We investigate attosecond streaking measurements, where a spectrogram is described by an ensemble of electron wave packets. Such a description may be required for processes more complex than direct photoemission from an isolated atom; an ensemble of wave packets may also be needed to describe the role of shot-to-shot fluctuations or a non-uniform spatio-temporal profile of attosecond light pulses. Under these conditions, we examine the performance of conventional (FROG) analysis of attosecond streaking measurements.
©2009 Optical Society of America
The invention of the “attosecond streak camera” [1, 2] enabled the first time-resolved measurements with attosecond resolution [1, 3, 4, 5]. In these pump-probe measurements, an electron wave packet is freed by an extreme-ultraviolet (XUV) pulse in the presence of an near-infrared (NIR) laser pulse. Provided that the fields of these two pulses are precisely controlled, a series of photoelectron spectra measured for different delays between the pulses, referred to as a spectrogram, contains the information about both the time-domain intensity and phase of the electron wave packet [6, 7]. Extracting this information has proved to be a challenging task [8, 9]. In combination with the versatile and reliable retrieval algorithms adapted from frequency-resolved optical gating (FROG) [10, 11], attosecond streaking remains a key technique for the characterization of attosecond pulses and experimental investigations of ultra-fast electron processes. As the precision and accuracy of these measurements constantly improve, attosecond streaking spectroscopy is no longer limited to studying isolated atoms – it was successfully applied for studying photoionization on metal surfaces , and there are no fundamental obstacles for performing such measurements on molecules. As attosecond streaking is applied to more complex systems with the quest of obtaining new insights into electron dynamics, it becomes increasingly important to understand the practical limitations of this technique. This is the main purpose of this paper.
The reliable reconstruction of an electron wave packet from a streaking measurement may pose difficulties for two reasons: On the one hand, the physical assumptions underlying currently used models may not always hold. As an example, it is common to assume that the NIR field does not affect the interaction of an XUV pulse with a system. On the other hand, the most fundamental assumptions about the light-matter interaction may be satisfied, but a model may still give an oversimplified description of a measurement. In both cases, more elaborate models can be devised, but are likely to preclude the usage of such a powerful technique as the FROG reconstruction. This motivates us to investigate the performance of current retrieval schemes under realistic measurement conditions.
All the streaking models that can be considered for the FROG reconstruction are built on the premise that the properties of an emitted electron wave packet can fully be described by a single complex-valued time-dependent function  (discussed below). A single wave packet – one with a fixed set of physical properties – may not be sufficient to describe a realistic streaking measurement, which includes fluctuations and variations in the physical parameters relevant to the experiment. First of all, the electron spectra are usually accumulated over many laser shots, while the electron wave packet fluctuates from shot to shot, so that a measured spectrogram represents a result of averaging over an ensemble of wave packets. Second, attosecond pulses may have a non-uniform spatio-temporal profile . In this case, different temporal profiles are observed at different positions within the beam, and it is their averaged contribution that is recorded in the spectrogram. Even if it were possible to generate highly stable attosecond pulses with a perfect spatio-temporal profile, models assuming a single wave packet may be inaccurate for treating photoionization of systems more complex that an isolated atom. A good example is the laser-dressed photoemission of electrons from the conduction band of a solid: electrons can be emitted from different energies within the band, which results in an ensemble of wave packets with different central energies .
Our approach to investigating the limitations and robustness of the current streaking model – which assumes the spectrogram to be composed of a single wave packet and laser pulse, as opposed to an ensemble of them – is to first identify several key physical parameters that describe the electron wave packet. We consider these parameters independently, and assume that there is some uncertainty associated with each of them. This uncertainty may stem from shot-to-shot fluctuations of the laser field, the non-uniform spatio-temporal profile of the XUV pulse or fromintrinsic properties of a systemunder investigation. We treat all these uncertainties on equal footing. For each parameter, we consider an ensemble of electron wave packets where only the chosen parameter is varied. A streaking measurement is simulated from this ensemble by using an established model [7, 14] to evaluate a spectrogram for each of the wave packets in the ensemble, and then averaging these spectrograms over the ensemble.
Although these uncertainties can originate from various sources, and are generally coupled to one another, we would like to understand the effect that each of them has on the streaking measurement. For this purpose, we independently consider the role of uncertainty in each of the parameters describing an XUV pulse. For every parameter, we show typical signatures that it leaves on a spectrogram, give a typical value for which the retrieved electron wave packet noticeably deviates from the true one, and explain the effect that the uncertainty has on the measurement and the retrieval.
