A theoretical prediction for generating isolated attosecond pulse in water window utilizing instantaneous frequency change of two-color driving laser pulse

Abstract

This work reports on the theoretical generation of isolated soft-X ray 106 attosecond pulse within the water window spectral region, through interacting a chirped two-color femtosecond laser pulse with a hydrogen atom. To construct this pulse, a chirped 3.5 optical cycle (9.33 fs) laser pulse with a wavelength of 800 nm and an intensity of is used as the main field. A 37.32 fs laser pulse with a wavelength of 1600 nm and intensity of , one-quarter of the main field intensity, is employed as the control field. A tangential hyperbolic function is used to impose chirp on the main field. The study simultaneously utilizes the lowest possible intensity and the highest possible pulse duration for both the main and the control field to reach the water window spectral region in high-order harmonic generation up to the 320th harmonic of the driving field.

Introduction

The 2023 Nobel Prize in physics was awarded to three scientists in attosecond physics¹. “The ability to generate attoseconds of light has opened the door on an extremely tiny timescale, and it also opened the door to the world of electrons” said Eva Olsson. These pulses make it possible to look at the events inside the atom². The photon energy of coherent attosecond pulses generally falls in the extreme ultraviolet (XUV) and soft X-ray regions of the electromagnetic spectrum. They serve as a coherent, bright, and directional source of ultra-short X-ray radiation, a subject that has long been a dream for researchers³. Specifically, the “water window” spectral region refers to a frequency range between the absorption edge of carbon and oxygen (284 eV and 543 eV, respectively), which includes soft X-rays with wavelengths between 2.3 nm and 4.4 nm⁴.

Attosecond is the natural time scale of electron dynamics⁵. Therefore, attosecond pulses can be nice tools for recording electron movies⁶. They, in addition to providing the possibility of direct observation of atomic phenomena, made it possible to control and manipulate the chemical and physical processes on this scale^7,8. This unprecedented time resolution has been a bridge towards the observation of phenomena in attophysics and attochemistry, that were not only measurable, but also not visible before^7–9. Isolated attosecond pulses (IAPs) are central for theoretical and experimental advances such as direct imaging of the ionization process and propagation of the generated electronic wave packet^10,11, the evolution of electron density during chemical reactions¹², measuring the attosecond time delay of the photoionization process^13,14, and direct observation of charge migration in atoms and molecules^15,16. Also, reconstruction and control of ultrafast dynamics of multi-electron wave packets¹⁷, correlation dynamics^18,19 and quantum electron decoherences²⁰, tracking the charge carriers in solids including semiconductors²¹ and dielectrics^22,23, and optical solitons²⁴ and the transition to petahertz electronics²⁵ are only a small part of the remarkable advances due to the widely use of attosecond pulses in basic science such as physics, chemistry, and biology.

High-order harmonic generation (HHG) is a phenomenon that provides the necessary frequency comb for generating attosecond pulses^26,27. This phenomenon directly results from the interaction of high-intensity femtosecond laser pulses (typically with an intensity range of to ) with matter²⁸. As a result of the interaction of an intense laser pulse with the coulomb potential of the atom, it will be possible for an electron to tunnel through the potential barrier. After a back-and-forth movement in the oscillating field, the separated electron, during the recombination, will release the acquired kinetic energy plus the ionization potential in the form of a photon, which is the same high-order harmonic radiation generated. This process is repeated for every optical half-cycle^29,30. If the driving pulse is limited to a few optical cycles (generally 1–2 cycles), a single isolated attosecond pulse will be generated³¹.

Perhaps the first option that comes to mind for generating an isolated attosecond pulse is to reduce the duration of the driving laser pulse to 1 to 2 optical cycles (below 5 fs), but since the generation of these extreme short intense laser pulses has its own difficulties from a practical point of view, in some cases, reducing the effective duration of the driving pulse to half an optical cycle seems to be a more suitable option³¹. This confinement to a single recombination event is done through temporal gating of the HHG process³². Therefore, there are mainly two methods for generation of isolated attosecond pulses, all of which are based on the confinement of the HHG process to one half cycle: (i) amplitude gating the driving laser field (AG method) and (ii) temporal gating the recombination process. There are different types of gating methods such as electron quantum path control in HHG via chirp variation of strong laser pulses³³, polarization gating (PG)³⁴, color gating (CG)³⁵, ionization gating (IG)^36,37, or combination of AG and PG³⁸, and or PG and TG named double optical gating (DOG) method^39,40.

