2.1.

Zoneplate Design

The common zoneplate equation defines the $n$’th zone boundary for a standard zoneplate:

Here, $p$ and $q$ are the imaging conjugates and $r_p$ and are $r_q$ are the distances from the object and image point to a given location in the pupil plane. The wavelength is $\lambda$. Figure 1(a) illustrates the geometry.

Fig. 1

(a) Illustration of ray path and statement of symbols used in standard zone pattern definition. (b) Illustration, showing the holographic approach to defining a zone pattern.

For the purposes of fabrication by electron-beam lithography, the authors have written code to generate zoneplate patterns in the graphic-database-system (GDS) format from basic inputs such as wavelength, imaging conjugates, and numerical aperture. The algorithm minimizes position errors in the output file. A web application for zoneplate design is available to the public in Ref. 6. Zoneplates can be specified via a web interface and the corresponding GDS-file can be downloaded. The site allows calculating resolution and bandwidth requirements for a given configuration and features educational content on the design, fabrication, and applications of zoneplate lenses.

A holographic approach is used to define the more complex zone patterns required by some complementary imaging modes. The concept is illustrated in Fig. 1(b). For a standard zoneplate, we consider an object point $P$ and an image point $Q$. In the following example, we discuss two separate off-axis points $P_1$ and $P_2$, imaged to the same image point $Q_3$. The corresponding waves in the pupil plane are defined in Eq. (2).

Waves ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$ can be combined in different ways in order to obtain an amplitude-only pattern that reflects the geometry given in Fig. 1(b). Two approaches are discussed as follows.

The first approach is to sum the waves and calculate the intensity distribution $I$ in the pupil plane.

This method is akin to the method of recording holograms in light-sensitive film. For e-beam fabrication, the resulting intensity is then binarized with a threshold intensity value to obtain a discrete zone pattern. The threshold level can be chosen to maximize a performance metric, such as diffraction efficiency into the twin first orders.

The second approach is to sum the products ${\psi }_1{\psi }_3$ and ${\psi }_2{\psi }_3$ and threshold its imaginary part $f$.

By itself, the phase of the product ${\psi }_1{\psi }_3$ in the pupil plane corresponds to a standard, off-axis zoneplate, imaging $P_1$ to $Q_3$. The latter applies to ${\psi }_2{\psi }_3$, imaging $P_2$ to $Q_3$ as well. Figures 2(a) and 2(b) show function $I$ and the matching binarized pattern. Function $f$ and its binarized version are shown in Figs. 2(c) and 2(d). The plots show examples of patterns typically obtained from geometries as presented in Fig. 1(b). The equivalence of the two binarized patterns can be explained by the matching center positions of the clear zones. The resulting zoneplates will, however, be different in grating efficiencies in the diffracted orders, and may produce slightly different image qualities. Pattern optimization for nonstandard zoneplates is an ongoing topic of investigation. An important aspect to consider is the printability of different patterns in a given lithographic process.

Fig. 2

(a) Continuous function $I$ from Eq. (3). (b) Binarized pattern from (a). (c) Continuous function $f$ from Eq. (4). (d) Binarized pattern from (c).

In order to create GDS-files for nonstandard zoneplates with the above approach, the continuous function $I$ or, respectively $f$, is calculated in the pupil plane at sub-nm resolution. This is done in up to 2500 individual, tiled segments of the pupil to keep the data volume low. A contouring algorithm is applied to interpolate an outline curve for each clear zone. The outline is then approximated by a polygon, with vertices matching a tolerance of $\lambda /100$ for the optical path length. The polygons from all segments are combined in a single GDS. For zoneplates that contain features smaller than 50 nm, a bias can be applied to the size of the polygons to correct for the size of the e-beam that is used for patterning.

2.2.

