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A technique for broadband spectral amplitude control of light pulses produced in high-order harmonic generation (HHG) is presented. It has been shown elsewhere that broadband spectral phase control in HHG is achievable using a computerized feedback loop scheme by coherently adding a filtered region of the HHG emission to the intense IR driving pulse with optimal attenuation and time delay parameters. In the present study, further computational evidence of the capabilities of this control scheme is provided by considering the spectral amplitude in a broadband region of the HHG spectrum as the control target for the production of isolated attosecond pulses. Different spectral widths and central photon energies are examined, such as a spectral width of 30 eV centered at 36 eV, well in the plateau, and a width of 20 eV centered at 60 eV in the cutoff region. An iterative procedure of the method is implemented and optimal isolated single cycle pulses at a central photon energy of 36 eV are obtained. This control scheme is a fundamental tool that can be implemented for amplitude and phase shaping of any suitable spectral region in HHG.

A method for controlling the spectral phase and amplitude of broadband coherent light from high-order harmonic generation (HHG) is presented. The method considers a feedback loop scheme [

It is well known that the broadband emission from HHG is a source for coherent attosecond light pulses, which can span photon energies from the ultraviolet to the soft X-ray region [

In a recent work [

As mentioned above, the feedback loop is based on using a filtered spectral width of the HHG spectrum to slightly reshape the HHG driving pulse, which is accomplished by a coherent combination of a weak XUV pulse with the strong IR driving field. Combining attosecond XUV pulses with the driving IR field has been studied for HHG [

The control technique basically includes a variable intensity attenuator, a variable delay line and a spectrum analyzer, which are managed by a computer. The experimental realization is limited by the existence of the optical elements at the different bandwidths and photon energies that are of interest. In this respect, the present work attempts to demonstrate the feasibility of the control method by analyzing different significant examples, since the observed effect cannot be addressed in general due to the intrinsic characteristics of a coherent control tool based on feedback loop schemes. Otherwise, the observed results are based on a conceptually clear physical mechanism. The manipulation of the spectral amplitude/phase of the harmonics is based on the coherent combination of several harmonic generation zones with appropriate delays in between and with optimal intensities, such that the resulting spectral amplitude/phase is modified in a broad spectral region, or equivalently, the addition of a weak filtered region of the emitted spectrum to the IR intense driving pulse produces a small reshape of the IR field that conveniently modifies the amplitude and phase of the electron quantum trajectories generating the different photon energies in the HHG process.

As schematically illustrated in ^{14} W/cm^{2} interacts with a gas medium–helium in the present simulations (ionization potential 24.59 eV), producing high-order harmonics with a photon energy cutoff at _{1} pulse is modified by a variable light attenuator (A_{1}) that scans for optimal intensities, and then it passes through a variable delay line (D_{1}) that scans for the position with respect to the IR pulse where the XUV_{1} pulse is to be added. Coherently combined, the resulting IR + XUV_{1} pulse is then sent to a second helium gas target that produces a second high-order harmonics signal. The output from the second gas target is filtered to select the same bandwidth as in the first interaction region, and the spectral amplitude of this last generated XUV pulse is reconstructed by using a spectrum analyzer. The information of the spectrum is then examined by a numerical algorithm that performs a linear regression from the data of the spectral amplitude versus photon energy and calculates its normalized chi-square (^{2}) function, such that the

Hence, the problem of fitting a set of _{i}_{i}^{2} function is used to measure how well the model agrees with the data. The ^{2} function is defined here as
^{2} function will obviously be zero and the fit is perfect. Due to the evident deviations of the spectral amplitude shape from the straight line this will not happen, and the numerical algorithm will thus find the parameters ^{2}. A linear regression and a value of ^{2} is computed for each intensity and position of the XUV_{1} pulse, and after normalizing ^{2} with ^{2} value provides the combination of IR andXUV_{1} pulses that produce the flattest spectral amplitude, _{1}, XUV_{2}, ..., XUV_{n}

(Color online) Schematic illustration of the numerical experiment. An intense IR laser pulse interacts with a first gas target—helium in this case. The output is filtered in a particular spectral region—and occasionally split in several pulses, in order to modify the XUV intensity, which is achieved by using variable light attenuators _{i}_{i}_{i}_{i}

The simulations have been performed by numerically solving the atomic response in the single-atom non-adiabatic strong-field approximation [

In _{1} pulse resulting from the first gas target is combined with the IR intense pulse and sent to the second interaction region by scanning its intensity from ^{6} to ^{7}, in steps of ∆^{6}, where _{XUV}_{IR}/α_{XUV}_{IR}_{1} pulse corresponds to an attenuation parameter as high as ^{7}. As commented above, the fitness function considered for the spectral amplitude control is the normalized ^{2} function obtained from a linear regression of the spectral amplitude versus photon energy in the second interaction region. The normalized ^{2} parameter is minimized by the numerical algorithm in order to find the values of the attenuation ^{2} is indeed extremely efficient to achieve optimal isolated attosecond pulses by spectral amplitude shaping. As it can be observed from

(Color online) Optimization of the HHG spectral region 21–51 eV obtained by considering a single feedback loop. The scan in ^{6} to ^{7}. The value of the normalized ^{2} function is 0.9998 for the original spectrum and 0.8929 for the optimized one. In (_{1} pulse are shown in (_{1} pulse has been rescaled to allow its visualization. In this case the optimal values of the fitness parameters are ^{7} and