All these methods are based on confinement of the recombination process or its temporal gating to a single event at the single-atom level, but in practice, the HHG process is performed in a medium containing a large number of atoms and molecules^41,42. Although, HHG can support the necessary frequency bandwidth of a coherent radiation to generate an IAP, in practice, this coherency is disturbed due to the destructive propagation effects^43,44. The propagation through an ionizing gas medium can affects spatial and temporal, and consequently phase structure of the fundamental laser beam and harmonic radiation generated^45,46. Therefore, if the macroscopic effects are managed in such a way that all HHG frequency components remain in phase during propagation, then the constructive superposition of harmonics can lead to an efficient IAPs generation^47,48. Exploiting time gated phase matching or macroscopic ionization gating method is an important and widespread approach for generating macroscopic high-flux isolated attosecond pulse⁴⁹.

This work focuses on extending the HHG spectrum to the water window spectral region through the interaction of chirped two-color femtosecond laser pulse with hydrogen atom, aimed to generate soft x-ray isolated attosecond pulses. Although the most straightforward approach for generating IAPs is to use CEP-stabilized intense few-cycle laser pulses, this method has its own difficulties from a practical point of view. On the other hand, polarization gating (PG) technique is an appropriate alternative that, by modulating the polarization state of driving laser pulse, generates a shorter isolated attosecond pulse than using CEP-stabilized intense few-cycle driving laser pulses. However, it will come at the cost of a significant reduction in the conversion efficiency of the HHG process⁵⁰.

Color gating (CG) method, by highlighting the role of the central part of the pulse can be a productive method in the generation of a single isolated attosecond pulse^51,52. Adding second color to the main field can increase phase matching between harmonics, increase the generation efficiency^53,54. From an experimental perspective, we have been looking for the least challenging method to maximize the HHG spectrum driven by the longest and at the same time the weakest laser pulses, such that the harmonic spectrum can cover the entire water window spectral region. The wavelength of the control field is chosen to be twice that of the main field. Here, we have only focused on finding the microscopic single-atom response and neglected the macroscopic propagation effects. The solving procedure is based on numerical solution of one dimensional time dependent Schrodinger equation (1D TDSE)^55,56. Numerical solution of TDSE is most accurate method for finding microscopic single-atom response to the driving laser field⁵⁷. We promote the cut-off frequency of the HHG spectrum and simultaneously decrease the isolated attosecond pulse duration step-by-step by adding the proper chirp and control field, with appropriate wavelength, duration, and intensity, to the main field.

Theoretical method

To generate high-order harmonics through interacting a high-intensity laser pulse with an atom, the time-dependent Schrödinger equation (TDSE) by considering a nonperturbative response to the driving field should be solved. The solving procedure is based on following approximations: (i) electric dipole approximation and (ii) Born–Oppenheimer approximation. The HHG quantum theory, assumes that the harmonic field is radiated by electric dipoles, i.e., the moving electrons in the laser field around the stationary nucleus of an atom.

Of course, for some intensity values, due to the doubled wavelength of the control field compared to the main field, (an 800 nm Ti: sapphire laser pulse), we are very close to the border of the region where the dipole approximation is valid^58–61. Non-dipole effects in strong-field ionization have been subject to a number of works by various groups^62–65.

HHG is mostly done in noble gas atoms. Of course, liquids and solids have also been explored^66–69, but this is beyond the current discussion. Here, we simulate the attosecond pulse generation process for a hydrogen atom exposed to a strong laser field. The HHG can be simulated for noble gases in a similar way to the one presented here in single active electron approximation, although more precise methods are also used^70,71.

The TDSE of a hydrogen atom subjected to a driving laser field in the atomic unit (a.u.) is given by:

1

where is the Coulomb potential in the atomic unit with as the softening parameter. The value of is chosen in such a way that the nuclear potential results in the ground state energy of the hydrogen atom, which is in one dimension. The term is the interaction term in the dipole approximation.

Starting from the ground state wave function of the hydrogen atom, one can calculate the wave function in the next times by using the standard second-order split-operator method^72–75. The ground state wave function can be obtained from one of the two methods of numerical solution of the field-free time-independent Schrödinger equation through the diagonalization of the Hamiltonian matrix or the numerical solution of the field-free time-dependent Schrödinger equation by the imaginary time propagation method⁷⁶. To prevent unwanted reflections of the wave function from the boundaries, it is necessary to multiply the calculated wave function in each time step in a mask function.

