**1. Introduction**

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The word "hologram" (from the greek *"holos"*: whole, complete and *"graphos"*: writing, drawing) means "total recording". Holography is a well known technique originally proposed in 1948 by Gabor, who also coined the name, as a new microscopy alternative. He realized that the interference of two mutually coherent waves, one called the reference wave and the second one - the object wave, allows for recording of information consisting of both amplitude and phase of diffracted or scattered beam from an object (Gabor, 1948). This coding of the amplitude and phase of the object beam into an interference pattern allowed him to demonstrate that from this complicated holographic pattern, ultimately the image of the original object can be obtained. Several years after the appearance of Gabor's paper, Baez (Baez, 1952) suggested extension of this idea to the X-ray region, but it remained as an interesting proposal till the early 1960s, when holography started to be widely applied. It was after the paper by Leith and Upatnieks, who proposed the off-axis holography - scheme which overcomes many of Gabor configuration drawbacks (Leith & Upatnieks, 1962). Since that time holography was widely used in numerous applications, some of them requiring increased spatial resolution. On this path, reducing the illumination wavelength is a direct way to improve spatial resolution both in nanopatterning (Solak et al., 1999; Wachulak et al., 2008a) and holographic imaging, described herein. This is the reason why short wavelength sources such as synchrotrons, extreme ultraviolet (EUV) and soft X-ray (SXR) lasers, high harmonics generation sources (HHG), etc., became an interesting alternative for high resolution imaging.

This chapter is devoted to 2-D and 3-D holographic imaging using a capillary discharge EUV laser. The chapter is organized as follows. In section 2 recent developments in high resolution holographic imaging will be briefly presented including different imaging techniques and short wavelength sources. In section 3 some general information about Gabor in-line EUV holography will be presented with detailed analysis of the resolution limitations due to coherence of the EUV source and digitization process. Starting from section 4 through 6 recent developments in holographic 2-D and 3-D imaging will be

Two and Three Dimensional Extreme

done in the field, only some aspects of it.

**3. Gabor in-line EUV holography** 

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 307

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

The acquisition of holographic images is a two step process consisting of recording and reconstruction phase. The holographic recording in Gabor's in-line configuration is depicted in Fig. 1a. During the recording step the interference pattern between two mutually, collinear and coherent beams is stored in the recording medium. The two interfering beams are the reference beam (black dashed lines) and the object beam (green solid lines). The recording medium is a material used to record the interference pattern that can provide a linear mapping between the incident intensity and some kind of change in the medium such as the reflection, transmission or height modulation. If the object and reference wavefronts

then the interference between these two complex fields occurring at the location of the recording medium can be expressed as the intensity distribution of sum of two fields:

> 2 2 \* \* (,) (,) (,) (,) (,) (,) (,)

= + + ⋅+ ⋅

fact capable of storing the intensity and the phase information simultaneously.

*Ixy rxy oxy rxy oxy rxy oxy*

(,) (,) (,) (,) (,) (,)

=+ =+ + =

*rxy oxy r xy oxy o xy rxy*

The first two terms are the intensities of both interfering beams, while the last terms depend also on their phases. That is why the recording medium, sensitive only to the intensity, is in

The linear mapping of the recording medium can be described as transmission change of the recording medium (for example photographic film) as a function of the incident intensity:

(,) (,) (,) *<sup>j</sup> <sup>x</sup> <sup>y</sup> oxy oxy e <sup>f</sup>* = (1)

(,) (,) (,) *<sup>j</sup> <sup>x</sup> <sup>y</sup> rxy rxy e <sup>y</sup>* = (2)

(3)

[ ][ ] <sup>2</sup> \*

are both expressed as a two complex fields having amplitudes and phases:

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 concludes the chapter.
