**3.2 Sampling considerations during the digitization process**

The information about the highest spatial frequencies in the hologram has to be properly preserved in the recording medium and subsequently retrieved to reconstruct the image of the object with the highest possible resolution. For the simplest case of point objects corresponding Gabor holograms are Fresnel Zones (FZ). This simple hologram has a maximum spatial frequency defined by the outermost zone width D*r* .

The resolution of the recording medium has to be better than the highest spatial frequency component in the hologram *rec* D £D*r* , in order to faithfully reconstruct the object. Similarly pixel size in the AFM scans has to be equal or smaller than the highest spatial frequency component in the hologram D £D *AFM r* . Given the relationship between the *NA* and D*r* (Attwood, 1999), the minimum number of samples required for a given spatial resolution is:

$$N\_{samples\,\,line\,line} = \frac{s}{\Delta r} \tag{14}$$

where *s* is the size of digitized hologram. The numerical aperture defined by the outermost zone width can be expressed as:

$$NA = \frac{\lambda}{2\Delta r} \tag{15}$$

Substituting Equation (7) yields to a resolution:

$$
\Delta = 2a\Delta r\tag{16}
$$

Moreover, using Equation (7) and digitization NA, expressed in terms of trigonometric functions sin arctan 2 *<sup>d</sup> p <sup>s</sup> NA z* <sup>é</sup> æ öù <sup>ê</sup> <sup>ç</sup> <sup>÷</sup> ÷ú <sup>=</sup> <sup>ç</sup> <sup>ê</sup> <sup>ç</sup> <sup>÷</sup> ÷ú <sup>ê</sup> ççè ø÷÷ú <sup>ë</sup> <sup>û</sup> , hologram scan size can be expressed as:

> 2 tan arcsin *<sup>p</sup> a s z* <sup>é</sup> æ ö *<sup>l</sup>* <sup>ù</sup> <sup>=</sup> <sup>ê</sup> <sup>ç</sup> <sup>÷</sup> ÷ú <sup>ç</sup> <sup>ê</sup> è ø <sup>ç</sup> <sup>D</sup> ÷÷ú <sup>ë</sup> <sup>û</sup> (17)

if 0 1 *<sup>a</sup><sup>l</sup>* £ £ <sup>D</sup> . Finally the number of sample points obtained with the AFM in single scan line has to be equal to:

Two and Three Dimensional Extreme

using compressed nitrogen.

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

protrusion, marked by an arrow, in the upper part of the tip.

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 313

The recording was done in a 120 nm thick layer of PMMA (MicroChem 950,000 molecular weight) spin-coated on top of a silicon wafer. Exposures of approximately 100 seconds were necessary with the experimental set up utilized in this work. After the exposure, the photoresist was developed using a standard procedure: the sample was immersed in a solution of MIBK - methyl isobutyl ketone (4-Methyl-2-Pentanone) with IPA (isopropyl alcohol) 1:3 for 30 seconds and rinsed with IPA for 30 seconds. Finally the sample was dried

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

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.

$$N\_{\text{samples}/line} = \frac{2a \cdot s}{\Delta} = \frac{4az\_p}{\Delta} \tan\left[\arcsin\left(\frac{a\lambda}{\Delta}\right)\right] \tag{18}$$

In case of small angle approximation it will be equal to:

$$N\_{\text{samples}/line} = \frac{4a^2 z\_p \lambda}{\Delta^2} \tag{19}$$

The total number of sampling points for two dimensional interference pattern can be expressed as:

$$N\_{\text{total}} = \left(N\_{\text{sample}/line}\right)^2 = \frac{16a^4 z\_p^2 \lambda^2}{\Delta^4} \tag{20}$$

Equation (20) shows that the number of points in the digitalization, necessary to attain a given image spatial resolution <sup>D</sup> , scales as <sup>4</sup> *Ntotal* <sup>~</sup> - <sup>D</sup> . This imposes a practical limitation in Gabor's scheme if the distance *<sup>p</sup> z* is not kept small. Due to the fact that the number of points per scan line in the AFM is often limited to 1024, 2048 etc., the NA is practically limited and the best option to increase the NA is to decrease the distance *<sup>p</sup> z* .

### **4. Holographic 2-D imaging using EUV lasers**

Two holograms of an atomic force microscope tip were obtained using a compact table top EUV laser in a Gabor's in-line configuration, as described in detail in (Wachulak et al., 2006). This configuration is very easy to set up and robust, moreover, it requires neither optics nor critical beam alignment.