As we are interested in using chirped two-color field, we construct the driving field as the superposition of a main and a controlling field, given as follows:

2

with and as the amplitude and the central frequency of the fields, respectively. stands for the envelope function, which is assumed to be as follows:

3

where is the full width at half maximum (FWHM) of the pulse.

In Eq. (2), delta function, , describes the time variation of the instantaneous frequency of the pulse which is chosen here as a tangent hyperbolic function:

4

where is the duration of each optical cycle. The parameter characterizes the rate of change of instantaneous frequency. The wavelength of the main field is taken to be 800 nm (for Ti: sapphire laser), and for the control field it is assumed to be doubled (1600 nm); therefore . The power spectrum in the frequency domain is calculated through a Fourier transform of time-dependent dipole moment. However, it has been proven that the dipole acceleration, , is a better parameter:

5

Ehrenfast’s theorem is used to calculate the dipole acceleration as:

6

The integration above is taken over the time in which the problem is solved. The attosecond pulse intensity is obtained from the superposition of selected harmonics given as:

7

where .

Result and discussion

Chirped single-color laser field

As a pioneering idea reported by Carrera and Chu, chirp can be considered as one of the effective factors in extending the HHG spectrum. The temporal chirp is the instantaneous frequency change of a pulse that will be accompanied by a temporal dispersion and, consequently, changes in pulse duration. Here, our definition of chirp does not include changes in pulse duration and is limited only to the instantaneous frequency change of the pulse.

In this work, describes changes in the instantaneous frequency of the pulse, connected to the temporal chirp of a femtosecond pulse (Eq. 4). Yet, it does not fully represent the impact of an actual chirp on a pulse. If one intends to model a chirped pulse accurately, the functions and should be interdependent. This is beyond the scope of this work. What we are describing here with is essentially a numerical way of changing in the instantaneous frequency of the pulse, conceptually similar to how self-phase modulation affects a femtosecond pulse.

A crucial consequence of the way the chirp was modelled, is a significant red shift (i.e., shift toward longer wavelengths) and broadening the pulse spectra as beta increases. In other words, we do not merely redistribute the frequency in time within a fixed spectral bandwidth (i.e. as a typical temporal chirp would do) but, also effectively generate new frequencies that did not exist initially in the pulse. Figures 1, 2 introduce the reader to the framework and peculiarities of this temporal chirp model. This figures demonstrate the red-shifting effect of the applied chirp. In Fig. 1, the first and the second rows show the function, and the third and fourth rows show the modulation of the pulse by the chirp. The parameter, in the Eq. (4), has been changed from 1 to 3 and the chirped pulse has been compared with the unchirped () one. As the figure shows, increasing the beta parameter causes stretch on the central part of the pulse and simultaneously compression on its leading and trailing edges.

Fig. 1.

First row: as a function of time for various parameters; Second row: modulation of pulse with chirp. Throughout the text, and in all figures, o.c. is 2.67 fs, the duration of each optical cycle of an 800 nm Ti: Sapphire laser pulse.

Fig. 2.

The HHG spectra generated through the interaction of a 3.5 optical cycle single color laser pulse with the intensity of without chirp () and with chirp (, , ) with a hydrogen atom.

The accepted definition of temporal chirp in the optics and ultrafast community does not inherently imply spectral broadening or narrowing. Generally, it does not. The spectral broadening effects observed are primarily due to the specific choice of the tangential hyperbolic function used to model the instantaneous frequency of the driving field. This very peculiar function simultaneously impacts the spectral phase and amplitude of the field’s Fourier transform (Eq. 2), depending on the value of the β parameter (Eq. 4).

Figure 2 depicts the HHG spectrum generated by unchirped () and chirped driving laser pulse with . Here, we have only applied a single color field in the interaction, i.e. we have used only the first term in Eq. (2). As the Fig. 2 shows, by increasing the chirp parameter from 0 to 3, the order of cutoff frequency goes from 80 to 200.