Zoneplate Nanofabrication

Zoneplates for the SHARP microscope are structured in silicon-nitride membrane windows on silicon chips. Figure 3(a) shows an image of SHARP’s zoneplate holder with three chips installed. Each chip is kinematically mounted, using three ruby balls that are glued into lithographically etched features on the chips and register to V-shaped grooves in the zoneplate holder. There are more than 200 zoneplates installed in the SHARP microscope. Figure 3(b) shows a magnified view of one of the zoneplate chips. Each chip holds approximately 70 zoneplates, grouped on three, 4-mm-long, silicon-nitride membranes, designed to accommodate a large number of zoneplates without compromising the robustness and flatness of the membrane.

Fig. 3

(a) Photographic image of SHARP’s zoneplate holder with three zoneplate chips installed. (b) Magnified image, showing one of the zoneplate chips. (c) Scanning electron microscopic image of zoneplates and holes on a chip.

The individual chips are cut from a $500\text{-}\mu \mathrm{m}$ silicon waver with 100 nm of silicon-nitride deposited to both sides. Optical lithography is used to mask the locations of the three membranes and the holes for the ruby balls. After removing the upper silicon-nitride layer in a dry-etch process, the underlying silicon is etched all the way down to the lower silicon nitride layer to form the 3 membranes. The zone patterns are electroplated in gold using electron-beam lithography. The gold patterns are approximately 40-nm tall. To improve efficiency and to reduce the potential effects of carbon contamination, open elliptical holes are etched through the membrane for the transmitted, illuminating beam. E-beam lithography and dry etch are used to punch holes in the membrane next to each zoneplate. Figure 3(c) shows a scanning electron microscopic image of the central part of one of the chips. The image shows 68 individual zoneplates and corresponding holes for the illumination, grouped on three membranes. All zoneplates for additional imaging modes addressed in this article are designed for a 6 deg central ray angle and have a mask-side NA of 0.0825, matching the mask-side NA of the ASML 3300 scanner (0.33 wafer-side NA).

4.1.

Directional Differential Interference Contrast

DIC is a microscopic technique originally developed by Nomarski.¹² It is sensitive both to phase and amplitude variations on small length scales in the object plane. Two images of the object are projected onto the image plane with an offset below the resolution limit. Introducing a $\pi$ phase to one image creates a destructive interference for regions of the image that are spatially uniform in amplitude and phase. Superimposed object points with an initial phase difference show constructive interference, with a maximum value occurring when the relative phase difference is $\pi$. Variation in the amplitudes from superimposed points also affect the intensity in the image, thus contributing to contrast. DIC in the x-ray spectral range has been accomplished using twin zone plates.¹³ Soft x-ray microscopy in DIC with a single optical element was demonstrated by Chang et al.¹⁴ The DIC-zoneplates discussed in the following are conceptually similar to the design used in Ref 14. Figures 7(a) and 7(b) illustrate zone patterns for DIC in the $x$- and $y$-directions.

Fig. 7

(a) Cartoon of an $x$-directional DIC-zoneplate lens. (b) Cartoon of a $y$-directional DIC-zoneplate lens. (c) Scanning electron microscope (SEM) image of an $x$-directional DIC-zoneplate lens, showing the phase shift in the pattern along the horizontal center of the aperture at the upper edge where the zones are smallest.

Figures 8(b) and 8(e) show SHARP image data of 175-nm contacts imaged using $x$- and $y$-directional DIC-zoneplates. The size of the contacts is close to the resolution limit of the $0.334\times \mathrm{NA}$ zoneplates. Thus, the image of a contact closely resembles the point spread function. Figures 8(a) and 8(d) show the phase of the simulated complex electric field in the image plane, multiplied by the amplitude, for a point-like object. A conventional image of 175-nm contacts and the phase of the corresponding simulated electric field are shown in Figs. 9(a) and 9(b) as a reference. The point spread functions of the DIC-zoneplates have two maxima, and in the center, the intensity is zero. The two maxima differ in phase by $\pi$, as can be seen in Fig. 8(a). The point spread function resembles the discrete derivative operator $D_x=[\begin{array}{ccc}-1& 0& 1\end{array}]$: the $x$-derivative is applied to the image. In the center of the images, the power is balanced in the two maxima. Outer parts of the object field display some imbalance in power. The amplitude of a wave emerging from an ideal point source is constant in power. The 175-nm contacts are expanded objects that project a diffraction pattern into the pupil. For plane-wave illumination, the distribution of power within the pupil depends on the location of the contact within the object field. For contacts in the centerline of the object field, perpendicular to the direction of the DIC, the power is distributed evenly between the two different sides of the pupil. With increasing distance to the centerline, the imbalance in power across the two sides of the pupil causes an increasing bright-field contribution added to the DIC image.

