**4.2 Results of 2-D holographic imaging with EUV laser**

The exposure was adjusted to be in region of linear response of the photoresist and consequently a relief pattern height printed in its surface was equivalent to the interference intensity pattern of the hologram. The developed photoresist surface was mapped with the AFM to generate digitized holograms. Fig. 4a shows the hologram with low numerical aperture (NA = 0.038) recording, digitized area ~300x300 m2, and pixel size corresponding to 270 nm, while Fig. 4c depicts digitized hologram with higher numerical aperture (NA = 0.172). In this case the area scanned is 42x42 m2 with pixel size equivalent to 41 nm. The holograms were reconstructed by numerically simulating illumination with a short wavelength EUV readout wave. The amplitude and phase distributions of the field in the image plane were obtained calculating the field emerging from the hologram illuminated by a plane reference wave and numerically back-propagating the fields with a Fresnel propagator (Schnars & Juptner, 2002). The reconstructed image was found by taking two dimensional inverse fast Fourier transform (2D-IFFT) of the product of spatial frequency Fresnel propagator and the 2D-FFT of the hologram. This calculation allowed to obtain the amplitude and phase distribution of the field in the image plane. Fig. 4b and Fig. 4d are, respectively, the reconstructed images of corresponding holograms shown in Fig. 4a and Fig. 4c. The inset in Fig. 4b is a magnified region showing the end of the tip. In both reconstructed images a triangular profile of the AFM tip is clearly revealed. In case of Fig. 4d it was evident after the reconstruction that the tip was broken and partially contaminated, as can be observed in protrusion, marked by an arrow, in the upper part of the tip.

Fig. 4. Hologram recorded in surface of photoresist a) and reconstruction b) with low NA= 0.038 and, consequently, hologram c) and reconstruction d) with higher NA= 0.172.

Two and Three Dimensional Extreme

0.172, pixel size 41 nm.

were summarized in Table 1.

Spatial coherence 2 2 *c p c*

*aR z R l* +

Rayleigh criterion based *a* = 0.61

reconstruction distances ranging from 118 to 132 μm.

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 315

consequently this analysis indicates that the reconstructed image has the spatial resolution equivalent to Δ=380 nm. Fig. 6b corresponds to higher numerical aperture recording (NA = 0.172). This plot clearly shows the maximum in the correlation values for X = 2 at all

Fig. 6. Correlation values for different object-hologram distances zp as a function of the wavelet scale X, a) recording with NA=0.038, pixel size 270 nm, b) recording with NA=

> Temporal coherence

> > *a*

*l*

*z z l*

æ ö <sup>ç</sup> <sup>÷</sup> -ç <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>÷</sup> çç <sup>+</sup> <sup>÷</sup> è ø

1 *<sup>p</sup> p c*

Table 1. Coherence limitation to the resolution, AFM sampling limitations and experimentally obtained resolutions estimated based on the correlation analysis.

This analysis indicates that the spatial resolution of the reconstruction is 4 pixels. In this case, the pixel size was 41 nm, setting the spatial resolution of the image to Δ=164 nm. Based on the recording NA, the spatial resolution limited by Rayleigh criterion Δ= (0.61 λ)/NA gives value of ΔR=166 nm that compares very well with the results of wavelet analysis. The results are in good agreement with theoretical calculations and coherence limitations and

2

*zp* = 4 mm 337.8 nm 64.1 nm 270 nm 329.4 nm 380 nm

*zp* = 120 m 30.4 nm 29.2 nm 41 nm 50 nm 164 nm

AFM pixel size D =D *AFM r*

AFM sampling limit D= D2*a r*

Experimental resolution

**4.3 Resolution estimation using wavelet decomposition and image correlation method**  An optimum resolution in the reconstructed image was assessed by correlation with a synthesized image – template, used as a reference image. This reference image was constructed as a binary template that has by definition 1 pixel resolution. From the reference image a set of lower resolution images (wavelet components) was generated by wavelet decomposition of the reference image, each one having a spatial resolution relative to the reference image given by Y = 2X, where Y is the relative resolution between the images in the wavelet decomposition and X is the order of the wavelet. In this case "haar" wavelet was applied. The reconstructed images were obtained by running the Fresnel propagator code with slightly different zp around 4 mm and 120 μm, respectively. Then all reconstructions were correlated with the set of decreasing resolution wavelet components, shown in Fig. 5. To perform the correlation, higher wavelet orders were resized to have the same image size. The correlation coefficients between wavelet components and the reconstructed images for slightly different zp provided a quantitative resolution of the reconstructions, relative to the synthesized reference images. This procedure also allowed for selection of an optimum reconstruction distance zp. Fig. 5 shows a binary template that constitutes the order zero (X=0) wavelet component and the first four wavelet components with decreased resolution given by a factor 2X.

