Goos-Hänchen effect observed for focused x-ray beams under resonant mode excitation

: We have coupled a nano-focused synchrotron beam into a planar x-ray waveguide structure through a thinned cladding, using the resonant beam coupling (RBC) geometry, which is well established for coupling of macroscopic x-ray beams into x-ray waveguides. By reducing the beam size and using specially designed waveguide structures with multiple guiding layers, we can observe two reﬂected beams of similar amplitudes upon resonant mode excitation. At the same time, the second reﬂected beam is shifted along the surface by several millimeters, constituting a exceptionally large Goos-Hänchen e ﬀ ect. We evidence this e ﬀ ect based on its characteristic far-ﬁeld patterns resulting from interference of the multiple reﬂected beams. The experimental results are in perfect agreement with ﬁnite-di ﬀ erence simulations.


Introduction
Planar waveguide for hard x-rays are nowadays well established [1][2][3][4][5][6]. They can be simply realized by a thin film structure consisting of a low density (guiding) layer sandwiched in between layers of high density, the so-called cladding layer. Planar x-ray waveguide can hence be fabricated by a thin film deposition techniques with sub-nm control. Planar x-ray waveguide can be used for the definition of nanometer-sized highly coherent x-ray beams, for mode filtering [7], and for x-ray holographic imaging [8]. More complex structures can be realized by extending and generalizing the sequence of thin film layers. In [9], multiple guiding layers separated by thin claddings were used to exploit coupling between several guided beams. Several coherent beams were extracted at the end of the waveguide, and evidenced by measuring the far-field interference pattern. More recently, waveguide arrays with multiple guiding layers (WGA) where tailored in the guiding layer thickness and position, to achieve a quasi-focusing in the field behind the waveguide exit [10].
Two different geometries are typically used to couple a synchrotron beam into a waveguide. Either the front coupling scheme, where the beam is coupled through the front face directly into the guiding layer [11,12], or the resonant beam coupling (RBC) scheme, where the beam enters from the side through a thinned cladding, usually the top face [13][14][15][16]. In this case, the guided modes are resonantly excited by shining a parallel beam onto the waveguide under grazing incidence using a precisely controlled incidence angle α i for each modes. In order to couple into the guiding layer, the evanescent tail of the parallel beam is used, which "reaches through" the thinned top cladding. The illuminated surface area of the planar waveguide (footprint) is typically a few millimeters long. Resonant mode excitation manifests itself in the plateau of total in form of sharp dips (cusps) at a set of α i . At these angles, photons "get trapped" under the resonance conditions in the guiding layer propagating parallel to the surface over an active coupling length [9]. Similar resonant effects -albeit with lower quality factor -can be observed in thin film samples with more general layer sequence (which do not form a waveguide as such) [17]. For infinite samples and beams, the cusp arises since photons are more likely to get absorbed, when photons are coupled into the structure, rather than being reflected at the top. If the footprint reaches the edge of the RBC, the guided beam may also exit at the side. The lateral shift of a waveguided radiation before being reflected can also be regarded as a special manifestation of the Goos-Hänchen effect, well known in the optical regime [18][19][20][21][22][23][24][25]. For visible and near-infrared radiation, it was also shown that this effect can become particularly strong for resonant grating waveguide structures, and can be accompanied by generation of multiple reflected beams [26,27].
In this work, we study the coupling of finite (sub-μm) x-ray beams into RBCs with three guiding layers in the [Ni/C] 3 /Ni structure. Using specially designed structures and resonant mode excitation, the Goos-Hänchen effect results in a enormous shift in the reflected beam of several millimeters. Since part of the incident beam is directly reflected without coupling, one can then face a peculiar situation with two (or even more) reflected beams of almost equal amplitude. In other words, optical simulation shows that multi-guide RBCs can be used as coherent beam splitters, with possible future applications in interferometry or holography. Furthermore, such devices could possibly also serve to split and delay ultra-short x-ray pulses. The goal of the present work is hence to shed light on this novel phenomenon, using both optical simulations and an experimental proof-of-concept.
For conventional use of RBCs, the standard simulation tool is a transfer matrix algorithm similar or the well known Parratt formalism [28]. The reflected and transmitted intensity, as well as the internal and external standing electromagnetic field can then be plotted as a function of the structural parameters of the RBCs (layer thickness, composition and density, interface roughness) as a function of α i . Tacitly, such simulations assume infinite beams and samples. For the present purpose, we hence have to turn to alternative techniques, namely finite-difference (FD) simulations of beam propagation, as presented in the next section devoted to optical design, simulation and fabrication of multi-guide RBCs.

