signal. A phase-locking mechanism causes part of the second-harmonic signal generated at the entry surface to become trapped and dragged along by the pump and to refract negatively, even though the index of refraction at the second-harmonic frequency is positive. These circumstances culminate in the creation of an anomalous state consisting of a forward-moving second-harmonic wave packet that has negative wave vector and momentum density, which in turn leads to non-specular reflections at intervening material interfaces. The forward-generated second-harmonic signal trapped under the pump pulse propagates forward, but has all the attributes of a reflected pulse, similar to its twin counterpart generated at the surface and freely propagating backward away from the interface. This describes a new state of negative refraction, associated with nonlinear frequency conversion and parametric processes, whereby a beam generated at the interface can refract negatively even though the corresponding index of refraction at that wavelength is positive.

Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities

We present a detailed study of soliton compression of ultrashort pulses based on phase-mismatched second-harmonic generation (SHG) (i.e., the cascaded quadratic nonlinearity) in bulk quadratic nonlinear media. The single-cycle propagation equations in the temporal domain including higher-order nonlinear terms are presented. The balance between the quadratic (SHG) and the cubic (Kerr) nonlinearity plays a crucial role; we define an effective soliton number—related to the difference between the SHG and the Kerr soliton numbers—and show that it has to be larger than unity for successful pulse compression to take place. This requires that the phase mismatch be below a critical level, which is high in a material where the quadratic nonlinearity dominates over the cubic Kerr nonlinearity. Through extensive numerical simulations we find dimensionless scaling laws, expressed through the effective soliton number, that control the behavior of the compressed pulses. These laws hold in the stationary regime, in which group-velocity mismatch effects are small, and they are similar to the ones observed for fiber soliton compressors. The numerical simulations indicate that clean compressed pulses below two optical cycles can be achieved in a β-barium borate crystal at appropriate wavelengths, even for picosecond input pulses.

Induced Diffraction in Phase-Mismatched Second-HarmonicGeneration

We show analytically that in phase-mismatched second-harmonicgeneration, an effective diffraction is induced at the second-harmonic(SH) frequency. Numerical simulation results agree with the analyticalpredictions. Compared to the case of linear propagation, the effect ofthe overall diffraction at the SH frequency becomes doubled due to theinduced diffraction, which causes an interesting result that the SHbeam width will be larger than that of the fundamental field.

Induced group-velocity dispersion in phase-mismatched second-harmonic generation

We show theoretically and experimentally that, in phase-mismatched second-harmonic generation, an effective group-velocity dispersion is induced at the second-harmonic frequency. Experimental results agree quite well with the theoretical predictions. Although large phase mismatch is required to induce appreciable group-velocity dispersion in birefringent nonlinear crystals, similar effects can be observed in quasi-phase-matched structures at much lower phase mismatch. Some implications of the induced dispersion for pulse propagation in quadratic media are discussed.

Magnetic-field-induced changes in the photorefractive sensitivity of lithium niobate

The effect of a magnetic field on the average photorefractive sensitivity of an undoped LiNbO3 crystal is studied by phase-mismatched second-harmonic generation. The experimental data obtained show the photorefractive sensitivity to reverse sign as the external magnetic field exceeds B1=−0.38±0.04 T. The magnetic field is oriented perpendicular to the crystal optical axis and to the plane of laser radiation polarization. The variation of the photorefractive sensitivity is associated with paramagnetic iron centers, whose photoionization probability depends on the direction of their magnetic moment relative to the optical axis.

Femtosecond nonlinear polarization evolution based on cascade quadratic nonlinearities

We experimentally demonstrate that one can exploit nonlinear phase shifts produced in type I phase-mismatched second-harmonic generation to produce intensity-dependent polarization evolution with 100-fs pulses. An amplitude modulator based on nonlinear polarization rotation provides passive amplitude-modulation depth of up to ~50%. Applications of the amplitude and phase modulations to mode locking of femtosecond bulk and fiber lasers are promising and are discussed.

High-energy pulse compression by use of negative phase shifts produced by the cascade ?^(2):?^(2) nonlinearity

We report a simple optical pulse-compression technique based on quadratic nonlinear media. Negative nonlinear phase shifts are generated by phase-mismatched second-harmonic generation, and the phase-modulated pulses are then compressed by propagation through materials with normal dispersion. Millijoule-energy pulses from a Ti:sapphire amplifier are compressed from 120 to 30 fs, and calculations indicate that compression ratios of >10 are realistically achievable by use of this approach with optimal materials. The insertion loss of the compressor can be less than 10% of the pulse energy, and scaling to higher pulse energies will be straightforward.

Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses

By eliminating the group-velocity mismatch and enhancing the group-velocity dispersion via pulse-front tilting, we obtained the formation of temporal solitary waves in phase-mismatched second-harmonic generation. Experimental and numerical results match well in a wide range of intensities and $\ensuremath{\Delta}k$'s. In a 7 mm BBO crystal, 58 fs, 527 nm pulses are obtained starting from 200 fs pump, at the same wavelength.