# Split-And-Delay Unit for FEL Interferometry in the XUV Spectral Range
## 2 Split-And-Delay Unit for XUV Interferometry  
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 1a. 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 1a 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 \(2\pi\). 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; 31]. 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; 33]. However, the setup utilizing a transmission grating is difficult to align, requires additional mirrors and has a low efficiency of  
Figure 1: Comparison of a conventional double-mirror SDU (**a**) and a lamellar-grating SDU (**b**). The insets with blue background show simulated light intensity distributions in the focal (XY) and the propagation (YZ) planes for nine different phase delays ranging from 0 to \(2\pi\). (**a**) The double-mirror SDU generates two spatially separated beams which are superimposed by the focusing optics at a small angle with their wavefronts tilted in opposite directions. As a result, their relative phase varies continuously along the Y-axis. Hence, a detector with infinitely high resolution would be required to discriminate single interference states. (**b**) Each grating of the SDU diffracts the incident beam.