**3. Gabor in-line EUV holography**

The acquisition of holographic images is a two step process consisting of recording and reconstruction phase. The holographic recording in Gabor's in-line configuration is depicted in Fig. 1a. During the recording step the interference pattern between two mutually, collinear and coherent beams is stored in the recording medium. The two interfering beams are the reference beam (black dashed lines) and the object beam (green solid lines). The recording medium is a material used to record the interference pattern that can provide a linear mapping between the incident intensity and some kind of change in the medium such as the reflection, transmission or height modulation. If the object and reference wavefronts are both expressed as a two complex fields having amplitudes and phases:

$$o(\mathbf{x}, y) = \left| o(\mathbf{x}, y) \right| e^{j\phi(\mathbf{x}, y)} \tag{1}$$

$$r(\mathbf{x}, y) = \left| r(\mathbf{x}, y) \right| e^{j\psi(\mathbf{x}, y)} \tag{2}$$

then the interference between these two complex fields occurring at the location of the recording medium can be expressed as the intensity distribution of sum of two fields:

$$\begin{aligned} I(\mathbf{x}, \boldsymbol{y}) &= \left| r(\mathbf{x}, \boldsymbol{y}) + o(\mathbf{x}, \boldsymbol{y}) \right|^2 = \left[ r(\mathbf{x}, \boldsymbol{y}) + o(\mathbf{x}, \boldsymbol{y}) \right] \left[ r(\mathbf{x}, \boldsymbol{y}) + o(\mathbf{x}, \boldsymbol{y}) \right]^\* = \\ &= \left| r(\mathbf{x}, \boldsymbol{y}) \right|^2 + \left| o(\mathbf{x}, \boldsymbol{y}) \right|^2 + r^\*(\mathbf{x}, \boldsymbol{y}) \cdot o(\mathbf{x}, \boldsymbol{y}) + o^\*(\mathbf{x}, \boldsymbol{y}) \cdot r(\mathbf{x}, \boldsymbol{y}) \end{aligned} \tag{3}$$

The first two terms are the intensities of both interfering beams, while the last terms depend also on their phases. That is why the recording medium, sensitive only to the intensity, is in fact capable of storing the intensity and the phase information simultaneously.

The linear mapping of the recording medium can be described as transmission change of the recording medium (for example photographic film) as a function of the incident intensity:

Two and Three Dimensional Extreme

spatial resolution is often given by:

is thus limited to:

Ultraviolet Holographic Imaging with a Nanometer Spatial Resolution 309

code that reconstructs the image using a Fresnel propagator. This calculation backpropagates the object beam wavefront to obtain the amplitude and phase distribution of the field in the image plane (Goodman, 1996; Schnars & Juptner, 1994, 2002). To evaluate Fresnel-Kirchhoff integral, the product of spatial frequency representation of the hologram, obtained through a two dimensional fast Fourier transformation, and a quadratic phase free space Fresnel propagator in the spatial frequency domain was computed. Using this method

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

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

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

> *a NA*

where *a* Î< > 0.3,1 depending on the method used to measure the resolution and coherence

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

*<sup>c</sup> sc*

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

*<sup>R</sup> NA*

=

*sc*

2 2

*c p*

2 2 *c p*

*c*

*aR z R l* +

+

*R z*

of the source (Heck et al., 1998), *λ* is the wavelength of the illumination.

two beams is Q*sc* then the recording numerical aperture is equal to:

<sup>⋅</sup>*<sup>l</sup>* D = (7)

D = (9)

(8)

the reconstruction of the hologram was performed numerically.

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

image that can be processed numerically in order to reconstruct the object.

resolution of ~ 10nm for e-beam exposure (Hoole et al., 1997, Yamazaki et al., 2004).

$$\mathbf{t}(\mathbf{x}, \mathbf{y}) = \mathbf{t}\_0 + \kappa \cdot I(\mathbf{x}, \mathbf{y}) \tag{4}$$

where 0*t* is the uniform transmittance of the film introduced by the constant exposure and *k* is a linear factor relating the transmission to the incident intensity *Ixy* (,) . By substituting equation Equation (3) into Equation (4) the transmission of the recording medium can be thus expressed as:

$$t(\mathbf{x}, \mathbf{y}) = t\_0 + \kappa \cdot \left| \left| r(\mathbf{x}, \mathbf{y}) \right|^2 + \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + r^\*(\mathbf{x}, \mathbf{y}) \cdot o(\mathbf{x}, \mathbf{y}) + o^\*(\mathbf{x}, \mathbf{y}) \cdot r(\mathbf{x}, \mathbf{y}) \right| \tag{5}$$