Simulation, design and fabrication
The IMD software [29] was used to simulate x-ray intensity inside a RBC structure as demonstrated earlier [9,13,14]. Figure 1(a) shows the single layer schematic of a RBC, serving as a reference for the multi-guide RBCs structures. The low density guiding layer is sandwiched in between two high density cladding layers. The top cladding is sufficiently thin to enable coupling of the beam in reflection geometry. Here we use Ni (5 nm) /C (50 nm) /Ni (50 nm) on a GaAs substrate. Figure 1(b) shows the simulated x-ray reflectivity as a function of incident angle α i in the range from 0.1 • to 0.3 • , for 13.8 keV photon energy. Note that both the incoming beam size and the RBC length L are treated as infinite. The sharp dips in the x-ray reflectivity between the critical angles α C c of C and α Ni c of Ni evidence the excitation of the T E 0 , T E 1 , T E 2 and T E 3 modes inside the cavities. The corresponding calculated electric field intensity distribution as a function of α i and the depth in x direction can be conveniently illustrated in form (d-f) The corresponding plots for a multi-guide RBCs structure with three C guiding layers and four Ni cladding layers. In the schematic (d) the incoming beam with a beam size FW H M is coupled into the multi-guide RBCs structure, illuminating the surface over a size s = FW H M/sin(α i ). If FW H M and s are sufficiently small, the RBCs structure can exhibit several reflected beams, which is the central phenomenon studied in this work. Important parameters characterizing the 1 st and 2 nd reflected beams are the path length of 1 st reflected beam l 1 (red dash line), the path length of 2 nd reflected beam l 2 (black dash line), the beam offsets on the surface o (green dash lines), and the distance between two beams p (blue dash lines). Note that the simulation shown in (e,f) assume an infinite beam and structure. of two-dimensional contour plots, as represented in Fig. 1(c). The characteristic antinodes of the x-ray standing waves corresponding to electric modes (T E m , m = 0, ..., 3) of RBC structure can be easily located. Figures 1(d-f) show the equivalent plots for the multi-guide RBCs with three guiding layers [Ni (5 nm)/C (50 nm)] 3 /Ni (50 nm) on GaAs substrate. The coupling of the modes results in a splitting and lifting of degeneracy, as first discussed in [9].
The simulations shown above have been calculated for an infinite beam. Such simulations can only be expected to describe experiments well, if the beam footprint on the surface s is much larger than the beam offsets o arising from the Goos-Hänchen effect, associated with the multiple reflections and coupling into modes, as schematically illustrated in Fig. 1(d). If the illuminated spot size on the surface s and o become of comparable size, we must expect deviations. In particular, we anticipate the formation of several reflected beams which are well separated in positions. With respect to the schematically sketched geometry, the path length difference Δ between 1 st and 2 nd reflected beams is given by where d RBC denotes the total thickness of the multi-guide RBCs, and p the distance between the two parallel reflected beams. Of course, this geometric optical argument is overly simplistic, but it should certainly give the correct order of magnitude, in particular to estimate the time delay Δt = Δ/c between the two beams, which is on the order of several attoseconds given the grazing incidence angles of mode excitation (a few mrad) and a typical total film thickness of the RBCs (several 100 nm). In order to observe such phenomena, we must collimate or focus the beam down to FW H M values which are on the order of the characteristic RBCs sizes along the vertical direction (x). Let's consider typical values for the experimental design: at the angle of the T E 0 mode α i = 0.133 • , a beam size of FW H M = 600 nm would result in s 260 μm, which is smaller than the beam offset o, estimated based on d RBC and the fact that multiple-reflections and coupling in and out of modes will results in a more significant beam offset o. In order to make these speculations precise, we need to simulate the propagation, coupling and reflection of finite beams in the RBCs structure. To this end, we use FD calculations of the parabolic (paraxial) wave equation [30]. Figure 2 presents the results of FD simulations, carried out for the multi-guide RBCs sketched in Fig. 1(d), with the theoretical design parameters tabulated in Tab. 1, for (the experimental) photon energy 13.8 keV, and incoming Gaussian beam with beam size FW H M = 600 nm. The near-field distributions are shown