Terahertz Sideband Research Articles

Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field

In a semiconductor, optically excited electron-hole pairs, driven by a strong terahertz (THz) field, can recombine to create THz sidebands in the optical spectrum. The sideband spectrum exhibits a “plateau” up to a cutoff frequency of 3.17Up, where Up is the ponderomotive energy. In this letter, we predict that the bandwidth of this sideband spectrum plateau can be broadened by applying an additional direct-current (DC) electric field. We find that if applying a DC field of EDC=0.2ETHz (where EDC and ETHz are the amplitudes of the DC field and THz field, respectively), the sideband spectrum presents three plateaus with 5.8Up, 10.05Up and 16Up being the cutoff frequencies of the first, second and third plateaus, respectively. This bandwidth broadening occurs because the DC field can increase the kinetic energy that an electron-hole pair can gain from the THz field. This effect means that the bandwidth of the sideband spectrum can be controlled flexibly by changing the DC field, thereby facilitating the ultrafast electro-optical applications of THz sideband generation.

Geometric diffusion of quantum trajectories.

A quantum object can acquire a geometric phase (such as Berry phases and Aharonov–Bohm phases) when evolving along a path in a parameter space with non-trivial gauge structures. Inherent to quantum evolutions of wavepackets, quantum diffusion occurs along quantum trajectories. Here we show that quantum diffusion can also be geometric as characterized by the imaginary part of a geometric phase. The geometric quantum diffusion results from interference between different instantaneous eigenstate pathways which have different geometric phases during the adiabatic evolution. As a specific example, we study the quantum trajectories of optically excited electron-hole pairs in time-reversal symmetric insulators, driven by an elliptically polarized terahertz field. The imaginary geometric phase manifests itself as elliptical polarization in the terahertz sideband generation. The geometric quantum diffusion adds a new dimension to geometric phases and may have applications in many fields of physics, e.g., transport in topological insulators and novel electro-optical effects.

Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors

In a semiconductor illuminated by a strong terahertz (THz) field, the electron-hole pairs excited by linear polarized near-infrared (NIR) laser can recombine to emit high-order THz sideband. Previous experimental results have shown, under the same condition of excitation intensity, the polarization direction of the NIR laser could affect the sideband intensity. In this letter, we theoretically investigate the effect of the NIR laser polarization direction on high-order terahertz sideband generation in bulk GaAs.

Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene.

In a semiconductor illuminated by a strong terahertz (THz) field, optically excited electron–hole pairs can recombine to emit light in a broad frequency comb evenly spaced by twice the THz frequency. Such high-order THz sideband generation is of interest both as an example of extreme nonlinear optics and also as a method for ultrafast electro-optical modulation. So far, this phenomenon has only been observed with large field strengths (~10 kV cm−1), an obstacle for technological applications. Here we predict that bi-layer graphene generates high-order sidebands at much weaker THz fields. We find that a THz field of strength 1 kV cm−1 can produce a high-sideband spectrum of about 30 THz, 100 times broader than in GaAs. The sidebands are generated despite the absence of classical collisions, with the quantum coherence of the electron–hole pairs enabling recombination. These remarkable features lower the barrier to desktop electro-optical modulation at THz frequencies, facilitating ultrafast optical communications.

Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields

High-order terahertz (THz) sideband generation in semiconductors is a phenomenon with physics similar to that of high-order harmonic generation but in a regime of much lower frequency. Our previous paper [] found that the electron–hole pair excited by a weak optical laser can accumulate a Berry phase along a cyclic trajectory under the driving of a strong elliptically polarized THz field. Furthermore, the Berry phase appears as the Faraday rotation angle of the emission signal under short-pulse excitation in monolayer MoS. In this paper, the theory of the Berry phase in THz extreme nonlinear optics is applied to biased bilayer graphene with Bernal stacking, which has similar Bloch band features and optical properties to monolayer MoS, such as the time-reversal related valleys and the valley contrasting optical selection rule. However, the biased bilayer graphene has much larger Berry curvature than monolayer MoS, which leads to a large Berry phase of the quantum trajectory and in turn a giant Faraday rotation of the optical emission (∼1 rad for a THz field with frequency 1 THz and strength 8 kV cm−1). This surprisingly big angle shows that the Faraday rotation can be induced more efficiently by the Berry curvature in momentum space than by the magnetic field in real space. It provides opportunities to use bilayer graphene and THz lasers for ultrafast electro-optical devices.

Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields

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.

Effects of excitation frequency on high-order terahertz sideband generation in semiconductors

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 (Up), 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 3.17Up 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 3.17Up above the band gap, the HSG spectrum presents no plateau feature but instead sharp peaks near the band edge and near the excitation frequency.

Intersubband optical absorption in a step asymmetric semiconductor quantum well driven by a terahertz field

The nonlinear optical absorption in a three-subband step asymmetric semiconductor quantum well driven by a strong terahertz (THz) field is investigated theoretically by employing the intersubband semiconductor-Bloch equations. We show that the optical absorption spectrum strongly depends on the intensity, frequency, and phase of the pump THz wave. The strong THz field induces THz sidebands and Autler-Townes splitting in the probe absorption spectrum. Varying the pump frequency can bring not only the new absorption peaks but also the changing of the energy separation of the two higher-energy levels. The dependence of the absorption spectrum on the phase of the pump THz wave is also very remarkable.

Frequency and amplitude modulation in terahertz-sideband generation in quantum wells

A narrow-band optical beam incident on a quantum well modulated at terahertz frequencies results in the formation of terahertz sidebands, i.e., optical signals at the input optical frequency plus integer multiples of the modulation frequency. The terahertz sidebands originate in the modulated complex optical susceptibility. In this study, we analyze theoretically terahertz-sideband generation in the context of frequency and amplitude modulations of the incident optical beam. We find that, at the optimum bias field, the frequency and amplitude modulations contribute almost equally to the maximum first-order sideband. The theoretical approach could offer simple yet useful tools for further engineering and optimization of modulator-type sideband-generation or wavelength-conversion devices.

A Review of Procedures for Summing Kapteyn Series in Mathematical Physics

Since the discussion of Kapteyn series occurrences in astronomical problems the wealth of mathematical physics problems in which such series play dominant roles has burgeoned massively. One of the major concerns is the ability to sum such series in closed form so that one can better understand the structural and functional behavior of the basic physics problems. The purpose of this review article is to present some of the recent methods for providing such series in closed form with applications to: (i) the summation of Kapteyn series for radiation from pulsars; (ii) the summation of other Kapteyn series in radiation problems; (iii) Kapteyn series arising in terahertz sideband spectra of quantum systems modulated by an alternating electromagnetic field; and (iv) some plasma problems involving sums of Bessel functions and their closed form summation using variations of the techniques developed for Kapteyn series. In addition, a short review is given of some other Kapteyn series to illustrate the ongoing deep interest and involvement of scientists in such problems and to provide further techniques for attempting to sum divers Kapteyn series.

1 THz Modulation in InGaAsP Multiple Quantum Wells for 40 Gb/s Applications

The terahertz (THz) rate modulation of quantum well (QW) electrooptic modulators necessitates a new way of thinking about how the modulation field modulates light; specifically, an incident narrow-band (with respect to the modulation frequency) signal once modulated acquires frequency components separated from the input signal center frequency by multiples of the modulation frequency. In this paper, we discuss the design of the QWs comprising the modulator to maximize the output at such THz sidebands of the incident optical frequency in InGaAsP QW based devices. We present a theoretical treatment of the case in which the THz modulation frequency is out of resonance between any exci- tonic levels near those exploited for the optical modulation, thus enabling an adiabatic treatment of the modulated optical susceptibility. We show that THz sideband conversion efficiencies of ~1% may be possible.

Nonlinear optical properties of semiconductor quantum wells under intense terahertz radiation

The nonlinear optical properties of semiconductor quantum wells driven by intense in-plane terahertz electric fields are investigated theoretically by employing the extended semiconductor Bloch equations. The dynamical Franz-Keldysh effect of the optical absorption near the band edge is analyzed with Coulomb correlation among the carriers included. The in-plane terahertz field induces a variety of behavior in the absorption spectra, including terahertz replicas of the (dark) 2p exciton and terahertz sidebands of the 1s exciton. The dependence of these interesting features on the intensity, frequency, and phase of the terahertz field is explored in detail.