This interference pattern, stored in the recording medium during the reconstruction step, acts as a complicated diffraction grating. If the intensity of the interference pattern during the recording step is linearly translated into a recording medium transmission then the reconstruction step is similar to the one depicted in Fig. 1b. The hologram is placed in the same geometry and illuminated by the same wavefront as in the recording step, now called reconstruction wavefront. This can be expressed as:

$$u(\mathbf{x}, \mathbf{y}) = r(\mathbf{x}, \mathbf{y}) \cdot t(\mathbf{x}, \mathbf{y}) = r(\mathbf{x}, \mathbf{y}) \left| t\_0 + \kappa \cdot \left| r(\mathbf{x}, \mathbf{y}) \right|^2 \right| + \kappa \cdot r(\mathbf{x}, \mathbf{y}) \cdot \left| o(\mathbf{x}, \mathbf{y}) \right|^2 + \tag{6}$$

$$\kappa \cdot o(\mathbf{x}, \mathbf{y}) \cdot \left| r(\mathbf{x}, \mathbf{y}) \right|^2 + \kappa \cdot o^\*(\mathbf{x}, \mathbf{y}) \cdot \left| r(\mathbf{x}, \mathbf{y}) \right|^2$$

Fig. 1. Holographic recording scheme a) using Gabor's in-line configuration. The interference pattern between flat wavefront (reference beam) and beam diffracted from the object (object beam) is stored in the recording medium. Optical reconstruction b): the transmission hologram is illuminated by the reconstruction beam (the same as reference beam) and the image of the object appears where the object was initially located during the recording phase.

The first term is a background, the second is very small due to Gabor holography requirement for the object with high transmittance, thus the <sup>2</sup> *oxy* (,) 0 . The wavefront diffracted from complicated transmission diffraction grating (hologram) converges behind the hologram generating the real image (fourth term \* ~ (,) *o x y* ) while divergent wavefront generates the virtual image in front of the hologram where the object was placed during the recording step (third term ~(,) *oxy* ). The reconstructed intensity image is then \* *U x*(,) (,) (,) *<sup>y</sup>* = ⋅ *u x <sup>y</sup> u x <sup>y</sup>* .

In recent experiments a high spatial resolution photoresist, usually utilized in electron beam lithography, was used for the recording of large numerical aperture (NA) holograms (Wachulak et al., 2006, 2007, 2008b). After developing, the holographic interference pattern was translated into a relief pattern in the surface of the photoresist and digitized using an atomic force microscope (AFM). Digitized hologram was used as an input for a numerical

where 0*t* is the uniform transmittance of the film introduced by the constant exposure and *k* is a linear factor relating the transmission to the incident intensity *Ixy* (,) . By substituting equation Equation (3) into Equation (4) the transmission of the recording medium can be

> <sup>0</sup> *txy t rxy oxy r xy oxy o xy rxy* (,) (,) (,) (,) (,) (,) (,) *k* <sup>é</sup> <sup>ù</sup> = +⋅ + + ⋅ + ⋅ <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup>

This interference pattern, stored in the recording medium during the reconstruction step, acts as a complicated diffraction grating. If the intensity of the interference pattern during the recording step is linearly translated into a recording medium transmission then the reconstruction step is similar to the one depicted in Fig. 1b. The hologram is placed in the same geometry and illuminated by the same wavefront as in the recording step, now called

0

*uxy rxy txy rxy t rxy rxy oxy*

⋅ ⋅ +⋅ ⋅

*k k*

(,) (,) (,) (,) (,) (,) (,)

(,) (,) (,) (,)

= ⋅ = +⋅ +⋅ ⋅ +

*oxy rxy o xy rxy*

Fig. 1. Holographic recording scheme a) using Gabor's in-line configuration. The interference pattern between flat wavefront (reference beam) and beam diffracted from the object (object beam) is stored in the recording medium. Optical reconstruction b): the transmission hologram is illuminated by the reconstruction beam (the same as reference beam) and the image of the

The first term is a background, the second is very small due to Gabor holography

diffracted from complicated transmission diffraction grating (hologram) converges behind the hologram generating the real image (fourth term \* ~ (,) *o x y* ) while divergent wavefront generates the virtual image in front of the hologram where the object was placed during the recording step (third term ~(,) *oxy* ). The reconstructed intensity image is then \* *U x*(,) (,) (,) *<sup>y</sup>* = ⋅ *u x <sup>y</sup> u x <sup>y</sup>* . In recent experiments a high spatial resolution photoresist, usually utilized in electron beam lithography, was used for the recording of large numerical aperture (NA) holograms (Wachulak et al., 2006, 2007, 2008b). After developing, the holographic interference pattern was translated into a relief pattern in the surface of the photoresist and digitized using an atomic force microscope (AFM). Digitized hologram was used as an input for a numerical

object appears where the object was initially located during the recording phase.

requirement for the object with high transmittance, thus the <sup>2</sup>

2 2 \* \*

thus expressed as:

reconstruction wavefront. This can be expressed as:

<sup>0</sup> *t x*(,) (,) *y* = +⋅ *t Ix k y* (4)

( ) 2 2

2 2 \*

*k k*

(5)

(6)

*oxy* (,) 0 . The wavefront

code that reconstructs the image using a Fresnel propagator. This calculation backpropagates the object beam wavefront to obtain the amplitude and phase distribution of the field in the image plane (Goodman, 1996; Schnars & Juptner, 1994, 2002). To evaluate Fresnel-Kirchhoff integral, the product of spatial frequency representation of the hologram, obtained through a two dimensional fast Fourier transformation, and a quadratic phase free space Fresnel propagator in the spatial frequency domain was computed. Using this method the reconstruction of the hologram was performed numerically.
