**3.1 Coherence limitation to the spatial resolution in EUV holography**

A photoresist was used to record the hologram. The interference pattern was converted into a height modulation after the developing process. To reconstruct the hologram a digitization is necessary. The digitization converts the photoresist height modulation into a gray-scale image that can be processed numerically in order to reconstruct the object.

The spatial resolution of the holographic recording is dictated by a numerical aperture *NA n* =⋅ Q sin( ) , where Q is a maximum half-angle of cone of light that can enter the imaging system and *n* is an index of refraction of a medium. Also, the spatial resolution depends on the resolution of the recording medium. Consequently, the highest spatial frequency that can be recorded in the recording medium sets a limit to the *NA*. To avoid this limitation the holograms were recorded in a high resolution photoresist, PMMA, that has the spatial resolution of ~ 10nm for e-beam exposure (Hoole et al., 1997, Yamazaki et al., 2004).

The spatial resolution of the hologram is also limited by spatial and temporal coherence of illumination source and by digitization process. The coherence limitations to the hologram *NA* manifest themselves when a path difference between radiation diffracted by the object and reference beam exceeds either the longitudinal or transverse coherence lengths. The spatial resolution is often given by:

$$
\Delta = \frac{a \cdot \lambda}{NA} \tag{7}
$$

where *a* Î< > 0.3,1 depending on the method used to measure the resolution and coherence of the source (Heck et al., 1998), *λ* is the wavelength of the illumination.

The limitation to the resolution set by the spatial coherence can be understood by the scheme depicted in Fig. 2. The reference beam and the beam diffracted from the point object will interfere only within coherence area depicted as a circle, with coherence radius *Rc* . Beyond that region one can assume that the interference will not occur. If the angle between two beams is Q*sc* then the recording numerical aperture is equal to:

$$NA\_{sc} = \frac{R\_c}{\sqrt{R\_c^2 + z\_p^2}}\tag{8}$$

where *<sup>p</sup> z* defines a distance from the object to the recording medium. The spatial resolution is thus limited to:

$$
\Delta\_{sc} = \frac{a\lambda\sqrt{R\_c^2 + z\_p^2}}{R\_c} \tag{9}
$$

Two and Three Dimensional Extreme

to the wavelength of illumination.

zone width can be expressed as:

functions sin arctan

line has to be equal to:

Substituting Equation (7) yields to a resolution:

2 *<sup>d</sup>*

*<sup>s</sup> NA*

*p*

*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>

**3.2 Sampling considerations during the digitization process** 

maximum spatial frequency defined by the outermost zone width D*r* .

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 311

It is important to notice, that according to Equation (7) improvements to the resolution can be done either by decreasing the wavelength of illumination using an output of a short wavelength EUV source, or by increasing the numerical aperture in the recording and reconstruction steps. Decreasing distance between the object and recording medium allows for storing in the photoresist surface finer interference fringes, thus improving the resolution. During reconstruction, the AFM scan has to be large enough and with sufficiently fine sampling to read these fringes, to be able to reconstruct the information about corresponding spatial frequency components in the final, reconstructed image. The experiments, described in this chapter, will show, that increasing the recording and reconstruction NA allows to reach the resolution in EUV holographic imaging comparable

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

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:

> *samples line* / *<sup>s</sup> <sup>N</sup>*

> > *NA*

where *s* is the size of digitized hologram. The numerical aperture defined by the outermost

2

 2 D= D*a r* (16) Moreover, using Equation (7) and digitization NA, expressed in terms of trigonometric

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

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

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

*s z*

*r*

, hologram scan size can be expressed as:

*a*

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

*<sup>r</sup>* <sup>=</sup> <sup>D</sup> (14)

*<sup>l</sup>* <sup>=</sup> D (15)

(17)

Fig. 2. Schematic description of spatial coherence, temporal coherence and scanning size limitations to the hologram spatial resolution.

The temporal coherence limits the recording numerical aperture as well. This can be seen in the scheme in Fig. 2 as well. The optical path difference *d* between the reference and diffracted beams has to be smaller than coherence length of the illumination source *<sup>c</sup> d* £ *l* in order to observe the interference. This limits region where the interference takes place, that can be defined by Q*tc* angle. The numerical aperture is thus restricted to:

$$NA\_{tc} = \sqrt{1 - \left(\frac{z\_p}{z\_p + l\_c}\right)^2} \tag{10}$$

and similarly the resolution:

$$\Delta\_{tc} = \frac{a\lambda}{\sqrt{1 - \left(\frac{z\_p}{z\_p + l\_c}\right)^2}}\tag{11}$$

Another limitation to the resolution is digitization process of the hologram. The resolution of reconstructed image is affected by scanning size *s* because for the object located at the center of scanning area, the finest interference fringes, carrying information about the highest spatial frequencies, are located at the edges of the scanned area. This can be related to the numerical aperture of the digitization as:

$$\text{NA}\_d = \frac{s}{\sqrt{4z\_p^2 + s^2}}\tag{12}$$

and similarly the spatial resolution:

$$
\Delta\_d = \frac{a\lambda\sqrt{4z\_p^2 + s^2}}{s} \tag{13}
$$

Fig. 2. Schematic description of spatial coherence, temporal coherence and scanning size

can be defined by Q*tc* angle. The numerical aperture is thus restricted to:

*NA*

*tc*

*tc*

The temporal coherence limits the recording numerical aperture as well. This can be seen in the scheme in Fig. 2 as well. The optical path difference *d* between the reference and diffracted beams has to be smaller than coherence length of the illumination source *<sup>c</sup> d* £ *l* in order to observe the interference. This limits region where the interference takes place, that

1 *<sup>p</sup>*

*a*

1

*d*

*d*

*<sup>s</sup> NA*

=

Another limitation to the resolution is digitization process of the hologram. The resolution of reconstructed image is affected by scanning size *s* because for the object located at the center of scanning area, the finest interference fringes, carrying information about the highest spatial frequencies, are located at the edges of the scanned area. This can be related

*p*

*z s*

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

*a zs s l* +

+

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

*p c z*

*z l* æ ö <sup>ç</sup> <sup>÷</sup> = -ç <sup>÷</sup> <sup>ç</sup> <sup>÷</sup> <sup>÷</sup> çç <sup>+</sup> ÷÷ è ø

> *p p c*

*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 2

2

(10)

(11)

(12)

D = (13)

limitations to the hologram spatial resolution.

to the numerical aperture of the digitization as:

and similarly the spatial resolution:

2 2 <sup>4</sup>

and similarly the resolution:

It is important to notice, that according to Equation (7) improvements to the resolution can be done either by decreasing the wavelength of illumination using an output of a short wavelength EUV source, or by increasing the numerical aperture in the recording and reconstruction steps. Decreasing distance between the object and recording medium allows for storing in the photoresist surface finer interference fringes, thus improving the resolution. During reconstruction, the AFM scan has to be large enough and with sufficiently fine sampling to read these fringes, to be able to reconstruct the information about corresponding spatial frequency components in the final, reconstructed image. The experiments, described in this chapter, will show, that increasing the recording and reconstruction NA allows to reach the resolution in EUV holographic imaging comparable to the wavelength of illumination.
