We propose a quasi-random-dot-array binary Gabor zone plate (QBGZP) with focusing properties of single order foci only. These features are verified with simulations and experiments in the visible light region. Moreover, we find that the performance of QBGZP, which is composed of hexagon patterns, is determined by the ratio of hexagon circumcircle diameter to the outermost zone width. The QBGZP offers a potential alternative for focusing and imaging in the soft x-ray and extreme ultraviolet region.
©2013 Optical Society of America
In the soft x-ray and extreme ultraviolet region, the Fresnel zone plate (FZP) is widely used for focusing and imaging [1–4]. The FZP transmittance function in radial direction takes only binary values of 0 and 1 between adjacent concentric rings. Consequently, higher-order foci and additional background noise are unavoidable by using the plate [5–7]. The Gabor zone plate (GZP) can be used to solve this problem by suppressing all higher-order foci and has a sinusoidal radial transmittance function [8,9]. Unfortunately, such a perfect GZP is hard to fabricate because of the difficulties of obtaining the required transmittance function with a sinusoidal distribution [10–12].
In the past two decades, significant progresses have been made to design and fabricate more efficient and higher resolution FZPs (especially FZPs for x-rays) [13–18]. Today, FZP with resolution of sub-10nm is available in the soft x-ray region, while in hard x-ray region zone-doubled FZP with a resolution of 15nm and focusing efficiency of 7.5% at 6.2 keV has been fabricated . In higher photon energy regions, S. Tamura et al have achieved a multilevel-type FZP having a typical efficiency of 51% at 70 keV . However, little attention has been paid to GZP. To realize the transmittance function of GZP, T. D. Beynon et al proposed a design of binary Gabor zone plate (BGZP). After that, C. M. Choy and L. M. Cheng described a new model of a high-efficiency cosine-approximated BinGZP to approximate BGZP, this device has a focusing efficiency at least 23% higher than that of BGZP . A drawback of both kinds of BGZP is that they consist of a large number of strips suffering sharp corners with unequal sizes. When both plates applicable to soft x-ray are fabricated using electron beam lithography which is mostly used to fabricate FZPs in x-rays, these sharp corners are hard to achieve precisely, because electron beam scattering effect tends to smooth the sharp corners and therefore decreases the precision of plate fabrication. To deal with this problem, L. Wei et al have designed an Annulus-sector-element coded Gabor zone plate (ASZP) with characteristic right angles annulus sectors to replace the strips of BGZP .
In this letter, we propose a quasi-random-dot-array binary Gabor zone plate (QBGZP) to realize the GZP. QBGZP is composed of hexagon patterns of the same size, some of them are transparent while others are opaque for incident light, the distribution of transparent hexagon is random along azimuthal direction but statistically regular in the radial direction. Since the characteristic angles of hexagon are larger than right-angle, the smoothing of corners by electron beam scattering should have a weaker influence on plates fabrication. And for the same size of all the hexagon patterns, conveniences would be brought about in some ways.
2. Design of the QBGZP
The radial transmittance function of a negative-cosine GZP is:9,21]. Similarly, we construct a QBGZP with its radial transmittance function approaching that of GZP as follows. Firstly, consider the zone plate ring on the circle of radius consists of seamless connection primitives that are very small. Then we choose randomly of them transparent while others opaque for incident light, according to the area encoding principle, the transmittance at radius is . A QBGZP with the same transmittance function to that of GZP is acquired when satisfies the condition that equals in Eq. (1).
However, the primitives of a real device have finite sizes and certain shapes, which will result in a difference between the transmittance function of QBGZP and GZP. Figure 1(a) shows a QBGZP ring composed of hexagon primitives. In cases of a large difference, the excellent focusing properties of QBGZP would vanish (see later simulations). Therefore, the sizes of primitives should be kept within a certain range to make the transmittance function close to that in Eq. (1). Figure 1(b) shows such a 50 zones QBGZP composed of hexagon primitives, whose properties are very similar to an ideal GZP. The primitive pattern could be other shapes such as square, rectangle and circle. However, if the pattern fills the plane seamlessly, the corresponding QBGZP would reach a maximum transmission of 50% and achieve a relatively high efficiency. Consider the smoothing of corners by electron beam scattering in fabrication process, a pattern with larger characteristic angles (i.e. the hexagon) is a better choice.
To validate the plate design, distribution of the QBGZP’s diffraction field is numerically calculated. Consider a plane wave of unit amplitude and wavelength normally incident on a QBGZP, the complex amplitude of diffraction field is:Fig. 2 . As expected, Fig. 2(a) reveals that the intensity distribution of QBGZP is free from higher-order foci on the optical axis, and the focal distance is 77.46 ± 0.12cm when the theoretical value is 77.41cm given by , the relative error is within 0.22%. Compared with the same parameters GZP diffraction pattern, there are no differences when properly normalized. Figure 2(b) shows that the transverse intensity distribution of QBGZP on the focal plane matches that of GZP very well, and their resolutions (Rayleigh criterion) both are 60 ± 0.5μm which is quite close to (is the width of the outermost zone, is the total number of plate zones). Although the QBGZP is asymmetric because it is composed of quasi-randomly distributed primitives, it does not distort the intensity distributions on the optical axis and the focal plane as shown in Fig. 2(a) and 2(b). And for a QBGZP, the transverse intensities around the optical axis at f/3, f/5, f/7, etc. are suppressed by a factor of 104.
