**2. Developments in high resolution EUV and soft X-ray holographic imaging**

The potential for holography in the EUV region was recognized very early. However realization of this goal become very difficult, thus only in the early 1970s the first images of very simple objects were obtained for the first time (Giles, 1969; Aoki & Kikuta, 1974). The main obstacle to record and reconstruct good quality holograms in this region of e-m spectrum was lack of sufficiently bright and coherent sources. It was not until 1987 that the high resolution x–ray imaging was realized by use of 2.5–3.2 nm SXR radiation from the National Synchrotron Light Source (NSLS), where spatial resolution of 40 nm was demonstrated. Fourier transform holography at the NSLS achieved spatial resolution of 60 nm (McNulty et al., 1992). Gabor holography with an early X-ray laser pumped by two beams of the fusion-class NOVA laser at Lawrence Livermore National Laboratory demonstrated a spatial resolution of 5 μm (Trebes et al., 1987).

Other experiments utilized synchrotron light to image biological samples, nano structures, and magnetic domains (Jacobsen et al., 1990; Lindas et al., 1996; McNulty et al., 1992). Lensless diffractive imaging, based on iterative phase retrieval, following the proposal by Sayre, (Sayre et al., 1998) have demonstrated SXR imaging with 50 nm spatial resolution utilizing λ = 1.5 nm source (Elsebitt et al., 2004). The first experimental demonstration of lens-less diffractive imaging using coherent soft X-rays generated by a tabletop SXR source allowed for image acquisition with spatial resolution of 214 nm (Sandberg et al., 2007) later improved to 72 nm (Sandberg et al., 2008). High resolution scanning X-ray diffraction microscopy proved to be a useful imaging technique employing coherent, short wavelength radiation, reaching spatial resolution better than 70 nm (Thibault et al., 2008).

The practical demonstration of EUV and SXR holography proved to be difficult in particular because the lack of sufficiently bright and coherent sources at short wavelengths, and to the fact that coherent EUV and SXR laser sources have historically been restricted to large user facilities. The first demonstration of a coherent table-top holographic imaging achieved 7 µm spatial resolution with a spatially-coherent HHG source (Bartels et al., 2002); this resolution has been extended to a resolution of 0.8 µm in later experiments (Morlens et al., 2006). Time resolved holographic imaging was also implemented with HHG sources to study the ultrafast dynamics of surface deformation with a longitudinal resolution of less than 100 nm and a lateral resolution of less than 80 µm (Tobey et al., 2007). Holography was also used to demonstrate 100 nm-resolution holographic aerial image monitoring based on lens-less Fourier transform holography at EUV wavelengths, using synchrotron-based illumination (Hun Lee et al., 2001). Femtosecond EUV radiation provided by the free-electron laser

presented (Wachulak et al., 2010). In section 4 a 2-D holographic imaging using a compact EUV laser, the experimental details, results and resolution estimation using a wavelet decomposition and correlation method will be discussed. The resolution of the 2-D holographic imaging was further improved by increasing the recording/reconstruction numerical aperture, leading to spatial resolution comparable to the illumination wavelength, approximately 46nm (section 5). A novel method of resolution and feature size assessment, based on a Gaussian filtering and correlation was applied and the results were compared with well established, knife-edge resolution test. Finally in section 6 a 3-D holographic recording and reconstruction, that allowed for successful 3-D information retrieval from a single high numerical aperture EUV hologram, will be presented. Section 7

**2. Developments in high resolution EUV and soft X-ray holographic imaging**  The potential for holography in the EUV region was recognized very early. However realization of this goal become very difficult, thus only in the early 1970s the first images of very simple objects were obtained for the first time (Giles, 1969; Aoki & Kikuta, 1974). The main obstacle to record and reconstruct good quality holograms in this region of e-m spectrum was lack of sufficiently bright and coherent sources. It was not until 1987 that the high resolution x–ray imaging was realized by use of 2.5–3.2 nm SXR radiation from the National Synchrotron Light Source (NSLS), where spatial resolution of 40 nm was demonstrated. Fourier transform holography at the NSLS achieved spatial resolution of 60 nm (McNulty et al., 1992). Gabor holography with an early X-ray laser pumped by two beams of the fusion-class NOVA laser at Lawrence Livermore National Laboratory

Other experiments utilized synchrotron light to image biological samples, nano structures, and magnetic domains (Jacobsen et al., 1990; Lindas et al., 1996; McNulty et al., 1992). Lensless diffractive imaging, based on iterative phase retrieval, following the proposal by Sayre, (Sayre et al., 1998) have demonstrated SXR imaging with 50 nm spatial resolution utilizing λ = 1.5 nm source (Elsebitt et al., 2004). The first experimental demonstration of lens-less diffractive imaging using coherent soft X-rays generated by a tabletop SXR source allowed for image acquisition with spatial resolution of 214 nm (Sandberg et al., 2007) later improved to 72 nm (Sandberg et al., 2008). High resolution scanning X-ray diffraction microscopy proved to be a useful imaging technique employing coherent, short wavelength

The practical demonstration of EUV and SXR holography proved to be difficult in particular because the lack of sufficiently bright and coherent sources at short wavelengths, and to the fact that coherent EUV and SXR laser sources have historically been restricted to large user facilities. The first demonstration of a coherent table-top holographic imaging achieved 7 µm spatial resolution with a spatially-coherent HHG source (Bartels et al., 2002); this resolution has been extended to a resolution of 0.8 µm in later experiments (Morlens et al., 2006). Time resolved holographic imaging was also implemented with HHG sources to study the ultrafast dynamics of surface deformation with a longitudinal resolution of less than 100 nm and a lateral resolution of less than 80 µm (Tobey et al., 2007). Holography was also used to demonstrate 100 nm-resolution holographic aerial image monitoring based on lens-less Fourier transform holography at EUV wavelengths, using synchrotron-based illumination (Hun Lee et al., 2001). Femtosecond EUV radiation provided by the free-electron laser

radiation, reaching spatial resolution better than 70 nm (Thibault et al., 2008).

demonstrated a spatial resolution of 5 μm (Trebes et al., 1987).

concludes the chapter.

FLASH was used for digital in-line holographic microscopy to image particles, diatoms and critical point dried fibroblast cells with 620 nm spatial resolution at 8 nm wavelength (Rosenhahn et al., 2009). Digital in-line SXR holography (DIXH) was used to image immobilized polystyrene and iron oxide particles with spatial resolution of 850 nm at wavelength range of 3.7-5.6 nm to take advantage of selective contrast in this wavelength range (Rosenhahn et al., 2008). Holographic measurement scheme to monitor the X-rayinduced explosion of microscopic objects was performed by a femtosecond time-delay X-ray holography, inspired by Newton's "dusty mirror" experiment, allowed to see the changes in EUV induced explosion of 140 nm diameter polystyrene beads (Chapman et al., 2007). By combining HHG holography with iterative phase retrieval algorithm, usually employed in diffractive lens-less imaging, reconstructed hologram spatial resolution was improved to ~53 nm (Sandberg et al., 2009). Holograms can be also obtained in very short exposure times. Using uniformly redundant arrays (URA) instead of a single or multiple reference pinholes in Fourier type holography the throughput of the imaging system might be sufficiently large to acquire a hologram with a single 15 fs EUV pulse and reconstruct with spatial resolution approaching 50 nm (Marchesini et al., 2008). Naturally, the body of knowledge related to this topic is so immense, that we are not able to mention all the work done in the field, only some aspects of it.
