To understand condensed phase dynamics, chemical, and biological processes at the most fundamental level, one would ideally like to observe structural dynamics as they occur at the atomic level [1, 2, 3]. In the last few years, enormous strides have been made in making such experiments possible, using ultrafast laser excitation pulses to trigger structural changes and either x-ray or electron probes to follow the induced atomic motions [4, 5, 6, 7]. The required time resolution is on the order of 100 fs given by the approximate time it takes two atoms to move far enough apart to break a bond, or the periods of the most highly damped motions along structural relaxation coordinates [1]. Laser pulses shorter than this time scale are readily available. The technical challenge is making sufficiently intense x-ray or electron pulses on this timescale to secure sufficient signal in a few shots, which is mandated by sample constraints. To obtain the highest time resolution, one must not only be able to produce such pulses but also to characterize their durations and determine their exact timing with respect to the excitation pulse inducing the structural changes [1, 3]. The latter issue is referred to as the t=0 problem. Both the pulse duration and the t=0 position need to be determined within 10–100 fs accuracy.

For x-rays produced by free electron lasers, these technical issues have only recently been overcome [8, 9]. By using electro-optic sampling of the relativistic electron beam for the x-ray pulse source, a time resolution of ≈100 fs has been achieved. For the 30–100 kV electron pulses used in electron diffraction, one can in principle use streak cameras or laser ponder-motive scattering to characterize the pulses. However, streak cameras are problematic for this application due to the physical length of the streaking plates needed to measure sub-picosecond pulses [10]; the duration of those pulses rapidly changes during propagation and hence while passing the streaking plates. The best time resolution for high electron number pulses would be around 1 ps. The t=0 position cannot be identified with streak camera technology at all. We have demonstrated pondermotive scattering with a high intensity laser pulse as a viable option for electron pulse characterization [11]. The drawback of this approach is its pulse energy requirement in the 10 mJ range. This requirement defeats one of the major advantages of using electron diffraction; namely that it is a table top experiment. While it is possible to calibrate the electron gun at a major facility and maintain the same conditions for the electron pulse afterwards, each experiment requires different probe conditions that lead to changes in temporal characteristics of the electron pulses and timing. Given the high sensitivity of electron pulses to space charge effects [10, 12], an in situ method to characterize them is needed. Here we describe a new approach that solves this issue using laser pulse energies that are well within reach for conventional small scale femtosecond lasers and even compact all fibre systems.

The ponderomotive force is proportional to the gradient of the light intensity [13]:

where e and m_e are the charge and mass of an electron, respectively, λ is the wavelength of the laser and I is the laser intensity. In a single propagating laser pulse, as used in our earlier experiments [11], the ponderomotive force is due to the gradient of the envelope of the laser pulse, which varies on length scales given by the laser pulse duration longitudinally and the focal spot transversely. In order to reduce the energy required to create a strong scattering signal, we use two counterpropagating laser pulses that overlap at their intersection with the electron beam in space and in time. The use of standing waves of light to scatter electrons was proposed by Kapitza and Dirac in 1933 [14] and has been realized to scatter [15] and to diffract electrons [16] at relatively low energy (30 eV and 380 eV, respectively). Possible applications to short pulse generation have also been discussed [17]. For counterpropagating Gaussian laser pulses propagating in x-direction, the intensity is

where I ₀ is the peak intensity of each laser pulse, k=2π/λ, ω=2πc/λ, and $2\sqrt{2\ln 2}{w}_t$ and $2\sqrt{2\ln 2}{w}_f$ are the laser pulse duration and the beam waist, respectively. The x-component of the ponderomotive force takes the following form on the x-axis:

The envelope of this force in the yz-plane is the same as for the intensity. The first two terms describe the laser pulses travelling in opposite directions and the last describes the standing wave. Since λ≪w_tc the derivative is dominated by the latter contribution.

Under otherwise identical conditions, the maximum of this force is over 100× stronger than the lateral force used in our previous study, if a beam waist of 10 µm FWHM and λ=800 nm are assumed. The lateral forces (F_y and F_z) are not enhanced by the standing wave and are therefore negligible in the present study. The direction and magnitude of the force varies from zero to its maximum over λ/2. Since the diameter of the electron beam is much larger than λ/2, we expect equal scattering in the positive and negative x-directions. The transverse coherence length of the electrons [18] is smaller than λ/2 and therefore we do not expect diffraction of the electron beam.