for two cases: Figs. 2(a-d) on-mode and Figs. 2(e-h) off-mode, where on-mode refers to the excitation of the T E 0 mode at α i = 0.133 • , and off-mode refers to α i = 0.145 • . The total length of the RBCs (and the FD simulation) along z is L = 10 mm. Under on-mode conditions, three reflected beams are observed, exiting the RBCs surface at different offsets along z. An interference zone [31] is also formed between the primary incoming beam and the 1 st reflected beam as shown in Fig. 2(b). The guided modes are resonantly excited in the different channels at different positions along z. When the guided modes form in the top channel, the 2 nd or 3 rd reflected beam comes out from the surface. Figure 2(c) shows a cross-section through the reflected beam in the near-field (red line, orthogonal to the reflected beam). Figure  2(d) shows the corresponding normalized far-field pattern (intensity profiles). For comparison, the off-mode case for α i = 0.145 • is shown in Figs. 2(e) and 2(f). From these simulations, we can conclude that the experimentally observable fingerprint of the multi-reflections is the characteristic lineshape of the reflected beam in the far-field, in particular the cusps visible in Fig. 2(d). Note that these cusps should be observed for constant α i as a function of the coordinate on the detector, or in a so-called detector-scan along 2θ when using a point detector. To sum up, FD simulations show that multi-guide RBCs structures can result in multiple reflected beams, linked to mode excitation with finite-size (sub-μm) beams. The phenomenon occurs only at the proper α i values required for mode excitation. Since the near-field distribution is often not accessible experimentally, it is important to evidence this effect by far-field measurements, which is possible due to the characteristic lineshape associated with the multiexit beam behavior. Next, we turn to the fabrication of the structures designed according to the FD simulation results.
For the purpose of fabricating the multi-guide RBCs sample, we have used direct-current magnetron sputtering system [32,33] at the Institute of Precision Optical Engineering at Tongji University, China. The fabricated RBCs consists of three amorphous C layers, and four polycrystalline Ni layers, following the theoretical design parameters as shown in Tab. 1. The in total 7 layers were deposited on GaAs substrates alternately, under the base pressure 3.0 × 10 −4 Pa. The sputter gas was Ar with purity of 99.999%, and the gas pressure was kept constant at 1.50 ± .02 mTorr (0.1995 Pa). After the fabrication, the x-ray macroscopic reflectivity (XMR) was obtained with macroscopic beam size, using an in-house Cu K α source (λ = 0.154 nm), equipped with a reflectometer. The 1-D XMR curve [as shown in appendix A] was fitted by using the Genetic Binda algorithm of IMD [29] with individual layer thickness values of the Ni and C layers, as shown in Tab. 1. Afterwards, the 2-D far-field pattern of RBCs sample was carried out at the GINIX (Goettingen Instrument for Nano-Imaging with Xrays) experiment setup, installed at the P10 beamline at the PETRA III synchrotron facility in Hamburg (DESY) [34]. Using the energy of 13.8 keV, the far-field patterns were recorded by a Eiger 4 M pixel detector (Dectris).    Figure 4 presents the schematic of the experimental setup of GINIX at beamline P10, and illustrates the data reduction. The synchrotron is focused onto the RBCs structure by a Kirkpatrick-Baez (KB) mirror system. The illuminating beam size at the focal distance f is determined by the gap of the horizontal slit hg (in x direction). With entrance slit larger than the KB acceptance hg ≥ 0.4 mm, the beam size at the sample plane SP is around 295 nm in x direction. Decreasing hg from 0.4 mm to 0.05 mm, the measurements have been carried out after careful alignment (at hg = 0.05 mm), with (successively) larger spot sizes (FW H M), broadened by diffraction. The transmitted primary beam and the desired reflected beam were recorded in the far-field by the Eiger 4 M pixel detector (Dectris), as exemplified in Fig. 4(b), with pixel size 75 μm, placed at D = 5.4 m behind the focal plane. In Fig. 4(c), a close-up of the 2-D lineshape of the reflected beam is shown under off-mode condition at α i = 0.170 • , while Fig. 4(d) shows the on-mode lineshape for the T E 2 mode at α i = 0.187 • . The 2-D farfield patterns are integrated in y direction to yield the corresponding 1-D profiles, see the Figs. 4(e) and 4(f), respectively. In both 1-D and 2-D representations, the far-field pattern of the on-mode case differs from the reflected beams signal of the off-mode