Focus on Nonlinear Terahertz Studies

Resulting from the availability of improved sources, research in the terahertz (THz) spectral range has increased dramatically over the last decade, leading essentially to the disappearance of the so-called 'THz gap'. While most work to date has been carried out with THz radiation of low field amplitude, a growing number of experiments are using THz radiation with large electric and magnetic fields that induce nonlinearities in the system under study. This 'focus on' collection contains a number of articles, both experimental and theoretical, in the new subfield of THz nonlinear optics and spectroscopy on various systems, among them molecular gases, superconductors, semiconductors, antiferromagnets and graphene.

We investigate the nonlinear wave-mixing in gases between intense, short optical pulses and long-wavelength fields (mid infrared and terahertz). We show numerically that the beating between the sum- and difference-frequency generation components can be isolated in the spectrogram of the interaction, and can be used to sample the electric field oscillations of the long-wavelength pulses. This, in turn, could be employed as a possible characterization method that provides information on the real electric field amplitude. Our numerical model is supported by spectrally resolved measurements of the four-wave mixing signals obtained from the interaction of intense, single-cycle terahertz fields (λ > 15 μm) and optical pulses (λ ≃ 800 nm, 50 fs duration) in air.

The quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition of the Dirac–Feynmann path integral. The quantum trajectories are a key concept in understanding extreme nonlinear optical phenomena, such as high-order harmonic generation (HHG) and high-order terahertz sideband generation (HSG). In contrast to HHG in atoms and molecules, HSG in semiconductors can have interesting effects due to nontrivial 'vacuum' states of band materials. We find that, in a semiconductor with non-vanishing Berry curvature in its energy bands, the cyclic quantum trajectories of an electron–hole pair under a strong elliptically polarized terahertz field can accumulate a Berry phase. Taking monolayer MoS₂ as a model system, we show that the Berry phase appears as a Faraday rotation angle in the pulse emission from the material under short-pulse excitation. This finding reveals an interesting transport effect in the extreme nonlinear optics regime.

We report the observation of a nonlinear terahertz response of split-ring resonator arrays made of high-temperature superconducting films. Intensity-dependent transmission measurements indicate that the resonance strength decreases dramatically (i.e. transient bleaching) and the resonance frequency shifts as the intensity is increased. Pump–probe measurements confirm this behaviour and reveal dynamics on the few-picosecond timescale.

We theoretically investigate the effects of the excitation frequency on the plateau of high-order terahertz sideband generation (HSG) in semiconductors driven by intense terahertz (THz) fields. We find that the plateau of the sideband spectrum strongly depends on the detuning between the near-infrared laser field and the band gap. We use the quantum trajectory theory (three-step model) to understand the HSG. In the three-step model, an electron–hole pair is first excited by a weak laser, then driven by the strong THz field, and finally recombined to emit a photon with energy gain. When the laser is tuned below the band gap (negative detuning), the electron–hole generation is a virtual process that requires quantum tunneling to occur. When the energy gained by the electron–hole pair from the THz field is less than 3.17 times the ponderomotive energy (U_p), the electron and the hole can be driven to the same position and recombined without quantum tunneling, so that the HSG will have large probability amplitude. This leads to a plateau feature of the HSG spectrum with a high-frequency cutoff at about 3.17U_p above the band gap. Such a plateau feature is similar to the case of high-order harmonics generation in atoms where electrons have to overcome the binding energy to escape the atomic core. A particularly interesting excitation condition in HSG is that the laser can be tuned above the band gap (positive detuning), corresponding to the unphysical 'negative' binding energy in atoms for high-order harmonic generation. Now the electron–hole pair is generated by real excitation, but the recombination process can be real or virtual depending on the energy gained from the THz field, which determines the plateau feature in HSG. Both the numerical calculation and the quantum trajectory analysis reveal that for positive detuning, the HSG plateau cutoff depends on the frequency of the excitation laser. In particular, when the laser is tuned more than 3.17U_p above the band gap, the HSG spectrum presents no plateau feature but instead sharp peaks near the band edge and near the excitation frequency.