As mentioned above, the size of primitives is a key factor to the QBGZP’s focusing properties. When it is too large, the transmittance function of QBGZP will largely deviate from that in Eq. (1) and distort the focusing properties. On the contrary, small primitive sizes would make the QBGZP transmittance function approach that in Eq. (1), but it is hard to achieve technically, which is the main obstacle for fabricating plates applicable in the short wavelength region.
In order to determine an appropriate size for the hexagon primitives, we commit simulations to find how the focusing properties of QBGZPs are affected by the sizes of the primitives. To compare QBGZPs of different sizes, we fix the radius of their first ring and define , where is the diameter of the hexagon circumcircle. Figure 3 shows the simulation results, where one can find that the focusing properties of QBGZPs for different sizes vary with a similar trend versus . From Fig. 3(a), we obtain the maximum focusing efficiency (the light flux within focal spot area over that incident on the zone plate) around 5.80% which is a little less than that of GZP (6.25%). Consequently, its application in the visible light region is obviously limited, however, in the range of short wavelength where ordinary glass lens do not work, QBGZPs play an important role as focusing or imaging devices. In Fig. 3, it is also shown that, with increasing , that focusing efficiencies and resolving powers both have a decreasing trend. More specifically, we find two phases: (1) , which denotes that the diameter of hexagon circumcircle is within half the outermost zone width, both focusing efficiencies and resolving powers decrease slowly with increasing. Note that reducing does not improve QBGZPs’ focusing properties much in this region, and therefore is not worth pursuing for very high performance when is below 0.5; (2) , the focusing efficiencies and resolving powers decrease quickly, therefore we should avoid in the design of QBGZP.
Comparing with a photon sieve under the condition of the same smallest size of plate, the QBGZP demonstrates better on efficiency because the efficiency of a photon sieve is about 3% when equals 1.5 and a Connes-type transmission window is used to modify the pinhole density per zone, where is the diameter of a pinhole and is the width of corresponding zones. Since the radial transmittance function of QBGZP mimics the sinusoidal function, it has no higher-order foci inherently. However, QBGZP is put at a disadvantage on resolving powers which overcomes the spatial resolution limit for a photon sieve.
A QBGZP applicable to visible light is fabricated in the following steps. First, creating a pattern of the QBGZP described in the caption of Fig. 4 . Then the pattern is converted into LEDIT format. Finally, we produce a QBGZP using electron beam lithography. The QBGZP is composed of a large number of hexagon primitives (gold) which are distributed quasi-randomly on the quartz substrate. Gold primitives are opaque for the incident light while the quartz substrate is transparent.
A demonstration experiment shown in Fig. 4 has been carried out to measure the intensity distribution along the optical axis. The incident light is an expanded 633nm partially polarized laser beam, and a charge coupled device (Princeton Instruments, PIXIS: 1024) behind the zone plate measures the intensity on the optical axis. Experiment results are shown in Fig. 5 . From the results we find that higher-order foci which inevitably exist in conventional FZP are successfully suppressed as the theory predicts. Experiments with the incident light of linear and elliptical polarizations are also carried out. The results indicate that the QBGZP suppresses higher-order diffraction independently of polarizations.
The QBGZP for soft x-ray is different from that for visible light. In addition to much smaller sizes, another problem needs to be considered in the former plate is that x-ray can partially penetrate metal material, which generates a constant phase modulation of the penetrated radiation. Therefore, the QBGZP in the x-ray range could be considered as a superposition of two complementary QBGZPs, and intensity of the double-QBGZP’s diffraction field is affected by the constant phase. However, its far field diffraction pattern is similar to that of the original QBGZP (no x-ray penetrates), and it is still free from higher-order diffraction on the optical axis.
We proposed a QBGZP to realize a GZP in the paper. It is composed of the same size primitives without sharp corners, which weaken the influence of electron beam scattering in fabrication of x-ray GZPs. In the visible light region, our simulations and experiments show that the QBGZPs have single order foci only suppressing all the higher-order foci, and the focusing properties are close to that of GZP when the circumcircle diameter of primitives is within half the outermost zone width. Analysis shows that the constant phase modulation at x-rays would not produce higher-order foci, and therefore the QBGZP become a potential alternative for the soft x-ray and extreme ultraviolet region.
The authors thank Prof. Dino Jaroszynski and Dr. Bo Zhang for help in polishing the manuscript. This research was supported by the National Natural Science Foundation of China (Grant No. 11174259, 10905051), and the Development Foundation of China Academy of Engineering Physics (Grant No.2011B0102022, 2011B0102021 and 2009A0102003).
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