1 Introduction

Present day optical laser fields can drive highly non-linear processes in quantum systems. These are studied both for their own unique properties, and because of the ultrashort pulsed high frequency radiation that can result from the laser/matter interaction. An example is the generation of very high order harmonics of the fundamental frequency when atoms in a gas jet are exposed to an intense laser pulse [1]. The generated harmonic radiation is coherent and of even shorter duration than the driving field and thus presents a novel and unique source of ultrashort pulses of extreme ultraviolet (XUV) radiation with a large potential for applications [2]. The generation process has been observed to continue to orders above 300 in atomic gases, and is highly non-perturbative [3]. The most striking manifestation of this non-perturbative behavior is the existence of a plateau in the harmonic spectrum in which many harmonics up to a characteristic cutoff energy all have approximately the same strength [4, 3]. In this paper, the time-frequency behavior of high order harmonics generated in neon by a near-infrared laser field is calculated and analyzed. We discuss the influence on the time-frequency characteristics of both single atom and phase matching effects.

The interest in the time-frequency behavior of a highly non-linear process, as well as in developing harmonic radiation as a feasible laboratory scale XUV source, drives efforts to experimentally characterize the temporal and spectral structure of harmonic pulses. For visible and UV radiation, several techniques allow for complete characterization of the amplitude and phase of an ultrafast pulse (see for instance [5]). Such techniques can be applied to the high harmonics or the intense mid-infrared laser source developed by DiMauro and coworkers [6]. While the high order harmonics from a Ti:Sapphire laser are not at present as easy to experimentally characterize, there is no doubt that characterization techniques built on non-linear processes with high harmonics will be developed in the near future. In a proof-of-principle experiment [7], the pulse length of the fifth harmonic of a 810 nm laser has been measured via a fourth order process.

The time-frequency behavior of a high order harmonic is influenced by a number of factors. First, the harmonic dipole radiation is predicted by single atom calculations to have a phase that depends on the intensity of the driving laser field [8]. The intensity dependence of the dipole phase links the coherence properties of the harmonic radiation to the dynamics of the atom ionizing in the strong field [9]. Through the time-dependence of the intensity during the pulse, the dipole phase introduces a large negative chirp on each harmonic and therefore a broad spectral profile. The spatial dependence of the intensity in the laser focus imposes a large curvature on the phase front of the harmonic beam which makes it strongly divergent. Second, the focusing of the driving laser beam has an effect on the temporal and spectral profiles of the harmonic field. Since atoms at different radii in the focus experience very different peak intensities, they will over the course of the pulse emit radiation with different chirps. The resulting harmonic pulse thus consists of many contributions each with different time-frequency characteristics. And finally, as the generated radiation propagates through the medium, phase matching is crucial to the final outcome and may favor certain phase behaviors compared to others.

In this paper we show calculated time-frequency representations (TFRs) of very high order harmonics, the 45th and the 89th, in neon at an intensity of 6×10¹⁴ W/cm² and a pulse length of 150–200 fs. The 45th harmonic represents a typical harmonic which is deep in the plateau region of the harmonic spectrum, whereas the 89th harmonic is representative of the cutoff region. We first calculate spectograms for the single atom harmonic emission. For the 45th harmonic we also calculate the TFR for the fully phase matched harmonic pulse emitted from a macroscopic collection of atoms after propagation through a gas jet. Both the single atom and the propagated 45th harmonic exhibit an overall negative chirp. The propagated harmonic also has several (weaker) chirp components to it. We can spatially resolve the time-frequency behavior, and find that at different radii in the farfield profile, the coherence properties of the harmonic pulse are very different. We also calculate a polarization gate FROG trace (see for instance [5]) and show that this is visually less intuitive than the TFR trace due to the complicated time structure of the harmonic. Before showing the results, we outline briefly our theoretical methods.

2 Theoretical approach

A time-frequency representation (or spectrogram) of an electric field E(t) is a simultaneous representation of the temporal and spectral characteristics of the pulse. We define it here as:

where ω is the frequency and τ is a time delay which defines the position of the window or gate function W(t-τ). For every τ, S(ω, τ) represents the instantaneous spectrum of the pulse at the time specified by W(t-τ). As the window function slides through all delays the TFR trace illustrates how the spectrum of the pulse changes with time. As the electric field E(t) we use either the single atom or the macroscopic harmonic pulse. The window function is a Gaussian with a full width at half maximum (FWHM) duration which is approximately 65% of the length of the harmonic pulse.

