Generation of ultra-short light pulses
by a rapidly ionizing thin foil

D. Bauer    R. R. E. Salomaa Department of Engineering Physics and Mathematics, Helsinki University of Technology, P.O.Box 2200, FIN-02015 HUT, Finland    P. Mulser Theoretical Quantum Electronics (TQE), Darmstadt University of Technology, Hochschulstr. 4A, D-64289 Darmstadt, Germany
December 15, 2020

A thin and dense plasma layer is created when a sufficiently strong laser pulse impinges on a solid target. The nonlinearity introduced by the time-dependent electron density leads to the generation of harmonics. The pulse duration of the harmonic radiation is related to the risetime of the electron density and thus can be affected by the shape of the incident pulse and its peak field strength. Results are presented from numerical particle-in-cell-simulations of an intense laser pulse interacting with a thin foil target. An analytical model which shows how the harmonics are created is introduced. The proposed scheme might be a promising way towards the generation of attosecond pulses.
PACS number(s): 52.40.Nk, 52.50.Jm, 52.65.Rr


I Introduction

In recent years several mechanisms generating harmonics of electromagnetic radiation have been discovered. Among these harmonics from gases [1], harmonics from a laser pulse propagating through underdense plasma [2], and harmonics production from the plasma-vacuum boundary when a laser pulse impinges on a solid target [3] are the most prominent ones. Especially the high-order gas-harmonics, exhibiting a “plateau” instead of a rapid decrease with the harmonic order, seem to be a promising source for xuv “water-window”-radiation.

Apart from the effort to make progress towards shorter wavelengths, another goal is to achieve shorter pulse durations because the temporal resolution in pump-probe experiments clearly depends on the pulse length. One scheme proposed to generate attosecond pulses is based on phase-matching pulse trains which are produced by a laser pulse focused into a jet of rare gases [4]. Another method makes use of the fact that the efficiency of gas-harmonics generation is sensitive to the ellipticity of the incident laser light [5].

The method to generate an ultra-short low order harmonic laser pulse as proposed in this paper is based on the time-dependent electron density of the target material in laser pulse-solid interaction. The mechanism is thus entirely different from those mentioned above which are based on phase-matching of nonlinear single atom-responses [4], relativistic (and thus nonlinear) electron trajectories under the influence of the surrounding plasma [2] or the oscillating vacuum-target-interface owing to the -nonlinearity in the Lorentz force. However, the key idea of the method proposed in this paper is similar to the one in Ref. [5], namely to control the time duration of effective harmonics production by the incident pulse itself. In Ref. [5] it is the ellipticity which is the relevant parameter governing harmonics creation while in the present paper it is the ionization rate which determines the rise in the free electron density of the target. The harmonic pulse duration is of the order of this risetime and thus can be tuned appropriately by varying the intensity of the incident pulse.

It turned out that although our method is capable of generating very short pulses of third and fifth harmonic light (about 2 fundamental laser cycles in length) the mechanism is not appropriate for generating particularly high order harmonics efficiently.

In a recently published conference proceeding [6] similar studies of “ionization harmonics” are presented, but the authors did not focus on the generation of ultra-short pulses there.

In section II we introduce our simple one-dimensional analytical model to study how the laser pulse propagation is influenced by the ionizing thin foil. In section III we discuss our particle-in-cell simulation results. Finally, we conclude in section IV.

