Optical elements based on Fresnel zones are used in a range of applications, from X-ray telescopy to microscopy and recently also in the manipulation of de Broglie matter waves. In 1992 Beynon and co-workers presented a binary Gabor type zone plate (henceforth referred to as the Beynon Gabor zone plate). Because this zone plate has no higher order foci, it is in principle a very attractive candidate for focusing of de Broglie matter waves and in some cases X-rays. So far the Beynon Gabor zone plate investigations presented in the literature have concentrated on the intensity distribution along the optical axis and in the focal plane. Here we present a detailed numerical investigation of the Beynon Gabor zone plate, including an investigation of the off-optical axis, off focal plane intensity distribution for point source illumination. We show that at integer fractions of the focal length, the beam becomes nearly toroidal (doughnut-shaped). This offers potentially interesting new possibilities for de Broglie matter wave and X-ray optics, for example in STED-like applications. We further show that the increased intensity at the focal point predicted in the literature for a particular Beynon Gabor zone plate transmission function configuration is an artifact due to the lack of sampling nodes. We support our calculations with experimental measurements in the visible light range, using a Beynon Gabor zone plate fabricated with electron beam lithography.
© 2013 OSA
Fresnel zone plate based optical elements of various forms are used extensively in X-ray optics, see for example [1, 2]. Fresnel zone based optical elements are also used in astronomy and it has been suggested that the ingenious Photon Sieve invented by Kipp and co. workers  would be particularly suited for being taken into space . Matter waves have proven a new application area for Fresnel zone based optical elements . First focusing experiments of molecular beams with zone plates were carried out using metastable helium , followed by focusing of neutral helium . In 2008 the first neutral helium microscopy images were obtained. Here a neutral beam of helium atoms was focused down to the micron scale and used to scan an object. A Fresnel zone plate was used to focus the beam . Fresnel zone plates have also been used to map the supersonic expansion of molecular beams [9, 10] and it has been suggested that the Poisson spot phenomenon (a configuration where the first Fresnel zones are blocked) can be used to test the quantum mechanical properties of large molecules . The first experimental realization of the Poisson spot for matter waves was done in 2009 using a deuterium beam .
Most photon based applications (be it light houses or X-rays) mainly use phase modulated Fresnel zone based optical elements (phase zone plates) , since they can in principle transmit the entire incident intensity into a single focus. However, they can be very difficult to fabricate for short wavelengths. Therefore Fresnel zone based optical elements modifying just the light intensity (amplitude zone plates) are often used as an alternative. An example of such an intensity modulating optical element (an amplitude zone plate) is the so-called Gabor zone plate (GZP) [13, 14]. Note! The GZP is not to be mistaken for a Gabor Phase zone plate (GPZP) . The GZP and the GPZP are both non-binary zone plates. The GZP has replaced the fully open or fully blocked of a Fresnel zone plate (FZP) with a sinusoidal varying amplitude-transmittance along the radius, where the GPZP is similar to the GZP but has a 100% transmittance and the amplitude-transmittance is replaced by a phase-shifting transmittance. For de Broglie matter wave applications only a simple intensity manipulation is possible: the focusing elements must be of the binary type (either an area is blocked or completely open) since the atoms or molecules typically do not penetrate solid materials. A binary type focusing element would also be an advantage for hard X-rays due to the very small wavelength and high energies making it difficult to fabricate a suitable phase or amplitude zone plate or a grazing incidence mirror . The simplest binary type Fresnel zone based optical element is the classical Fresnel zone plate (Fresnel-Soret zone plate). A FZP has alternating blocked and open zones. Unfortunately it also has higher order foci, which leads to a decrease in the signal to noise ratio in the first order focus. In 1992 Beynon and co-workers presented a methodology for making a binary Gabor type zone plate, the Beynon Gabor zone plate (BGZP), by introducing an azimuthal variation in the (binary) transmission function . Because the BGZP has no higher order foci, it is in principle a very attractive candidate for de Broglie matter wave optics and hard X-rays. Various BGZP configurations have been discussed in the literature [18, 19, 20]. The fabrication is challenging for most configurations, because of the complex azimuthal transmission function, but a paper presented this year in this journal, demonstrates how a Beynon Gabor zone plate can be made in a Kipp-inspired sieve configuration , which is easier to fabricate (though at the cost of intensity in the focus). Here we present a detailed numerical investigation of various Beynon Gabor zone plate configurations, including an investigation of the off-axis, off focal plane intensity distributions. We combine our numerical investigations with experimental measurements performed with a BGZP for visible light, fabricated in our own laboratory using electron beam lithography. The paper begins with a theoretical introduction to Fresnel zone based optical elements, section 2. Then follows a section on the numerical simulations, section 3 and the results of the numerical simulations, section 4. In section 5 the fabrication method for the BGZP used in the experiments presented here is described, and the experimental results presented. The paper finishes with a conclusion, section 6.
