**6. Conclusion**

22 Will-be-set-by-IN-TECH

If we regard a 3D scene as *M* sources of amplitude *Ai* at (*xi*, *yi*, *zi*), the linear nature of EM

If we wish to sample *h*(*u*, *v*) over the region *umin* ≤ *u* ≤ *umax*, *vmin* ≤ *v* ≤ *vmax* to form a *P* × *P*

<sup>2</sup> + 

In Algorithm 2 we present a version of OSPR that generates *N* full-parallax 3D holograms

(*n*) *uv* , *<sup>n</sup>* <sup>=</sup> <sup>1</sup> ··· *<sup>N</sup>*, for a given set of *<sup>M</sup>* point sources *Ai*, *<sup>i</sup>* <sup>=</sup> <sup>1</sup> ··· *<sup>M</sup>*, at positions (*xi*, *yi*, *zi*) in

where �{·} represents the real part

**Algorithm 2:** The OSPR algorithm modified to calculate *N P* × *P* pixel full-parallax 3D

To test this algorithm, we consider the calculation of *N* = 8 holograms of resolution 512 × 512 and size 2 mm × 2 mm centered at the origin of our plane *P*, giving a pixel size of Δ = 4 *μ*m and hence a viewing angle of around 9 degrees under coherent red illumination *λ* = 632 nm. The 3D scene used was a set of *M* = 944 point sources that formed a wireframe cuboid of

The simulated RPFs produced were calculated by propagating Huygens wavelets from the

perpendicular to the line from the center of the cube to the pinhole), and recording the

and *K*<sup>2</sup> = (0.39, −0.39, 1.92) - are shown in Figures 12(a)-(b) together with experimental results

*uv* in turn through a pinhole aperture *K* onto a virtual screen (a plane

. Simulated views of the hologram from two positions - *K*<sup>1</sup> = (0.20, −0.39, 1.95)

dimensions 12 cm × 12 cm × 18 cm, located at a distance of 1.91 m from the plane.

*xy* |

(*n*) *uv* of size *<sup>P</sup>* <sup>×</sup> *<sup>P</sup>* pixels

where �{·} represents the imaginary part

with *r* =

*vmin* + *v*

(*u* − *xi*)

<sup>2</sup> <sup>+</sup> (*<sup>v</sup>* <sup>−</sup> *yi*)

*<sup>P</sup>* <sup>−</sup> *yi*

*vmax* − *vmin*

with *ri* as equation 49 and where <sup>Φ</sup>(*n*)

2; as before, the time-averaged percept is *Vxy* =

<sup>2</sup> + *z*<sup>2</sup>

<sup>2</sup> + *z*<sup>2</sup>

*<sup>i</sup>* (48)

*<sup>i</sup>* (49)

*<sup>i</sup>* is

propagation results in the total field hologram *h*(*u*, *v*)

*ziAi jλr*<sup>2</sup> *i* exp 2*jπ <sup>λ</sup> ri* 

*<sup>P</sup>* <sup>−</sup> *xi*

**inputs** : *M* point sources of amplitude *Ai* at position (*xi*, *yi*, *zi*), *M*, *N*

*umax* − *umin*

*M* ∑ *i*=1

*umin* + *u*

**output**: *N* binary phase holograms *h*

 *g* (*n*) *uv* 

uniformly distributed in the interval [0; 2*π*]

 *g* (*n*) *uv* 

<sup>−</sup>1 if *<sup>m</sup>*(*n*) *uv* <sup>&</sup>lt; <sup>0</sup> 1 if *<sup>m</sup>*(*n*) *uv* <sup>≥</sup> <sup>0</sup>

holograms *huv* for a given set of *M* point sources *Ai*.

*h*(*u*, *v*) =

hologram *huv*, then *ri* becomes:

*ri* =

the image plane (*x*, *y*, *z*).

**for** *n* ← 1 **to** *N*/2 **do**

Let *<sup>m</sup>*(*n*) *uv* <sup>=</sup> �

(*n*) *uv* <sup>=</sup> <sup>∑</sup>*<sup>M</sup> i*=1 *Zi Ai jλr*<sup>2</sup> *i* exp *j*Φ(*n*) *<sup>i</sup>* <sup>+</sup> <sup>2</sup>*j<sup>π</sup> <sup>λ</sup> ri* 

Let *<sup>m</sup>*(*n*+*N*/2) *uv* <sup>=</sup> �

(*i*)

intensity distribution on the screen <sup>|</sup>*F*(*i*)

from Cable (2006) in Figures 12(c)-(e).

(*n*) *uv* <sup>=</sup>

Let *h*

Let *h*

*N* holograms *h*

1 *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup> i*=1 *F*(*i*) *xy* 2

**end**

*h*

This chapter has described a number of technical innovations that have enabled the realization of a real-time, phase-only holographic projection technology.

By defining a new psychometrically determined optimization metric that is far more suited to human perception than the conventional MSE measure, a method for the generation of phase-only holograms which results in perceptually pleasing video-style images was demonstrated. This allows the realization of phase-only holographic video projection systems which, for the first time, overcome the twin barriers of the computational complexity of calculating diffraction patterns in real time and the poor quality of the resultant images.

Using these techniques, the chapter has demonstrated algorithms and methods for the generation of 2D and 3D images in the Fraunhofer and Fresnel regimes. As shown in simulation and by preliminary experiment, the RPFs produced by the calculated holograms exhibit a substantial improvement in quality and a reduction in computation time on the scale of orders of magnitude compared to the other techniques demonstrated thus far.

A number of commercially available products, notably from Light Blue Optics Inc. (2010), now employ variants of this technology. This chapter, and the information contained herein, contains a thorough description of state-of-the-art holographic projection technology and provides a complete reference to enable an interested reader to simulate, construct and characterize a 2D or 3D phase-only holographic projector.

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**14** 

*1Poland 2USA* 

**Two and Three Dimensional Extreme Ultraviolet Holographic Imaging with** 

 **a Nanometer Spatial Resolution** 

*1Military University of Technology, Institute of Optoelectronics* 

*and Computer Engineering, Colorado State University* 

*2NSF ERC for Extreme Ultraviolet Science & Technology and Department of Electrical* 

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

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

**1. Introduction** 

resolution imaging.

P. W. Wachulak1 and M. C. Marconi2