The single atom dipole radiation spectrum is calculated by numerically solving the time dependent Schrödinger equation (TDSE) for a neon atom interacting with an intense laser pulse, within the single active electron approximation (SAE) [10]. The time profile of the qth harmonic is found by multiplying the spectrum with a window function centered around qω ₀, where ω ₀ is the driving frequency, and Fourier transforming into the time domain.

The macroscopic harmonic pulse results from the solution of Maxwell’s wave equation in the paraxial and the slowly varying envelope approximations in a neon gas subject to a focused laser beam [11]. As a source term for the non-linear part of the polarization field we use the atomic dipole moments calculated within the SAE approximation. These dipole moments are calculated as a function of the intensity by performing a series of SAE calculations using a quasi constant driving field.

The TFR trace of the propagated harmonic pulse, S_p(ω, τ) is calculated for each radial point in the farfield harmonic profile E(r, t) and integrated over the radial coordinate:

to simulate an experimental time-frequency representation as closely as possible.

The following section contains results calculated as described above. First we show single atom TFRs of the 45th and the 89th harmonics, and then several TFRs of the 45th harmonic after propagation through a non-linear medium.

3 Results

Figure 1 shows TFRs of the 45th (left) and the 89th (right) harmonic emission from a neon atom using a driving laser pulse of length 216 fs and peak intensity 6×10¹⁴ W/cm². The driving field has a wavelength of 810 nm. The TFR traces the instantaneous value of the harmonic frequency (shown in units of the driving frequency, ω ₀, relative to a zero point at the harmonic frequency), which changes almost linearly with time from a positive to a negative value. Both harmonics exhibit negative linear chirps, and the chirp of the 45th harmonic is much larger than that of the 89th harmonic. In the following we first concentrate on the 45th harmonic. The FWHM duration of the harmonic pulse can be found from the TFR trace to be approximately 25 fs. The Fourier transform limited FWHM bandwidth of a 25 fs pulse is 18 THz, which corresponds to 0.05ω ₀. The presence of the negative chirp broadens the spectral profile to 0.3ω0. Both the time profile and the spectrum are somewhat asymmetric, with more weight in the latter part of the pulse and thereby - via the negative chirp - on the low frequency side of the spectrum. The time on the x-axis is really a measure of the delay between the harmonic pulse and the window function. In this case it also measures the time profile of the pulse relative to t=0, since the position of the peak of the window function is well known. In an experiment, an absolute time scale can often not be defined.

 

Fig. 1. Time-frequency representations of the 45th (left) and the 89th (right) harmonics emitted from a neon atom subject to a 216 fs laser pulse with a peak intensity of 6×10¹⁴ W/cm².

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From the slope of the TFR trace it is possible to calculate the rate of change of the instantaneous frequency (the chirp rate). The chirp rate contains information about the dipole phase behavior in the following way. According to the well known semi-classical interpretation of high harmonic generation, there are several components, so-called quantum path contributions, to the harmonic dipole moment [8, 12]. Interference between the different contributions can give rise to the spectral and temporal asymmetry of the TFR trace of the 45th harmonic in Fig.1. The phase of each quantum path contribution i is linearly proportional to the intensity with a proportionality constant -α_i. These phases each give rise to a negative chirp. Since the driving laser pulse intensity is approximately quadratic in time close to its peak, these negative chirps are approximately linear during the time when the harmonic is generated. The chirp rate β_i for each of these negative linear chirps is proportional to α_i, the peak intensity I ₀, and 1/T ², where T is the pulse length of the driving pulse [9]. In this particular case we have used a laser field with a cos² envelope, which gives the following expression for β_i:

From the rate of change of the harmonic frequency it is therefore possible to estimate which quantum path contributions dominate the dipole moment.

The SAE calculations presented in this work are fully quantum mechanical and are performed using a realistic description of the atom. Nevertheless, analyses like those presented in [12] and [9] show that the semi-classical framework, in which the harmonic dipole moment consists of several contributions each with a simple phase behavior, is still largely valid for harmonics above the ionization threshold. The semi-classical and the SAE calculations mainly differ on the weight of each contribution.