THz sideband generation at telecom wavelengths in a GaAs-based quantum cascade laser

Terahertz sideband generation is demonstrated by using the nonlinear intracavity interaction between far-infrared and near-infrared modes in a GaAs waveguide. A low power near-infrared beam at 1.32μm is coupled into a quantum cascade laser operating at 104μm (2.9 THz), which acts as both the THz source and the nonlinear medium. The conversion efficiency of the resulting sidebands is found to be approximately 4.5×10−6.

Terahertz/optical mixing in symmetric semiconductor quantum wells embedded in optical microcavities

A quantum well (QW) in the simultaneous presence of a terahertz field polarized in the growth direction and an incident optical field near an excitonic resonance results in substantial frequency mixing between the terahertz and optical fields. In particular, a response at new frequencies given by the input optical frequency plus or minus multiples of the terahertz frequency occurs-the terahertz sidebands. In a symmetric QW, the dominant contribution to terahertz-sideband formation is the high-frequency modulation of the overlap integral of the relevant conduction- and valence-subband envelope functions that determine the strength of the interband dipole moment. terahertz-sideband generation is shown to be strongly enhanced in a high quality-factor optical microcavity. Numerical values of the reflected intensity into the first terahertz sideband normalized with respect to the reflected intensity at the fundamental as large as /spl sim/10% are estimated. This suggests that terahertz-sideband generation in semiconductor microcavities is a promising option worthy of exploration for wavelength conversion for wavelength-division multiplexing applications.

Time-resolved, nonperturbative, and off-resonance generation of optical terahertz sidebands from bulk GaAs

We have investigated terahertz (THz) sideband generation from bulk GaAs using intense pulses of coherent THz (or far-infrared) radiation from a free electron laser. In contrast to previous studies, sidebands appeared inside the band gap and therefore were not resonantly enhanced from real states. Also, using picosecond pulses and changing the temporal overlap between the THz and near-infrared pulses allowed us to monitor the evolution of the THz sidebands directly in the time domain; this suggests a convenient method for characterizing THz pulses using a conventional Si photodetector. In addition to an expected second-order sideband, we detected a first-order sideband, which has previously been observed only in an asymmetrically coupled double quantum well system where the inversion symmetry was intentionally broken. Finally, the THz power dependence clearly revealed a deviation from perturbative behavior, indicating the entrance into the strong-field regime.

Optical absorption and sideband generation in quantum wells driven by a terahertz electric field

Linear optical absorption and terahertz sideband generation in quantum wells in the presence of a strong growth-direction oriented terahertz electric field is investigated. It is shown that the electric field not only produces multiple optical absorption peaks due to resonant coupling between the valence subbands---the Autler-Townes splittings---but also produces splittings in the generated sidebands. Due to the proximity of subbands in typical quantum wells, the terahertz field can couple more than two levels simultaneously and particular effects of that coupling are discussed.

Picosecond dynamics of terahertz-sideband generation in far-infrared illuminated quantum wells

Optical excitation of a quantum well illuminated by a strong far-infrared (FIR) field is known to give rise to sidebands on the optical spectra separated from the fundamental by multiples of twice the FIR frequency Ω. In this study, we consider the picosecond dynamics of terahertz (THz) sideband formation using picosecond optical excitation pulses. We find that the emission at the THz sidebands largely tracks the temporal evolution of the excitation pulse. Applications to wavelength-division multiplexing are discussed.

Dynamic Franz–Keldysh effect: perturbative to nonperturbative regime

We study theoretically the dynamic Franz-Keldysh effect for quantum wells excited by a cw optical field and a terahertz (THz) frequency driving field. By monitoring the THz sidebands one can show that for, increasing THz-field amplitudes, nonperturbative effects become predominant, and one must account for THz-field-induced nonlinearities to all orders.

Terahertz sideband generation in quantum wells viewed as resonant photon tunneling through a time-dependent barrier: An exactly solvable model

The propagation of light through quantum wells in the vicinity of an excitonic resonance subjected to a THz electric field is considered. The treatment makes explicit the relationship between THz sideband generation and resonant photon tunneling through a time-dependent resonant-tunneling structure. A method for the coherent control of excitons in quantum wells is also suggested.