2. Attosecond streaking spectroscopy
The attosecond streak camera, a direct extension of the streak camera principle to attosecond time scales, relies on the photoionization of an atom by a pair of synchronized light pulses – an NIR laser pulse of femtosecond duration, and an XUV photoionizing pulse with a duration that is shorter than the laser period. As the electron is ejected from the atom by the XUV field, its momentum is progressively modified by the slowly varying electric field of the laser pulse. The recorded photoelectron spectrum is thus modified (streaked) due to the presence of the NIR laser field, and this effect depends on the segment of the laser pulse that follows the photoionization.
In general, electron wave packets of any type or any origin can be streaked by a laser field. For example, streaking measurements have already been performed with Auger electrons emitted from krypton, leading to the first time-resolved measurement of the Auger effect , and with electrons emitted from a tungsten surface , allowing for the first experimental determination of electron transport times in a solid.
Streaking measurements can be analyzed by composing a spectrogram from a series of streaked electron spectra recorded for different delays between the laser and the attosecond pulse. Such a data set contains both temporal and spectral information about the physical process under study. In many cases, laser-dressed photoelectron spectra observed in the polarization direction of the laser and XUV fields can be described by the following model [7, 15],
where S(p,Δts) is the spectral intensity for a delay Δts between the XUV and NIR fields. The matrix element d(p) describes the transition from a bound state to a continuum state with momentum p. Expression (1) assumes atomic units, which are used throughout the paper.
When model (1) is applicable, we refer to the complex-valued function χ(t) as the electron wave packet. In the most studied case of direct photoionization of isolated atoms, χ(t) is just a replica of the complex envelope of the XUV field, 𝓔 X(t). The “gate” G(p, t) is defined as
where AL(t) is the vector potential of the NIR laser pulse. Physically, the wave packet χ(t) can be understood in terms of the momentum-space wave function ψ̃(p, t) of the free electron :
where W is the ionization potential of the system, ΩX is the central energy of the XUV pulse and the moment t is large enough so that the numerator on the right-hand side of (3) does not depend on time.
By making the substitution G(p, t)→G(pc, t) [16, 17], where is the electron’s central momentum, the gate function is rendered momentum-independent, and the right hand side of (1) is just the expression for a FROG spectrogram, where the wave packet and laser pulse form a probe and gate pair. It has been demonstrated that both the wave packet and NIR fields can be fully characterized by feeding the streaking spectrogram to a retrieval algorithm tailored for attosecond measurements [8, 16]. We stress the fact that, in general, this technique reconstructs the temporal profile of the electron wave packet χ(t), which is not necessarily a replica of the XUV field. For example, in processes such as resonant photoemission or Auger decay, the simple relation between χ(t) and 𝓔X(t) no longer holds [15, 18].
The retrieval algorithm [8, 11, 16] is an iterative approach to solving the two-dimensional phase retrieval problem. It works by imposing alternating constraints between the frequency and time domains in order to find an XUV-NIR pulse pair that yields a spectrogram that matches the measured one. A guess is first given for the XUV and NIR pulses, which is used to compute a complex spectrogram according to (1) and (2). The frequency-domain constraint consists of replacing the modulus of the calculated spectrogram with the square-root of the measured spectral power density. The resulting complex spectrogram is Fourier-transformed back into the time domain, from which an optimization strategy selects the best probe-gate pair for the next iteration.
3. Effects of the uncertainties
As mentioned in the Introduction, a single wave packet may not be sufficient to describe a realistic streaking measurement due to fundamental properties of the physical system under study, or due to experimental imperfections. A more realistic description should make use of an ensemble of wave packets to represent a spectrogram. We will now introduce our methodology.
We have identified several key parameters that can largely describe an ensemble of electron wave packets. To investigate the role of the uncertainty in each of these parameters, we compute spectrograms from an ensemble of pulses having different values of the parameter in question, while keeping all other parameters fixed. We then sum the spectrograms calculated from each pulse in the ensemble, feed the resulting spectrogram to the attosecond FROG algorithm, and compare the retrieved pulse with the original one. Although uncertainties in different parameters are in general correlated with one another, we chose to study them independently in order to better understand their individual effects.
To further avoid unnecessary complexity, we assume the parameters to follow a uniform distribution. Although realistic uncertainties arise from a bell-shaped distribution, assuming a uniform distribution here merely results in a scaling factor between realistic uncertainties and our quoted values for uncertainty, as well as minor differences in higher-order moments between the distributions.