The three-step model well explains the broadening of the harmonic spectrum with increasing wavelength of the driving laser pulse. According to three-step model, in the final step of the interaction, the photon is emitted. The maximum energy of the emitted photon is given by:

8

where is the ionization potential of the atom and , with , , , and as the electric field amplitude, electron charge, electron mass, and frequency of the field, respectively, is the average kinetic energy gained by the electron during the round trip, known as ponderomotive potential. According to the expression, by time stretching the driving laser pulse and reducing its instantaneous frequency, the electron will gain more kinetic energy in the laser field. This happens because the time-of-flight of the electron and its opportunity to achieve kinetic energy increases. The result of this excess energy manifests itself in a more energetic XUV photon and an extension of the frequency range of the harmonic spectrum.

By superimposing the cutoff region harmonics (the end of the plateau and the beginning of the cut-off region), attosecond pulse can be constructed. Figure 3 shows attosecond pulses generated with various β parameters. These pulses were obtained through the use of a 3.5 optical cycles single-color driving laser pulse with intensity of . The impact of chirp on reducing the pulse duration as well as the number of side lobs are obvious. The attosecond pulses shown in Fig. 3a,b, have been obtained by superposing 60th to 100th and 90th to 100th harmonics, respectively. In Fig. 3c, the pulse is formed by adding 130th to 150th harmonics, and in Fig. 3d, it is formed by 180th to 200th harmonics. As expected, by extending the spectrum and coming in a more energetic frequency range, a shorter single attosecond pulse is generated. The shortest isolated attosecond pulse obtained from the interaction of a 3.5 optical cycles chirped single-color driving laser pulse with is “109 as”.

Fig.3.

Attosecond pulses generated by using a 3.5 optical cycles single-color driving laser pulse with the intensity of with (a) , (b) , (c) , and (d) . The shortest duration for an attosecond pulse is 109 as for .

Chirp-free two-color laser fields with various control field intensities

Now, a weaker field, called control field, is added to the main field which is stronger. Before continuing the discussion of two-color fields, we will talk a little about their physics and the reason for their effective role in extending the harmonic spectrum and increasing the efficiency of the generation process. As Fig. 4 shows, the combination of the main field with a control field shortens the oscillation amplitude at the head and tail parts of the pulse and simultaneously increases significantly the amplitude of the field in its central part. Therefore, the pulse can effectively play the role of a high-energy few-cycle pulse. Less spectral modulation of the driving field in its sidebands can help to generate clean isolated attosecond pulses⁷⁷.

Fig. 4.

Time profile of the (a) main field, (b) control field, and (c) two-color field . In this figure, the duration (FWHM) of main pulse is chosen equal to T = 9.33fs and the control pulse duration is four times. The total duration of the main pulse is 16 cycles, and for the control pulse, this value is equal to 64 cycles ().

The generation of an isolated attosecond pulse with a duration of less than 100 as requires a spectral range with a width of more than 40 eV³⁸. Two-color fields can significantly extend the spectrum, and therefore provide the necessary ultrabroad supercontinuum to generate isolated attosecond pulses with durations of the order of the atomic unit of time and even much shorter. Therefore, it is promising to generate short isolated attosecond pulses. But the issue that is important here is the phase compensation of the formed attosecond pulse. The duration of the generated attosecond pulse itself does not fall within this range. Since the kinetic energy of the electron depends on the return time, the photon energy also changes with time, which is the origin of the chirp of attosecond pulses. The short trajectory is positively chirped, whereas the long trajectory is negatively chirped. Experimentally, it is possible to suppress the long trajectory by phase matching and spatial filtering. Therefore, it is possible to generate attosecond pulses by using the short trajectory⁷⁸. Furthermore, duration of the control pulse can be much longer than the main pulse. Thus, the result of the superposition of two pulses would be a multicycle pulse. Increasing the number of cycles in a pulse increases the energy content of the pulse, therefore increases the conversion efficiency of HHG³¹.

In the following, by adding control fields, to the main field, we try to extend the HHG spectrum as much as possible to cover the entire water window with the lowest intensity and at the same time the longest femtosecond laser pulses. For the moment, we ignore chirp. For the control field, we chose the central wavelength to be twice that of the main field. Figure 5 shows the constructed two-color field with various amplitudes of , and . As expected, the main field modulation is decreased by decreasing the control field amplitude.

Fig. 5.

The constructed unchirped driving two-color field by considering six values for control field amplitudes of (a) , (b) , (c) , (d) , (e) , and (f) with a wavelength of twice that of the main field.

Figure 6 shows the HHG spectrum generated by an unchirped two-color field, with various control field intensities.

Fig. 6.