In the units used in Figure 1, the chirp rate of the 45th harmonic is -0.012 ω ₀/fs. This corresponds to a proportionality constant α of 40×10^-14 cm²/W. This finding is typical for harmonics in neon calculated within the SAE approximation. The semi-classical model predicts the quantum path with α≈25×10^-14 cm²/W to be by far the most dominant [12]. From this quantum path the SAE in general predicts only a very small contribution.

The chirp rate of the 89th harmonic - which as mentioned above belongs to the plateau region - is approximately 2.6 times smaller than the chirp rate of the 45th harmonic. This is in excellent agreement with the semi-classical model which predicts that only one quantum path, with α≈13×10^-14 cm²/W, dominates the cutoff harmonics. The time-frequency behavior of these harmonics is in general simpler than the plateau harmonics, due to the absence of interference between different quantum path contributions.

The following figures show the 45th harmonic generated in and propagated through a 1 mm long gas jet of neon atoms. The driving laser pulse has a peak intensity of 6×10¹⁴ W/cm² and a pulse length of 150 fs. The laser focus is long compared to the gas jet, with a confocal parameter of 5 mm.

 

Fig. 2. (a) TFR trace of the 45th harmonic in neon after propagation through a 1 mm long gas jet. In (b) we show the logarithm of the spectrogram shown in (a).

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The time-frequency representations in Figure 2 show that the time structure of the macroscopic harmonic is much more complicated than of the single atom emission. The TFR trace of the 45th harmonic is shown both using a linear scale (a) and a logarithmic scale (b). A range of chirps can be seen, including a weak positive chirp. The harmonic pulse can therefore not be considered for applications as having one single linear chirp. The chirp that dominates the TFR is still approximately linear, though. This chirp rate can be found from the figure to be ≈-0.014 ω ₀/fs and is the cause of the large spectral bandwidth of 0.4ω ₀. Through Eq.(3) and using α=40×10^-14 cm²/W, this corresponds to an “average” peak intensity experienced by the atoms which is between 3 and 4×10¹⁴ W/cm². This reflects the fact that due to bad phase matching conditions, the radiation that exits the medium often does not originate from the center of the laser focus [13]. The origin of the many weaker chirp components are discussed in further detail below.

By including or excluding different parameters and effects in the calculation, it is possible to study their influence on the time-frequency behavior. In Figure 2, the effects of ionization are excluded from the calculation. Including ionization in the form of depletion of the medium over time, and spatial defocusing of the fundamental due to the presence of free electrons, does not change the TFR trace much. The overall chirp is somewhat smaller when ionization is present, since defocusing of the laser beam decreases the intensity in the gas jet. The spectrum is blue-shifted compared to the spectrum free of ionization, due to the depletion of the medium towards the end of the negatively chirped pulse.

When the phase of the atomic dipole is excluded, however, the TFR trace changes dramatically. In this case, the main part of the harmonic pulse is unchirped, and there are faint contributions to the time-profile exhibiting a range of both positive and negative chirps, many of which are also found as the weaker chirp contributions in Fig.2 (b). Phase matching and propagation through space thus complicates the time-frequency behavior, even in the absence of any explicitly time-dependent phase contributions.

Figure 3 shows a TFR trace of the same harmonic pulse as shown in Fig.2, using a different gate function. The figure simulates an experimental representation of a high-order harmonic by using the polarization gate FROG method [5], where the intensity |E(t-τ)|² of the harmonic pulse is used as the window function. Since the time-structure of the harmonic pulse is very complicated, as demonstrated in Fig.4, the FROG trace looks very non-intuitive and does not indicate the negative chirp found above. The full information about the time-frequency behavior of the radially integrated harmonic pulse can of course be obtained from the FROG trace using a phase retrieval algorithm.

 

Fig. 3. Polarization gate FROG trace of the 45th harmonic shown in Fig. 2.

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Fig. 4. Time profile of the 45th harmonic after propagation through the gas jet.