For our simulations, we assume single and twin pulses described by
with a duration of τX=200as, and a temporal chirp of β≡φ″(t)=40×106 as-2. These time-domain parameters lead to a bandwidth of ≈10.5eV and a group delay dispersion γ= ″(ω)≈-6244as2, where (ω) is the spectral phase at angular frequency ω. We centred the wave packets at an energy εc=p 2 c/2=60eV. Moreover, we consider two types of temporal profiles: a single isolated attosecond pulse, and twin attosecond pulses separated by half a laser period T L/2≈1.2fs, given respectively by (4) and (5).
The laser field used for calculating the streaked spectra is a bandwidth limited Gaussian pulse, given in terms of its vector potential A(t)=A 0 exp(-ln(4)t 2/τ2 L)cos(ω L t), where τL=3.3fs. The vector potential amplitude A 0≈-0.1878a.u. corresponds to an intensity of 5.0×1012W/cm2, and the angular frequency ωL≈2.616rad/fs corresponds to a wavelength of 720nm.
For a single pulse, we investigate uncertainties in the central energy, bandwidth, and group delay dispersion (GDD). We also look at the effect of timing jitter between the laser field and the electron wave packet. We call this uncertainty the streaking delay uncertainty.
The uncertainties pertaining to isolated attosecond pulses each leave their own signature on the measured spectrogram, as illustrated in Figures 1(a), (b), (c) and (d) (in comparison with the pure spectrogram shown in Fig. 1(e)). Figures 1 and 2 below serve as an informal “Table of Contents” for this paper, and will be referred to in subsequent sections. The overall effect of the uncertainties for a single pulse is a smearing and distortion of the spectrogram, and in some cases (especially apparent in Figs. 1(a) and 1(d)) a loss of the spectral broadening and narrowing effect at zero-crossings of the vector potential – which is crucial for determining the chirp of the wave packet. For an isolated electron pulse, we found that the smearing of the spectrogram equates to an overestimation of the bandwidth, and therefore an underestimation of the pulse duration. Figure 1(f) is a plot of the pulse duration retrieved by the attosecond FROG algorithm as a function of the bandwidth δ̄ of the spectrogram (that contains the uncertainty).
We define the spectrogram bandwidth as the average standard deviation of the streaked spectra:
For each of the parameters considered, Fig. 1(f) suggests that the retrieved pulse duration is smaller when the spectrogram is more smeared out. Therefore, a major effect of the uncertainties pertaining to a single pulse is an error in the retrieved pulse duration.
While some attosecond experiments strive to obtain a perfectly isolated attosecond pulse, practical experimental limitations yield an attosecond pulse accompanied by a smaller satellite pulse. It has been experimentally demonstrated that the satellite pulse is much weaker than the main pulse after the spectral filter for most laser CEP settings . In fact, currently available ultrashort laser pulses ensure that at most two half-cycles of the laser field produce measurable attosecond bursts. Therefore, we consider the case of twin attosecond pulses, separated by half a laser period T L/2≈1.2fs, to be a realistic scenario for ongoing attosecond experiments.
For double pulses, the spectrogram has a fringe pattern, which is a result of the self-interference of an electron ejected at adjacent laser half-cycles. The fringes are mainly defined by the parameters associated with the two electron pulses:
• the fringe position is proportional to the relative phase between the two pulses;
• the fringe spacing is inversely proportional to the time difference between the pulses;
• the fringe visibility (or contrast) is related to the relative intensity between the pulses.
As we will see, the fringe pattern is sensitive to changes in these parameters (relative phase, relative intensity and relative timing), and its smearing affects the retrieval of the satellite pulse, defined as the smaller of the two pulses. The signatures of the uncertainties related to double pulses are shown in Fig. 2. The main effect in this case is a smearing and loss of visibility in the fringe pattern.
The smearing of the fringe pattern mainly affects the retrieval of the satellite pulse. Figure 2(f) plots the relative intensity of the satellite pulse as a function of the contrast of the fringe pattern in the unstreaked spectrum,averaged over the uncertainty. For each parameter, we notice that the intensity of the retrieved satellite pulse is correlated to the fringe contrast: A reduced fringe contrast leads to an underestimation of the satellite, despite the clear shift of its spectral components away fromthose of the main pulse. Moreover, the uncertainties in the relative phase and relative intensity between the pulses appear to alter the spectrogram in a similar manner – reducing the overall fringe contrast – whereas the uncertainty in the relative timing has a more complicated effect, as will be discussed later on.