The setup is shown in Fig. 1. The electron gun has been described elsewhere [1]. It has been modified to use the 500 nm, 50 fs output of a non-collinear optical parametric amplifier to drive the electron gun via two-photon photoemission. These pulses are significantly shorter than the initial laser pulse and their wavelength has been tuned close to the threshold above which no photoemission is observed. This minimizes the excess energy of the emitted electrons by allowing only the highest energy electrons to escape, thereby reducing longitudinal as well as transverse spread of the resulting electron pulses.

 

Fig. 1. (a) Optical setup. The laser pulse is split by a beam splitter (BS). A small part of the beam is used to produce visible light using a non-collinear optical parametric amplifier (NOPA), which, after a delay line, is used to drive the electron gun. The other part of the beam is split again after a focusing lens to form the grating at the focus intersecting the electron beam. One of the beams is sent through a delay line for compensation of the relative delay between the pulses. (b) Schematic view of the experiment geometry. When the laser pulse hits the photocathode, it creates an electron pulse which is accelerated by a DC electric field. The electron pulse is stripped of its outer electrons by a pinhole before it is scattered by the ponderomotive force of the intensity grating created by the two laser pulses.

Download Full Size | PDF

The two laser pulses used to create the ponderomotive force profile need to overlap in space as well as in time with each other and with the electron pulse to produce a scattering signal. An elliptical pinhole is placed in the electron beam at 45° with respect to the electron and laser beams. Both laser beams are aligned on this pinhole. The temporal overlap between the electron pulse and the laser pulses is determined for each of the two arms by observing electron beam deformations by plasma generation on a metal surface [1]. The delay between the pulses is adjusted to compensate the differences.

Images of the electron beam on the detector are shown in Fig. 2. The images taken at different electron pulse-laser pulse delays τ are analysed to give a time trace S(τ) [11]:

where X and Y are the horizontal and vertical coordinates on the detector screen (origin at electron beam center) and D_τ (X,Y) is the number density of electrons detected in the image at time delay τ. A number of these time traces for different electron numbers per pulse are shown in Fig. 3.

 

Fig. 2. Detector images (animation, 528 kB) of the electron beam. The counter in the bottom right corner indicates the relative delay between the laser and electron pulses. The black spot in the electron beam is caused by detector damage. Conditions: 10,000 electrons per pulse, 135 µJ laser pulse energy, images integrated over 4000 pulses. [Media 1]

Download Full Size | PDF

To illustrate that this time trace is a direct cross-correlation of the electron and laser pulses, we assume that the displacement that electrons experience due to the ponderomotive force is negligible over the size of the laser focus and therefore the force acting on an electron can be integrated along a straight line. If this condition is not met, the linearity of the signal with respect to the laser intensity could be compromised. The cumulative displacement along the trajectory would be reduced, leading to saturation and a broader time trace. For the parameters used in this study, the condition is not strictly met, however, experimental verification of the linearity of the signal vs. laser power did not reveal any signs of saturation. Since the distance from the beam crossing point to the detector is much longer than the laser spot size, the deflection of an electron at the detector is

where T_d is the propagation time to the detector. Here we assume that the direction of propagation of the electron is parallel to the z-axis. In practice, lateral velocity components are inevitable. Numerical simulations with 10,000 electrons per pulse show that under the present experimental conditions, the average transverse velocity just before the interaction with the grating is about 4×10^-4 v where v is the velocity of the electrons in z-direction. The offset caused by this transverse velocity over the width of the grating is not a significant fraction of the grating pitch π/k.

According to Eq. 4 with the lateral envelope of the laser beam, F_x can be written as a product of a constant F ₀ and of four functions each of which only depends on one of the variables t, x, y, z:

Equation 5 can be rewritten as the sum of |X| over all electrons in a pulse. If the electron beam on the detector in absence of the grating is small, we can set X=ΔX. Using the electron density in the electron pulse at the beam crossover position ρ(r,t)=ρ_xy(x,y)ρ_t (t+τ-z/v) and Eq. 6 and Eq. 7, we obtain

S(τ) can therefore be identified as a convolution of the temporal profiles of the electron pulse and the laser pulse and of the transverse spatial profile of the laser pulse as it is crossed by the electron pulse. As opposed to our earlier experiments [11], the scattering potential is not moving through the electron beam, thereby eliminating the transverse spatial profile of the electron pulse from the convolution.

 

Fig. 3. Time traces with Gaussian fits. Vertical offset for clarity. While a Gaussian appears to be a good fit for traces of pulses with relatively few electrons, there are systematic deviations between the data and the fit for pulses with more electrons. The fits have widths of 1005 fs, 757 fs, 546 fs and 404 fs FWHM, respectively.

