Tilted femtosecond pulses for velocity matching in gas-phase ultrafast electron diffraction

Recent advances in pulsed electron gun technology have resulted in femtosecond electron pulses becoming available for ultrafast electron diffraction experiments. For experiments investigating chemical dynamics in the gas phase, the resolution is still limited to picosecond time scales due to the velocity mismatch between laser and electron pulses. Tilted laser pulses can be used for velocity matching, but thus far this has not been demonstrated over an extended target in a diffraction setting. We demonstrate an optical configuration to deliver high-intensity laser pulses with a tilted pulse front for velocity matching over the typical length of a gas jet. A laser pulse is diffracted from a grating to introduce angular dispersion, and the grating surface is imaged on the target using large demagnification. The laser pulse duration and tilt angle were measured at and near the image plane using two different techniques: second harmonic cross correlation and an interferometric method. We found that a temporal resolution on the order of 100 fs can be achieved over a range of approximately 1 mm around the image plane.


Introduction
Recent improvements in electron pulse technology have resulted in tabletop sources capable of delivering femtosecond pulses to a target [1,2]. This has enabled ultrafast electron diffraction (UED) experiments on solid samples with a resolution on the order of 200 fs [2]. In these experiments, thin (submicrometer) samples were used to capture diffraction patterns in transmission mode. Another important application of UED is investigating ultrafast chemical reactions on isolated molecules. For these experiments, the target is a gas beam with a diameter typically between 0.1 and 1 mm. The group velocity mismatch between the laser and electrons results in a blurring of the resolution as the pulses traverse the gas jet [3]. The group velocity mismatch limits the resolution of most experiments to several picoseconds [4][5][6], with the highest resolution of 850 fs achieved using a microjet with a diameter of only 0.1 mm [7]. A laser pulse with a tilted intensity front was used to match the velocity of the laser and electrons on the surface of a solid sample [8]. However, the matching has only been demonstrated at a single plane along the propagation direction; therefore, it is not clear that it can be applied to the problem of gas UED, in which the target is extended. For example, electrons with a kinetic energy of 100 keV travel with a speed of 0.55 c, where c is the speed of light in vacuum. A laser pulse propagating at an angle of 57°with respect to the electron beam, with a pulse front tilted at the same angle, would match the speed of the electrons. However, the problem is that the velocity must be matched throughout the length of the gas target and both the tilt angle and duration of the laser pulse change as it propagates. In this paper, we investigate this issue in detail and show experimentally that, with the appropriate optical design, it is possible to achieve a resolution on the order of 100 fs.
A tilted laser pulse has an intensity front that is tilted with respect to the direction of propagation. The component of the laser group velocity in the direction normal to the intensity front depends on the tilt angle [9]. This dependence can be exploited to match the normal laser velocity to the velocity of either the electrons or the electromagnetic waves traveling more slowly through a medium. Several applications of tilted pulses have been reported in the literature. Tilted pulses have also been used to match the group velocity of a pump laser to the phase velocity of terahertz waves to increase the efficiency of optical rectification [10][11][12]. In xray laser generation, a tilted laser front-generated by step mirrors or a slightly misaligned laser compressor-was used to provide optimum preplasmas for a traveling wave pumped by a second laser pulse [13][14][15]. The angular dispersion used to generate a tilted pulse front also results in a rotation of the pulse front as the beam focuses and defocuses [16,17]. Femtosecond laser pulses with such rotating fronts have been used recently for electron wakefield acceleration and the generation of attosecond pulses [18][19][20].
For gas-phase UED experiments, the laser beam is focused to a spot size below 0.5 mm to achieve high fluence and minimize the group velocity mismatch. Thus, in addition to a large tilt angle, a large demagnification factor is required. In our configuration, a grating provides the angular dispersion. The grating surface is imaged onto the target with a demagnification factor (M) of 12.7. At the image plane, the diffracted components will recombine into a short pulse, as long as the diffracted beam is normal to the grating surface. However, a longer pulse duration is expected both before and after the image plane. We use an optical setup with a long Rayleigh length, about 2 mm, to lessen the broadening of the pulse duration as it traverses the target. Optical aberrations might also result in increased pulse duration across the beam, even at the image plane. For gas-phase UED experiments, the pulse duration must remain short over the length of the target. We measured the tilt angle and pulse duration as a function of the distance from the image plane and at several positions laterally across the beam. We used a simple technique that consists of measuring the interference between the tilted pulse and a known reference pulse [21]. These results have been compared to those from a previously demonstrated technique using second harmonic generation (SHG) between the tilted pulse and a reference pulse [22]. The interferometric technique is more convenient for in situ measurements because it requires only a detector (or screen) to be placed at the position of the measurement.

