**5.2 Gaussian filtering and correlation method for resolution and feature size assessment**

To obtain a global assessment of the image spatial resolution, a correlation analysis on the reconstructed holograms was performed (Wachulak et al., 2008c). This method is based on the correlation between the reconstructed holographic image and a 2-D set of templates with calibrated resolution and diameter of the nanotubes, all generated from a master binary template. The set of master binary templates is derived from the original image, depicted in Fig. 10a, by skeletonizing the image, (Yatagai et al., 1982). The result of skeletonization is shown in Fig. 10b. It represents the shape of the nanotube, but has thickness of only 1 pixel. Then the skeleton is convolved with a set of circular templates of different diameters representing different diameters of the CNTs. One of those templates after the convolution is shown in Fig. 10c. From these master binary templates, a sub-set of templates with variable and calibrated resolutions was obtained by applying a Gaussian filter of variable

Two and Three Dimensional Extreme

reconstructed images.

**6.1 Experimental details** 

object used.

**6. Holographic 3-D imaging using EUV lasers** 

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 319

can be recorded, establishing a limit for the spatial resolution of ~12 nm that practically does not play a role in this experiment. However, the resolution is affected by photoresist resolution (estimated to be ~20 nm for photon exposure (Solak et al., 1999)) and by resolution of the AFM that is defined by radius of curvature of the tip equal to 10 nm. The latter is not a fundamental limitation, as it can be overcome using a high resolution AFM tips. Taking all these factors in the convolution, the best possible resolution in our experiment is estimated to be ~39 nm, slightly better than the observed resolution. An additional possible factor degrading the resolution is roughness of Si membrane. Variations in the membrane's thickness can introduce a random background noise that degrades the image resolution. Detailed modeling indicates that a surface roughness of ~20 nm, which is similar to a measured roughness of the wafer that contains the membrane, would degrade the resolution to 45-46 nm adding a noise background similar to the noise measured in the

The possibility of volume three dimensional imaging by numerical sectioning obtained from a single high numerical aperture hologram (Wachulak et al., 2007) will be discussed in this section. Three dimensional images were obtained from Gabor holograms recorded in the photoresist after exposure using a table-top EUV laser. Digitized holograms were numerically reconstructed over the range of image planes by numerically sweeping the

The experimental set up is schematically illustrated in Fig. 11. The same source was used in this experiment as in the previous ones. The test object used in holographic volume imaging experiment consisted of a tilted metallic surface covered with opaque spherical markers.

Fig. 11. Diagram of the 3-D EUV holography experimental set up showing details of the test

This test object was fabricated by covering a semicircular hole 1.5 mm in diameter made in an 80 μm thick Mylar sheet with a 100 nm thick aluminum foil. The hole was partially covered

reconstruction distance, resulting in numerical optical sectioning of image depths.

width. Fig. 10d shows one example of these templates, where the edges of the nanotube in Fig. 10c were blurred by the filtering process. Then the 2-D correlation between all the templates with variable resolution and variable nanotube diameter has been computed resulting in a 2-D correlation map. The global maximum in this map points out to the template that is the most similar to the reconstructed hologram. Both, resolution and nanotube diameter in the reconstructed image are the same as in the template that maximizes the correlation coefficient in the correlation map. Fig. 10e depicts the 2-D correlation map that indicates the CNT diameter equal to 70.6±5 nm and the spatial resolution 47.5±5 nm, where error is assigned as a one step in calculation of the surface shown in Fig. 10e.

Fig. 10. Procedure to generate the templates with calibrated resolution. Filtered image a) with the low frequency background fluctuations removed. Skeletonizing b) of the image. Binary template c) obtained by convolution of the skeleton template (b) with a circular template. Template with degraded resolution d) obtained by convolution of the binary template (c) with a 2-D Gaussian filter. Correlation coefficients e) of the reconstructed image with different templates plotted as a function of nanotube diameter and template resolution. The nanotube diameter and resolution of the image are determined to be 70.6±5 nm and 47.5±5 nm by localizing a global maximum in the correlation plot.

The obtained resolution exceeds by ~19-21 nm the best possible obtainable resolution of 27 nm set by 2 *NA l* (Jacobsen et al., 1990). Several factors degrade the resolution. It was verified that the spatial and temporal coherence of the laser source is not a limitation in this experiment, because *zp<<lc, Rc* and NA~1. The longitudinal coherence length of the EUV laser is approximately *lc* ~ 470 m, <sup>4</sup> *l l* 10- D » , and the spatial coherence radius is approximately 0.25 mm at the location of the recording, 0.75 m from the source (Liu et al., 2001). With these parameters, the spatial resolution set by the laser coherence properties is approximately 29 nm. Another limiting factor could arise from the digitalization because the AFM maps the hologram into a 1024×1024 matrix over a surface of approximately 9.9×9.9 µm2, giving a pixel size of 9.7 nm. This pixel size sets the maximum spatial frequency that can be recorded, establishing a limit for the spatial resolution of ~12 nm that practically does not play a role in this experiment. However, the resolution is affected by photoresist resolution (estimated to be ~20 nm for photon exposure (Solak et al., 1999)) and by resolution of the AFM that is defined by radius of curvature of the tip equal to 10 nm. The latter is not a fundamental limitation, as it can be overcome using a high resolution AFM tips. Taking all these factors in the convolution, the best possible resolution in our experiment is estimated to be ~39 nm, slightly better than the observed resolution. An additional possible factor degrading the resolution is roughness of Si membrane. Variations in the membrane's thickness can introduce a random background noise that degrades the image resolution. Detailed modeling indicates that a surface roughness of ~20 nm, which is similar to a measured roughness of the wafer that contains the membrane, would degrade the resolution to 45-46 nm adding a noise background similar to the noise measured in the reconstructed images.