### **4.1 Experimental details**

In this experiment a table top discharge pumped capillary Ne-like Ar laser, radiating at 46.9 nm wavelength, was used for hologram recording. The laser was configured to produce 0.1 mJ pulses at repetition rate of 1 Hz. With a ratio λ/Δλ ≈104, the capillary discharge laser has a longitudinal coherence length of *lc* ≈ 470 μm. Using Gabor's geometry, shown in Fig. 3, two holograms with two different NA were obtained by changing the distance between the object and the recording medium zp. Two selected distances were zp ≈ 4 mm (NA = 0.038) and zp ≈ 120 μm (NA = 0.172). The temporal coherence limits the image spatial resolution to Δ ≈ 64 nm for zp = 4 mm and Δ ≈ 30 nm for zp = 120 μm. However, spatial coherence imposes a more severe limitation. In both cases this is the limiting factor to the maximum attainable resolution, Δ ≈ 340 nm for the smaller NA at zp ≈ 4 mm and Δ ≈ 30 nm in the second case.

Fig. 3. Experimental set up used to record the EUV holograms in Gabor's configuration.

tan arcsin *<sup>p</sup>*

42 2

*<sup>l</sup>* = = D (20)

16 *<sup>p</sup>*

*a z*

÷ú <sup>ç</sup>

(18)

(19)

<sup>é</sup> æ ö *<sup>l</sup>* <sup>ù</sup> <sup>⋅</sup> = = <sup>ê</sup> <sup>ç</sup> <sup>÷</sup>

DD D <sup>ê</sup> çè ø÷÷ú <sup>ë</sup> <sup>û</sup>

2

*a z*

*<sup>l</sup>* <sup>=</sup> <sup>D</sup>

/ 2 4 *<sup>p</sup>*

> 2 / 4

The total number of sampling points for two dimensional interference pattern can be

Equation (20) shows that the number of points in the digitalization, necessary to attain a given image spatial resolution <sup>D</sup> , scales as <sup>4</sup> *Ntotal* <sup>~</sup> - <sup>D</sup> . This imposes a practical limitation in Gabor's scheme if the distance *<sup>p</sup> z* is not kept small. Due to the fact that the number of points per scan line in the AFM is often limited to 1024, 2048 etc., the NA is practically

Two holograms of an atomic force microscope tip were obtained using a compact table top EUV laser in a Gabor's in-line configuration, as described in detail in (Wachulak et al., 2006). This configuration is very easy to set up and robust, moreover, it requires neither optics nor

In this experiment a table top discharge pumped capillary Ne-like Ar laser, radiating at 46.9 nm wavelength, was used for hologram recording. The laser was configured to produce 0.1 mJ pulses at repetition rate of 1 Hz. With a ratio λ/Δλ ≈104, the capillary discharge laser has a longitudinal coherence length of *lc* ≈ 470 μm. Using Gabor's geometry, shown in Fig. 3, two holograms with two different NA were obtained by changing the distance between the object and the recording medium zp. Two selected distances were zp ≈ 4 mm (NA = 0.038) and zp ≈ 120 μm (NA = 0.172). The temporal coherence limits the image spatial resolution to Δ ≈ 64 nm for zp = 4 mm and Δ ≈ 30 nm for zp = 120 μm. However, spatial coherence imposes a more severe limitation. In both cases this is the limiting factor to the maximum attainable resolution,

2 4

*samples line*

( )

*total sample line*

limited and the best option to increase the NA is to decrease the distance *<sup>p</sup> z* .

Δ ≈ 340 nm for the smaller NA at zp ≈ 4 mm and Δ ≈ 30 nm in the second case.

Fig. 3. Experimental set up used to record the EUV holograms in Gabor's configuration.

*N*

*N N*

*a s az <sup>a</sup> <sup>N</sup>*

/

*samples line*

In case of small angle approximation it will be equal to:

**4. Holographic 2-D imaging using EUV lasers** 

expressed as:

critical beam alignment.

**4.1 Experimental details** 

The recording was done in a 120 nm thick layer of PMMA (MicroChem 950,000 molecular weight) spin-coated on top of a silicon wafer. Exposures of approximately 100 seconds were necessary with the experimental set up utilized in this work. After the exposure, the photoresist was developed using a standard procedure: the sample was immersed in a solution of MIBK - methyl isobutyl ketone (4-Methyl-2-Pentanone) with IPA (isopropyl alcohol) 1:3 for 30 seconds and rinsed with IPA for 30 seconds. Finally the sample was dried using compressed nitrogen.