Theory
A laser pulse with a tilted front can be generated by propagation through a prism [23,24] or diffraction from a grating [24]. In the case of diffraction from a grating, the tilt angle γ is given by [22,24] is the angular dispersion, θ out is the angle of the diffracted beam with respect to the grating normal, λ 0 is the central wavelength of the laser pulse, and M is the demagnification factor. If no demagnification is involved, M will be 1. The angular dispersion depends on the diffracted order and the grating constant where d is the grating constant and k is the diffraction order. In our experiments, we used the first-order diffracted beam (k = 1). If the beam is demagnified using imaging optics, M is a function of position and is given by the ratio of the diameter of the laser beam on the grating to the diameter of the laser beam at the position where the tilt is measured. The angular dispersion Ψ generated by the grating results in a temporal chirp such that pulse duration increases as the pulse propagates away from the grating. The pulse duration τ at a distance z from the grating is given by where τ 0 is the original laser pulse duration (60 fs in our experiments) [17,22]. To recreate a short pulse at a target position, the grating surface must be imaged at this position. It has been shown that a nonzero value for θ out will result in temporal aberrations [22], as the distance to the image plane will be different across the beam. The requirement of θ = 0 out prevents us from using the experimental setup shown in references [15,18]. In [22], the pulse duration on the image plane was measured for M = 1. Here, we investigate the case of large demagnification and measure the pulse duration both across the beam and as a function of distance to the focal plane. Figure 1 shows the experimental setup used to generate and measure the tilted pulses. The tilt angle and pulse duration were measured using a cross correlation between the tilted pulse and a (nontilted) reference pulse using SHG or by directly measuring the interference between the two pulses. For the SHG method, the two pulses overlapped in a thin barium borate (BBO) crystal placed at the image plane (see inset in figure 1). Due to the tilt in one of the pulses, the two pulses spatially overlapped only along a narrow strip. The width of the spatial overlap region depended upon the duration of the pulses. A second harmonic signal was generated in the region where the two pulses overlapped. This region was imaged on a charge-coupled device (CCD) camera, while light at the fundamental frequency was blocked with a filter. The tilt angle was measured by recording the horizontal shift in the overlap region as a function of the delay of the reference pulse. The BBO crystal and CCD were translated together along the direction of the optical beam to measure the tilt angle and pulse duration as a function of distance from the image plane. For the interferometric measurement, the CCD camera was placed directly at the position where the pulses overlapped. Instead of measuring the width of the region where SHG was observed, the region of interference was measured directly.