2. Theoretical background
Diffractive optical elements based on zone plates can be described as holograms : the effect of a monochromatic point source of wavelength λ [nm] shining on the hologram plate from a distance u is to form an image of the object recorded in the hologram at the distance v = (1/f ± 1/u)−1 where f is the focal length associated with the hologram.Eq. (1) yields a 0th order containing 25% of the intensity incident on the GZP and a 1st and −1st order (the focus and virtual focus) each getting one sixteenth (6.25%) of the incident intensity . Thus only 3/8 of the incident intensity is in fact transmitted. The rest of the incident radiation is absorbed or reflected.
The classical Fresnel zone plate (FZP), as mentioned earlier, is made of alternating open and blocked zones. Its construction consists in opening the zones that participate constructively to the interference at the focus point. The hologram analogy is not straight forward due to the nature of the abrupt change in transmittance, its hologram representation would be produced by a plane wave shining on an infinite number of point scatterers distributed along the optical axis at odd fractions of the focus (f/1, f/3, f/5, ...). The (binary) transmission function can be expressed as:
It can be shown that the Fourier series representation of Eq. (2) yields a 0th order containing 25% of the intensity incident on the FZP (similar to the GZP) and a 1st and −1st orders (the first focus and first virtual focus) each getting 10.13% of the incident intensity. The rest of the transmitted intensity (50% of the incident intensity) goes into higher order foci and virtual foci at odd fractions of the focal length (±f/3, ±f/5, ..). The other 50% of the incident intensity is reflected or absorbed .
In 1992 Beynon and co-workers presented a new type of binary zone plate, the Beynon Gabor zone plate (BGZP)  which has the same radial transmittance as the Gabor zone plate but is not rotationally invariant. Thus the BGZP has a transmittance that varies with the azimuthal angle ϕ, but for each radius r, the average transmittance tav(r) must be equal to that of the GZP, i.e.
Figure 1 shows the radial transmittance for the GZP and BGZP zone plates in comparison to that of a classical FZP zone plate.
Many different transmittances can be designed that fulfills the requirement in Eq. (3). In the original paper  a 40 zone BGZP was presented with the transmittance distributed symmetrically over 40 sectors. It is explicitly mentioned in the original paper that this symmetry requirement is ”not necessary for on-axis imaging”.
The original BGZP paper does not discuss the absolute intensity of the focus in details, but only shows that the on-axis intensity distribution along the optical axis around the focal point is identical to that of a GZP. Two years after the original Beynon paper, a paper  was published by another group, claiming an improvement in focusing efficiency of at least 23% in comparison with the transmission function used in the original BGZP paper. The improvement, it was claimed, decreased with the number of zones. The transmission function was constructed by dividing each zone into N sectors, and fitting a full cosine (from −π to π) curve with argument linear in r to each sector. The focusing efficiency was claimed to be ≈ 10 – 16 % compared to a claimed ≈ 8 – 12% for the original BGZP transmission function configuration.
In 2003 a paper was published  investigating how modified versions of the BGZP transmission function can change the intensity distribution in the focal plane, off-axis. The paper discusses how circularly symmetrical intensity distributions and other shapes can be generated in the focal plane.
Two very interesting recent papers in this journal also discuss transmission functions of BGZPs: The first paper proposes a configuration with a transmittance having an azimuthal variation that only approximately fulfills the requirement in Eq. (3) [19, 20]. The paper shows that this actually gives an increased focusing efficiency (independent of numbers of zones) compared to that of a standard GZP (6.25%). The intensity at the focal point is 10.1% similar to that of a classical FZP. The other paper combines the BGZP with the photon sieve idea  at the expected cost of reduced intensity at the focal point. Both of these papers mentions the very high efficiency claimed in  but without discussing it further.
3. Numerical simulations
To study the effects of the variations of the transmission function t(r, ϕ), we performed several simulations based on the Fresnel-Kirchhoff diffraction formula. The field at r⃗p, a point in the plane of detection, is explicitly given by :26].
The numerical analysis was performed using an adaptive implementation in C of the 2D Symmetrized Simpson rule in polar coordinates, with Nvidia CUDA libraries for parallel computations. The number of integrand evaluations was doubled in both r and ϕ dimensions until the intensity at each point in the plane of detection was changing by less than 0.1 %, representing the converging criteria. Each core of the 512 CUDA cores of the graphic processing unit (Nvidia GeForce GTX 580 with 1.536GB of GDDR5) was performing the diffraction integration for one point in the detection plane at a time.
