## Abstract

It is proposed in this paper that an x-ray Gabor zone plate can be realized by properly arranging annulus-sector-shaped nanometer structure apertures along each zone. This provides a new coding methodology which can be used to fabricate a binary zone plate with single order foci only. Numerical simulation results show good agreement with the physical design.

© 2011 OSA

## 1. Introduction

Zone plates (ZPs) are generally utilized in an x-ray microscopy system (XMS) as an x-ray lens for focusing soft and hard x-rays, and image-formation [1–4]. The spatial resolution of a zone-plate-based microscope mainly depends on the outmost zone width of the objective zone plate. In x-ray region, it has been reported that the spatial resolution of a zone-plate-based XMS has achieved 12 nm [5], which is the highest two-dimension resolution microscopic imaging based on an XMS.

However, the conventional ZPs have three drawbacks: low efficiency, strong chromatic dispersion, and multi order foci. The efficiencies of conventional ZPs are typically around 10% in the first order [4], and vary depending on the material, thickness, etc., due to the phase shifting caused by the ZP material. To make matters worse, chromatic dispersion and multi order foci produce strong background to deteriorate the imaging quality. For many years, the ZPs research focus on improving efficiency and restraining the strong background.

An order-sorting aperture (OSA) is a frequently-used method to remove the background in an XMS. Phase zone plate (PZP) is an alternative to solve the problems of the efficiency and background. In 1888, L. Rayleigh introduced the concept of a PZP to improve the poor efficiency of the ZPs; the first order efficiency of a pure PZP is raised to 40.5% and the 0th order transmitted rays decrease greatly. In recent years, PZPs have grown a lot. A multilevel zone plate which introduced a four-level phase-shifting profile in every ring can increase its first order efficiency to 81.5% in theory, and suppress the background effectively in hard x-ray region (7 keV) [6]. It’s also reported that a multilayer Laue lens, which is in fact a one-dimensional ZP with multilayer coating can focus to 16 nm in one-dimension at 20 keV [7]. In addition, PZPs can also accomplish varieties of applications in x-ray beam shaping [8]. Blazed ZPs can also improve the efficiency and restrain the background, which can produce first order almost with 100% efficiency in theory, but its wedge-shape zones are difficult to fabricate in x-ray region.

Another way to restrain the background of high orders is the Gabor zone plate with a sinusoidal transmittance along the radial direction [9, 10]. It has only one pair of foci and behaves actually as a conventional refractive lens. But no such kind of device has been applied in x-ray region for fabrication difficulties. Beynon *et al.* [11] had proposed a concept of so called binary Gabor zone plate (BGZP) where a spatial symmetrical two-dimensional binary pattern was introduced by using a nonlinear function to fabricate the Gabor zone plate. Soon after this, Choy and Cheng [12] suggested using of a linear function instead of the nonlinear one to get higher diffraction efficiencies.

In this paper we propose a different binary methodology to realize a GZP. It is called the annulus-sector-shaped-element binary Gabor zone plate (ASZP), which can provide a cosinusoidal transmittance to realize a single order x-ray focus. Compared with the multilevel and blazed ZPs, our ASZP can’t produce so high first order efficiency. But the ASZP can completely get rid of all the high orders in soft x-ray region without an OSA. In the following, we will introduce in detail how to design such an ASZP, and simulate the imaging properties of the ASZP.

## 2. Design of the ASZP

Considering a plane wave of amplitude ${A}_{0}$ and wavelength $\lambda $ incident on a diffractive zone plate of radius$R$, the far field diffraction pattern can be obtained with Fresnel equation:

As we know, the GZP demands a cosine variation of *t* [11], which can be written as:

The *r* can be written as

Using the Taylor expansion when$n\gg 1$,

It’s seen the Gabor condition cannot be fulfilled by the Eq. (8) and (7) when $n\ge 1$, but this doesn’t affect our following simulation result. We will use Eq. (8) instead of a GZP transmittance function shown in Eq. (3) when $n\ge 1$.