Download Full Size | PDF

It is obvious from Fig. 3 that the width of the time trace and hence the electron pulse duration increases with the electron number density. Particularly for pulses containing large numbers of electrons i.e. long electron pulses whose shape dominates the corresponding time trace, the shape of the traces is non-Gaussian. The actual temporal shape of the electron pulse cannot be directly deconvolved from noisy experimental time-traces unless the other contributions to the convolution are small. However, the variance σ ² _ρ of the temporal shape ρ_t(t) of the electron pulse can be extracted from the experimental trace without prior knowledge of the non-Gaussian pulse shape by

where σ ² _trace, σ ² _t and σ ² _f are the variances of S(τ), f_t(t) and f_z(z), respectively.

A common measure for the pulse duration is the full width at half maximum with $\mathrm{FWHM}=2\sqrt{2\ln 2}\sigma$ for a Gaussian. This conversion does not apply to other pulse shapes but it is a useful value to estimate the effect of the pulse shape on the temporal resolution of a pumpprobe experiment. Therefore, we calculate the electron pulse duration T as $T=2\sqrt{2\ln 2}{\sigma }_{\rho }$ . The contributions that need to be deconvolved from the trace are the temporal laser pulse shape (210±10 fs FWHM sech²) and the spatial profile of the laser pulse at the focus in z direction $(41\pm 10\mu m\stackrel{\wedge}{=}310\pm 80\mathrm{fs}\mathrm{FWHM}$ Gaussian for 55 keV electrons). Figure 4 shows the electron pulse duration as a function of the number of electrons per pulse. The sensitivity of the pulse duration to the electron number density clearly demonstrates the importance of this new in-situ measurement as a routine diagnostic tool.

 

Fig. 4. Electron pulse duration T vs. number of electrons per pulse; General Particle Tracer (GPT) simulations and measurements obtained in the described experiments.

Download Full Size | PDF

The measured pulse durations are compared to simulations performed with the General Particle Tracer [19] software package. The parameters used in the simulations correspond to the current setup of the electron gun: Electrons are emitted from a gold photocathode with a 50 fs, 500 nm, 200 µm FWHM diameter laser pulse. In order to account for the two-photon process, the effective pulse duration used in the simulations is 35 fs and the diameter is 141 µm. An excess energy distribution of 0.1 eV width is assumed. The electrons are accelerated over a 6 mm gap to 55 keV. The anode is a 150 µm diameter pinhole and the electrons propagate freely for 26mm after that. A 30 µm diameter pinhole strips the electron pulse followed by another 7mm of free propagation before the laser interaction. Note that in our femtosecond electron diffraction experiments [20], the propagation distance to the sample is 9mm shorter than in this setup. This is not a limitation of this pulse characterization technique but was chosen to minimize the complexity of the setup.

The observed electron pulse durations are in good agreement with calculations both with respect to duration and dependence on electron number. The relatively small systematic deviation may be caused by imperfections of the experiment, e.g. pedestals in the temporal profile of the laser pulse. Also, the finite size of the undisturbed electron beam on the detector (just under 1/4 the size of the scattered electron distribution) leads to a slight broadening of the time traces. Shorter, cleaned up laser pulses, tighter focusing of the laser as well as focusing of the electron beam on the detector will shorten the system response of the pulse characterization and reduce errors in the pulse shapes for the deconvolution. The intrinsic system response is determined mostly by the electron transit time across the focus (310 fs presently). A 10 µm focus and 20 fs laser pulse would give a system response of 80 fs and 10 fs accuracy to the t=0 position with pulse energies of 10 µJ.

In conclusion, we have demonstrated a method for the full characterization of femtosecond electron pulses. Using grating enhanced pondermotive scattering, the scattered electrons are imaged to give direct visualization of the laser-electron pulse interaction with modest laser input power. This new approach permits routine characterization of both duration and t=0 position of electron pulses for pump-probe experiments. This information is critical to attaining the highest time resolution possible in the use of femtosecond electron pulses for diffraction and imaging applications in atomically resolved structural dynamics [1]. Equally important, this method enables in-situ measurements directly at the sample position. The measurements largely confirm n-body simulations performed with the General Particle Tracer n-body code. This illustrates that the many body effects and assumptions on the initial electron distribution are taken into account reasonably well, and can be used with some confidence in advancing electron source brightness.