In summary, we have reconstructed the near-field distribution of anN i/CX-ray waveguide array (WGA) from the measured far-field data. To this end, we have used two different supports (the tight support and the loose support). Phase re-trieval of one-dimensional structures is known to be problematic. Despite the fact that the experiment has used a focused beam and a two-dimensional detec-tion scheme, and hence falls into the (nominal) category of two-dimensional, the variation of the signal is essentially one dimensional. For this reason we had anticipated that we could require as much support information as possi-ble, and have therefore used the stronga prioriinformation of position of the seven waveguide channels (the beamlets). However, in the case of the WGA a tight support did not turn out to be necessary, and the loose support actu-ally gave smaller errors in the reconstruction. The reconstruction results are quite robust. The coarse pattern of the reconstructed field was similar in both cases. Contrarily, for the periodic WGM, the tight support turned out to be nec-essary, which is not surprising given the known difficulty associated with phase retrieval of periodic structures. Therefore, the constraints have to be tightened.
Note that we also performed one-dimensional phase retrieval based on detec-tor data summed over the columns. As expected, these reconstructions were less stable.
Using the two-dimensional phase retrieval, we could validate the concept of tai-lored near-field distributions, put forward before on the basis of analytical the-ory and numerical simulations. According to this concept, the multi-beam in-terference pattern is controlled by variation of both sevenCguiding layer thick-nesses and eightN icladding layer thicknesses in the experiment. This leads to beam intensity modulations in the free space behind the waveguide exit, which are distinctly different from those obtained for a WGM with constant sevenC guiding layer thicknesses and eightN i cladding layer thicknesses. In particu-lar, quasi-focal spot sizes in the sub-50 nm range can be generated. In future,
3.5 Discussion and Conclusion 77
such tailored near-fields exhibiting large structural diversity can be used for co-herent imaging, for example by ptychography [82–84], which has been shown to benefit from a highly structured illumination wavefield. Note that, not only for imaging applications but also as a more powerful probe reconstruction for inspection of the WGA near-field, ptychography is an obvious extension for fu-ture work. Finally, we suggest that fufu-ture generalizations of the WGA concept could include design of twin peaks for differential phase contrast, or emission of radiation directed away from the optical axis (off-axis), similar to the optics of distributed antenna in other spectral ranges.
Acknowledgments
We thank Mike Kanbach for help in waveguide processing, Markus Osterhoff and Aike Ruhlandt for help during the beamtime, Michael Sprung for continu-ous advice and perfect working conditions at the P10 beamline, and Nan Wang and Prof. Dr. Michael Seibt from the IV. Physical Institute of Göttingen Univer-sity to help the TEM measurement. We gratefully acknowledge the German Re-search Foundation (DFG) for funding through Grant No. SFB 755, and the China Scholarship Council (CSC) of P. R. China for financial support.
4 The Goos-Hänchen effect observed for focused x-ray beams under
resonant-mode-excitation
Qi Zhong, Lars Melchior, Jichang Peng, Qiushi Huang, Zhanshan Wang and Tim Salditt
Reproduced fromOptics Express, 25(15), 17431–17445 (2017), with permission (c) Optical Society of America.DOI: 10.1364/OE.25.017431
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 macro-scopic 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 effect. We evidence this effect based on its characteristic far-field patterns resulting from interference of the multiple reflected beams. The experimental results are in perfect agreement with finite-difference simula-tions.
4.1 Introduction
Planar waveguide for hard x-rays are nowadays well established [13–18]. They can be simply realized by a thin film structure consisting of a low density (guid-ing) 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 deposi-tion 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 fil-tering [79], and for x-ray holographic imaging [11]. More complex structures can be realized by extending and generalizing the sequence of thin film lay-ers. In [30], 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 lay-ers (WGA) where tailored in the guiding layer thickness and position, to achieve a quasi-focusing in the field behind the waveguide exit [80].
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 [25,27], or theresonant beam coupling(RBC) scheme, where the beam enters from the side through a thinned cladding, usually the top face [35,42–44]. In this case, the guided modes are resonantly excited by shining a parallel beam onto the waveguide un-der grazing incidence using a precisely controlled incidence angleαi for each modes. In order to couple into the guiding layer, the evanescent tail of the paral-lel beam is used, which “reaches through” the thinned top cladding. The illumi-nated surface area of the planar waveguide (footprint) is typically a few millime-ters long. Resonant mode excitation manifests itself in the plateau of total in the form of sharp dips (cusps) at a set ofαi. At these angles, photons “get trapped”
under the resonance conditions in the guiding layer propagating parallel to the surface over an active coupling length [30]. 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) [85]. For infinite sam-ples 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 [46,47,49–51,53,54,86]). 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 [87,88].
In this work, we study the coupling of finite (sub-µm) x-ray beams into RBCs with three guiding layers in the [N i/C]3/N istructure. 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 in-cident 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 de-lay ultra-short x-ray pulses. The goal of the present work is hence to shed light