1. Introduction

The coherent control of nonlinear interactions has been an area of active research in recent years. Most theoretical works on nonlinear interactions assumed that atoms or molecules do not possess permanent dipole moments (PDM). In fact, PDMs do exist in some quantum systems, such as polar molecules [1–4] and some solid materials [5–8], and must be considered in such dipole systems. The properties of highly nonlinear optical processes can be greatly modified in dipole systems [9–13]. For instance, extension of harmonic spectrum was shown in a dipole gas and both odd and even harmonics can be found in the harmonic spectrum [13]. In our previous work, it was demonstrated due to the effects of PDMs in dipole systems, that the adiabatic energy difference is changed, which leads to enhancement of multiphoton excitations in low intensity scheme [14].

In fact, the effects of PDMs can be modified by the temporal shape of electric field. If only a few optical cycles are contained within the laser pulse envelope, the carrier-envelope phase (CEP) will dramatically affect the temporal shape of electric field [15–17]. Because intense laser-matter interactions depend on the electric field of the pulse, the CEP is important for a number of nonlinear processes [18–22]. Recently, it has been reported that the enhanced ionization of the asymmetric molecule (HeH²⁺) is much stronger when the electric field at the peak of the few-cycle pulse is antiparallel to the PDMs than that when the field at the peak is parallel to the PDMs [23]. This result was confirmed by a stronger harmonic emission obtained when the electric field at the peak of the few-cycle pulse is antiparallel to the PDMs [24]. The corresponding physical picture of enhanced ionization in these phenomena can be illustrated in dressed state interpretation. The dressed ground state moving up to cross with excited states enhances the ionization mechanism, and this ionization mechanism is weaker for a 180° change in the CEP because of the ground state moving down and avoiding excited states [23,24].

Phase effects can also be capable of providing important control over the excitations of atoms or molecules when two-color pulses are applied. In nondipolar medium, studies have found that the higher spectral components depend crucially on the relative phase of the two-color few-cycle pulses: continuum and distinct peaks can be achieved through the control of the relative phase [25,26]. By controlling the relative phase of the two-color large area pulses (20π corresponds to a intensity of 1.8×10¹³ W/cm²), the ultrafast optical four-wave mixing (FWM) can be obtained in two-level systems [27], and the relative phase dominating the efficiency of the coupling to the anti-Stokes Raman component is determined by the sign of the total ac Stark shift [28]. Very recently, coherent propagation effects in two- and three-level systems with two-color pulses were studied, it is shown that population transfer and frequency down-conversion are both relative-phase-sensitive [29]. In dipole systems, the interaction of a two-level polar molecule with two laser pulses, where one laser’s frequency is tuned to the energy level separation (pump laser) while the second laser’s frequency is much smaller compared to the energy level separation (probe laser), has been investigated. It was found that the excited state population can be greatly affected by the CEP of the probe laser [30,31].

Recently, the spectral features of pulses propagating in dipole medium have attracted much attention of researchers [32–37]. However, to our knowledge, there has been no investigation on two-color few-cycle ultrashort pulses controlling FWM during the course of propagation in dipole medium. In this work, we apply two-color few-cycle ultrashort laser pulses with ω_l and 2ω_l combination scenario in a polar molecule medium. It is found due to the effects of PDMs, that the enhancement of ultrafast FWM can be produced even without high field intensities, which can not occur in the nondipolar medium. Moreover, the conversion efficiency of FWM can be controlled by the combination of CEPs because the temporal shape of electric field is modified by the interference between the two-color few-cycle pulses, and when the oscillation peak antiparalleling to the PDMs reaches a maximum, the intensity of FWM generated wave can be enhanced more than 10 times.

This paper is organized as follows. In Sec. II, we present a theoretical model of the propagation of two-color few-cycle laser pulses with a two-level polar molecule medium. In Sec. III, we investigate the relationship between ultrafast FWM and the corresponding input electric field shapes with different combinations of CEPs in this model. Finally, we offer some conclusions in Sec. IV.

