Role of the Carrier-Envelope Offset Phase of Few-Cycle Pulses in Nonperturbative Resonant Nonlinear Optics

Abstract

We study the influence of the carrier-envelope offset phase of few-cycle pulses on nonperturbative resonant extreme nonlinear optics in a semiconductor. If the Rabi frequency becomes comparable to the light frequency, the different Rabi sidebands interfere around twice the laser center frequency, giving rise to a signal which strongly depends on the carrier-envelope offset phase. This signature should be measurable in GaAs samples. DOI: 10.1103/PhysRevLett.89.127401 PACS numbers: 78.47.+p, 42.50.Md, 42.65.Re The rapid development of yet shorter and shorter laser pulses [1] has now led us into a regime in which the phase between the rapidly oscillating light frequency and the electric field envelope has become a relevant quantity [2 -4]. This carrier-envelope offset (CEO) phase, , of a single few-cycle pulse can significantly influence the outcome of an experiment. It has to be distinguished from the well-known relative optical phase, i.e., the phase between two different beams or pulses.In order to fix the CEO frequency f , recent work [5,6] has, for example, used the interference of the fundamental frequency of a laser pulse, which was spectrally broadened by self-phase modulation in a (photonic crystal) fiber of a few millimeter length, with the second harmonic generated with the help of a separate crystal. In Ref. [7], the same idea was used, except for the fact that the fundamental spectrum did already cover one octave, hence no need for additional broadening. Somewhat similar to this, Ref. [8] proposed to use the interference of the third harmonic, generated at a silicon wafer surface, with the second harmonic generated in a separate crystal. All the optical nonlinearities used in these and other [9] cases are off-resonant and within the perturbative regime; i.e., an expansion in terms of nonlinear optical susceptibilities is meaningful. Furthermore, in most of these cases, the pulses have to propagate over a considerable distance within the apparatus and, hence, the difference between the phase velocity and the group velocity can change the CEO phase within the measurement setup. Currently, this is the main obstacle in measuring .A nonperturbative and, hence, distinctly different way to determine the CEO phase would be via x-ray generation in extreme nonlinear optics in atoms [1,10]. In a recent review [11] on the implications of the CEO phase on metrology [12,13], the authors state in their outlook: ''. . . If an experimental technique can be developed that is sensitive to the carrier-envelope phase and works with the direct output of a mode-locked oscillator (i.e., without that amplification that will be necessary for extreme nonlinear optics), it may in turn benefit optical-frequency synthesis because it may create a simpler technique for determining/controlling the comb offset frequency. '' In this Letter, we show that resonant extreme nonlinear optics in a solid, exemplified by carrier-wave Rabi flopping [14], shows a dependence on the CEO phase , which potentially allows one to determine . The ...

An Iterative Method for Extreme Optics of Two-Level Systems

“…For each slice, this approach offers a simple analytical formula for the ratio of level population amplitudes, resulting in equally simple analytical formulae for population inversion, induced dipole moment, and spectrum of the radiation emitted by this dipole. A good quality of the method is proved by showing that the resulting photon-emission spectra reproduce the numerically calculated ones available in literature [8,9,[13][14][15][16][17][18][19]. Moreover, the method indicates weak points of the square--wave solution of Tritschler et al [23] and is shown to be particularly useful in the area of extreme nonlinear optics [23], i.e., when few-cycle, strong pulses stimulate significant population dynamics in a two-level system on a timescale of half the optical cycle.…”

“…The latter is known [8,9,12,[24][25][26][27] to be a relevant quantity determining the response of the system in the regime of few-cycle pulses. Traditionally, there has been solved in different coupling regimes either a set of two linear differential equations for b k with no RWA applied (e.g.…”

“…Over the last decade, for example, the model has succeeded in explaining the main features of propagation of strong few-cycle pulses through atomic and semiconductor media [2][3][4][5][6][7][8][9][10][11][12], e.g., carrier wave Rabi flopping, third-harmonic generation in disguise of second harmonic, and carrier-envelope phase effects, to name a few. It has also made a basis for the description of high-order harmonic generation from a single atom [13], a symmetric molecular ion [14][15][16] and a double-well quantum structure [17][18][19] with emphasis on the strongly non-perturbative picture of the phenomenon and the occurrence of peaks in the spectrum of coherently scattered light at the positions of even harmonics.…”

Carrier-envelope phase-dependent transmitted spectra in inversion-asymmetric media with permanent dipole moments

“…It should be noted that the period of these CEP-dependent signals is 2π . This is different from the interference case in symmetric media [15,20,21], in which the CEP-dependent period is π . To illustrate the underlying physics, we calculate the dipole-induced emitted field spectra (Ê em (ω) ∝ FFT − ∂P x ∂t [44]) from the medium at the same propagation distance as in figure 1.…”

“…Previous investigations in inversion-symmetric media have shown that these CEPdependent signals are originated from the interference effects between different harmonic components. The CEP-dependent period in inversion-symmetric media is π , which means that the sign of the electric field cannot be determined in these phenomena [15,20,21].…”