4. Results of the numerical simulations
4.1. Focusing efficiency
We started out by investigating the simulations presented in , claiming a significant increase in intensity for a particular transmission function. We reproduced the original simulations and it became evident that the number of sampling points used in the numerical evaluation of the diffraction integral was insufficient, leading to aliasing. Using our direct integration method, we tried to reproduce the pixellized zone plates in . Our results indicate that the conclusions in  were based on Fast Fourier Transforms (FFT) suffering from aliasing. We were therefore unable to reproduce the claimed improvement in focusing efficiency. Their observation of significant changes in the intensity when varying the number of sampling points indicates that their result had not converged. Our simulation used a high resolution pixellized zone plate, mapping every zone j with 399 pixels along the radial direction and 1299 pixels in the azimuthal direction. A 40-zone BGZP would therefore be represented with 40×399×1299 pixels, while  used 2048×2048 for an entire 40-zone zone plate. In our case, the number of sampling points was checked to be sufficient for the results to have converged. We then found that the actual focusing efficiency for the transmission function in  is 6.25% for a sufficiently large number of zones similar to that of the classical GZP.
As a further test, we constructed a Beynon-type GZP in a similar way to the procedure used in , but fitting two sinusoidal curves for each sector to express the GZP transmittance:
The center of the sector i (i = 1, 2,...,N), ranging from −π/N to π/N is expressed as:
The transmittance of the sector i is expressed:18] makes the maximum of the cosine (linear in r) slightly shifted inward with respect to the maximum of the GZP transmittance function (linear in r2). However this difference tends to zero for the outermost zones. Our simulation shows that the two sinusoidal curves in Eqs. (5) for the transmittance also gives a focusing efficiency of 6.25%. We also tested the original transmission function used by Beynon et. al. . Originally they used 40 sectors and 40 zones. Figs. 2(a)–(c) show our calculations of the amplitude of probability in the focal plane of Beynon-type zone plates with 40 zones and 6, 20 and 30 sectors with λ = 1Å and f = 0.4773 m. As expected the central spot of the diffraction pattern (the focus), shown in Fig. 2(d), is identical for any number of sectors and the focusing efficiency is 6.25% similar to a GZP zone plate.
4.2. The Doughnut: off-axis simulations at fractions of the focal length
Because the BGZP is binary, we would expect higher diffraction orders similar to those of a FZP. Since previous papers on BGZP have concentrated on the intensity distribution along the optical axis and in the focal plane, we decided here to investigate what actually happens to the higher diffraction orders. Our simulations show that higher order diffraction orders do occur, as expected, but the higher diffraction orders do not meet on the optical axis and hence do not create higher order foci at fractions of the main focus as is the case for a FZP.
Instead a doughnut-like image (a doughnut with ”hills” and ”valleys”) is formed in transverse planes at axial distances of f/2 and f/3. The azimuthal intensity distribution in the doughnut (the number of hills and valleys) is determined by the number of sectors in the transmission function. Increasing the number of sectors reduces the width of the doughnut and increases its maximum intensity. This is illustrated in Fig. 3.
5. Experimental results
5.1. Zone plate fabrication
In order to test our theoretical calculations we fabricated a real BGZP, designed to have a focal length of 1m, when using a Helium-Neon laser with a wavelength of 632.8 nm. We used the transmittance represented by the two sinusoidal curves in Eqs. (5) described in section 4 with 60 sectors and 40 zones. The BGZP was fabricated using electron beam lithography (EBL), and the resulting structure can be seen in Fig. 4. The BGZP is equipped with a central middle-stop to block the 0th order on the optical axis, so that the diffraction pattern can be observed more clearly. Other fabrication methods such as commercially produced optical photo-masks were tested, but the resolution was too low (about 1.5 – 2 μm). EBL allows for a much higher pattern resolution. The drawback of EBL, in particular for the high resolution, ”low voltage” devices common in research laboratories, is that it is a time consuming technique, and therefore not well suited for large structures. However, using a new fast-writing technique recently developed in our laboratory , we were able to write a BGZP, 16 mm i diameter, with a pattern resolution of 200 nm, in less than 9 hours. The zone plates were made on boro-silicate microscopy cover-slips. 80 nm chrome was evaporated onto the cover-slip to make it optically opaque  using an electron beam evaporator (EBE). Then 150 nm of poly(methyl methacrylate) (PMMA) was spin coated on top of the chrome, and the substrate was soft-baked for 5 minutes at 175°C. The zone plate was patterned by EBL, and developed in 1:3 Methyl Isobutyl Ketone (MIBK):Isopropyl alcohol(IPA) for 2 minutes, and finally rinsed in IPA to stop over-developing. Using the patterned PMMA as a mask, the exposed chrome was etched for approximately 20 seconds using chrome-etchant heated to 40°C to give an etch rate of 40Å/s. Finally the PMMA mask was stripped using 2-propanone (Acetone). For a detailed description of the fabrication, and of how to do fast large area patterning with EBL see .
