Carrier-envelope phase dependence of the duration of generated solitons for few-cycle rectangular laser pulses propagation
Introduction
The problem of laser pulse propagation in the resonant region has been extensively studied in atomic gases and semiconductors. Interesting phenomena such as self-induced transparency (SIT) [1], [2], [3], electromagnetically induced transparency (EIT) [4], Rabi flopping [5], [6], and solitonlike propagation [7], [8] have been observed. In the last decade, ultrashort laser pulses consisting of only few cycles of the electromagnetic field [9], [10], [11], [12], [13], [14], [15], [16], have expanded the horizon of modern optics and offer new applications in metrology, ultrafast spectroscopy, and material processing. It is well established that when pulse duration is down to a few cycles, the carrier-envelope phase (CEP), will become an important parameter [17]. The CEP strongly affects many processes including the inversion in two-level system [18], [19], [20], [21], higher frequency spectra generation [21], two-photon resonant propagation [22], and so on. As for temporal envelopes, most attention is paid to Gaussian or hyperbolic secant pulses. However, with the development of the pulse-shaping technology, few-cycle rectangular pulse with both steep leading edge and steep trailing edge can be realized in the radio-frequency, microwave and Terahertz regimes [18], [19], [23], [24]. The most important feature which is different from Gaussian and hyperbolic secant pulse is that the spectrum of rectangular pulse contains not only central frequency but also separated high and low frequency sidebands. In fact, the asymptotic theory of ultrawideband dispersive pulse propagation in a Lorentz model dielectric was first introduced by Sommerfeld and Brillouin in 1914 [25], [26], which established the physical essence of precursor fields. They investigated a step-modulated input pulse propagating through a dielectric medium as a function of distance and found that the transmitted field consisted of three parts: a Sommerfeld precursor (high frequency components) arriving first, a delayed Brillouin precursor (low frequency components), and an even more delayed main signal (resonant frequency components) [27]. Recently, there has been considerable interest in studying precursors [28], [29], [30], [31], [32], [33], [34], [35], [36], [37] including the separation of optical precursors and main fields [28], [29], [30], the enhancement of precursors [34], [35], and so on. They most focused on precursors though the main fields are also very important parts during the pulse propagation.
In this paper, we investigate few-cycle rectangular laser pulse propagation and especially consider the evolution of the main fields by employing the symmetric double quantum wells whose resonant intersubband (IS) transition is in the Terahertz regime. It is shown that a SIT soliton can be generated and separated from Sommerfeld–Brillouin precursors during the course of pulse propagation. We find that the duration of the SIT soliton depends on the CEP of the incident rectangular pulse, which results from the phase-dependent distribution of high and low frequency sidebands.
Section snippets
Model and basic equations
We consider an n-type modulation-doped symmetric double-coupled semiconductor GaAs/AlGaAs quantum wells as shown in Fig. 1. The structure comprises two GaAs symmetric square wells of l = 5.5nm width and 219 meV height, coupled by Al 0.267Ga0.733 As barrier with width d = 1.1 nm. There are only two lower energy subbands that contribute to the system dynamics: n = 0 for the lowest subband with even parity and n = 1 for the excited subband with odd parity. The Fermi level is below the subband minimum, so
Phase dependence of SIT solitons
In this section, we employ a standard predictor–corrector finite-difference time-domain (FDTD) approach to solve the full wave Maxwell equations and the effective nonlinear Bloch equations, which avoids invoking the slowly varying envelope approximation (SVEA) and the rotating wave approximation (RWA). In what follows, the initial conditions are S1(0) = S2(0) = 0 and S3(0) = − 1. We consider a rectangular form for the incident laser pulse, which is described by
Conclusions
In summary, we have investigated the phase-sensitive phenomena of few-cycle rectangular laser pulses propagating in semiconductor quantum wells. The phase-dependent low frequency components result in the phase-dependent multi-photon transitions which affect the generation of the SIT soliton. As a result, the duration of the generated SIT soliton depends sensitively on the CEP. In other words, the information of the CEP of the incident rectangular laser pulse is encoded on the SIT soliton.
Acknowledgments
The work was supported by the National Natural Sciences Foundation of China (grant nos. 10874194 and 60978013), and the National Basic Research Program of China (973 Program) (grant no. 2006CB806000). Yueping Niu is also sponsored by the Shanghai Rising-Star Program (grant no. 11QA1407400).
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