3.1. Central energy
We first consider variations in the central energy of the wave packet. Changes in central energy might be intrinsic to the experiment, e.g. electrons ejected from a conduction band in a metal will contain an inherent energy uncertainty; or they can also occur due to fluctuations or variations in the laser intensity, which affects the kinetic energy of the recolliding electrons (and hence the cutoff of the harmonic spectrum). The spatial chirp of the laser beam might also result in variations of the central energy, due to the strong dependence of the ponderomotive energy on wavelength.
At moments tM when the the vector potential reaches an extremum A M=A L(tM), the streaking effect mainly results in a shift of the electron spectra, given approximately by
which is a function of the electron’s central energy εc=p 2 c/2. Figure 3(a) shows streaked spectra centred at different energies, taken at a delay Δt s=0 corresponding to the largest spectral shift. The amount of spectral shift is plotted against the central electron energy in the inset of Fig. 3(a), where the solid line is the theoretical curve given by (7).
The incoherent sum of these streaked spectra clearly produces an artificially broader spectrum, which smears the spectrogram along the energy axis. As a result, the retrieval algorithm overestimates the bandwidth, leading to an underestimation of the pulse duration. The pulse retrieved by the attosecond FROG algorithm is significantly shorter (28%) than the original one, and its temporal phase is somewhat distorted, the chirp being overestimated by 155%.
3.2. Bandwidth and group-delay dispersion
An uncertainty in the bandwidth of the electron wave packet can happen due to overall changes in the laser waveform, particularly the intensity, and can also result from spatial variations in the cutoff energy. The bandwidth of the wave packet directly correlates with the bandwidth of the streaked spectra, but has little impact on the averaged spectrogram. Figure 4(a) shows plots of streaked spectra taken at a delay Δt s=-T L/4≈-0.600fs, where the vector potential is zero, but changes approximately linearly in time. In this case, the streaked spectra are not shifted, but broadened or narrowed depending on the sign of their chirp and the sign of the laser electric field, as described by (10) below.
Figure 4(a) illustrates the effect of the electron’s bandwidth on streaked spectra. These spectra are taken at a delay Δt s=-0.600fs, where spectral narrowing due to the streaking field is the most pronounced. The inset of Fig. 4(a) shows a plot of the streaked bandwidth versus the unstreaked bandwidth. The solid line is the theoretical curve calculated from (12).
In our simulations we assume a rather large bandwidth uncertainty, corresponding to pulses with durations ranging from 186as to 3.3fs. Despite such large bandwidth variations, the quality of the resulting spectrogram was sufficient for an accurate retrieval, yielding errors in pulse duration and temporal chirp of 11% and 9% respectively. Figure 4(b) shows that the retrieved phase diverges from the original one where the intensity is low. This is also the case in most of the other retrievals, and is caused by the fact that the retrieved phase becomes ambiguous at low intensities.
The group delay dispersion of XUV pulses is an inherent property of the harmonic generation process , which has been shown by measurements  to be accurately described by the three step model . The GDD is directly determined by the laser waveform that generates the harmonics. Thus, any changes in the laser field over its spatial profile or from shot to shot, will affect the chirp of the attosecond XUV emission, and hence that of the streaked electron.
Figure 5(a) shows streaked spectra taken at the same delay as in Fig. 4. The inset shows a plot of the streaked bandwidth δs, given by (12), as a function the GDD γ. Just as with the bandwidth uncertainty, the uncertainty in the harmonic GDD has hardly any effect on the streaking measurement: different GDD’s will merely change the spectral width, but not the centre. After processing a spectrogram with a GDD uncertainty of 100000as2, the FROG algorithm was able to recover the correct pulse duration and temporal chirp to within 10% and 21% respectively. Figure 5(b) shows a slight artefact in the phase at ≈0.37as. This is due to the fact that the intensity is low in that region, where the phase is more ambiguous.