Experimental setup
The laser system delivered 60 fs pulses at a central wavelength of λ = 800 nm 0 and pulse energy of 2 mJ with a repetition rate of 5 kHz. The laser beam was attenuated and then evenly split into two beams with a polarizing beam splitter (BS1 in figure 1). One beam was diffracted from a reflection grating to introduce angular dispersion. The laser beam incident on the grating was slightly elliptical with a full width at half maximum (FWHM) of 7.0 mm in the vertical direction and 6.0 mm in the horizontal direction.
The diffraction grating was a gold-coated holographic grating with a grating constant of d = 150 mm −1 . This small grating constant was used to achieve the desired angular tilt with a large demagnification factor. Using the grating formula, in out 0 , we calculated that an incident angle of θ =°6.9 in would result in the diffracted beam being normal to the grating surface. In this configuration, the diffraction efficiency of the grating into the first order was 80%. The reference beam traveled through a variable delay line to adjust the time of arrival at the BBO crystal. Both beams were focused onto the BBO with a 23 cm focal length lens (L 1 ). The distance between the grating and the lens was = S 315 cm 0 . Using the lens maker's equation, , where ϕ 0 and ϕ i are the laser beam diameters on the grating and on image plane, respectively. The beam size on the image plane was elliptical with FWHM of 0.56 and 0.47 mm. The tilt angle (γ) was 56.8°. The value of M (and thus the tilt angle) could be adjusted to match electron pulses with different kinetic energies by changing the distances. The BBO crystal used for SHG had a thickness of 0.20 mm. A FGB39 band pass filter was used to block the fundamental wave and transmit the second harmonic. The surface of the BBO crystal was imaged using lens L 2 (2.4 cm focal length) onto a CCD detector with four times magnification. Data were recorded for several positions before and after the image plane by moving the BBO crystal, lens L 2 , and the detector together along the direction of beam propagation for the SHG measurements. For the simpler interferometric measurements, only the detector position had to be varied. A small vertical angle was introduced between the tilted and reference pulses that created interference fringes. The pulse duration was also measured at five different positions across the beam by changing the delay between the pulses.  and z L1 is the either distance from lens L 1 to the detector in the interferometric method or to the BBO crystal in the SHG method. The experimental results from the interferometric and SHG techniques are in good agreement. The tilt angle decreased with increasing Δz because the beam diameter was increasing. The minimum beam diameter occurs before the image plane. The zero of Δz was defined as the i replaces d. We treat the light field on the image plane as an image of the light field immediately leaving the grating, such that the phase modulation imprinted by the grating on the beam has a spatial frequency increased by a factor of M.
Assuming a Gaussian beam away from the focus, the divergence angle (α) after the focal plane can be approximated by α = focal plane to the image plane and Δ R z ( ) is the laser beam radius as a function of Δz. We expect this to be a good approximation because S is significantly larger than the Rayleigh length (2 mm). The demagnification relative to the beam size at the image plane is where R i is the radius of the beam at the image plane. The tilt angle near the image plane is There is good agreement between the theoretical and experimental results. The plotted line in figure 2 corresponding to the theory does not include any fitted parameters. The measured tilt angles from the interferometric method are consistently larger than those measured with the SHG method. We attribute this to the uncertainty in finding the position of the image plane (position 0 in figure 2). This uncertainty is on the order of ±200 μm for the interferometric method. With the SHG method, the position of the image plane can be determined more accurately because the overlap region is imaged onto the CCD with magnification. For Δz values that are small compared to S, the tilted angle (γ) varies at a rate of approximately −1.5°/ mm. A change in tilt angle on the order of 1°would not significantly affect the velocity mismatch or temporal resolution of the experiment. For example, if the tilt angle were off by 1°, the velocity difference would be only 0.015 c.

Pulse duration
In the SHG measurement, the pulse duration (τ T ) of the tilted pulse was obtained from the FWHM of the spatial overlap of the two beams ΔL (figure 3), where τ R is the pulse duration of the reference pulse. The tilted pulse duration τ Δz ( ) at different positions can be deduced from R c a m e r a 2 2 2 [22]. Here, τ camera depends on the pixel size of the CCD camera. Our camera has a pixel size of 5.2 μm, corresponding to a temporal resolution of . For the interferometric method, τ T can also be extracted from the measured width of the interferometric cross correlation (ΔL). Assuming that the beams have a Gaussian temporal profile, the electric fields of the reference (ε R ) and tilted (ε T ) pulses at a specific z coordinate can be written as describes the time delay of the tilted pulse front. The interference intensity at the detector is where c.c. denotes the complex conjugate. The first and second terms make a constant contribution (C 0 ) to laser intensity, whereas the third and fourth terms represent interference. Substituting equation (6) into equation (7), we obtain  Figure 4 shows the experimental and calculated pulse widths [τ T (Δz)] over a region of approximately 3 mm before and after the image plane. There is good agreement between both experimental methods, and between the experiment and theory. The calculation was completed using equation (3) with the displacement and tilted angle as described in section 4.1. The pulse duration reaches a minimum of 66 fs at the image plane, compared to the initial pulse duration of 60 fs. The small broadening of the pulse duration maybe be due to aberrations or dispersion in the optical system. The pulse duration increases to 300 fs at a distance of 3 mm after the image plane. The pulse duration increases more rapidly for displacements toward the focus because the beam diameter becomes smaller and thus, the tilt angle and dispersion are larger.
At each position, the pulse duration was measured at five lateral positions across the beam, and it was found to vary by approximately 5% across the beam with the minimum near the beam center, which means that aberrations do not cause significant broadening transversely in our configuration. Figure 5 shows the pulse duration as a function of the lateral position from the beam center and as a function of distance for a region near the image plane.