case. Namely, it shows characteristic cusps (minima). Note that due to mirror imperfections, also the off-mode reflected beam exhibits vertical stripes, but under on-mode additional and more pronounced minima are observed. Figure 4(g) presents the intensity distribution, as a function of α i and detector coordinate x, after integration of the intensity along y. In this plot the angular increment is Δα i = 0.001 • , while the pixel size (x direction) is 75 μm. Typically, α i was scanned in the range from -0.1 • to 0.5 • . Except for α i 0, where the incoming beam can pass above the RBCs surface, the RBCs with its GaAs substrate significantly attenuates the primary beam. Next, we compare the experimental pattern to the FD simulations.To this end, we start from the integrated intensity patterns as shown in Fig. 4(g), but now transform the detector coordinate along x to α f − α i . In this way the reflected intensity as function of α i corresponds to a horizontal stripe, with a width given by the divergence of the reflected beam. Figure 5(a) shows the experimental result, for the relevant range including in particular the region α C c ≤ α i ≤ α Ni c , where four different modes are observed. The angular resolution in α f − α i is about resolution 8 × 10 −4• . The blank gaps due to the inter-module gaps of the detector [see Fig. 4(g)] had been filled with values from neighboring pixels. In order to obtain data of similar structure, we treat the FD results in an analogous manner. First the reflected beam is simulated by near-field propagation, as shown in Fig. 5(b) for the case FW H M=600 nm and α i = 0.133 • . The geometric parameters of the multi-guide RBCs structure were taken from the XMR fitting structure in Tab. 1. For the simulation, the pixel sizes in x and z directions were 1 nm and 0.1 μm, respectively. The amplitude and phase of the reflected beam were then collected at a certain reference plane at about 2.5 mm behind the focus or center of the RBCs structure, indicated by the red line at z = 3 mm in Fig. 5(b). Figure 5(c) exhibits the amplitude distribution along x at z = 3 mm, for all α i . In this plot, the angular resolution is Δα i = 10 −4• . From this simulated amplitude data as shown in Fig. 5(c) and the corresponding phase (not shown), we then compute the far-field intensity distribution by performing a discrete Fourier transform (FFT algorithm). Note that in order to perform the FFT, we first have to transform the data along the vertical red line in Fig. 5(b) to a plane orthogonal to the reflected beam. This is accomplished by a multiplicative phase f actor = exp(ik x sin(α i )), resulting in a properly tilted coordinate x for the output field. Figure 5(d) presents the results, i.e. the simulated far-field intensity, again as a function of α i and α f − α i , so that it can be directly compared to the experimental result of Fig. 5(a). The angular resolution in α f − α i is 3.03 × 10 −4• . Direct observation illustrates that the characteristic stripe patterns observed in the simulated and experimental far-field intensities [Figs. 5(a) and 5(d)] are nearly identical. Note that the observed far-field stripe patterns can be directly attributed to the multi-reflections in the near field for the different on-mode angles α i . In other words, they indicate the presence of several reflected beams by interference. Figure 6 shows an extended comparison of the measured far-field patterns and the simulated far-field patterns, presented for different beam sizes, as controlled by the entrance slits. Namely, results are shown of Figs. 6(a, e) hg = 0.1 mm, 6(b, f) hg =0.2 mm, 6(c, g) hg = 0.3 mm, and 6(d, h) hg = 0.4 mm, respectively. For all slits and beam sizes, the experimental results and the simulations are in good agreement. Note that the beam size is an important control parameter, since the multiple reflected beam can only be observed for finite (typically sub-μm) beam sizes. From the FD simulations, the beam sizes FW H M are 600 nm for the case hg = 0.05 mm, 300 nm for the case hg = 0.1 mm, 130 nm for the case hg = 0.2 mm, 110 nm for the case hg = 0.3 mm and 90 nm for the case hg = 0.4 mm, respectively. Note that the relationship between beam size and hg is further elucidated in appendix B.

Discussion and conclusion
The comparison between the simulated and experimental reflectivity clearly shows that the multi-guide RBCs structure can result in two spatially offset reflected beams, when modes are excited (on-mode condition). This requires finite-size illumination (sub-μm) beams. In the far-field the beams interfere due to their divergence, but in the near-field the beams are well separated.
