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The macroscopic build-up of the high-order harmonic signal depends on the free electron density in the generation medium. The free electrons affect the harmonic yield and spectral shape through modifying the refractive index and the phase matching conditions. These dependences allow studying the He+ ion channel formation in a He gas jet. The evolution of an ion channel created by an ultrashort laser pulse via optical field ionization was monitored using the harmonic signal generated by a collinear propagating second laser pulse. From the measured high harmonic signal as function of the delay we are able to gain information about the free electron density. Under our experimental condition, the ion channel has been fully formed 300 fs after the first laser pulse, resulting in an enhancement of harmonic yield of the second laser pulse by two orders of magnitude.
©2009 Optical Society of America
1. Introduction
Irradiating atoms with high intensity laser pulses results in a strong ionization via optical field ionization [1]. The generated free electrons are pushed out from the central region of the laser beam leaving behind an ion channel. This channel formation plays an important role in several experiments such as laser electron acceleration [2], laser wake field accelerators [3, 4], and laser based betatron radiators [5–7]. These experiments typically use strongly focused laser beam with relativistic intensity and the interaction length can be extended via self-guided laser propagation in the created ion channel. The plasma dynamic is extensively studied with different methods using probe beams in the optical [8–10] or extreme ultraviolet (XUV) [11] spectral range.
On the other hand, high order harmonic (HH) generation by short laser pulses relies on non-relativistic intensities. In neutral noble gases, the generated HH spectrum was recently extended into the soft-x-ray spectral range beyond 1 keV [12, 13] reaching 3–4 keV [14] with a photon flux apt for x-ray absorption experiments [15, 16]. For further more demanding applications it will be necessary to increase the photon yield and/or extend the spectral range up into the hard-x-ray regime. These extensions are restricted because the spectral extension requires laser pulses with very high peak intensity and consequently most of the atoms are already ionized at the leading edge living behind a lot free electrons. The dispersion of the created free electrons reduces the coherence length substantially, restricting the harmonic signal generated from atoms or mainly ions by interacting with the peak intensity to very low level. The strong ionization on the leading edge of the pulse can be minimized by using media with the highest possible ionization potential. The phase matching problem can be solved by applying either quasi phase matching [17–20] or non-adiabatic phase matching [20, 21, 12, 14] techniques as recently demonstrated in neutral gases.
So far the shortest wavelength radiation relying on HH has been achieved by using He, having the highest ionization potential among all neutral atoms. Only ions have a higher ionization potential, consequently HH generation from weakly ionized atoms is extensively studied nowadays. One or two-times ionized atoms were prepared by capillary discharge [23, 24] or in solid plasmas by a laser pre-pulse [25, 26]. From these ions HH radiation was generated with high efficiency in the XUV. The reason for the observed limitation for photon energies below 300 eV is the short coherence length due to the fact that an essential part of free electrons remain in the same volume as the ions. A further extension of the wavelength range requires a combination of ions with phase matching schemes and a method to spatially separate the ions from the free electrons prior to HH generation.
Here we describe a new possibility of extending the coherence length of high order harmonic generation by formation of an electron free channel containing dominantly He+ ions. He+ is the most promising candidate, because it has the highest ionization potential among all singly ionized atoms. We will show, applying a double pulse scheme, the phase matching conditions can be controlled by the pulse to pulse separation. The first pulse forms an ion channel for the second one. Under optimized conditions we are able to demonstrate efficient HH radiation from He+ ions. Due to sensitivity of the HH yield from the coherence length, our method allows also the monitoring of the evolution of the on-axis free electron density.
2. Experimental setup
In the experiment, the laser pulses are delivered from a two-stage Ti:sapphire amplifier system running at 1 kHz repetition rate. The output pulses have energy up to 3 mJ and the pulse duration is in the order of 15 fs. The system is described in detail elsewhere [14, 27]. In the laser system we have incorporated an acousto-optic programmable dispersive filter (DAZZLER, Fastlite). By applying an appropriate control signal we were able to generate double pulses with a variable delay [28, 29] and identical pulse energies of 0.75 mJ. Both pulses are linearly polarized in the same direction. A higher energy is possible, but then the pulse shape and delay is strongly influenced by the saturation of the amplifier. Thus for a reliable double pulse operation the maximum energy is limited to the above mentioned value.
The pulses are focused into the He gas jets by a high reflecting mirror with a focal length of 200 mm. The interaction between the laser beam and the helium gas takes place within a nickel tube. Its axis is aligned perpendicularly to the laser beam and the interaction length is about 0.4 mm, much shorter than the confocal parameter (2.3 mm) of the laser beam, avoiding self-channeling. The beam diameter in the focus is measured to be 20±5 μm (FWHM) yielding an on-axis peak intensity of about 1×1016 W/cm2 for both pulses assuming Gaussian pulse shape in time and space. For these parameters the estimated on-axis HH cut-off energy will be in the order of 1.9 keV [1]. For a more realistic pulse shape, the estimated peak intensity and consequently the cut-off energy is about 20% smaller [27].