The HHG spectra obtained from the unchirped driving two-color field with various control field amplitudes given in Fig. 5. For a better comparison, each spectrum has been increased by 1 unit compared to the previous one.

As mentioned before, ponderomotive energy, the average kinetic energy gained by the electron during its reciprocating movement in the continuum space outside the atom, is proportional to the intensity and second power of the wavelength of the driving laser field . The cutoff photon energy is calculated from the equation of . Therefore, an increase in intensity or wavelength (or in other words, a decrease in frequency) can both be a reason for increasing time of flight of the electron flight and kinetic energy achieved. Here, only total field intensity changes, because of the increase in the intensity of the control field. Therefore, the only factor that contributes to the extension of the electron propagation time, and an increase in the cut-off frequency will be the increase in the incident field intensity.

In Fig. 7, corresponding attosecond pulses are shown, where by using a two-color field without a chirp, an isolated attosecond pulse with a duration of 116 as is obtained. Note that in this step, we have ignored chirp for the fields.

Fig. 7.

The attosecond pulses generated by driving two-color fields without chirp and with various controlling field amplitudes of (a) , (b) , (c) , (d) , (e), and (f). The selected harmonies are, in order (a)185th to 200th, (b)140th to 150th, (c)110th to 125th, (d)105th to 120th, (e)80th to 100th and (f)80th to 100th harmonies.

Of course, by changing the harmonic interval, cleaner but longer attosecond pulses can also be achieved, which are shown in Fig. 8. In this figure, isolated pulses for control pulse intensity of and are plotted in which 120th to 130th and 95th to 105th harmonics have been used, respectively.

Fig. 8.

The re-plot of attosecond pulses generated by driving two-color fields without chirp and with previous controlling field amplitudes used in Fig. 7, as (c) and (f). The selected harmonies this time are (c) 120th to 130th and (f) 95th to 105th harmonies.

Our motivation for dealing with other intensities for the control field is to investigate the possibility of generating isolated attosecond pulses using weaker control fields, as in practice, it is easier and has fewer technical difficulties. Introducing a weaker control field to the driving field that still produces isolated attosecond pulses in the water window range, would be a great success. As Fig. 6 shows, the cutoff frequency of the HHG spectrum, when a control field with an amplitude of one-quarter of the main field is used, is placed in the energy range of the water window (cutoff order reaches 200). We note that the order of cutoff harmonics corresponding to the water window region is approximately between 180 and 350th harmonic.

Chirp-free two-color laser fields with various control field durations

Until now, in all these cases, the pulse duration of the control field has been considered equal to the main field. At the moment, by changing the pulse duration of the control field up to four times of main field, the HHG spectrum and attosecond pulse are investigated. The results are shown in Figs. 9, 10. As the figures show, no significant change in cutoff frequency, nor the attosecond pulse duration are seen, by changing the control field duration. However, one can see a flat segment in the HHG spectrum which is extended by using a broader control field. In the other words, a broader supercontinuum is generated using a broader control field.

Fig. 9.

The HHG spectra obtained by using two-color unchirped fields with various control field durations. For a better comparison, each spectrum has been increased by 2 units compared to the previous one.

Fig. 10.

Isolated attosecond pulses generated using unchirped two-color driving laser pulses with various control pulse durations: (a) equal, (b) doubled, (c) three times, and (d) four times of the main field. In each of the four cases, the 185th to 200th harmonics have been selected.

We summarize that so far, we could generate an isolated 202 as pulse with combination of a 9.33 fs pulse with the intensity of and a 37.32 fs pulse with the intensity of constituting an unchirped two-color driving laser pulse.

The time window selected for attosecond pulse extraction will include 15 harmonics from 185 to 200th, corresponding to an energy width of 23.27 electron volts, and the cutoff photon energy corresponding to the 200th harmonic will be 310.21 electron volts. It is noted that the water window includes the photon energy range between 282 and 533 electron volts.

Chirped two-color laser field

Now we will check the effect of our chirp model on a two-color driving laser field. Using four values of β = 0, 1, 2, and 3 for the chirp parameter, the effects are investigated. It was found that the chirp only on the main field can impose a positive and effective impact on extending the harmonic spectrum.

