For excitation of the model semiconductor GaAs with optical pulses which are both extremely short (5 fs) and extremely intense (ഠ10 12 W cm 22 ), we can meet the condition that the Rabi frequency becomes comparable to the band gap frequency -a highly unusual and previously inaccessible situation. Specifically, in this regime, we observe carrier-wave Rabi flopping, a novel effect of nonlinear optics which has been predicted theoretically and which is related to the failure of the area theorem.

In contrast with traditional nonlinear optics, a peak at the spectral position of the second harmonic of a laser can also be generated in an inversion-symmetric medium in the regime of extreme nonlinear optics. We describe the underlying mechanism of such third-harmonic generation in disguise of secondharmonic generation and compare theory with the optical as well as the radio-frequency spectra measured in recent experiments on thin ZnO films. The peak at twice the carrier-envelope offset frequency in the radio-frequency spectra is shown to be an unambiguous signature of such a process. DOI: 10.1103/PhysRevLett.90.217404 PACS numbers: 78.47.+p, 42.50.Md, 42.65.Re Inversion symmetry has strict consequences in traditional nonlinear optics, but more relaxed ones in the regime of extreme nonlinear optics [1]. Consider an incident pulse with electric field E t Ẽ E t cos ! 0 t . E E t is the envelope of the pulse, the cosine term is the carrier wave with a carrier frequency ! 0 and a carrierenvelope offset (CEO) phase [2 -4]. Second-harmonic generation (SHG), i.e., a contribution with a carrier frequency 2! 0 , is forbidden in an inversion-symmetric material. In traditional nonlinear optics, the spectral width of the envelope of the generated wave is much smaller than ! 0 . Thus, a peak at a spectrometer frequency 2! 0 cannot occur. In contrast with this, in extreme nonlinear optics, the spectral width of the envelope can approach ! 0 or even exceed it. Equivalently, the electric field itself governs the behavior rather than the light intensity. Thus, the envelope of third-harmonic generation (THG) with carrier frequency 3! 0 can, e.g., lead to a low-frequency sideband (or a strong contribution) at spectrometer frequency 2! 0 -even for an inversion-symmetric material. This phenomenon is called ''THG in disguise of SHG'' in what follows.Indeed, a number of theoretical studies have dealt with this problem [5][6][7][8][9][10]. To the best of our knowledge, however, no corresponding experimental results in the regime of extreme nonlinear optics have been discussed.How could one unambiguously distinguish THG in disguise of SHG from usual SHG? The laser pulse itself has phase by definition, the usual SHG has phase 2 (even if it originates from, e.g., a 4 process), and the third harmonic has phase 3 -although it may exhibit a peak at spectrometer frequency 2! 0 . Thus, the beat note of the usual SHG with the fundamental has a difference phase , that of the THG in disguise of SHG and the fundamental has a difference phase 2 . For pulses out of a mode-locked laser oscillator, oscillates with the CEO frequency f [2 -4]. Hence, usual SHG would lead to a beat note at frequency f in the radio-frequency (rf) spectrum, THG in disguise of SHG to a beat note at 2f . Thus, a peak at frequency 2f is an unambiguous experimental signature of THG in disguise of SHG.In this Letter, we (i) describe the mechanism in more detail that can lead to THG in disguise of SHG in an inversion-symmetric medium. Next (ii), we apply this physics to a ...

We study the linear-and nonlinear-optical lineshapes of metal nanoparticles ͑theory͒ and metallic photonic crystal slabs ͑experiment and theory͒. For metal nanoparticle ensembles, we show analytically and numerically that femtosecond second-or third-harmonic-generation ͑THG͒ experiments together with linear extinction measurements generally do not allow to determine the homogeneous linewidth. This is in contrast to claims of previous work in which we identify a technical mistake. For metallic photonic crystal slabs, we introduce a simple classical model of two coupled Lorentz oscillators, corresponding to the plasmon and waveguide modes. This model describes very well the key experimental features of linear optics, particularly the Fano-like lineshapes. The derived nonlinear-optical THG spectra are shown to depend on the underlying source of the optical nonlinearity. We present corresponding THG experiments with metallic photonic crystal slabs. In contrast to previous work, we spectrally resolve the interferometric THG signal, and we additionally obtain a higher temporal resolution by using 5 fs laser pulses. In the THG spectra, the distinct spectral components exhibit strongly different behaviors versus time delay. The measured spectra agree well with the model calculations.

We observe a triplet around the third harmonic of the semiconductor band gap when exciting 50 -100 nm thin GaAs films with 5 fs pulses at 3 10 12 W=cm 2 . The comparison with solutions of the semiconductor Bloch equations allows us to interpret the observed peak structure as being due to a twoband Mollow triplet. This triplet in the optical spectrum is a result of light-induced gaps in the band structure, which arise from coherent band mixing. The theory is formulated for full tight-binding bands and uses no rotating-wave approximation. DOI: 10.1103/PhysRevLett.92.217403 PACS numbers: 78.20.Bh, 42.50.Md, 42.65.Re, 78.47.+p When an intense light field at frequency ! 0 excites band-to-band transitions in a semiconductor, additional energy gaps can evolve within the original bands [ Fig. 1(b)]. These so-called light-induced gaps [1][2][3][4][5] are the analog of the induced doublets, split by the Rabi frequency R , in a resonantly excited two-level system [ Fig. 1(a)]. Here, two out of the four possible optical transitions are energetically degenerate. Hence, the optical spectrum consists of three peaks -the famous Mollow triplet for atoms [6,7] or excitons [8,9]. In the semiconductor band case, the light-induced gap in, e.g., the conduction band arises because the original conduction band and the one-photon sideband of the valence band lead to an avoided crossing. The corresponding Hopfield coefficients [1] determine the amount of conduction band admixture. It is crucial to note that this admixture can be finite even far away from the fictitious crossing point. This statement becomes particularly important for Rabi energies h R approaching the photon energy h! 0 , while it can be neglected for small Rabi energies. The latter case is tacitly assumed in previous graphical representations of calculated light-induced gaps. Importantly, as a result of this finite admixture, optical transitions between the induced bands are possible throughout an appreciable fraction of momentum space. Consequently, the transitions acquire considerable spectral weight. Again, two out of the four sets of possible optical transitions are energetically degenerate and a triplet results, which we will refer to as the two-band Mollow triplet in what follows.Light-induced gaps in semiconductor bands have not unambiguously been observed yet. In order to observe the splitting, it clearly has to be larger than the damping. As typical electron dephasing times under relevant conditions are on the order of 10 fs or less, the Rabi oscillation period has to be shorter than this value. This brings one close to Rabi frequencies approaching the frequency of light -a regime which has previously been referred to as carrier-wave Rabi flopping [10]. In this regime, a splitting in the third-harmonic spectra has been observed [11]. However, as the measurements have been compared only with calculations on the basis of two-level systems [11], skepticism has been expressed whether this interpretation was correct indeed. Furthermore, later theoretical work [12...

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 ...