Damage threshold of grating
The full characterization of the pulse duration and the good agreement with theory give us confidence that this method can be applied successfully to UED and other experiments. With the current setup, the main limitation to the intensity that can be reached at the target (the image plane) is the damage threshold of the grating. A tighter focusing geometry would lead to higher intensity, but would also result in a reduced Rayleigh length that would make it more difficult to have velocity matching over an extended target. Thus, the best way to increase the intensity is to keep the same configuration but reduce the size of the beam incident on the grating. To determine how far the intensity could be increased with our current configuration, we measured the damage threshold of the grating.  The laser pulse was focused to an area of 4.5 × 10 −3 cm 2 on the grating. The grating surface was imaged on a CCD camera with three times magnification. The pulse energy was increased in small steps and the grating was exposed to the laser for 40 min at each step. Damage was defined as any change in the diffracted laser beam. For a pulse energy of 0.54 mJ, a damaged spot was observed after 2 min of irradiation. For an energy of 0.50 mJ, a damage spot was visible after approximately 40 min. For a pulse energy of 0.40 mJ, no damaged was observed even after 2 h of irradiation. Thus, we conclude that the grating can be safely operated at a fluence of 90 mJ cm −2 , with the damage threshold at or below 110 mJ cm −2 . For a pulse duration of 60 fs, safe operation would mean a maximum intensity of 1.5 × 10 12 W cm −2 on the grating and 2.4 × 10 14 W cm −2 on the image plane.

Implication for gas-phase UED experiments
Our results indicate that it is be possible to reach a temporal resolution on the order of 100 fs for gas-phase UED experiments using tilted laser pulses to compensate for the group velocity mismatch. For example, let us consider the broadening that can be expected for a typical gas jet with a diameter between 0.5 and 1 mm. For a UED experiment without pulse tilting, the velocity mismatch would limit the resolution to several picoseconds [25]. With pulse tilting, there are two contributions to the broadening: the increase in the laser pulse duration and the remaining velocity mismatch due to the change in the tilt angle of the laser as it traverses the target. In our measurements, we started with a laser pulse duration of 60 fs at the output of the laser, and measured a pulse duration of 66 fs at the image plane of the grating. The average pulse duration over a 0.5 mm distance around the image plane was 71 fs, and it increased to 78 fs when averaged over 1 mm. For the effect of the remaining velocity mismatch, we consider that the angle changes at a rate of 1.5°/mm. If the tilt angle is off by 1°, the velocity mismatch between the laser and the electrons will be 0.015 c. This will lead to a broadening of the resolution of about 60 fs mm −1 . The total resolution of the experiment can be calculated as = + + T t t t total Laser GVM Electron , where the three terms on the right represent the laser pulse duration, broadening due to group velocity mismatch, and the electron pulse duration, respectively. For an electron pulse duration of 100 fs, the overall temporal resolution will be 126 fs for a target diameter of 0.5 mm and 140 fs for a target diameter of 1 mm. For longer electron pulses, the resolution will be determined mainly by the electron pulse duration. If shorter electron pulses are available, a resolution of 50 fs is within reach by using shorter laser pulses and keeping the target diameter below 0.5 mm.
In many photochemistry experiments, the photon energy required for excitation is in the ultraviolet range. Here, we discuss how changing the wavelength will affect the overall time resolution. Take the third harmonic, λ = 267 nm, as an example. To keep the tilt angle the same, the grating constant d must be tripled to compensate for the wavelength change, according to equations (1) and (2). Substituting this change into equation (4), we obtain that the tilt angle variation around the image plane will be the same as for the 800 nm pulse. The change in the pulse duration is described by equation (3), where the wavelength dependent term inside the square root is proportional to λ 6 Ψ 4 or λ 6 d 4 . Thus, at the shorter wavelength, the spreading in the pulse duration will be less severe if the same tilt angle is used. Therefore, the time resolution broadening due to the variation of the tilt angle around the image plane will be independent of wavelength, whereas the broadening due to pulse duration will be reduced for shorter wavelengths.

Conclusion
We experimentally studied the generation and measurement of a high-intensity tilted front femtosecond laser pulse for velocity matching in UED experiments. The tilted pulse was generated by imaging the surface of a diffraction grating onto the target position. We measured the tilted angle and pulse duration in the range of ±3 mm around the image plane using the SHG method and an interferometric method. The interferometric method is better suited for in situ measurements, as it requires only a measurement of the interference pattern at the target position. With an input pulse duration of 60 fs, we measured a pulse duration of 66 fs on the image plane, and the duration stayed below 100 fs for a distance of ±0.5 mm around the image plane. We showed that the pulse duration does not vary significantly across the beam. Our optical configuration is well suited for applications that require a large tilt angle and high fluence, and it can be used to reach an intensity above 10 14 W cm −2 on the image plane while preserving the pulse duration and tilt angle through the target. For UED experiments, tilted pulses could be used in combination with methods to produce femtosecond electron pulses [1,26,27] to break the 100 fs resolution barrier in gas-phase experiments.