The physical mechanism by which the optimal pulse is achieved, and in particular how much of the ionization in the process is due to tunneling, considering a tiny change in the electric field of the IR pulse, how much is due to ionization caused by the XUV_{1} field, or by the effect of time modulation of the atomic potential and bound wave function is difficult to elucidate. However, some insight of the physics behind the described process can be obtained. _{1} pulse is added lies in the temporal late wing of the IR pulse, which in this case corresponds to _{1} pulse nea the center of the IR pulse. Observing the reduction of the intensity generated in the HHG process from the central part of the IR pulse (_{1} pulse induce a phase reshaping in the HHG quantum trajectories that result in a strong cancellation of the HHG signal. This conjecture is supported by the fact that small variations on the amplitude of the IR field can produce a substantial change on the phase of the quantum trajectories, as it is described in detail in [_{1} pulse in the late wing of the IR pulse induces an enhancement of the signal in the spectral region that is optimized (21–51 eV) together with a suppression of the contribution of the long trajectories in that region, which is a relevant consequence of the optimization procedure. The overall effect is a confinement of the HHG signal at

(Color online) Time-frequency analysis of the original (

In order to investigate the influence of the position at which the attosecond XUV_{1} pulse is added with respect to the IR driving field on the optimization procedure, the previous simulations have been repeated by allowing the scan of the position angle ^{7}. As shown in _{1} pulse to the higher values of the IR electric field results in a higher enhancement of the optimal spectral amplitude with respect to the original spectrum. Also, since the value of the normalized fitness function ^{2} for the optimal spectrum is not as low in this case (^{2}= 0.9893) as it was in the previous simulations (^{2}= 08929), _{1} pulse in the optimal configuration induces a considerable suppression of the long trajectories in the enhanced region. Note that the suppression of the long trajectories is however not as complete in this case as it was in the case of

(Color online) Optimization of the HHG spectral region 21–51 eV obtained by considering a single feedback loop. The scan in ^{7}. The value of the normalized ^{2} function is 0.9998 for the original spectrum and 0.9893 for the optimized one. In (

The results shown in ^{6}^{7}), no further optimization is observed in this case by considering additional iteration loops. In order to investigate the capabilities of the multiple iteration procedure, simulations have been performed by considering weak perturbations of the IR driving pulse and by allowing the iteration procedure to run until a convergence of the fitness function is obtained. In the simulations shown in ^{9} to ^{10} in steps of ∆^{8}, and the angular position has been scanned from ^{9} and ^{2}= 0.6789. Once the optimal values have been determined from the first iteration, further iterations are performed by scanning only near the first iteration optimal region, which substantially improves the performance of the iterative method. In the present simulation the value of the fitness function converges at approximately the 5^{th}^{th}^{10} and ^{2} = 0.4927. In ^{th}^{9}^{10}). The intensity of the optimal XUV pulse after the 5^{th}^{2} function for the increasing iterations. Clearly, as the iteration procedure evolves, the value of the fitness function improves such that at the 5^{th}^{2} function becomes basically constant after the 5^{th}

(color online) Time-frequency analysis of the original (

(Color online) Optimization of the HHG spectral region 21–51 eV obtained by considering several iterations of the feedback loop for ^{9} to ^{10}(^{11} to ^{12} (^{th}^{th}^{10} and ^{th}^{2} parameter as a function of the iteration number. From approximately the 5^{th}^{2} and the intensity profile of the resulting XUV pulse do not change substantially. (^{th}^{th}^{12} and ^{th}^{2} parameter as a function of the iteration number. From approximately the 8^{th}

From the simulations that have been carried out, it can in general be concluded that the optimized configuration is not critically sensitive to the value of the intensity of the XUV_{1} pulse, and also that weak perturbations of the IR pulse seem to be generally more effective. This supports the fact that the short trajectories contribute most on the optimal configuration, since they are more robust against changes in the intensity compared with the long trajectories [^{11} to ^{12} in steps of ∆^{11}. The corresponding results are shown in

(Color online) Optimization of the HHG spectral region 50–70 eV obtained by considering a single feedback loop. The scan in ^{7} to ^{8}. The value of the normalized ^{2} function is 0.8771 for the original spectrum and 0.2309 for the optimized one. In (^{8} and

Finally, the optimization of the spectral amplitude has also been considered in a spectral region near the cutoff, as a second significative example, which shows the generality of the control technique. The filtered spectral region in this case is centered at 60 eV with a bandwidth of 20 eV, and the results are shown in ^{7} to ^{8}, in steps of ∆^{7}, and from ^{8} and _{1} pulse with respect to the IR driving pulse lies near the late temporal wing, as shown in _{1} pulse–the corresponding field amplitude is shown in the inset of _{1} optimal pulse is centered, clearly showing again a selection of the short trajectories around the HHG region that is being optimized (50–70 eV).

(Color online) Time-frequency analysis of the original (

In conclusion, a feedback loop scheme has been proposed for shaping the spectral amplitude in a broad spectral region of HHG. In previous results, it was shown that the same control technique is suitable to reshape the spectral phase in a broad region of the spectrum [

Support from the Spanish Ministry of Economy and Competitiveness through “Plan Nacional” (FIS2011-30465-C02-02) is acknowledged.

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^{+}under simultaneous irradiation of fundamental laser and high-order harmonic pulses