2. Theoretical model

The level scheme of the medium we investigate in this work is presented in Fig. 1. Consider a two-level polar molecule medium where |1> and |2> represent the ground and the excited states, with corresponding energies $E_1=-\frac{1}{2}ħ{\omega }_0$ and $E_2=-\frac{1}{2}ħ{\omega }_0$, respectively. ω ₀ is the transition frequency between the two levels.

 

Fig. 1. Schematic energy level diagram.

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The total Hamiltonian describing the interaction of the fields with the polar molecules is given by

Here $\underset{\_}{\mu }$ _ij are the dipole moment matrix elements. We assume that the laser fields are linearly polarized, and the transition moment $\underset{\_}{\mu }$ ₂₁ and permanent moments $\underset{\_}{\mu }$ _jj are taken to be aligned with the direction of polarization of the laser fields. The hyperbolic secant two-color pulses can be expressed as

where E ₀, ω_l, τ_p and ϕ_i are the electric field amplitude, the fundamental laser field frequency, the durations and the CEPs of each pulse, respectively.

The time evolution of the system is governed by the usual density-matrix equations and the equations of the time dependent ρ₂₁, ρ₂₂ and ρ₁₁ can be expressed as follows:

where T₁ and T₂ are the dephasing time and the excited-state lifetime, respectively. d=μ₂₂-μ₁₁ is the difference in the PDMs between the ground and the excited levels. dE(t) represents the effect of quantum interference resulting from the effects of PDMs. We should note for polar molecules, that the enhancement of ultrafast FWM can be achieved even without high intensity pulses, which can not occur in nondipolar systems. With proper combination of the CEPs, the intensity of the FWM generated wave can be enhanced more than 10 times in the polar molecules. In fact, PDMs can enlarge the multiphoton transitions in one specific orientation. The CEPs of the two-color pulses can modify the temporal shape of electric field. Therefore, PDMs and the CEPs play very important roles in the ultrafast FWM as we will show later.

The electromagnetic radiation is treated classically and the Maxwell’s equations take the form

Here H, P, μ₀ and ϵ₀ are the magnetic field, the macroscopic nonlinear polarization, the permeability and the permittivity of free space, respectively. The macroscopic nonlinear polarization is related to the ensemble average of the expectation value of the dipole moment operator for polar molecules.

We employ a standard finite-difference time-domain approach for solving the full-wave Maxwell equations and predictor-corrector method to solve the density-matrix equations without any standard approximation. The time and space increments, Δt and Δz, are chosen to ensure cΔt ≤ Δz [38].

3. Results and discussion

The numerical simulations are based on molecular parameters characteristic of the S ₀→S ₁ electronic transition in 1-[p-(N, N-dimethylamino) phenyl]-4-(p-nitrophenyl)-1, 3-butadiene which was involved in previous experimental [39] and theoretical [14,40] investigations of the effects of PDMs on pulse interactions. In atomic units, the molecular parameters are ω ₀=8.59×10-2, μ₂₁=3.93, and d=1.18×10. The decay times for molecular system can be chosen as T ₁=T ₂=1.00 ps [41]. The molecular density is taken as N=1.00×10²⁴ m^-3. All the molecules initially are in their ground states. The pulse durations are τ_p1=τ_p2=10 fs. A relative low peak intensity of 3.71.1011 W/cm² is applied in the numerical calculations. The results to follow can, of course, be obtained for other combinations of laser pulses and dipolar mediums.

Figure 2 presents the spectra produced by the two-color few-cycle ultrashort pulses at the propagation distance of z=60 μm, and the corresponding temporal shapes of the synthesized electric fields of input two-color pulses. In order to investigate the effects of PDMs on nonlinear interactions, we assume that there exists a nondipolar molecule medium with the same parameters as the above molecular parameters except for d=0. The propagation spectrum in the nondipolar molecule medium is shown in Fig. 2(a). In comparison to the initial spectrum (dashed line), there is no change and higher spectral components can be hardly presented. When the effects of PDMs are considered, the situation can be changed fundamentally. Figures 2(b), 2(d) and 2(f) show the propagation spectra in the polar molecule medium (d=1.18×10 a.u.). It can be found due to the effects of PDMs, that higher spectral components are produced in which the two peaks at frequencies 5ω_l and 3ω_l are the FWM generated waves, i.e. the ultrafast FWM can be enhanced with low intensity pulses propagating in polar molecule medium.

