To produce a double pulse sequence from a single pulse, one requires two key components: a beam splitter to divide the initial pulse in two and a translation stage to introduce a phase shift to one of them. Common tools used to split beams in the visible wavelength range are semi-transparent mirrors (amplitude beam splitters). These devices divide the incident wavefront in two replicas—one transmitted and one reflected—each carrying half the energy (for a 50:50 split) and preserving the shape of the original beam. One of the replicas is delayed in time and then both of them are recombined in another (or the same) beam splitter. The advantage of amplitude beamsplitters is that they inherently allow for collinear beam mixing. Collinear propagation puts the pulse replicas in a single interference state as their phase difference is solely defined by the introduced delay. Phase-resolved signal detection thus becomes easy, provided that the delay stage is capable of sub-wavelength movement control. However, amplitude beam splitters are not available below 140 nm [29
] and for shorter wavelengths, one has to rely on reflective optics.
A common design of a reflective split-and-delay unit for XUV wavelengths is a double split mirror shown in Figure 1
a. Its reflective surface is divided into two halves, one of which can be displaced along the normal and thus delay part of the incident wavefront. The two partial beams are later overlapped by tilting the parts of the split mirror or by focusing optics as shown in the figure. Since the two half-beams are spatially separated, they are superimposed at a small skew angle with their wavefronts tilted in opposite directions. This means that the relative phase between the beams varies continuously along the intersection plane. The simulated light distribution created by the split mirror in the focus is shown in Figure 1
a as a function of phase delay between the two partial beams. As the delay changes, the interference pattern continuously “scrolls” inside the intensity envelope defined by the transverse focus size. One can see from the fringe spacing that the phase difference between the beams changes across the focal spot on the scale of
. In experiments demanding high photon fluxes, the XUV beams are focused into spots that do not exceed several microns in size. Therefore, to select a region with a fixed phase relation between the two light fields required for interferometry, one would need a detector with a submicron spatial resolution. This is a very challenging task. Instead, the signal is generally integrated over the whole focal volume and the phase information is lost due to phase averaging. This is a major obstacle for phase-resolved measurements with split mirror SDUs.
Limitations of the double split mirror can be overcome using diffractive optics. The idea to use a diffraction grating as a beam splitter is not new [30
]. For example, a Michelson interferometer based on a transmission grating was successfully employed to characterize femtosecond UV pulses and was proposed for applications in XUV wavelength range [32
]. However, the setup utilizing a transmission grating is difficult to align, requires additional mirrors and has a low efficiency of the order of several percent [32
]. An alternative approach using reflective optics was suggested by Strong and Vanasse in their lamellar-grating interferometer developed for Fourier spectrometry in the far-infrared [17
]. The authors overcame the problem of the two-element split mirror—generation of two spatially separated beams—by using two interleaved multi-mirror arrays (lamellar gratings) shown in Figure 1
b. Each grating represents a sequence of narrow rectangular facets separated by gaps wide enough to house the facets of another grating. When interleaved, the gratings form a sequence of alternating lamellae with neighboring elements belonging to different gratings. Optical performance of the lamellar-grating beam splitter was thoroughly investigated [17
]. In short, each grating diffracts the incident beam in a number of diffraction orders. The partial beams from both gratings propagate collinearly in every order which ensures that their interference depends only on the longitudinal offset between the gratings. This property makes the lamellar-grating SDU a reflective analog of a Michelson interferometer. As an example, for a
phase delay, diffraction from each lamellar grating results in constructive interference in odd orders and destructive interference in even orders. Upon focusing, the angular distribution of the diffracted light translates into a sequence of spots in the focal plane separated by:
is the grating period,
is the focal length of the focusing optics and
is the mean wavelength of the incident light. The intensity of the zeroth order recorded as a function of grating displacement yields the interferogram of the light source. Provided that
is larger than the focus size and can be resolved by the detection system, a single order can be selected for the signal accumulation. Compared to a transmission grating setup, the lamellar-grating SDU has fewer optical elements and offers higher efficiency.
Experimental Design of the SDU
The SDU designed for XUV experiments utilizes two lamellar gratings of different but complementary design. The first grating is a 60 × 35 × 1 mm3 Si wafer with the central 10 × 20 mm2 area processed as a slotted grid with a circular diamond saw. The structure of 250 µm period comprises 150 µm wide slits separated by 100 µm wide reflective facets. The second grating comprises 100 µm wide, rectangular ridges projecting (protruding) from the substrate for 1.25 mm. The ridges are spaced 150 µm apart and fit into the slits of the first grating. When interleaved, the gratings form an alternating pattern of 100 µm facets separated by 25 µm gaps giving a fill factor of 0.8 for the assembled device.
The interleaved gratings are installed on a specially designed mount equipped with motors necessary for grating alignment and translation as shown in Figure 2
. The slotted grating is rigidly fixed to the mount while the ridged grating has three degrees of freedom driven by piezo actuators. One of them is used to translate the grating and thus delay the reflected beam. The two others control the grating rotation in two planes and keep both gratings parallel. The delay piezo stage has a travel range of 250 µm in the closed loop control mode, which translates in the delay ranging from −50 to +574 fs at the incidence angle of 22° with respect to the surface. We note in passing that the resulting peak fluence of unfocused FEL beams on the optics surface in this geometry is well below the melting and ablation thresholds of bulk Si.
The described SDU is designed for time-resolved ionization experiments of low-density gas targets by XUV photons with detection of ionization products, i.e., electrons and ions. It follows from Equation (1) that diffraction orders from an XUV beam generated by a grating with a 250 µm period in the focal plane will be separated only by some tens of microns. For example,
]. Therefore the detection system must have high spatial resolution in order to distinguish signals from individual orders. The ionization volume can be readily imaged with a resolution of few microns by an elongated velocity map imaging (VMI) spectrometer operated in the spatial imaging mode, which proved to be sufficient for 38 nm wavelength [22
]. Higher resolution required for shorter wavelengths can be achieved by extending the focal length of the mirror and using a spectrometer purposely designed for magnified spatial imaging of charged particles, i.e., an ion microscope [36