To obtain further insight into the role of bandwidth and GDD uncertainties, we model the streaking at the zero-crossing of the vector potential assuming a linearly chirped Gaussian wave packet
where τ is the FWHM duration and β=φ″(t) the temporal chirp of the unstreaked electron wave packet with a central momentum . When the duration of the XUV pulse is much shorter than the laser period, the vector potential of the streaking field is approximately linear,
with a slope given by the peak electric field of the half-cycle, E R=E(t R), which occurs at the moment t R. Using (8) and (9) together with (1), we find that the width of the streaked spectrum at the zero-crossing of the vector potential is given in atomic units by
This expression shows that the streaking field adds an additional chirp
to the electron wave packet while it is being ejected from the atom. This additional chirp is proportional to the value of the electric field E R. If E R has the opposite sign as β, then the chirp will be amplified and the streaked spectrum will be broader than the unstreaked one. If, on the other hand, it has the same sign, it may partially or completely cancel the effect of the chirp, and produce a spectrum that is narrower . Since adjacent zero-crossings of the vector potential correspond to different signs of E R, we expect the streaked spectra at these delays to have different bandwidths (one narrower and one smaller). In terms of the unstreaked bandwidth δ (in units of angular frequency) and the time-bandwidth product , where γ= ″(w) is the unstreaked GDD, expression (10) can be rewritten as
Integrating the streaked spectra over a range of uncertainty in bandwidth δ or GDD γ results in an averaged spectrum that resembles the original one.
3.3. Relative timing
Perfectly isolated single attosecond pulses are difficult to produce in practice, although several advances have been made [5, 21, 22, 23]. Attosecond pulses generated with an ultrashort laser pulse are often accompanied by a small “satellite” pulse. The spacing between the two XUV pulses is approximately equal to a laser half-period, but can change due to fluctuations in the laser CEP  or also because of the laser beam profile, which affects the Gouy phase shift . The relative timing can also change over the harmonic beam profile due to the influence of long and short recollision trajectories, which are emitted with different divergences. A streaking measurement performed with a sequence of two attosecond pulses exhibits a fringe pattern which is the result of spectral interference between the emitted electron pulses. The spacing of these spectral fringes is inversely proportional to the time separating the pulses, whereas the position of the minima and maxima is a function of the difference in classical action
accumulated between the electron pulses, with momenta p 1 and p 2 and separated by a time Δt X, from their moments of birth into the continuum (Δts for the electron with momentum p 1 and Δt s+Δt X for the one with momentum p 2) until the end of their interaction with the streaking field. This relative action, which translates into the position of the fringes in the spectrogram, is both a function of the time Δt X separating the pulses as well as the delay Δts between the XUV and NIR fields. Neglecting the A 2 L(t) terms, the relative phase ξ(Δts,Δt X) can be approximated as
Figure 6(a) shows plots of the fringe position obtained from (14) for different values of the relative timing Δt X. These curves represent the delay dependence of the minima and maxima of the fringes as a function of the streaking delay, and can be compared to the fringe patterns of the spectrograms in Fig. 6(b), 6c and 6(d). As the position and modulation rate (ΔωF=2π/Δt X) of the fringes are both functions of ΔtX, each streaked spectrum will be affected differently by uncertainties in the relative timing. Integrating over different values of the relative timing distorts the fringe pattern in a rather complicated manner, as shown by comparing the spectrogram in Fig. 2(a) to the undistorted one in Fig. 2(e). The modulation depth of the fringes quickly decreases with the relative timing jitter between the pulses, and the overall modulation rate of the fringe pattern is no longer uniform over the spectrogram. Figure 7(a) shows unstreaked spectra calculated for different values of the relative timing, further illustrating the fact that the fringes are quite sensitive to the relative timing.
The fringe pattern contains much of the information pertaining to the satellite pulse, and if it is excessively disturbed, the satellite cannot be properly recovered. With an uncertainty of 600as in the relative timing, the retrieval algorithm fails to properly recover the satellite pulse, whereas the main pulse seems to be accurately characterized (its duration is off by 6as) as shown in Fig. 7(b). Interestingly, the duration of the retrieved satellite is longer than the original, by an amount that is correlated with the uncertainty in the relative timing between the pulses. This suggests that information about the relative timing uncertainty is actually embedded in the averaged in the averaged spectrogram.
3.4. Relative phase
The CEP of attosecond pulses is determined by the laser field that generates the high-order harmonics. Within the three-step model , it is related to the amount of action accumulated by the recolliding electron during its excursion into the continuum from the time of its birth until its recollision time. The accumulated action – which depends on the strength, period and overall shape of the laser half-cycle that drives the electron back to the parent ion – gets directly mapped to the phase of the attosecond burst.