The generated harmonic signal was measured with a scanning x-ray spectrograph (248/310G, McPherson) equipped with a platinum-coated grating 300 lines/mm. The x-ray photons were detected with a photomultiplier (Channeltron® 4715G, Kore Technology) and the measured signal was filtered and amplified with a lock-in amplifier. The x-ray signal was optimized in the single pulse mode (1.5 mJ; 15 fs pulses) by adjusting the gas backing pressure to 250 mbar and the gas jet was moved approx. 1 mm after the focus. Putting the target behind the focus allows the exploitation of further phase matching contributions [13]. For vacuum separation between the source chamber and the spectrograph and for blocking the lower order harmonics we inserted a 100-nm-thick free standing Ti foil between the two chambers. With this setup we recorded the x-ray spectra for double pulses with a pulse to pulse distance in the range from 0 to 300 fs, which is the limit of our setup.
3. Ionized channel formation and measurement
Fig. 1. Ion channel formation and high-order harmonic generation by double laser pulse (red). The first pulse creates the channel (right) and the second pulse generates the harmonics (blue) in the channel (left). Red/blue color scale: positive/negative effective charge, respectively.
The mechanism for channel formation is illustrated in Fig. 1. The first (non-relativistic) laser pulse is intense to fully ionize the neutral He atoms via optical field ionization, but not strong enough to ionize them two times. The freed electrons are accelerated by the electric field of the laser pulse. The classically calculated electron trajectories depend significantly on the time of ionization within the electric field. Nevertheless, nearly all of the electrons will gain enough momentum in the direction of the laser polarization for moving away from the optical axis leaving behind a positively charged ion (He+) channel. After some adjustable delay, the second laser pulse is launched into the channel. The intensity (1×1016 W/cm2) is high enough for the generation of high-order harmonic radiation from the positively charged atoms.
3.a. Following the channel formation by observing the spectral modulation of the spectrum
The measured HH spectra as a function of the pulse to pulse delay are plotted in Fig. 2a. The harmonic signal dramatically drops at around 50–100 fs pulse separation as a consequence of the nonlinear dependence of the harmonic yield on the laser intensity and remains at low levels due to the reduced coherence length given by the large free electrons density from the singly ionized He atoms. However, later as the electrons leave the on-axis region, the harmonic signal grows due to the better phase matching condition because the generated HH intensity is proportional to the square of the coherence length and consequently inverse proportional to the square of the free electron density (see Eq. (2) and for a detailed discussion in the section “Discussion”). The free electron distribution affects not only the harmonic yield but also the shape of the spectra due to the wavelength dependence of the phase matching factor [20, 30]. A simple model for HH assuming a finite coherence length predicts a sinc squared dependence of the harmonic signal as function of the photon energy [12, 13]. The periodicity will be more pronounced for HH with few cycle laser pulses, because x-rays are only generated during one or two optical cycles with nearly the same - and better defined -free electron distribution [12, 13]. However, for multi-cycle pulses the short wavelength radiation is generated at different free electron densities resulting in non-periodic and weaker modulation of the spectrum. The a-periodic spectral modulation is clearly visible in our experimentally obtained spectrum for a single pulse (see green curve in Fig. 2b).
Fig. 2. The recorded high-order harmonic spectra (a) essentially depend on the distance between the two generating laser pulses. (b) The shape and periodic structure of the spectra are also affected by the applied delay (spectra are normalized in the inset). (c) The evaluated spectral periods (diamonds) follow the theoretical prediction (pink curve) and the free electron density also can be estimated.
Contrary, when a pre-pulse creates an ionized channel, the second pulse experiences only the dispersion from the free electrons created during the first pulse, because the ionization of He+ by the second pulse is much weaker. Consequently, the free electron density is much better defined and the spectrum will show a periodic structure with a modulation period related to the free electron density in the channel. In Figure 2b we present measured spectra where the periodic modulation is clearly visible. For the sake of clarity we added marks corresponding to the observed minima. The estimated period is 83 eV and 60 eV for a delay of 200 fs and 300 fs, respectively. For a delay of 50 fs, the free electron density and consequently the modulation period is too high to be efficiently displayed in the same plot. In the inset in Fig. 2b we plotted the spectra for a reduced energy range revealing the modulation period of 12.7 eV. The origin of the extra minimum in the measured spectra appearing in the range at 450 eV is attributed to the L absorption edge of the Ti filter. The minimum around 200–250 eV correlates with a dip in the diffraction efficiency of the used grating. As both of these minima were independent of the delay we neglected them in our evaluation. The evaluated period as a function of the delay is summarized in Fig. 2c. For a comparison we have added the theoretically predicted curve which will be described and discussed in Sec. 4.
