1. Introduction

The increased demand for high-energy broadband pulses in the terahertz (TH) frequency range (from tens to thousands of micrometers for corresponding wavelengths) has given rise to many new developments in the field and, after the first demonstrations [1, 2], developments of optical rectification (OR) methods have become very intensive among various groups of researchers (see e.g., [3] for the review). Despite the tremendous interest in and efforts towards developing the capabilities of OR methods, to the best of our knowledge optical-to-TH conversion efficiencies achieved by OR methods typically appear in the range 10⁻⁵ − 10⁻⁴, and a conversion efficiency ∼ 2.5 × 10⁻³ was achieved quite recently in a stoichiometric lithium niobate crystal through excitation by high-energy 100-mJ picosecond pulses [4].

It is generally accepted that the Manley–Rowe conversion limit (which allows, in principle, a conversion efficiency of at least ∼ 10⁻² from a wavelength of one to one hundred micrometers) and interaction process requirements (including strict velocity matching constraints and prerequisite low absorption) are necessary for the efficient generation of a TH wave. To overcome these difficulties, a few ideas have been proposed, such as using quasi-phase-matched (QPM) materials with a periodically inverted sign of the second-order susceptibility [5], cascading processes of TH generation [6, 7], and velocity matching under conditions of self-induced transparency (SIT) [8, 9]. In addition, the idea of using OR under excitation of nonlinear media with a tilted pulse front [10, 11] was proposed and successfully implemented.

It should be noted that most efforts to generate high-energy broadband TH pulses were directed toward the implementations for third-order nonlinear wave mixing in non-centrosymmetric media (or media containing permanent dipole moments). In general, this is a sensible choice because usually the second-order nonlinearity prevails over the third-order one. However, this disregards the consideration of centrosymmetric media, many of which (e.g., centrosymmetric quantum dots) are auspicious for applications in mid- and long-infrared photonic devices.

Fortunately, it is known that susceptibilities are normally enhanced when the mixing frequencies approach resonance; thus, one can make successful use of wave-mixing processes in the vicinity of resonances. The recently demonstrated experimental technique [12] for efficient four-wave mixing in K atomic vapors, exploiting one- and two-photon resonances, should be regarded as an example of enhanced difference-frequency generation (DFG) in a quantum ensemble. Therefore, natural questions about the role and significance of induced vibrational coherence for the generation of electromagnetic radiation in TH frequency range should be raised. Note that very recently such an attempt was made using non-depleted pump approximation [13] which, however, does not allow to evaluate the efficiency of THz generation.

In the present work, we numerically evaluate the efficiency of broadband TH radiation produced by four-wave DFG with ultrashort light pulses. A specific case with two pump pulses in the vicinity of the two- and one-photon resonances in an ensemble of three-level quantum systems (TLQSs) is considered. It should be emphasized that processes of four-wave mixing are considered under coherent four-wave light-matter interaction conditions when the time scale during which the coherences of the transitions in the TLQS are preserved exceeds the durations of both pump pulses.

2. Four-wave mixing at one- and two-photon resonances

We consider a four-wave mixing process in the medium with local centers of inversion, where the second-order susceptibility vanishes owing to inversion symmetry. In this case, emission of the TH wave at the frequency difference of the spectral components appears from the ordinary OR process by interacting in a suitable material of the ultrashort optical pulses oscillating at frequencies of ω_L and ω_P ≅ 2ω_L. In the absence of resonances, this process can be interpreted as ${ℰ}_{\mathit{TH}}~{\chi }^{(3)}{ℰ}_L^2{ℰ}_P^{*}\text{cos}(\phi )$, where ℰ_TH,L,P denote complex amplitudes of the fields at TH and two pump frequencies of the fundamental and injected (near to second-harmonic) fields, respectively. χ⁽³⁾ is the non-resonant third-order nonlinear susceptibility, and φ is the relative phase difference between the pump waves. Intrinsically, in this (off-resonance) case, a TH wave results from hyper-Raman scattering (HRS), the two-photon-excited ordinary Raman scattering, which is a very weak process and needs high field intensities to be applied. Again, the efficiency of HRS can be enhanced owing to the resonances, and we proceed with the case of coherent resonant interaction, i.e., when the following conditions are fulfilled for the optical pulses, frequency detunings Δ_i(i = 1, 2) and the irreversible polarization relaxation time T₂ : τ_L,P < T₂ and ${\tau }_{L,P}^{-1}>{\mathrm{\Delta }}_i$ (i = 1, 2). Exploring two-photon resonance-enhanced HRS in the case of coherent light–matter interaction, the energy of a two-photon resonant ultrashort pulse may be kept constant for a relatively long distance. Indeed, the evolution of the slowly varying complex envelope ℰ_L(z, τ) under coherent propagation at the two-photon resonance is given by [14, 15]

 

Fig. 1 (a) Evolution of the pump-pulse ”area” Θ_L(z) = ψ(z, ∞) under coherent propagation at a two-photon resonance. Results are plotted using solutions of the Eq. (2) for different initial conditions: Θ_L(0) = π, 3π, 5π, 7π. (b) Level configuration and excitation paths of the TLQS: a pump field excites the system to the state |3〉 by two photons (ω_L + ω_L) through off-resonant (not-shown) levels, whereas injected and generated fields excite the system by pair of photons (ω_TH + ω_P) through resonant level |2〉.