For a single pulse, the attosecond CEP has no effect on the measured spectrogram, as shown by formula (1). However, in the case of two pulses, the fringe pattern in the spectrogram carries the information pertaining to the relative CEP (relative phase) between the two attosecond pulses. This relative phase is generally non-zero due to the fact that, for ultra short laser pulses, the adjacent half-cycles will not convey the same amount of action to their recolliding electrons. The relative phase is therefore sensitive to the laser waveform, and a simple change in laser intensity or laser CEP, for example, will result in a different relative phase.
Changing the relative phase between the attosecond pulses translates into a uniform shift of the fringe pattern along the energy axis. This is illustrated in Fig. 8(a), which shows unstreaked spectra computed for different values of the relative phase. The delay-dependent behavior of the fringe positions (due to the relative action ξ(Δts,Δt X) accumulated by the electron in the streaking field) remains unchanged as a result of an uncertainty in the relative phase, and the net effect of this uncertainty is just a uniform loss of fringe contrast over the whole spectrogram, as shown in Fig. 2(b). This means that the spectrogram is slightly more robust against variations in the relative phase, as compared to relative timing.
For this simulation, we assumed an uncertainty of 3π/2 for the relative phase. Figure 8(b) shows that the power contained in the satellite pulse is underestimated. However, since the fringe pattern only loses contrast but remains otherwise undistorted, the characterization of the satellite pulse is accurate: its duration and temporal chirp were correctly retrieved to within 4% and 13% respectively. Additionally, the relative phase between the pulses was also retrieved, owing to the fact that the integrated fringe pattern is not shifted with respect to the original one. On the other hand, if the relative phase is completely unknown, then the fringe pattern disappears, and the satellite is unrecoverable. Despite the underestimation of the satellite, the main pulse is correctly retrieved.
3.5. Relative intensity
Since the ionization rate is a highly nonlinear function of the electric field amplitude, the relative contributions to harmonic generation by adjacent recollision events, i.e. the relative intensity between two consecutive attosecond bursts, depends on the laser pulse intensity.
The relative intensity between the attosecond pulses simply changes the fringe modulation depth in the spectrogram: the more the pulses are similar in power, the more the fringe pattern is modulated and vice versa. Although the modulation depth depends on the relative intensity, Fig. 9(a) shows that the positions of the fringes remain the same. Thus, integrating over different ranges of relative intensity hardly changes the spectrogram, apart from a very minor loss in fringe visibility as shown in Fig. 2. Even for a large uncertainty of 50% in the relative intensity, the retrieved pulses closely match the original ones, as evidenced in Fig. 9(b); the satellite being underestimated by 17% but otherwise properly characterized.
3.6. Streaking Delay
Between the initial moment of harmonic generation and the refocusing on the streaking target, the laser and XUV beams are decoupled in order to induce a delay between them. In some experimental setups [1, 4], the two beams remain collinear. In others, the beams might actually follow a different geometrical path, for example, to convert the NIR field to its third or fifth harmonic. Experimental instabilities, leading to changes in the spatio-temporal profile of the laser or the XUV beam can also have an impact on the streaking measurement. Uncertainty in the delay Δt s between the laser and XUV beams can occur because of shot to shot laser jitter or due to the spatio-temporal profile, where the timing of the attosecond pulse can depend on the radial position away from the axis. Moreover, it can be due to an intrinsic property of the experiment, as in the case of photoionization from different layers in a solid, leading to a timing uncertainty in the onset of streaking. Since the streaking spectrogram bares a close resemblance to the waveform of the streaking field – the spectral shift resembles the laser vector potential – the moment when the electron wave packet begins to be streaked will determine the position of the spectrogram with respect to the delay axis. This delay uncertainty will therefore induce an additional smearing effect on the spectrogram, horizontally along the delay axis.
In the simple case of a purely isolated attosecond pulse, this smearing will cause a bandwidth overestimation, and can be responsible for an underestimation of the duration of the attosecond pulse. This smearing effect is most pronounced at delays corresponding to the zero-crossings of the vector potential, where most of the information pertaining to the chirp of the wave packet can be found. At the peaks of the vector potential, the spectral shift is approximately constant, and the delay uncertainty has a minimal effect.
Figure 10(a) shows that, for a timing uncertainty of 900as, the retrieved pulse is 13% shorter than the original one, and the algorithm completely failed to retrieve its chirp due to the smearing of the spectra at the zero-crossings of the vector potential.