3.b. Measurement of channel formation by spectral intensity
Insight in the dynamic of the channel formation can be also gained from the different evaluation of the HH spectra as a function of the delay. The HH signal was integrated in several spectral ranges and the results are plotted in Fig. 3a. For zero delay we optimized the HH signal by adjusting the backing pressure and the focusing into the gas jet. Starting from the maximum, the signal completely disappears for a pulse separation of 50 fs. This feature can be well explained by comparing it with the high-order autocorrelation function of the laser pulses. The harmonic signal is proportional to the driving laser pulse energy to the power of 5 to 6 similarly to Ref. [30] and has been also verified in the spectral range with a single pulse experiment. A high order autocorrelation curve of the same order has been calculated (Fig. 3, dashed green lines), and is in reasonable agreement with the measurement. If no other effects will play a role it is expected that the signal will remain at the same level as it is at 50 fs for longer delays. However, the measured signal shows a different behavior. The integrated harmonic signal increases for longer delays and reaches a maximum at about 250–300 fs pulse separation. The gain in the harmonic yield is about 2 orders of magnitude compared to the signal for 50 fs delay.
In a next experiment we optimized the HH yield for a double pulse excitation with a pulse separation of 300 fs. For maximizing the signal we had to increase the backing pressure up to 520 mbar compared to 250 mbar for the single pulse excitation. For these measurement series we replaced also the grating (300 lines/mm) in our spectrograph by one with 1200 lines/mm and the Ti filter by a 300-nm-thick silicon film. The more dispersive grating has a blazing energy at around 0.5 keV ensuring a nearly constant diffraction efficiency over the full examined spectral range The lower diffraction efficiency will be partly compensated by the higher transmission of the Si filter especially in the range below 100 eV.
Despite these major changes of the experimental setup the observed HH yield as function of the delay is very similar as shown in Fig. 3b. Only the optimum conditions for HH generation are now reached for somewhat shorter delays of 200 to 250 fs. Nevertheless for both sets of data we observe the maximum enhancement for HH radiation in the range of 300 eV. Further the maximum for the different energy ranges will shift towards longer delays for lower energy photons (see Fig. 3). The observed higher photon yield at earlier times for higher energy photons can be easily understood. The higher energy x-rays are generated closer to the optical axis due to the higher intensity of the driving laser and near the axis the electron density will be more efficiently reduced at earlier times due to the higher drift velocity.
Fig. 3. Ionization channel formation in He gas at two gas pressure of a) 250 mbar and b) 520 mbar, respectively. Dashed green: high order autocorrelation curve of the laser pulse; Continuous lines: theoretically calculated HH intensities; marks: measured integrated signal in mentioned spectral ranges. The data set of 200–300 eV is missing in Fig. 3a because of the low diffraction efficiency of the grating in this range as mentioned in the text.
4. Discussion
The harmonic signal should be very low (as it is between 20 and 50 fs) independently of the delay between the two pulses if only the effect of the created number of the free electrons were considered. Below 20 fs delay when the two laser pulses overlap, there is a huge increase of the harmonic signal as a consequence of the strongly non-linear dependence of the harmonic yield from the laser intensity. Furthermore, the measured increase of the harmonic signal at longer delays can not be explained by a simple interplay of the two pulses. The additional signal can not be generated by the first laser pulse, because no mechanism is known, which modulates the yield as function of the delay in the observed way. It can only be generated by the second pulse. To understand the observed enhancement it is necessary to calculate the ionization rate of He [31] with the parameters used in the experiment. For our calculations we used a model [32] assuming the isolated single atom approximation to describe the ionization, which is justified in the case of rare gases and low ionization rates. For a higher gas density, higher ionization rates, or considering multiple ionization the interaction with the surrounding ions must be taken into account [31], which is not the case in our calculation yet. The electric field and the fraction of ionized atoms as a function of time are shown in Fig. 4a in the case of single pulse (0 fs delay). The ionization happens at the leading edge of the pulse, where the intensity is much lower than the peak intensity, restricting a further extension of the HH spectrum by a simple increases the laser pulse intensity.