5.2. Experimental set-up
The diffraction pattern of the BGZP was investigated for visible light using a helium-neon laser, with a wavelength of 632.8 nm. The experimental set-up is shown in Fig. 5. A polarization filter was used to control the beam intensity. For cleaning out spontaneous emission photons and getting a Gaussian beam profile (spherical source) the laser was passed through a spatial filter, consisting of a microscope objective (Beck x45/0.65 (Achromate)) and a pinhole aperture, 20 micron in diameter. A parallel beam was obtained and used for illuminating the zone plate by placing a lens in the beam path, after the spatial filter, with a distance of 200 mm (the focal length of the lens) to the pinhole aperture. Next the zone plate was placed in the beamline. Due to the parallel beam, the distance from the zone plate to the lens was not crucial and arbitrarily set to 150 mm. The diffracted beam was limited using an iris aperture. The diffraction pattern was acquired using a CCD sensor in a DSLR camera (Nikon D80), by removing the objective and recording the signal directly on the CCD chip. This simple detector set-up could be used since the central part of the diffraction pattern was smaller than the CCD sensor, allowing the pattern to be collected simultaneously without moving the detector in the plane. Further the camera shutter-speed could be optimized for gathering the relative intensity information, due to the linear response of the CCD under non-saturated conditions . The non-saturated condition was accomplished by adjusting the polarization filter, previously discussed, and tuning the shutter-speed of the camera. The detector set-up was mounted on an optical rail in order to scan along the optical axis of the zone plate. The whole experimental set-up was placed on an optical table to achieve vibrational damping.
5.3. The focus
The zone plate was illuminated by a parallel beam, and the focus was therefore located at the focal length of the zone plate (1 meter). In Fig. 6(a) the experimental measurement of the focus with the surrounding diffraction pattern is shown and in Fig. 6(b) the corresponding simulated pattern can be seen. The agreement is excellent. Note that the actual focus is overexposed in order to show the surrounding diffraction pattern simultaneously.
5.4. The doughnuts: images at fractions of the focal length
The simulated diffraction patterns in Figure 3 for the BGZP show that the higher order diffractions (2nd, 3rd etc.) do not meet on the optical axis. This means that there are no higher order foci. Instead, a doughnut shaped ring with ”hills” and ”valleys” is formed. The doughnut shape varies with the number of sectors per zone and the number of zones. When performing the experiment, the doughnut was found as predicted. A measurement of the doughnut at f/2 can be seen in Fig. 7. Note that here we also see some focusing in the center not predicted by the simulations. We contribute this to imperfections in the fabrication process. Such imperfections can lead to a ”generalized” zone plate as described by Chau , giving a focus at even fractions of the main focus.
We present a numerical and experimental investigation of the Beynon Gabor Zone Plate (BGZP). We tested several transmission functions fulfilling the Gabor zone plate transmission and show that the expected intensity in the focus for all of these is 6.25% similar to a normal GZP. In particular a previously predicted efficiency of up to 16%, often cited in the literature, is shown to be an artifact caused by aliasing. Because the BGZP is binary, we expect higher order diffraction orders. In a normal Fresnel zone plate higher order diffraction leads to higher order foci at fractions of the focal length (for a parallel incident beam) along the optical axis. We show that for the BGZP the higher order diffractions orders are indeed present as expected, but they do not meet at the optical axis. Instead a doughnut-like shape in transverse planes at axial distances that are fractions of the focal length (for a parallel incident beam). This is an interesting effect, which could prove useful for applications in matter wave manipulation or for STED microscopy  at photon energies where polymeric phase plates are not suitable for generating the doughnut shape. The exact shape of the BGZP doughnut can be manipulated by azimuthal variations in the transmission function.
This work was supported by the Norwegian Research Council through the NæringsPhD Program. We are grateful to Øyvind Frette for general advice and for letting us use his optical laboratory. The Beynon Gabor zone plate was fabricated at the UiB Nanostructures laboratory sponsored by Trond Mohn with Bergens Research Foundation, the University of Bergen and the Norwegian Research Council (Gaveforsterkning)
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