# Flat Field Soft X-ray Spectrometry with Reflection Zone Plates on a Curved Substrate
## 1 Introduction  
With the advent of novel soft X-ray sources that provide noticeably short pulses, the need for their temporal resolution in ultra-fast spectroscopy has been emerging rapidly over the past 10 years. Besides synchrotrons and free-electron lasers, relatively small state-of-the-art systems on the laboratory scale are meanwhile capable of producing femtosecond (fs) pulses, and the accessibility to a wide range of photon energies from regions near, or even below, the water window within (0.28-0.53) keV to about 1 keV could be verified in experiments. In contrast to large facilities like the X-ray free-electron laser (XFEL), with an emission up to \(\sim\)20 keV at an average rate \(>\)10\({}^{12}\) photons per pulse, however, compact sources for the laboratory fall short of that number of photons--by orders of magnitude. To maintain both the spatiotemporal pattern of the pulses and a sufficient count rate on the detector of the spectrometer, the efficiency of the optical components plays a crucial role. Conical diffraction, for instance, is known as a convenient concept that fulfills those criteria in the (extreme) ultraviolet regime. Beyond \(\sim\)0.2 keV, the varied line space grating (VLSG) proved as a valuable device for soft X-ray spectrometry in the past [1]. Further development led to a \"holographic\" VLSG, the off-axis reflection zone plate (RZP) that integrates two-dimensional focusing and wavelength dispersion in a single diffractive optical element. Manifold experiments demonstrated the appropriateness of this approach and good performance, especially in the domain of time-resolved measurements on ultra-short pulses [2]. The RZP [3], as a wavelength-dispersive optical element, has been successfully used in several designs of laboratory spectrometers for soft X-rays [4]. While demonstrating its evident advantages in comparison with conventional VLS gratings, like a high resolving power at the design energy and an excellent signal-to-noise ratio, RZPs on planar substrates suffer from a narrow energy range in parallelspectra registration, limiting the applications of this type of optics. Recent developments in theory, technology, and metrology of RZPs make it possible to fabricate RZPs on spherical substrates with a small radius of curvature down to 2 m. To compensate for the slope error of the substrate, an algorithm for diffractive wavefront correction [5] was used in the calculation of the groove structure. The development of theory and technology for RZPs on figured substrates opens new possibilities for considerable improvement of the instrumental performance in soft X-ray spectroscopy, pushing forward the frontiers in pulse duration, energy resolution, and the efficiency of wavelength dispersive optical elements. Successful results of tests enabled NOB Nano Optics Berlin GmbH (NOB) to start customer-oriented production of RZPs on spherical substrates for spectroscopic applications. The concept and theoretical calculations of the RZP properties on spherical or generally figured substrates were published, for example, in the paper [5] and several conference presentations. Unfortunately, up to now, all these theoretical results were not supported by experimental measurements, especially in the soft X-ray range. The novelty of our results lies in the first experimental measurements of properties of the RZP, fabricated on a spherical substrate in a practically useful soft X-ray spectrometer. The measurements became possible due to the latest achievements in technology, made by NOB, enabling the manufacturing of RZPs on a spherical surface with a relatively small radius. This paper is the result of joint efforts between the Institut fur angewandte Photonik e.V. (IAP) and NOB in the calculation of the theoretical structure, modeling of the optical layout of a novel spectrometer, and finally, the precise production of RZP structures on a spherical mirror.