Now we present in detail the design of the ASZP to satisfy the Eq. (8), but be different from that of Beynon *et al.* [11]. Figure 1(a)
shows an ASZP which is a 100-zone or 50-zone-pair system. For each zone pair, 100 transparent annulus-sector-shaped elements with the same field angle are closely arrayed in azimuthal direction on the Au foil, the details of which are shown in Fig. 1(b). The annulus-sector-shaped elements in different zone pairs have different radial lengths, which equals half of the zone-pair widths:$D/2$, where $D={r}_{n+1}-{r}_{n}$. To fulfill the cosine-approximated transmittance function shown in Eq. (8), such annulus-sector-shaped elements of the *n*th zone pair must obey the sinusoidal distribution:

*N*is the total number of the elements in the

*n*th zone pair, and

*r*is the radial coordinate of the center of the element (RCCE). ${f}_{n}(r)$ denotes the elements number at the RCCE

*r*in the

*n*th zone pair, which can be integrated in the interval $\left[{r}_{n}+D/4,{r}_{n}+3D/4\right]$:where the integral interval $\left[{r}_{n}+D/4,{r}_{n}+3D/4\right]$ ensures that the elements can’t exceed the nth zone pair (${r}_{n}<r<{r}_{n+1}$) for the radial lengths of the elements equal $D/2$.

For the coordinates of the elements are discrete distributed in our design, the number of the elements at the radial coordinate $r={r}_{i}$ can be expressed as a discrete formula in Eq. (9):

Considering a typical case: divide each zone pair uniformly into *N* sectors, then the field angles of all the elements are$2\pi /N$, which are illustrated in Fig. 1(c).

The transmittance function at the radial coordinate $r={r}_{0}$can be calculated by$t\left({r}_{0}\right)=l\left({r}_{0}\right)/2\pi {r}_{0}$, where $l\left({r}_{0}\right)$ denotes total length of the cumulative transmitting arcs along the azimuthal direction at the radial coordinate $r={r}_{0}$. From the Fig. 1(b), it’s seen the circle at the radial coordinate $r={r}_{0}$ can only overlap some elements with the RCCEs located in the interval $\left[{r}_{0}-D/4,{r}_{0}+D/4\right]$, so $l\left({r}_{0}\right)$ can be expressed as:

When $\Delta {r}_{i}\to 0$, the sum formula in Eq. (12) turns into an integral formula. Combined with Eq. (10), the transmittance at the radial coordinate $r={r}_{0}$ can be written as:

By integrating Eq. (13), we obtain the transmittance function of the zone plate:

## 3. Simulation of the imaging properties

To validate the design, the far field diffraction pattern of the ASZP is simulated using Eq. (2). In the x-ray region, the ASZP material of the opaque zone is partially transparent, so the black-white model as described above should be corrected, and then the screen function of the ASZP can be expressed as:

*d*is the thickness of the zone, $k=2\pi /\lambda $, and $\beta $ and $\delta $ are the x-ray optical constants of the ASZP zone material gold [13]. The focal length of the ASZP is given by$f={r}_{1}^{2}/\lambda $. The parameters of the ASZP and in the simulations are listed in Table 1 . Figure 2(a) shows the numerical results of the intensity distribution along the optic axis. The results match our anticipation: no high order foci appear. Note a conventional ZPs would give foci at$f/3$,$f/5$,$f/7$, etc..

Figure 2(b) shows the diffraction pattern along the radial direction on the focal plane, and it is symmetrical, which means that its focusing properties are not affected by the asymmetric structure of the ASZP. Its focusing efficiency is calculated to be 10.1%, which is higher than that (6.25%) of an ideal Gabor zone plate. The efficiency is comparable to the first order efficiency of the conventional ZPs. From Fig. 2(b), the width between the peak value and the first minimum is 0.419$\text{\mu m}$. According to the Rayleigh criterion, the angular resolution limit is $2.36\times {10}^{-6}$rad, which is quite close to that of an idealized ZP with the same outmost zone width, given by 1.22$\Delta {r}_{out}/f$ = $2.41\times {10}^{-6}$rad.

It is also necessary to validate if the asymmetry of ASZP zone-pair annulus-sector-shaped elements distort the off-axis images or not. For the off-axis case, if the source is off the axis to some extent, the Fresnel approximation cannot be applied [14], thus the diffraction patterns associated with the off-axis distances are simulated by using the Fresnel-Kirchhoff equation for the complex amplitude$U\left(P\right)$,

*r*is the distance from the point source

*p*to a point on the ASZP aperture surface and $r$ is its unit vector;

*s*is the distance from the observation point

*q*to a point on the aperture surface and $s$ is its unit vector. $n$ is the unit vector normal to the aperture surface, respectively. The corresponding coordinate system is shown in Fig. 3(a) .