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Further, we will proceed with the planar propagation of the waves in the z-direction and will normalize the z coordinate to the length 1/K₂ of two-photon absorption. Owing to previous studies [16, 17] of the coherent dynamics of two-photon excitation processes, the equations (scalar form) for the slowly varying amplitudes of the electric fields can be written as follows:

It should be noted that Eqns. (3)–(5) were written without taking into account one-photon absorption (amplification), i.e., assuming that processes involving amplification of the injected ℰ_P and generated ℰ_TH fields develop primarily owing to the induced coherence ρ₁₃ in the TLQS. For the same reason (coherent light-matter interaction), we have omitted exponential phase multipliers on the right-hand side of Eqns. (3)–(5) arising from a nonresonant contribution to the wave-number mismatch.

The system of the wave equations in Eqns. (3)–(5) along with the nonlinear response of the TLQS is given by the equations for elements of density matrix ρ̂:

3. Numerical examples of TH generation by four-wave DFM

Usual split-step procedure of separating the nonlinear equations (3–5) into dispersive and nonlinear parts inside each incremental step dz was used to evaluate the propagating fields. The Runge-Kutta procedure of the 4th-order was applied on the same step to solve the Eqns. (6)–(7).

In Fig. 2 we compare two cases of TH radiation generation dynamics, both of which can be regarded as characteristic for moderate regimes of interaction. Here, the duration of both the pump and injected pulses are taken to be equal: τ_L = τ_P =1ps, whereas the area of the two-photon resonant pump pulse were taken as θ_L(0) = 1.8π and θ_L(0) = 3.8π when producing the results shown in (a) and (b), respectively (for the rest of the parameters described in the figure caption). As a distinguishing feature between the two cases of TH wave generation dynamics depicted here, the coherent dynamics (Rabi oscillations) of an atomic transition exposed to the more intensive field in Fig. 2(b) should be noted. (This should not be confused with the long-damped oscillation tail coupled with the Raman active phonon-polariton mode [18]).

 

Fig. 2 Formation of the growing long-wavelength radiation at the initial conditions for the material variables ρ₁₃ = 0, n₃₁ = −1 and the parameters Θ_P(0) = π/4, τ_D = 1ps, β = 10, and Ω_TH/ω_L = 10⁻². TH radiation amplitudes are shown for (a) Θ_L(0) = 1.8π and (b) Θ_L(0) = 3.8π.

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To demonstrate the Rabi oscillations more clearly the amplitude of TH field at K₂z = 2 is also plotted in Fig. 3(a). Note that the appearance of a Rabi oscillation by increasing the area of the pump pulse could be used for broadband generation in TH frequency range. Finally, the dependence of the generation efficiency is plotted versus the time delay τ_D between pump pulses for the cases Θ_L(0) = 1.8π (blue squares) and Θ_L(0) = 3.8π (red circles) in Fig. 3(b). Maximum demonstrated photon conversion efficiency (30%) should be regarded as very optimistic for the proposed method of TH radiation generation. It should be noted that an additional possibility for optimization of DFG in the coherent resonant regime of four-wave mixing consists in proper choice of the delay time between two (pump and injected) pulses, as was mentioned in Sec. 2.

 

Fig. 3 (a) The TH field intensity at K₂z = 2 resulted from DFG with initial area of the pump pulse Θ_L(0) = 3.8π; (b) Photon conversion efficiency to TH frequency range versus delay τ_D for two values of Θ_L(0): 1.8π (blue squares) and 3.8π (red circles). Other parameters and initial conditions as in Fig. 2.

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4. Conclusions

In conclusion, the present study emphasizes that centrosymmetric media with embedded two-photon resonant molecular impurities appear promising for TH radiation generation by OR. It was demonstrated that coherent resonantly enhanced hyper-Raman scattering results in efficient (∼ 30%) photon conversion to the TH frequency range. It was also stressed that the generation of TH radiation in the proposed coherent regime of FW DFM is possible, even without overlapping pump and injected pulses. This feature of proposed method arise due to the finite memory of coherence induced in TLQS and allows for optimal generation of THz radiation by choosing the proper time delay between pulses. The good candidates to practical realization of the proposed method would be ensemble of semiconductor nanoparticles where the resonant interactions with optical light fields are strongly enhanced owing to the quantum confinement effect [19, 20]. Taking into account the enhanced optical transitions to bi-excitonic states, the intensity of the femtosecond pulse required to produce a sufficient area of the pump (Θ(0)_L,P ≥ 2π) in the above mentioned structures should be evaluated as only hundreds and even tens of megawatts per centimeter squared. Additionally, molecular (ammonia e.g.) vapors can be regarded as good candidates for experimental testing of the theoretical predictions presented in this theoretical study.