Ii One-dimensional modeling

We assume a linearly polarized laser pulse impinging perpendicularly on a thin foil target. In the following analytical and numerical calculations the whole setup is treated one dimensionally in space, i.e., the laser pulse propagates along and the electric field is in -direction. The foil will be ionized by the pulse. To calculate the pulse propagation through a medium with varying free electron density one has to solve the inhomogeneous wave equation


The Green’s function of this equation is where is the step-function, i.e., for and 0 otherwise. The solution of (1) can be written as the sum of the incident field and the radiation field produced by the current , i.e., , with


In order to model thin foils we now assume a delta-like current in space [7]. If the thin foil is located at the current is , with and the electron and ion density, and velocity, respectively, and the ion’s charge state. Integrating over one finds that the current per unit area equals that of a “real” physical thin foil of thickness (as long as there is no strong electron density gradient across the foil). Inserting the current into (2) and performing the spatial integration lead to


where . If we assume that the pulse hits the target at we finally get


for the electric field ( is the retarded time). The current itself depends on the electric field. Neglecting the ionic contribution to the current, we have


where .

Here it has been assumed that all newly created electrons are born with the appropriate fluid element velocity and that collisional as well as relativistic effects are negligible. Besides, we neglect in our analytical treatment energy subtraction from the pulse due to the finite ionization energy of the target material. How this energy loss as well as momentum transfer due to the velocity distribution of the ionization produced electrons can be incorporated in a fluid description is studied in [8]. All pulse intensities considered in this article do not cause relativistic electron motion.

Supposing an ionization rate applicable for pulse intensities under consideration has been chosen, the electron density in the foil is given by . When the target is fully ionized the electron density is . We finally end up with the following integral equation for the electric field ,




is the plasma frequency of the fully ionized target, , and and are the incident EM wave’s frequency and length, respectively. The dimensionless parameter determines how strong the propagation of the incident pulse is affected by the foil. For the foil is optically “thin”.

If the foil is not pre-ionized or ionization is not completed already during the very early part of the pulse, (6) remains nonlinear due to the electron-density shape-factor which depends on the electric field through the rate . Therefore one expects harmonics in the transmitted and reflected light.

In what follows we will restrict ourselves to study (6) in first order in (i.e., we assume a “thin” foil and iterate (6) once). At the position of the foil then


holds. Here, the difficulty is to calculate . The ionization rate depends on the absolute value of the electric field, i.e., the rate has two maxima per fundamental laser cycle. Supposed that the pulse envelope is sufficiently adiabatic the rate may be expanded in a Fourier-series with even multiples of the fundamental frequency only, and a slowly time-dependent envelope ,


Since the rate is a complicated functional of the field, in general all terms in the expansion (9) are present. However, if we assume the incident pulse (divided by its amplitude) to be an even function in time, , all coefficients in (9) vanish. This finally leads to

Here, , , and is the modified Bessel-function. Note that are slowly time-dependent due to their dependence on .

From (II) we can deduce that by ionization in first order of only odd harmonics will be produced: In (II) the term in curly brackets is composed of even harmonics, but multiplied with odd harmonics are created.

Harmonics production is enhanced when is increased. Therefore one expects that increasing the density or the thickness of the foil acts in favor of the creation of harmonics. However, one has to bear in mind that a perturbative treatment with respect to preceded here. Furthermore, increasing the density or the thickness simply makes the foil less transparent.

Iii Numerical results

A 1D2V particle-in-cell (PIC) code was used to simulate laser pulse-solid interaction. In order to incorporate ionization the rate equations governing the ionization state of the target were solved during each “PIC-cycle” [9]. For simplicity only one ionization state with the ionization energy of hydrogen (13.6 eV) was assumed. Landau’s tunneling-rate [10] was used which is a reasonable choice for the field strengths and frequencies under consideration. The short risetime of the electron density forces a tiny time step. Usually one fundamental laser cycle (wavelength) was sampled by 1000 temporal (spatial) gridpoints. About computer particles, sampling the physical charge densities of the thin foil were found to be sufficient. The ions were mobile (although this is unimportant for the effect under consideration) and 1836 times heavier than the electrons (hydrogen).

In Fig. 1 numerically computed spectra of the transmitted light are shown for 5 different peak field strengths , corresponding to intensities and  W/cm. All other parameters were held constant: wavelength  nm, foil thickness , incident -shaped laser pulse of duration  fs, and the density was the critical one with respect to the fundamental frequency, i.e.,  cm.