As seen in Fig. 11a, with increasing β, the frequency distribution of the pulse changes in such a way that the pulse becomes a little more elongated in the central part, and this means a change in the instantaneous frequency of the pulse. This changes in a way that causes a significant redshift (shift to longer wavelengths) with increasing beta. The applied chirp model not only redistributes the frequency in time in a constant spectral bandwidth as pulse duration is constant, but also effectively generates new frequency components that were not initially in the pulse.

Fig. 11.

(a) The chirped and unchirped two-color driving field and (b) HHG spectra generated by the two-color driving laser field with an intensity of the main field. Each spectrum has been increased by 2 units compared to the previous one.

In Fig. 11b, the HHG spectrum obtained by imposing chirp on the two-color driving laser pulse is plotted for control field with an intensity of one-quarter of main field. The maximum extension of HHG spectrum happens for . As can be seen from Fig. 11b, with increasing the chirp, the harmonic spectrum shows a significant extension.

The red shift in the central frequency of the driving pulse (i.e., the center of mass of the calculated spectrum) and the corresponding change in the spectral bandwidth affect the HHG spectrum by increasing β. In this way, the cutoff frequency changes from 200 to 260 and from 260 to 320, for and , respectively. Figure 11b reveals that for the β = 2, the extended spectrum sweeps approximately the entire frequency range of the water window. For values greater than , the spectrum deviates from its standard shape and the location of the cutoff frequency moves back. So, we ignore in the next calculations.

In Fig. 12, the isolated attosecond pulses corresponding to the chirped two-color driving field are shown. To generate the attosecond pulses, harmonics from 185 to 200th, 235th to 250th, and 285th to 300th have been used, for (a) , (b) , and (c) , respectively.

Fig. 12.

The isolated attosecond pulses generated by two-color driving fields with chirping parameter of: (a) , (b) , and (c),(d) .

In all three cases, the 15 harmonics of the plateau region leading to the cutoff frequency have been chosen. However, in practice, it was observed that by slightly changing the interval for and by superimposing the 280th to 300th harmonics, a shorter attosecond pulse can be achieved (Fig. 12 d). As seen from Fig. 12d, the minimum attosecond pulse duration is 114 as which corresponds to .

Chirped two-color laser field with various control field intensities

Here we inspect the effect of control field intensity on the HHG spectrum with the optimum chirp parameter . Six control field intensities of have been used. Figure 13 shows significant shrinkage in the HHG spectrum with decreasing the control field intensity. Here again, the red shift in the driving pulse frequency distribution plays the main role in explaining the reason for the increase in the harmonic spectrum. As we introduce a more low-frequency components by increasing the intensity of the field, the center of mass of driving laser pulse, , shifts significantly towards longer wavelengths. This increase in the cutoff frequency can be concluded from the extended electron propagation time in the continuum.

Fig. 13.

The HHG spectra corresponding control field intensity of of main field for . For a better comparison, each spectrum has been increased by 1 unit compared to the previous one.

Figure 14 shows corresponding isolated attosecond pulses. Here, we did not see a clear relationship between the intensity changes and the duration of the isolated attosecond pulse generated. Here, the shortest attosecond pulse of 110 as has been obtained for a control field with an intensity of that of main field.

Fig. 14.

The isolated attosecond pulses obtained corresponding to six values for control field intensity of (a) , (b) , (c) , (d) , (e) and (f) of the main field with chirping parameter of .

We note that in this section, the lowest intensity of the control field of , which is of the main field, with the highest value of effective chirp , in combination with the main field, results in an isolated 227 attosecond pulse. This attosecond pulse is obtained from the beginning part of the water window spectrum (carbon absorption edge).

Chirped two-color laser field with various control field durations

Finally, we investigate the effect of control pulse duration on the HHG spectrum and attosecond pulse with considering the chirp on the main pulse. We recall that utilizing longer femtosecond driving laser pulses that can still generate attosecond pulses in water window is a great advantage. Figure 15 shows the HHG spectrum obtained with changing the control pulse duration. It is not easy to detect the change of widening the control pulse on the spectrum, but its effect on the attosecond pulse, shown in Fig. 16, is somewhat visible. All pulses have a duration of around 100 attosecond. However, for a control pulse with a duration of four times the main pulse, the shortest attosecond pulse of 106 as is obtained.

Fig. 15.

The HHG spectrum generated by a two-color field and chirp for the main field with . The control field intensity is one-quarter of the main field. For a better comparison, each spectrum has been increased by 2 units compared to the previous one.

Fig. 16.