Fig. 8

(a) and (d) Magnified simulated false color images, showing the phase of the complex electric field in the sensor plane, multiplied by the amplitude, for a point-like object, imaged with $x$- and $y$-directional DIC-zoneplates. (b) and (e) SHARP image data showing 175-nm contacts, imaged in $x$- and $y$-directional differential interference contrast (DIC). (c) and (f) Image details of 400-nm half-pitch elbows, imaged in $x$- and $y$-directional DIC under coherent illumination.

Fig. 9

(a) Magnified simulated false color image, showing the phase of the complex electric field in the sensor plane, multiplied by the amplitude, generated by a point-like object, for a standard zoneplate. (b) SHARP image data showing 175-nm contacts, imaged using a standard zoneplate. (c) Image details of 400-nm half-pitch elbows, imaged with a standard zoneplate under coherent illumination.

Figures 8(c) and 8(f) show DIC image details of 400-nm elbows. A conventional bright-field image is shown in Fig. 9(c) as a reference. All images of the elbows were recorded with a pupil fill of $\sigma =0.01$. Figure 8(c) shows $x$-directional DIC imaging. The central vertical line of the elbow pattern shows clear, $x$-derivative characteristics. The signal is high at the edges and drops to a low value inside the clear line. With increasing distance from the center axis, in addition to signal at the edges of the lines, there is bright-field contribution visible in the clear lines. The granularity is minimized close to the center axis and enhanced in both vertical and horizontal lines away from the center axis. The $y$-directional DIC image shows similar properties.

4.2.

Isotropic Differential Interference Contrast

The radial Hilbert transform constitutes another differential contrast enhancing technique. It provides an isotropic measurement of the amplitude and phase gradient in a sample.¹⁵ The technique has been implemented, adding a vortex-like phase plate $H_p$ of the form $H_p$$(r,\phi )=\exp (iP\phi )$ to the back focal plane of an imaging system.¹⁶ The phase plate retards the phase by $P$–times $2\pi$ along its circumference. Combination of phase modification and focusing power in a single optical element is demonstrated in Ref. 15 for the soft x-ray spectral range. The latter concept is implemented in SHARP, using spiral zone plates of the orders $P=1$ and $P=3$ in an off-axis configuration.

Figure 10(a) illustrates the pattern of an on-axis spiral zoneplate of the order $P=1$. The pattern of an off-axis spiral zoneplate of the order $P=1$ is illustrated in Figure 10(b). Figure 11(b) shows SHARP image details of 175-nm contacts, imaged using a spiral zoneplate. The point spread function is ring shaped. The calculated change in phase along the circumference of the point spread function can be seen in Fig. 11(a). Figure 11(c) shows an image of 400-nm elbows, recorded with the spiral zoneplate. It shows relief-like contrast and edge enhancement in the sweet spot of the lens. The granularity caused by substrate roughness is enhanced compared to the bright-field image.

Fig. 10

(a) Cartoon of an on-axis spiral zoneplate of the order $P=1$. (b) Cartoon of a spiral zoneplate of the order $P=1$ in off-axis configuration.

Fig. 11

(a) Magnified simulated false color image, showing the phase of the complex electric field in the sensor plane, multiplied by the amplitude, generated by a point-like object, for a spiral zoneplate of the order $P=1$. (b) SHARP image data showing 175-nm contacts, imaged using a spiral zoneplate. (c) SHARP image details of 400-nm half-pitch elbows, imaged with the spiral zoneplate under coherent illumination.