We report a diffraction-limited photonic terahertz (THz) source with linewidth <10 MHz that can be used for nonlinear THz studies in the continuous wave (CW) regime with uninterrupted tunability in a broad range of THz frequencies. THz output is produced in orientation-patterned (OP) gallium arsenide (GaAs) via intracavity frequency mixing between the two closely spaced resonating signal and idler waves of an optical parametric oscillator (OPO) operating near λ = 2 μm. The doubly resonant type II OPO is based on a periodically poled lithium niobate (PPLN) pumped by a single-frequency Yb:YAG disc laser at 1030 nm. We take advantage of the enhancement of both optical fields inside a high-finesse OPO cavity: with 10 W of 1030 nm pump, 100 W of intracavity power near 2 μm was attained with GaAs inside cavity. This allows dramatic improvement in terms of generated THz power, as compared to the state-of-the art CW methods. We achieved >25 μW of single-frequency tunable CW THz output power scalable to >1 mW with proper choice of pump laser wavelength.

We investigate the response of multi-layer epitaxial graphene and chemical vapor deposition (CVD)-grown single-layer graphene to strong terahertz (THz) fields. Contrary to theoretical predictions of strong nonlinear response, the transmitted fields exhibit no harmonic generation, indicating that the nonlinear response is limited by fast electron thermalization due to carrier–carrier scattering. The fast electron heating gives rise to large THz transmission enhancement (> 15%) in single-layer CVD graphene at high THz fields (E_THz > 10 kV cm⁻¹). The nonlinear effects exhibit non-Drude behavior in the THz conductivity, where THz fields induce extreme non-equilibrium electron distributions.

The operation of state-of-the-art optoelectronic quantum devices may be significantly affected by the presence of a nonequilibrium quasiparticle population to which the carrier subsystem is unavoidably coupled. This situation is particularly evident in new-generation semiconductor-heterostructure-based quantum emitters, operating both in the mid-infrared as well as in the terahertz (THz) region of the electromagnetic spectrum. In this paper, we present a Monte Carlo-based global kinetic approach, suitable for the investigation of a combined carrier–phonon nonequilibrium dynamics in realistic devices, and discuss its application with a prototypical resonant-phonon THz emitting quantum cascade laser design.

We present a characterization of the combined spatial and spectral properties of the terahertz (THz) and mid-infrared emission from gas plasmas generated and driven by two-colour femtosecond optical pulses. For its use in nonlinear spectroscopy, the impact of the relatively complex spatial profile for both broadband (∼ 10 THz) and ultra-broadband (> 100 THz) emission needs to be considered, in particular for experiments based on z-scan techniques. Here we apply spatially resolved measurements based on both field autocorrelation and sum-frequency (up-conversion) detection. Based on these results, we present simulations of the ultra-broadband profile during its passage through a focal region. In addition to the inherent features of the emission profile due to the generation mechanism in the plasma filament, we also analyse the role of the semconductor (silicon) wafer typically placed after the plasma to discard the optical pump beams, whose photoexcitation also can play a role in the resultant THz profile.

We use a broadband microbolometer array to measure the full three-dimensional (3D) terahertz (THz) intensity profile emitted from a two-color femtosecond plasma and subsequently focused in a geometry useful for nonlinear spectroscopic investigations. Away from the immediate focal region we observe a sharp, conical intensity profile resembling a donut, and in the focal region the beam collapses to a central, Lorentz-shaped profile. The Lorentzian intensity profile in the focal region can be explained by considering the frequency-dependent spot size derived from measurements of the Gouy phase shift in the focal region, and the transition from the donut profile to a central peak is consistent with propagation of a Bessel–Gauss beam, as shown by simulations based on a recent transient photocurrent model (You et al 2012 Phys. Rev. Lett. 109 183902). We combine our measurements to the first full 3D visualization of the conical THz emission from the two-color plasma.

The ultrafast population dynamics in electrically modulated three-well terahertz quantum cascade lasers (QCLs) is studied by using the self-consistent Bloch–Poisson equations. In the modulation process, the non-equilibrium oscillations of the population inversion are found before the population equilibrium recovers. The equilibrium formation time increases nonlinearly with the period number. This phenomenon stems from the non-uniform distribution of the electric potential. In different periods, different responses to the electrical modulation are also explored. An in-depth understanding of electron transport in the cascade structure is obtained. Finally, we demonstrate the feasibility of a modulation frequency up to gigahertz in terahertz QCLs.

