As carrier-envelope phase contributes to high-order harmonic phase: effects on the electric field of attosecond pulses

In this work we report on a self-referencing technique for the measurement of the effect of the carrier-envelope phase (CEP) of the driving pulses on the phase difference between consecutive harmonics. By using a numerical model based on the nonadiabatic saddle-point method [1] we demonstrate that, in particular experimental conditions, it is possible to directly control the electric field of the attosecond pulses by controlling the electric field of the driving pulses. Harmonic emission has been produced focusing femtosecond light pulses (800-nm central wavelength), into an argon jet, located around the focus of the laser beam to enhance the contribution of the long quantum paths. The CEP of the infrared (IR) pulses has been stabilized as described in [2] and has been varied by introducing in the beam path a glass plate with variable thickness. Upon increasing the intensity of the IR pulses the harmonic peaks broaden and eventually overlap in the spectral region between consecutive odd harmonics, where distinct spectral peaks, whose position is CEP dependent, are formed. This is due to the fact that the temporal variation of the IR pulse intensity produces a harmonic chirp, which broadens the spectrum of each individual harmonic. This effect is more significant in the case of the long quantum paths. If the spectral broadening is larger than the frequency separation between consecutive harmonics, the high-frequency side of the qth-harmonic spectrum, generated on the leading edge of the IR pulse, overlaps the low-frequency side of the spectrum of (q+2)th harmonic, generated on the pulse trailing edge, thus giving rise to the observed interference effect. We have then varied the CEP, ?. Figure 1 shows the portion between 13th and 15th harmonics of nine spectra for different amounts ?z of glass in the beam path, corresponding to different CEPs in the range ?0<? <?0+?. Each horizontal line represents a spectrum measured at a fixed CEP. The positions of the interference peaks continuously shift by changing the CEP of the pulses. The beat pattern periodically changes for a CEP variation ??=?. The same behavior was observed for all the pairs of consecutive harmonics. Using the algorithm of Fourier transform spectral interferometry it is possible to retrieve, from the interference pattern, the phase difference, ??q, between the consecutive harmonics in the overlapping region. From this analysis we have obtained two important conclusions: (i) the delay ?T between the interfering field components is not appreciably affected by ?; (ii) a variation ?? of the CEP determines a variation of ?? given by 2 ??. The experimental results lead to the conclusion that the phase, ? q(?), of the harmonic field is related to the CEP of the driving pulses by the following expression: ?q(?) = ?q(?)-q?, as confirmed by the results of numerical simulations based on the use of the nonadiabatic saddle-point method [1]. Moreover such calculations demonstrate that the validity of the previous conclusion can be extended also to the contributions of the short paths. Using the nonadiabatic saddle point method, it is possible to calculate the temporal evolution of the electric field of the generated XUV radiation. We have first considered the contribution of the long quantum paths. We found that, while the synchronization between the attosecond pulses and the infrared field is not affected by ?, the CEP of the attosecond pulses is directly influenced by that of the IR pulse and it can be directly controlled by controlling the CEP of the fundamental pulses. On the contrary, neither the synchronization nor the CEP of the attosecond pulses generated by the contributions of the short quantum paths are significantly influenced by the CEP of the driving pulses. In conclusion, using a self-referencing technique, we have experimentally demonstrated that the harmonic phase is directly affected by the CEP of the driving pulses. By using the nonadiabatic saddle-point method, we have shown that, in particular experimental conditions, it is possible to directly control the electric field of the attosecond pulses