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The origin of the many different chirp components present in the 45th harmonic can be furhter elucidated upon resolving the time-frequency behavior spatially, by considering the TFR trace at individual radial points in the harmonic farfield profile. In general, these TFRs are complicated and often exhibit several chirp components from the range of chirps seen in Figure 2, with different chirps present at different radii. The dominating chirp of the 45th harmonic is found at many radii. Here we concentrate on two examples where the temporal phase behavior is relatively simple, for radii located at the inner and outer edges of the farfield profile, as shown in Figure 5(a) and (b), respectively. The TFR trace shown in (a), being calculated at the outer edge (at FWHM) of the farfield profile, represents the time-frequency behavior of the most divergent part of the harmonic beam. Clearly, this also corresponds to the most strongly chirped part of the time-profile. The chirp seen in Figure 5(a) corresponds to the dominant chirp found in Figure 2, with no contributions from smaller chirps.

 

Fig. 5. The time-frequency behavior of the harmonic radiation found at (a) the outer edge of the farfield profile, and (b) on-axis in the farfield profile.

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The time-frequency behavior of a point very close to the axis of propagation (b) shows that the harmonic pulse is clearly separated into two different equally strong spectral components. In addition to the large negative chirp observed above, there is now also a contribution from a very small (also negative) chirp. This slowly chirped part is present only very close to the propagation axis and therefore cannot be seen when the TFR is integrated over the entire farfield profile. The component with the larger chirp is found at most radii, as mentioned above, and therefore survives the radial integration.

This spatial separation can be explained by the intensity dependence of the harmonic phase [14, 9]. The same phase behavior that gives rise to a chirp in time makes the phase-front of the harmonic beam curved in space due to the focusing of the driving laser. The rapidly chirped part of the harmonic beam is therefore also strongly divergent, whereas the non-chirped part stays close to the propagation axis. These observations reinforce that the time-frequency behavior of high-order harmonics generated by a focused laser beam are closely connected to spatial effects, both through phase matching and the divergence imposed by the focusing of the laser.

The very slowly chirped component of the harmonic pulse also has its origin in the single atom dipole phase, as a quantum path contribution to the dipole moment with a small intensity dependence (with an α of less than 2×10^-14 cm²/W in the semi-classical model [12]). For the 45th harmonic in neon, a quantum path analysis [12] shows that the contribution from this latter quantum path is very small, which is the reason that it cannot be seen in the single atom TFR in Fig.1. The selection of only certain time-frequency behaviors provided by the spatio-temporal separation demonstrated in Figure 5 brings to light this small contribution since it is very localized in the radial coordinate.

By the selection of one part or another of the macroscopically sized harmonic farfield profile, it is thus possible to select the coherence properties of the harmonic pulse. The outer part of the harmonic beam as shown in Fig.5(a), for example, can be considered for applications as having only one negative chirp rather than a variety of different chirps. This would be advantageous when attempting to temporally compress a harmonic pulse by manipulating the timing of its frequency components. The central part of the harmonic beam, on the other hand, presents a much more collimated and spectrally confined source of XUV radiation.

The results in Figure 5 are in very good agreement with the results in argon presented in [14, 15]. Using an interferometric method, the farfield profiles of high order harmonics in argon were found to separate into a outer region with a broad spectrum, and an inner region having a spectrum with both a narrow and a broad component.

4 Summary

We have studied the time-frequency behavior of high-order harmonics generated in the interaction between a strong laser field and neon atoms, and find that they exhibit a clear negative chirp, both in the single atom response and in the macroscopic response. We also find that the (macroscopic) harmonics belonging to the plateau region, as exemplified by the 45th harmonic, exhibit very complicated time-frequency structures, with contributions from many different negative chirps. Even though one chirp component is clearly dominating, these harmonics cannot be considered as having only one chirp. However, since the different chirp components diverge differently in space, the time-frequency behavior separates in the spatial domain. Very close to the axis of propagation we find harmonic radiation which has only a very small chirp and therefore a narrow spectrum. Far from the propagation axis we find radiation that has a well defined linear chirp, and which could therefore possibly be recompressed to a very short duration.

The spatial and temporal behaviors are thus closely connected in harmonic generation from an ensemble of atoms, and it is important to take into account space, time and the interplay between them when considering harmonic generation from a focused laser beam.