The interaction of strong single-cycle THz pulses with the optically induced polarization in germanium quantum wells is studied. With increasing THz field strength, it is observed that the excitonic resonances shift toward higher energy and broaden before weak signatures of a splitting of the exciton line occur. In comparison with high-quality GaAs-based quantum wells, where a much clearer Autler–Townes splitting is observed, the germanium system response is significantly more broadened and shows signatures of a quasi-steady-state behavior due to the intrinsic fast dephasing times dominated by intervalley scattering.

Nonlinear carrier relaxation/recombination dynamics and the resultant stimulated terahertz (THz) photon emission with excitation of surface plasmon polaritons (SPPs) in photoexcited monolayer graphene has been experimentally studied using an optical pump/THz probe and an optical probe measurement. We observed the spatial distribution of the THz probe pulse intensities under linear polarization of optical pump and THz probe pulses. It was clearly observed that an intense THz probe pulse was detected only at the area where the incoming THz probe pulse takes a transverse magnetic (TM) mode capable of exciting the SPPs. The observed gain factor is in fair agreement with the theoretical calculations. Experimental results support the occurrence of the gain enhancement by the excitation of SPPs on THz stimulated emission in optically pumped monolayer graphene.

We investigate high-power terahertz (THz) generation in two-color laser filamentation using terawatt (TW) lasers including a 0.5 TW, 1 kHz system, as well as 2 and 30 TW systems both operating at 10 Hz. With these lasers, we study the macroscopic effect in filamentation that governs THz output energy yields and radiation profiles in the far field. We also characterize the radiation spectra at a broad range of frequencies covering radio–micro-waves to infrared frequencies. In particular, our 1 kHz THz source can provide high-energy (>1 μJ), high average power (>1 mW), intense (>1 MV cm⁻¹) and broadband (0.01–60 THz) THz radiation via two-color filamentation in air. Based on our scaling law, an ∼30 TW laser can produce >0.1 mJ of THz radiation with multi-gigawatt peak power in ∼1.5 m long filamentation.

We demonstrate the direct observation of non-equilibrium intersubband dynamics in a modulation-doped multiple quantum well sample subject to intense few-cycle terahertz (THz) pulses. The transmission spectra show a distinct dependence on the incident THz field strength and contain signatures of a multitude of nonlinear effects that can be observed owing to the large THz-pulse bandwidth. We focus our attention on a case of transient nonlinear refractive index caused by the efficient transfer of electronic population from the ground state to higher-excited states of the quantum well sample. By comparing the experimental results with a one-dimensional finite-difference model going beyond the slowly varying envelope approximation, we prove that, depending on the pulse shape, the leading part of the intense pulse efficiently transfers electrons from the ground state to higher lying excited states. For weak electric fields and small-population transfer, the linear Lorentz model holds. For strong electric fields, up to 55 and 20% of the ground-state electrons are transferred to the first and second excited subbands, respectively, which could lead to the observation of the optical gain.

Nonlinear transmission spectroscopy was performed on a doped Ge:Ga semiconductor using intense THz pulses with different cycle numbers. When single-cycle pulses were used, non-perturbative phenomena, such as the ionization of shallow impurities, competed with the conventional coherent transition, whereas the coherent transition was dominant when multi-cycle pulses were used.

Optical transitions between exciton states in semiconductors—intraexcitonic transitions—usually fall into the terahertz (THz) range and can be resonantly excited with narrowband, intense THz radiation as provided by a free-electron laser. We investigate this situation for two different quantum well structures by probing the near-infrared excitonic absorption spectrum near the band edge. We observe the dynamical Stark—or Autler–Townes—splitting of the 1s exciton ground state and follow its evolution for various THz photon energies and field strengths. The behavior is considerably more complex as compared to the atomic systems. At the highest field strengths, where the Rabi energy is of the same order of magnitude as the exciton level separation, the system cannot be described within the standard framework of a two-level system in rotating wave approximation. When the ponderomotive energy approaches the exciton binding energy, signatures of exciton field ionization are observed.

Table-top sources of intense multi-terahertz (THz) pulses have opened the door to studies of extreme nonlinearities in the previously elusive mid- to far-infrared spectral regime. We discuss two concepts of fully coherent coupling of phase-locked THz pulses with condensed matter. The first approach demonstrates two-dimensional multi-THz spectroscopy of the semiconductor material InSb. By phase- and amplitude-sensitive detection of the nonlinear optical response, we are able to separate incoherent pump–probe signals from coherent four-wave mixing and reveal extremely non-perturbative nonlinearities. While this class of interactions is mediated by the electric field component of the THz pulse, the second approach is complementary, as it demonstrates that, alternatively, the magnetic THz field may be exploited to selectively control the spin degree of freedom in antiferromagnetic NiO.

