5.2 Intensity dependence
5.2.1 Ionization yield
The integrated ionization yield depends nonlinearly on the laser intensity. This is intrinsic in any analytical formulas used to describe ionization both in the tunneling and in the multiphoton regime (Sec. 1.2.1), as well as in numerical models solving directly the time-dependent Schr¨odinger equation (TDSE). For few-cycle pulses, it is therefore expected that also the directional photoelectron asymmetries depend on the intensity in a characteristic way.
A first approach for varying the laser intensity generating the ATI emission is to move the gas source (a 30-µm thin glass capillary) along the beam propagation direction in the focal region. This allows measuring at smoothly changing intensities up to the peak intensity, reached in the center of the focus. The selection of electrons generated at a well-defined position in the focus is once again ensured by the two pair of slits used in the stereo-ATI setup (Fig. 3.7 on page 46).
To quantify the detected left-right asymmetry, a suitable parameter is defined as (L−R)/(L+R), where L and R are the ionization yields respectively in the left and right direction. This quantity oscillates around zero, and the contrast of the measured oscillation can be expressed by its amplitude (peak-to-valley). Figure 5.8(a) shows the measured contrast of the oscillations as a function of the propagation distance in the focus. At each position, it is also indicated the laser intensity estimated from the position of the cutoff in the ATI spectra, occurring at an energy of 10Up. Note the area
of relatively constant intensity in the central part of the focus, not surprising due to the astigmatism of the beam (Fig. 5.3). The highest contrast is observed in the outer part of the focus, where the intensity is lowest. By moving toward the central part of the focus, the contrast decreases, reaches a minimum and increases again, acquiring relatively constant values in the center of the focus. By sorting the data, one can plot the contrast as a function of the intensity [Fig. 5.8(b)].
It is worth noting that, due to the strong dependence of the contrast of asymmetry also on the pulse duration (Sec. 5.3), Fig. 5.8(b) cannot be used to calibrate the laser intensity in the experiments by measuring the asymmetry parameter. The best gauge in this respect remains the position of the cutoff in the ATI plateau. However, for a fixed pulse duration (as it is the case in Fig.5.8), the contrast reveals to be an extremely
−2 −1 0 1 2 0 0.1 0.2 asymmetry parameter −2 −1 0 1 2 5 10 15 propagation distance [mm] intensity [10 13 W/cm 2 ] 10 15 0.02 0.04 0.06 0.08 0.1 0.12 0.14 intensity [1013 W/cm2] asymmetry parameter 5 ( a ) ( b )
Figure 5.8: (a) Contrast of the left-right asymmetry (solid line) and estimated laser intensity (dashed line) as a function of the position in the focal range. (b) Asymmetry parameter as a function of intensity, as deduced from (a). The large error bars are due to uncertainties in estimating the intensity, not to poor signal-to-noise ratio. The contrast shows a minimum at intensities around 1014W/cm2. Note that the value of the parameter also strongly depends on the pulse duration.
the contrast in the central part of the focus [Fig. 5.8(a)] is actually a signature of small intensity variations, hard to recognize in the ATI plateau, but compatible with the aberrations of the focused beam.
To remove this source of intensity instability in the measurement, somewhat not under control, a second approach for studying the dependence of the contrast on the intensity is to change the pulse energy, while keeping fixed the position of the gas jet and of the slits. This can be done by attenuating the laser beam by means of thin pellicles positioned at tunable angles of incidence. The thickness of the pellicles (4µm) ensures that the pulse duration is not affected significantly, so that any observed difference in contrast can be attributed only to the different intensity. With this procedure the difference of intensity is known with high precision simply by measuring the pulse energy with a power meter.
Figure 5.9(a) shows the asymmetry parameter (L−R)/(L+R) as a function of the absolute phase (as usually, varied by means of the glass wedges) for different intensities3. The value of the phase was independently determined for each intensity on the basis of the high-energy electrons, as described in Sec. 3.5. The intensity was estimated by means of the position of the ATI cutoff for the spectra taken at to 7.5×1013W/cm2 [violet curve in Fig. 5.9(a)], and subsequently scaled for the others
3For technical reasons, the very low energy electrons (<4 eV) were discarded in these measure-
ments. This weakly affects the value of the asymmetry, but not its intensity dependence, under investigation here.
5 6 7 8 0.05 0.1 0.15 0.2 0.25 0.3 0.35 intensity [1013 W/cm2] asymmetry parameter 0 2 4 6 0.2 0.1 0 -0.1 -0.2
absolute phase [rad]
(left-right) / (left+right) 8.3e13 7.5e13 6.7e13 5.7e13 4.8e13
Figure 5.9: (a) Left-right asymmetry as a function of the absolute phase measured for different laser intensities, indicated on the plot. No intensity-dependent phase shift can be detected. (b) Contrast of the asymmetry as a function of the intensity, as deduced from (a). The increase at low intensities observed in Fig.5.8(b) is here confirmed.
according to the measured pulse energy. Note that no intensity-dependent phase shift can be observed in this intensity range, i.e. the maximum contrast always occurs for approximately the same pulse shape, namely at a phase slightly higher thanπ4. Figure 5.9(b) shows the measured contrast as a function of the intensity.
The main features observed in these experiments are currently object of theoret- ical investigations based on numerical solutions of the time-dependent Schr¨odinger equation [Chelkowski02]. The semiclassical model, on the other hand, is generally unsuitable for describing ionization asymmetries, obviously due to the influence of the Coulomb potential. However, it is still useful since it provides qualitative explana- tions of the physical processes. Figure 5.10 shows the calculated contrast of left-right asymmetry as a function of the laser intensity. The asymmetry parameter increases at low intensities, in qualitative agreement with the experimental data [Fig. 5.9(b)]. This is due to the dependence of the ionization rate [Eq. (1.9) on page 8] on the field strength, predicting higher nonlinearities at low intensities. The second remarkable characteristic of the contrast, namely the minimum appearing at intensities around 1014W/cm2 [Fig. 5.8(b)], can also be qualitatively explained by tunneling theory. In-
deed, at intensities between 1014 and 2×1014W/cm2 the ionization probability of a
xenon atom significantly grows and reaches saturation. Thus, ionization takes place mainly in the leading edge of the pulse (cf. Fig.1.4on page11), resulting in aneffective shorter pulse and increased contrast.
5 10 15 0.1 0.2 0.3 0.4 0.5 intensity [1013 W/cm2]
asymmetry parameter 10 ionization level [%] 20 30 40 50
Figure 5.10: Asymmetry parameter (solid line) and ionization level of the atom (dashed line) as a function of the laser intensity. The calculation assumes xenon atoms (Ip = 12.13 eV) and
a pulse duration of 5 fs FWHM. At high intensities, saturation starts playing a significant role, confining the ionization mainly in the leading edge of the pulse.