Then we calculated the ionization for a double pulse excitation, as in our experiments. The first laser pulse ionizes the He gas creating mainly He+ ions and a few He++ ions too (Fig. 4b). The second pulse, which is delayed by 60 fs, further ionizes the He+ ions. The ionization by the second pulse is accompanied by the generation of HH radiation. However, for the macroscopic buildup of the HH signal phase matching is necessary. The signal grows over the coherence length which inversely proportional to the free electron density in the generation medium [30]. The second pulse propagates in the highly ionized He gas, and due to short coherence length the generated harmonic signal remains low. As mentioned above, the electrons gain additional momentum in the electric field of the first laser pulse in the direction of the laser polarization. The resulting drift energy can be as high as several 100’s of eV [33]. After a certain time, most of the free electrons are pushed out from the region near the optical axis, leaving behind a more or less electron free channel around the axis (Fig. 1). The drift of the electrons away from the axis is terminated by electrostatic restoring forces due to the positive ions and possibly space charge effects. The channel formation and the remaining free electron intensity can be accurately calculated by particle-in-cell simulation (e. g. [3, 6]). For an estimation of the on-axis free electron density and the resulting coherence length we have used a simplified classical model. The transverse electron distribution is approximated by a Lorentz shaped distribution
with the scaling parameter Δr and the electron motion is supposed to be sinusoidal in the field of the ion channel with the period of 4τ, where τ is the time when the electrons reach the maximum distance r0 from the optical axes. We have chosen a Lorentzian distribution because it ensured a better agreement for longer delays with our experimental observations as summarized Fig. 2c, 3a and 3b. At shorter delays, below 100 fs, a spatial Gaussian distribution electron density provides a better fit, because of the Gaussian laser beam profile. However the electrons have a large velocity dispersion so the initial Gaussian distribution will broaden immediately in a non-uniform way turning to a Lorentzian like for longer delays. Our simple model describes the electron distribution very well in the time of the formation of the channel, which is our range of interest. However the behavior of the channel can be different at later times as observed previously in theory and experiments [8].
The observed periodicity of the HH spectrum ΔE is proportional to the coherence length which, while the harmonic intensity IHH scales with square of the coherence length [20, 30]. With these assumptions we can correlate our experimental results with the on-axis free electron density:
These equations describe very well the observed experimental results as shown in Fig. 2c, 3a and 3b, (continuous lines) with the following parameters of τ = 265±45 fs and r0/Δr = 5±2. Furthermore, the maximum excursions of the electrons can be estimated to r0 = 4.6 μm from the initial energy (up to 2 keV) of the electrons and from the corresponding speed of 2.6·107 m/s. This distance is realistic, because the effective beam diameter for harmonics above ~50 eV is narrower compared to the estimated channel diameter. So we can safely claim that the HH radiation from the second pulse is generated in the ionized channel mainly from He+ ions. Furthermore, the two orders of magnitude increased signal of the HH at around 250–300 fs can be only explained by an order of magnitude smaller free electron density at the time of generation compared to the first pulse. This observation in good agreement with the factor of 7.5 calculated from the measured spectral periodicity at 250 fs and 50 fs delay, respectively.
Fig. 4. Calculated ionization of He gas for (a) single and (b) and (c) double pulse excitation. For all pulses the intensity is in the order of 1×1016 W/cm2, expect for the second pulse in c) with an eight times higher one.
5. Summary
Applying short laser pulse with a peak intensity of 1016 W/cm2, an ion channel can be formed in He gas within 300 fs. The channel formation has been monitored by generating high order harmonics in the ion channel by a second laser pulse. The free electron density decreases by an order of magnitude within 250–300 fs to about 6×1017 cm-3, which can be concluded from the observed periodicity of the high order harmonic spectra and from the harmonic yield generated in the channel. The resulting longer coherence length allows an enhancement of the signal by two orders of magnitude. Launching a second a laser pulses with a peak intensity of 8×1016 W/cm2 into the He+ channel opens the way for extending the generated harmonic spectra into the hard-x-ray regime up to 15 keV. This possibility is provided by the higher ionization potential (54.4 eV) [34] of He+ ions compared to the ionization potential of 24.6 eV of the neutral He atoms. The ionization as a function of time is shown in Fig. 4c, and it can be clearly seen that the major fraction of the He+ ions will be further ionized by the second pulse at the time when it reaches its peak intensity. Further extension of the harmonic spectrum is possible by using ions with a higher ionization potential. A good candidate will be Li+ ions with an ionization potential of 75 eV. However, preparing Li atoms with a high density in the gas phase require a more complicated setup. Another option is Ne3+ with an ionization potential of 110 eV. But formation of an ion channel with multiple charged ionized atoms need further experimental and theoretical work.
Acknowledgments
This study has been sponsored by the Austrian Science Fund (grants No. F016 P03), the DFG grant SP 687/1-3, and BMBF grant 06WU266I.
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