The presence of a satellite pulse produces a fringe pattern in the spectrogram which contains essential information about relative quantities between the two pulses – e.g. timing, phase and uncertainty. We have seen that this fringe pattern is very sensitive, and uncertainties in the streaking delay also contribute to its smearing in addition to the effects previously mentioned for a single pulse. When the two electron pulses are separated by a time Δt X=T L/2, the fringe pattern varies rapidly at the peaks of the vector potential, and is roughly constant at the zero-crossings. Therefore, the fringes are mostly smeared out at the delays where the largest spectral shifts occur, and are better preserved near the zero-crossings, as shown in Fig. 2(d) (also for a timing jitter of 900as). This was also experimentally measured in .
Figure 10(b) shows that the delay uncertainty has a negative impact on the retrieval of the satellite pulse. Due to the smearing of the fringes, the retrieved satellite is underestimated, and the relative phase between the two pulses was not correctly recovered. As well, the bandwidth overestimation, due to the smearing effects previously described for a single pulse, lead to the main pulse’s duration to be underestimated by 18% and its retrieved temporal chirp to be off by 68%.
As in the case of conventional FROG spectrograms, attosecond streaking spectrograms should be viewed as an average over an ensemble of wave packets. While experimental imperfections are a common source of these uncertainties, fundamental properties of systems probed by an attosecond pulse may also call for an ensemble description. We have shown that the attosecond FROG reconstruction based on model (1) remains adequate, but care must be taken in interpreting the results.
We have investigated the effect of uncertainties of several key parameters on the streaking measurement and gauged their individual importance. Although these parameters are generally correlated to one another – e.g. a change in relative timing will almost certainly be accompanied by a change in relative phase, and a change in the central energy will undoubtedly occur with a change in bandwidth – we do not expect their contributions to mutually cancel each other, rather these should be compounded into the spectrogram. We have seen, for a single pulse, that each uncertainty produces an overall broader spectrogram; whereas for twin pulses, each uncertainty reduces the visibility of the fringe pattern. We found that uncertainties in certain parameters have a particularly detrimental effect on streaking measurements: the uncertainties in the central energy of the wave packet, its timing with respect to the laser field and the relative timing between twin pulses.
The reconstruction of a single pulse is generally robust, and mainly suffers when there is significant smearing of electron spectra, which can occur due to variations in the central energy of the electron wave packet, or the relative timing between the laser and XUV pulses, and has the effect of overestimating the bandwidth. As a consequence, the retrieved pulse duration is smaller than the actual one. We have also verified the role of uncertainties in the GDD and bandwidth, and found that they have a minimal impact on the spectrogram and the retrieved wave packet. Even when these parameters change by unrealistically large amounts, the wave packet can still be correctly characterized.
In the case of twin pulses, we found that the attosecond FROG reconstruction is greatly affected by any uncertainties that can smear the fringe pattern in the streaking spectrogram. This is because the fringe pattern is highly sensitive to changes in relative parameters between the pulses. Uncertainty in the relative timing between the pulses have the largest impact on the quality of the fringe pattern, since the relative timing has a twofold effect on the fringe pattern: it shifts the fringe position and changes their modulation rate. As a result, the fringe pattern loses its visibility and quickly becomes distorted, preventing the characterization of the satellite pulse, as observed in . Relative phase and relative intensity variations affect only the fringe contrast, and therefore allow for the proper characterization of both pulses. In spite of the presence of uncertainties in relative phase, timing and intensity, the information about the main pulse remains properly encoded in the spectrogram. Although these uncertainties hamper the characterization of the satellite, they hardly affect the reconstruction of the main wave packet. This alleviates, to some extent, the need to generate a perfectly isolated pulse: the temporal structure around the main wave packet can be recovered despite the presence of satellites.
Currently, there are no general methods that can analyze spectrograms which are significantly distorted by uncertainties. However, our results indicate that spectrograms contain enough information to identify, and maybe quantify, the most important uncertainties. For instance, Figures 1 and 2 show that the main uncertainties in the parameters of the electron wave packet each give their own fingerprint on the measured spectrogram. This calls for the development of new analysis techniques for attosecond streaking measurements on systems more complex than an isolated atom.
The authors are grateful to R. Ernstorfer and F. Krausz for providing valuable insight about the attosecond streak camera technique and the photoionization of metal surfaces. This work was supported by the DFG Cluster of Excellence: Munich-Centre for Advanced Photonics.
References and links
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