In fact, the FWM conversion efficiency can be controlled by the CEP combinations of two-color few-cycle pulses because the CEPs can modulate the synthesized electric field shape. For ϕ₁=ϕ₂=0, it can be seen in Fig. 2(c) that the electric field at the peak paralleling to the direction of polarization (or PDMs) is obviously larger than that at the opposite direction. The FWM generated waves at 3ω_l and 5ω_l are both enhanced in the polar molecule medium than in the nondipolar molecule medium due to the effects of PDMs (shown in Figs. 2(a) and 2(b)). When one of the CEPs is changed (ϕ₂=π/2), the magnitude of the electric field at opposite direction of the PDMs is enlarged, and the peak intensities at both directions are about same (shown in Fig. 2(e)). It can be seen in Fig. 2(d) that the spectral component at 4ω_l and other higher components, which are productions of complete multiphoton excitations, are enhanced. Although the ultrafast FWM at 5ω_l is a process of three-photon excitations, the magnitude of the FWM generated wave is about same with the spectral component at 4ω_l which is produced by a two-photon process. However, it should be noted that the FWM at 3ω_l is not enhanced. It is because that the FWM at 3ω_l includes a non-excited process. Figure 2(g) shows the temporal shapes of the synthesized electric fields in the situation ϕ₁=ϕ₂=π. The electric field at the peak antiparalleling to the PDMs reaches a maximum, which is evidently stronger than that at the other direction. Figure 2(f) presents the spectrum induced by the input two-color pulses of Fig. 2(g). The generated waves at 4ω_l and 5ω_l are further enhanced. The two-photon and three-photon excited processes are greatly enlarged, which suppresses the enhancements of other higher order processes.

 

Fig. 2. The spectra of two-color few-cycle ultrashort pulses at z=60μm. The peak intensity of the pulses is 3.71×10¹¹ W/cm². (a) represents the nondipolar molecule case of ϕ₁=ϕ₂=0. (b), (d) and (f) represent the polar molecule cases, (b) ϕ₁=ϕ₂=0, (d) ϕ₁=0 and ϕ₂=π/2, (f) ϕ₁=ϕ₂=π, and the corresponding input carrier of two-color few-cycle pulses (c), (e) and (g).

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Our numerical results have demonstrated when the electric field at the peak is antiparallel to the PDMs, that multiphoton excitations can be enhanced in the polar molecule medium. The previous result showed that the enhanced ionization is stronger when electric field at the peak of a single pulse is antiparallel to the PDMs in high intensity scheme [23]. However, in our case, it is because of the interference of two-color few-cycle pulses, that the temporal shape of the synthesized electric fields is specially modulated by the CEPs of the two pulses. Thus, the electric field at the peak antiparalleling to the PDMs can be much larger than that at opposite direction and the effects of PDMs is greatly enhanced, which is the essential physical mechanism of that the FWM conversion efficiency can be controlled by the CEPs in low intensity scheme. This CEP dependence of FWM can not occur in nondipolar systems in such intensity scheme, and it is also difficult to achieve for single-pulse scenario. The intensity of the FWM generated wave at 5ω_l when ϕ₁=ϕ₂=π can be 10 multiples of that when ϕ₁=ϕ₂=0, which is shown in Fig. 3.

 

Fig. 3. Plot of the dimensionless FWM intensity. The parameter values are same with those of Fig. 2.

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4. Conclusions

In summary, we have investigated the spectra of few-cycle two-color ultrashort pulses in the two-level polar molecule medium by solving the full-wave Maxwell’s equations and density-matrix equations. It was demonstrated due to the effects of PDMs, that the enhancement of ultrafast FWM can occur in the polar molecule medium even for low pulse intensities. Moreover, the conversion efficiency of FWM can be controlled by the combinations of CEPs of two-color few-cycle ultrashort pulses because the interference between the two pulses can substantially modify the temporal shape of electric field. Using proper combination of the CEPs, the FWM generated wave can be enhanced greatly. The procedure developed in this work is to achieve the coherent control of the nonlinear interactions in low intensity scheme, which might be helpful to control ultrafast multiphoton excitations during the course of propagation.