To accentuate these results, the near-field intensity distribution is shown in Fig. 7 for the on-mode condition, both for Fig. 7(a) the theoretical design parameters, and Fig. 7(d) the XMR fitting parameters as tabulated in Tab. 1. In both cases the desired beam-splitting is observed, see Figs. 7(b) and 7(e) the close-ups of the near-field profile, and Figs. 7(c) and 7(f) the corresponding far-field profiles, with the experimental data shown alongside in Fig.  7(f). The experimental 1-D profile (black) and the simulated far field curve (red), shown for the T E 2 mode at hg = 0.1 mm [corresponding to Fig. 6(a)], exhibit a similar lineshape with the identical cusps positions. Note that the deviations in the off-mode regions of the far field patterns can be attributed to the fact that the FD simulations assume an idealized Gaussian beam, while the experimental focal lineshape is quite different. Importantly, simulation and experiment give consistent results and both show that the two reflected beams are of nearly equal amplitude, as In summary, the multi-guide RBCs with three guiding layers in the [Ni/C] 3 /Ni structure, as can be fabricated by state-of-the-art thin film sputter deposition, can exhibit interesting new phenomenon when illuminated with finite-size (sub-μm) beams, which are completely obscured when simulating and measuring with standard macroscopic beams. In particular, multiple reflected beams can be observed exiting the structure at well controlled spatial offset. This effect could be exploited in interferometric applications or for off-axis holographic x-ray imaging. One could, for example, place an object onto the surface at a position illuminated only by the second beam. The associated phase shifts could then be probed by far-field interference with the first (reference) beam. While the present work is entirely focused on continuous-wave (cw) illumination (both simulation and experiment), one could also explore the effect of such RBCs on ultra-short and focused x-ray pulses, as generated by free-electron laser (FEL) or higher harmonic (HHG) radiation. In particular, such designed multi-guide resonant beam couplers could serve as time-delay beam splitters with attosecond delay, orders of magnitude smaller than current macroscopic pulse delay stages [35,36]. Multi-guide RBCs in combination with sub-μm beams may further be useful to probe surface and near-surface structure and dynamics. Finally, they offer novel opportunities for x-ray quantum optical experiments, extending the seminal work in this field performed with single guiding layers [37].
Appendix A: X-ray macroscopic reflectivity (XMR) XMR was performed for the multi-guide RBCs, using an in-house Cu K α source (λ = 0.154 nm), equipped with collimating multilayer mirrors and a fully motorized reflectometer [38]. The x-ray reflectivity was recorded with 0.1 mm beam as defined by the entrance slits. An angular range of 0 • to 8 • was scanned. Figure 8 shows the reflectivity as a function of α i in black line, after subtracting the diffuse (nonspecular) background as measured by an off-set scan (off-set angle of 0.1 • ), and after performing the illumination correction. The XMR curve was fitted (red line) in the region of 0.05 • to 5.5 • using the Genetic Binda algorithm of IMD with individual layer thickness values of the Ni/C layers as free parameters. The fitting results are tabulated in Tab. 1.  Fig. 9(c). These results deviate from those obtained by matching the lineshape of the FD simulations to the experimental data (where beam size was a free parameter), see values tabulated in Tab. 2. This can be attributed to a number of reasons: Firstly, finite source size is not included. Secondly, the slit size hg is a nominal value likely to have an offset with respect to the true gap. Finally, the sample may have been positioned slightly out of the focal plane in some scans. Nevertheless, the general trend of increasing FW H M with decreasing hg is preserved. Figure 9(d) shows a comparison of the plateau of the total reflection, between the simulations for infinite beam (IMD simulation, black) and the different finite beam sizes, as varied by hg as Fig. 10. Using similar simulations as shown in Fig. 7, Figures 10(a, c, e, g, i) near-field and 10(b, d, f, h, j) far-field profiles, calculated for increasing FW H M, from 130 nm to 1800 nm. the control parameter in FD simulations. The dips (cusps) become much sharper with decreasing hg, i.e. increasing FW H M. Compared to the IMD reflectivity, all dips of the simulated results (hg = 0.4 mm in red line, hg = 0.3 mm in light blue line, hg = 0.2 mm in green line, hg = 0.1 mm in dark blue line, hg = 0.05 mm in purple line) keep the antinodes corresponding to the on-mode case at the same positions, but the intensity differences between on-mode and off-mode decreases.