Isolated attosecond pulse generated by control pulse with an intensity of one-quarter of the main pulse and different durations: (a) equal, (b) twice, (c) three times, and (d) four times the main pulse.

Figure 16 shows that by doubling the control pulse duration, the duration of the generated attosecond pulse decreases from 114 to 106 as, and this value remains constant for three and four times the durations. In any cases, the important and interesting point is that by increasing the duration of the control pulse, and of course with an intensity equal to one-quarter of the main pulse, we were able to produce a soft X-ray 106 as pulse in oxygen K-absorption edge.

This was the shortest attosecond pulse that we were able to generate in this article according to the conditions we considered for the main and control pulse intensities, wavelengths, and durations. The soft X-ray energy range covers 100 to 3000 eV, which corresponds to the 155th to 465th harmonics of Ti: sapphire laser output which is 800 nm. Hence, cutoff spectral regions of all HHG spectra shown in Fig. 15 are located in all soft X-ray ranges. Also, as the water window spectral region refers to the wavelength range between 2.2 to 4.4 nm, the cutoff of Fig. 15 is located in this range.

Conclusion

In this work, we endeavored to generate soft-X ray isolated attosecond pulses with a duration around 100 as on oxygen K-absorption edge, using two-color field method and applying a proper chirp to the main field. Initially, we discovered that by applying tangential hyperbolic chirp, , to a single-color field with an intensity of 5 × 10¹⁴ W/cm², it is feasible to shorten the duration of the attosecond pulse. In this instance, we achieved a pulse duration of 109 as with a cut-off frequency of the HHG spectrum as falling within the water window spectral region. Subsequently, we employed a two-color field. The wavelength of the control field was determined to be twice that of the main field. By utilizing the two-color field without applying chirp, we were able to obtain a duration near to that of the previous attosecond pulse. The optimal intensities for the control field was found to be and of the main field. By selecting suitable harmonics within the cut-off region, we were able to achieve a pulse duration of 116 as. Increasing the wavelength of the control field did not alter the duration of the attosecond pulse significantly, but rather extended the plateau region of HHG. We then applied chirp on the two-color field, but only to the main field. This allowed us to increase the order of the cut-off frequency to approximately 320, which was a promising outcome. In fact, chirp itself does not cause the broadening of the harmonic spectrum, but the type of chirp modeling in this work causes a redshift in the driving pulse. The broadening of the spectrum is the net result of the longer propagation of electron trajectories in the continuum. In his work, not only have we redistributed the frequency in time over a fixed spectral bandwidth (what a typical chirp would do), rather, we have effectively produced new frequency components that were not present in the pulse in the beginning. Therefore, the redshift in the spectrum plays a key role in the increase of harmonic cutoff. By applying a proper chirp to the main field in the two-color structure , our lowest control field intensity, which is of the main field , is also located in the water window and at the carbon K-absorption edge. Since increasing the pulse duration does not have a significant effect on the structure of the harmonic spectrum and the duration of isolated attosecond pulse generated, by increasing the total pulse duration to 170 fs and the FWHM equal to 37 fs, we can still obtain the same output from the proposed two-color scheme. Therefore, in the final step, we have increased the duration of the control pulse in our chirped two-color HHG scheme. Naturally, it is easier to produce a pulse with a quarter intensity and a longer duration of four time. The highest intensity in our considered range, equal to one-fourth of the main field, is selected. Therefore, our final result was a soft-X ray attosecond pulse with duration of 106 as in approximately the end of the water window spectral region and in oxygen K-absorption edge. In summary, this work demonstrates the successful generation of isolated attosecond pulses with a duration of approximately 100 as in the water window using two-color fields with a chirp on the main field, where the wavelength of the control field is double, the pulse duration is four times, and the intensity is one-quarter of the main field.

Author contributions

M. S.: Supervising the project, Conceptualization, Editing the manuscript, Revising, M. D.: Making computational codes, conducting computations, Data curation, Preparation figures, Methodology, Data validation, writing original draft. S. N.: Providing computational machine, Conceptualization, Editing the manuscript.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Ethics approval and consent to participate

All authors agree to the ethics and consent to participate in this article and declare that this submission follows the policies of Scientific Reports. Accordingly, the material is the authors’ original work, which has not been previously published elsewhere. The paper is not being considered for publication elsewhere. All authors have been personally and actively involved in substantial work leading to the paper and will take public responsibility for its content.