The higher the field strength, the broader are the harmonics peaks in the spectrum. The pulse length of the harmonics radiation is closely related to the risetime of the electron density in the foil since as soon as the density remains constant harmonics production will stop. In Fig. 2 the normalized electron density is plotted vs time for the 5 field strengths of Fig. 1. A risetime covering fundamental periods for the weakest pulse and only one cycle for the strongest pulse can be inferred from the plot (each stair in the density corresponds to one half cycle). The density risetime is very sensitive to field strength and pulse shape. Increasing the field strength leads to a decreasing density risetime and hence to a shorter harmonics pulse length. However, the conversion efficiency decreases when the field strength of the incident pulse is increased since the nonlinearity switches off too soon. Furthermore the “harmonics” peaks are shifted and asymmetrically broadened if complete ionization occurs within only one fundamental cycle (or even less). In the limit of a step-like behavior of the electron density the spectrum resembles the Fourier-transform of the -function with no -harmonics structure at all.

In Fig. 1 the pulse length of the harmonic radiation can be estimated by fitting the peaks in the spectrum to a Fourier transformed “test envelope” . is the pulse duration of the th harmonic. For the 5 cases of Fig. 1 one finds for the pulse length of the 3rd harmonic times the fundamental period . A lower limit for certainly is itself because a shorter risetime of the electron density leads to a vanishing -structure in the spectrum. The power in the 3rd harmonic is about of the fundamental. This conversion efficiency is similar to the one in Ref. [2] while all other methods mentioned in the introduction are superior as far as power transferred to (short pulse) harmonic radiation is concerned.

One may object that the incident pulse intensity we took in our numerical simulations was already small (at least for “up-to-date” short pulse laser systems) so that the 3rd-harmonic pulse with only a millionth of its intensity is not acceptable at all. However, the incident pulse might be a stronger but defocused pulse so that the 3rd harmonic output, when focused, becomes considerable. Besides, using a shorter fundamental wavelength (and correspondingly a thinner and/or denser foil) would require a higher field strength to fully ionize the target within the same number of cycles. By examining the dimensionless parameters and one can estimate the “experimental parameters” , and in order to meet the desired harmonic pulse duration . The practical limit for found in the numerical simulations is about . If the incident light has already a rather short wavelength (e.g., if light, produced by one of the high-harmonics mechanisms described above are used) then the -limit can be shifted towards the attosecond-domain. However, in that case it would be certainly a challenging task to find the optimal parameters and for a manufacturable thin foil. In recent experiments with thin foil targets [12, 11] the thickness was about  nm and the electron density (when fully ionized) was of the order of  cm, i.e., by a factor 100 denser than assumed in our numerical simulations presented so far.

One may wonder if it is really essential to take a thin foil. When a sufficiently strong laser pulse impinges on a thick overdense target only a thin plasma layer is created anyway and our analytical treatment in section II should therefore apply as well. We thus expect also from thick targets short harmonic radiation, at least in reflection.

In Figure 3a spectra calculated from timerows of the transverse electric field in front (reflected light), in the middle, and at the rear edge (transmitted light) of a thicker target are presented. The target was one wave length thick, , and ten times overcritical,  cm. The incident pulse had an intensity  W/cm. The other parameters were  nm,  fs, square-sine in shape again. The spectra have a rather complicated structure owing to the complex spatial and temporal dependence of the free electron density. This can be seen in Figure 3b where the target density is shown as a contour plot vs time and space. During the pulse the target gets fully ionized only in a thin layer at the front. The electron density quickly rolls off and at the rear side not even the critical density is reached. In the spectra we clearly observe the well-known effect that higher frequency radiation penetrates easier the target than light of the fundamental frequency does since for third harmonic light the target density is only slightly overcritical while it is even undercritical for the 5th harmonic. This effect was utilized as an experimental method to determine the plasma density [11]. We observe also frequency shifts since the peaks are not located any longer at the odd harmonics’ positions. Obviously, thick targets are not as easily accessible by means of simple physical reasoning as it is the case for thin foils where the -current model described in section II works well.