Consider a typical case: the image-to-source magnification is 10, the object distance is 1.1*f*, and the image distance is 11*f*. Figure 3(b) gives the simulated results for a point source at different positions off the axis: 0 mm, 13 mm, and 17 mm. The x-coordinates corresponding to the peak intensities of all the off-axis images are translated to x=0 for easy comparisons. It’s seen that the images remain symmetrical as the off-axis distance increases. The results indicate that the asymmetry of the annulus-sector-shaped elements don’t restrict the ASZP’s use on the off-axis imaging. It’s also seen in Fig. 3(b) that as the off-axis distance increases, the curve becomes emanative, the peak intensity decreases, and the limbs increase. This means that the imaging quality of the ASZP has a decline as the off-axis distance increases, which has also been observed in the imaging of conventional ZPs [14].

## 4. Conclusion

We have proposed a new cosine-approximated Gabor zone plate called ASZP. The advantages of the ASZP are: it does not produce any high-order foci; it has a higher focusing efficiency than an ideal Gabor zone plate. In addition, the imaging properties of the ASZP are characterized in the x-ray region in detail by simulations. The results accord with our anticipation. The ASZP may become a potential alternative, to replace a conventional Fresnel zone plate, in the x-ray, extreme ultraviolet, and microwave applications.

## Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants No. 10727504, and No. 10876039, and by the Chinese Academy of Sciences under Grant No. KJCX2-YW-N36. XW's email is wang1@ustc.edu.cn.

## References and links

**1. **G. Andersen, “Large optical photon sieve,” Opt. Lett. **30**(22), 2976–2978 (2005). [CrossRef] [PubMed]

**2. **Y. Wang, W. Yun, and C. Jacobsen, “Achromatic Fresnel optics for wideband extreme-ultraviolet and X-ray imaging,” Nature **424**(6944), 50–53 (2003). [CrossRef] [PubMed]

**3. **L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, “Sharper images by focusing soft X-rays with photon sieves,” Nature **414**(6860), 184–188 (2001). [CrossRef] [PubMed]

**4. **A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics **4**(12), 840–848 (2010). [CrossRef]

**5. **W. Chao, J. Kim, S. Rekawa, P. Fischer, and E. Anderson, “Hydrogen silsesquioxane double patterning process for 12 nm resolution X-ray zone plates,” J. Vac. Sci. Technol. B **27**(6), 2606 (2009). [CrossRef]

**6. **E. Di Fabrizio, F. Romanato, M. Gentili, S. Cabrini, B. Kaulich, J. Susini, and R. Barrett, “High-efficiency multilevel zone plates for keV X-rays,” Nature **401**(6756), 895–898 (1999). [CrossRef]

**7. **H. Kang, H. Yan, R. Winarski, M. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. Macrander, and G. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. **92**(22), 221114 (2008). [CrossRef]

**8. **E. Di Fabrizio, D. Cojoc, S. Cabrini, B. Kaulich, J. Susini, P. Facci, and T. Wilhein, “Diffractive optical elements for differential interference contrast x-ray microscopy,” Opt. Express **11**(19), 2278–2288 (2003). [CrossRef] [PubMed]

**9. **G. S. Waldman, “Variations on the Fresnel Zone Plate,” J. Opt. Soc. Am. **56**(2), 215–217 (1966). [CrossRef]

**10. **M. H. Horman and H. H. M. Chau, “Zone plate theory based on holography,” Appl. Opt. **6**(2), 317–322 (1967). [CrossRef] [PubMed]

**11. **T. D. Beynon, I. Kirk, and T. R. Mathews, “Gabor zone plate with binary transmittance values,” Opt. Lett. **17**(7), 544–546 (1992). [CrossRef] [PubMed]

**12. **C. M. Choy and L. M. Cheng, “High-efficiency cosine-approximated binary Gabor zone plate,” Appl. Opt. **33**(5), 794–799 (1994). [CrossRef] [PubMed]

**13. **http://henke.lbl.gov/optical_constants/filter2.html

**14. **X. Wang and J. Wang, “Analysis of high-resolution x-ray imaging of an inertial-confinement-fusion target by using a Fresnel zone plate,” Acta Phys. Sin. **60**(2), 025212 (2011).