We have recently reported a study (Ishikawa 2010 Phys. Rev. B 82 201402) on a nonlinear optical response of graphene to a normally incident terahertz radiation pulse within the massless Dirac fermion (MDF) picture, where we have derived physically transparent graphene Bloch equations (GBE). Here we extend it to the tight-binding (TB) model and oblique incidence. The derived equations indicate that interband transitions are governed by the temporal variation of the spinor phase along the electron path in the momentum space and predominantly take place when the electron passes near the Dirac point. At normal incidence, the equations for electron dynamics within the TB model can be cast into the same form of GBE as for the MDF model. At oblique incidence, the equations automatically incorporate photon drag and satisfy the continuity equation for electron density. Single-electron dynamics strongly depend on the model and pulse parameters, but the rapid variations are averaged out after momentum-space integration. Direct current remaining after the pulse is generated in graphene irradiated by an intense monocycle terahertz pulse, even if it is linearly polarized and normally incident. The generated current depends on the carrier-envelope phase, pulse intensity and Fermi energy in a complex manner.

The coherent modulation of electronic and vibrational nonlinearities in atoms and molecular gases by intense few-cycle pulses has been used for high-harmonic generation in the soft x-ray and attosecond regime, as well as for Raman frequency combs that span multiple octaves from the terahertz to petahertz frequency regions. In principle, similar high-order nonlinear processes can be excited efficiently in solids and liquids on account of their high nonlinear polarizability densities. In this paper, we demonstrate the phononic modulation of the optical index of Si and GaAs for excitation and probing near their direct band gaps, respectively at ∼3.4 and ∼3.0 eV. The large amplitude coherent longitudinal optical (LO) polarization due to the excitation of LO phonons of Si (001) and LO phonon–plasmon coupled modes in GaAs (001) excited by 10 fs laser pulses induces effective amplitude and phase modulation of the reflected probe light. The combined action of the amplitude and phase modulation in Si and GaAs generates phonon frequency combs with more than 100 and 60 THz bandwidth, respectively.

We present the nonlinear response of superconducting niobium nitride (NbN) thin film and NbN metamaterial with different thicknesses under intense terahertz pulses. For NbN thin film, nonlinearity emerges and superconductivity is suppressed with increasing incident terahertz electric field, and the suppression extent weakens as the film thickness increases from 15 to 50 nm. As the variation in intense terahertz fields alters the intrinsic conductivity in NbN, a consequent remarkable amplitude modulation in NbN metamaterial is observed due to the strong nonlinearity. Absorbed photo density in either NbN film or NbN metamaterial is estimated and used to understand the mechanism of nonlinear response. With a thicker NbN film element of 200 nm, the resonance of the metamaterial shows similar nonlinear modulation accompanied by a lower loss and a higher quality factor compared with a thinner NbN film element of 50 nm, which demonstrates the innovative implementation of strongly enhanced nonlinearity with thick superconducting film elements and the potential for novel applications using nonlinear metamaterial.

Within a density-matrix formalism based on the Bardeen–Cooper–Schrieffer (BCS) model and the Bogoliubov–de Gennes equations we provide a description of the dynamics of the non-equilibrium superconducting pairing induced by a terahertz (THz) laser pulse in bulk and quasi-one-dimensional (1D) samples of conventional (BCS-type) superconductors. A cross-over from an adiabatic to a non-adiabatic regime takes place for short and intense THz pulses. In the non-adiabatic regime, the order parameter performs a damped oscillation. We discuss how the parameters of the THz pulse influence the amplitude and the mean value of the oscillation in bulk samples. It is demonstrated that for high intensities the non-adiabatic regime can be reached even for pulses longer than the oscillation period. For the 1D samples we find that the oscillation may attenuate with a different power law. This is analysed by comparing the THz-induced dynamics with the dynamics induced by a sudden switching of the pairing strength, which exhibits essentially the same behaviour. The numerical calculations show that the exponent of the power law depends critically on the density of states in the Debye window and therefore changes in an oscillatory way with the confinement strength. Irregularities in the decay of the oscillation are predicted when the 1D quantum wire is cut short to an elongated zero-dimensional quantum dot structure.