Fig. 5. Binary template (X=0) and wavelet components used to perform 2-D correlation with the reconstructed images to assess a relative resolution of the holograms.

Fig. 6 shows obtained correlation values for different images, reconstructed at different object-hologram distances zp and different wavelet components, as a function of the wavelet scale X. By slightly changing the reconstruction distance the correlation coefficient changes but the shape of the curves remains unchanged.

Fig. 6a corresponds to smaller numerical aperture hologram (NA = 0.038). This plot shows that the largest value of the correlation function for all zp corresponds to wavelet scale X = 0, and does not significantly change in the interval between X = 0 to X=1, decreasing faster for wavelet orders larger than 1. We conservatively assumed that the best correlation curve corresponds to X = 0.5, indicating that the resolution for this image is 20.5 = 1.41 relative to the reference image. The synthesized binary reference image has a pixel size of 270 nm and

**4.3 Resolution estimation using wavelet decomposition and image correlation method**  An optimum resolution in the reconstructed image was assessed by correlation with a synthesized image – template, used as a reference image. This reference image was constructed as a binary template that has by definition 1 pixel resolution. From the reference image a set of lower resolution images (wavelet components) was generated by wavelet decomposition of the reference image, each one having a spatial resolution relative to the reference image given by Y = 2X, where Y is the relative resolution between the images in the wavelet decomposition and X is the order of the wavelet. In this case "haar" wavelet was applied. The reconstructed images were obtained by running the Fresnel propagator code with slightly different zp around 4 mm and 120 μm, respectively. Then all reconstructions were correlated with the set of decreasing resolution wavelet components, shown in Fig. 5. To perform the correlation, higher wavelet orders were resized to have the same image size. The correlation coefficients between wavelet components and the reconstructed images for slightly different zp provided a quantitative resolution of the reconstructions, relative to the synthesized reference images. This procedure also allowed for selection of an optimum reconstruction distance zp. Fig. 5 shows a binary template that constitutes the order zero (X=0) wavelet component and the first four wavelet components with decreased resolution

Fig. 5. Binary template (X=0) and wavelet components used to perform 2-D correlation with

Fig. 6 shows obtained correlation values for different images, reconstructed at different object-hologram distances zp and different wavelet components, as a function of the wavelet scale X. By slightly changing the reconstruction distance the correlation coefficient changes

Fig. 6a corresponds to smaller numerical aperture hologram (NA = 0.038). This plot shows that the largest value of the correlation function for all zp corresponds to wavelet scale X = 0, and does not significantly change in the interval between X = 0 to X=1, decreasing faster for wavelet orders larger than 1. We conservatively assumed that the best correlation curve corresponds to X = 0.5, indicating that the resolution for this image is 20.5 = 1.41 relative to the reference image. The synthesized binary reference image has a pixel size of 270 nm and

the reconstructed images to assess a relative resolution of the holograms.

but the shape of the curves remains unchanged.

given by a factor 2X.

consequently this analysis indicates that the reconstructed image has the spatial resolution equivalent to Δ=380 nm. Fig. 6b corresponds to higher numerical aperture recording (NA = 0.172). This plot clearly shows the maximum in the correlation values for X = 2 at all reconstruction distances ranging from 118 to 132 μm.

Fig. 6. Correlation values for different object-hologram distances zp as a function of the wavelet scale X, a) recording with NA=0.038, pixel size 270 nm, b) recording with NA= 0.172, pixel size 41 nm.

This analysis indicates that the spatial resolution of the reconstruction is 4 pixels. In this case, the pixel size was 41 nm, setting the spatial resolution of the image to Δ=164 nm. Based on the recording NA, the spatial resolution limited by Rayleigh criterion Δ= (0.61 λ)/NA gives value of ΔR=166 nm that compares very well with the results of wavelet analysis. The results are in good agreement with theoretical calculations and coherence limitations and were summarized in Table 1.


Table 1. Coherence limitation to the resolution, AFM sampling limitations and experimentally obtained resolutions estimated based on the correlation analysis.

Two and Three Dimensional Extreme

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 317

Fig. 8. a,c) Holograms and b,d) reconstructed images of 50-80 nm diameter carbon

Fig. 9. The intensity lineouts through the reconstructions indicating a 10-90% intensity modulation over approximately 4.5 pixels. Figure a,b) corresponds to the reconstructed images in Fig. 8b,d) respectively. A mean value of multiple cuts obtained at different

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

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

locations yielded the spatial resolution of 45.8±1.9 nm.

**assessment** 

nanotubes. The holograms were obtained by scanning the photoresist surface with the AFM. Knife-edge test was applied to estimate the resolution of the reconstructed image in b,d).