Our experiences from several PIC runs with different target densities and thicknesses can be summarized as follows: as soon as there is a density gradient across the foil the spectra of the reflected and transmitted light get distorted (compared to the thin foil-spectra in Fig. 1). To avoid this density gradient the optimal -product is about which is, with present day technology, not easy to achieve. Using a higher incident pulse intensity in order to ionize the target more homogeneously reduces the density risetime and thus the conversion efficiency.

Apart from these “technical problems” one may argue that during the plasma formation process electron-ion-collisions might be important, especially for the relatively low field strengths about  W/cm. Since it is during the plasma formation where the harmonics are produced there might be serious distortions in the spectrum of the transmitted light. In order to take electron-ion-collisions into account we introduced a collision frequency into our 1D2V PIC-code. This leads to dissipation of energy due to friction of the oscillating charge-sheets (note that in a 1D PIC-code each “computer-particle” represents an actual charge sheet [9]). The dissipated energy is used to determine a “sheet-temperature” which, in turn, enters into . We found collisions causing mainly distortions at high frequencies but the 3rd and 5th harmonic-peaks were almost unaltered. For stronger incident pulses the effect of collisions is even less.

With our PIC-code we also examined the effects of energy subtraction due to the ionization energy of the target material (according to the model in [8]). The most prominent effect, as far as harmonics generation is concerned, is that ionization gets slowed down slightly (which can be compensated by choosing a higher incident pulse intensity).

It is worth mentioning that the observed effect of harmonics production due to the rise of the electron density in a thin foil may be used to measure the ionization time of the foil instead of presupposing an ionization rate. This would offer an opportunity to check the validity of ionization models experimentally.

Iv Conclusion

In summary, we have studied the spectrum of a perpendicularly incident laser pulse when transmitted through a rapidly ionizing foil. A simple analytical method was utilized to show what the underlying mechanism for the generation of short pulse odd harmonic radiation is. The pulse duration of the harmonic radiation is only a few cycles with respect to the frequency of the incident laser light. The pulse length is governed by the risetime of the electron density in the target and therefore it can be easily tuned through adjusting the peak field strength of the incident pulse. This might be a promising way towards the generation of attosecond pulses.


This work was supported by the European Commission through the TMR Network SILASI (Super Intense Laser Pulse-Solid Interaction), No. ERBFMRX-CT96-0043. One of the authors (D.B.) would like to thank Mr. G. Baier for stimulating discussions.


Figure 1: Spectra of the transmitted light for 5 different peak field strengths , corresponding to intensities (solid), , (dotted),  W/cm (solid again). All other parameters were held constant: wavelength  nm, foil thickness ,  fs incident -shaped laser pulse, and density  cm. The higher the field strength the broader are the harmonics peaks in the spectrum.
Figure 2: Normalized electron density vs time for the 5 field strengths of Fig. 1. A risetime covering fundamental periods for the weakest pulse and only one cycle for the strongest pulse can be inferred (each stair in the density corresponds to one half cycle).
Figure 3: (a) The spectra taken in front (solid), in the middle (dotted), and at the rear side (dashed) of the -foil, normalized to the fundamental peak of the reflected light. The incident pulse had an intensity  W/cm. The other parameters were  nm,  fs, square-sine in shape. For the fundamental frequency, the plasma layer at the front end is 10 times overcritical while higher frequency light can penetrate more easily. Compared to the clear harmonics structure in Fig. 1 the spectra are rather distorted. The ionization harmonics peaks appear up-shifted. (b) The target electron density vs space (scaled in fundamental wave length) and time (in fundamental cycles). At the front end the target is (for the incident light) 10 times overcritical while at the rear side the target is not fully ionized ().



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