1. Introduction

Frequently in x-ray microscopy the object under investigation has little amplitude contrast or is primarily a phase object [1]. In such cases, the contrast of the object can be enhanced using phase-sensitive techniques such as the Zernike method [2], phase contrast interferometric imaging [3], differential interference contrast method [4], and spiral zone plate imaging method [1]. Since glass lenses cannot be used for x-ray, and the diffraction efficiency of diffractive lenses is limited, it is desirable to use as few diffractive elements in a setup as possible. The technique of ref. [1] is based on the radial Hilbert transform – known to detect isotropically the gradient of the complex refractive index of an object [5, 6] –, and uses the first diffraction order of a single spiral zone plate (SZP) to achieve simultaneously spatial filtering on the object spectrum, and imaging on a CCD. Another x-ray microscopy technique that employs a single diffractive element – a Fresnel zone plate (FZP) – is conventional x-ray Fourier holography [7, 8]. The great advantage of this setup is that the FZP – rather than acting as a conventional focusing element where 0^th order is an undesirable diffraction order – is used as a beam splitter: it creates a spherical wave for reference wave, and a plane wave for illuminating the object, hence makes efficient use of the strong 0_th diffractive order. However, conventional x-ray Fourier holography does not produce edge enhancement, and is hence not efficient for displaying phase contrast.

Here we propose a Hilbert filtering setup, based on x-ray Fourier holography, but employing a Laguerre-Gaussian zone plate (LGZP) instead of a FZP. Laguerre-Gaussian (LG) spatial filtering was first proposed by Guo et al. [9], who demonstrated both theoretically and experimentally the advantage of LG spatial filtering over constant-amplitude radial Hilbert transforms. In ref. [9], LG spatial filtering was performed using an off-axis forked grating, which can be thought of as the interference pattern between an off-axis plane wave and a LG helical wave. However, since in our setup the 0^th-order diffracted wave serves to illuminate the object, we chose an on-axis geometry for LG spatial filtering. The LGZP can be thought of as a hologram that records the interference pattern between an on-axis spherical wave and a LG helical wave. The complex amplitude distributions of the paraxial spherical wave and the LG helical wave are

and

respectively, where (r,φ) are the polar coordinates, λ is the wavelength, f is the focal length, R is the radius of the entrance pupil of the system, and w is a constant determining the radius of the maximum intensity ring of the LG intensity distribution. In our calculations we chose w = R/√2, so that the maximum intensity ring is at R/2. The definition of the circular apodization function circ(x) is: circ(x) = 1 for x ≤ 1 and circ(x) = 0 for x > 1.

The LGZP transmittance function is

Figure 1(a) presents the schematic view of a binary LGZP, suitable for fabrication using available lithography techniques such as nano-imprint [10] or electron-beam [11] lithography, with a spatial resolution better than 40nm. Note that Fig. 1(a) is similar to the SZP in ref. [1], except that for the LGZP the duty cycle of the local gratings – representing the local transmitted intensity – varies with r.

2. Proposed setup and numerical results

Our proposed setup is presented in Fig. 1(b). It is similar to that of refs. [7, 8], except that the FZP is replaced by the binary LGZP of Fig. 1(a). First, a monochromatic x-ray plane wave illuminates the LGZP. The 0^th diffraction order after the LGZP is a plane wave which illuminates the object. The 1^st diffraction order is a convergent LG helical wave focused to a diffraction-limited doughnut spot at the focal plane [12]. At this focal plane a pinhole is placed, and the object is placed inside the pinhole, towards the side, so that it does not overlap the 1^st order doughnut spot. A CCD is placed at a distance z from the object. If z is large enough to satisfy the Fraunhofer condition, the CCD pattern will be the interference between the Fourier spectrum of the object and a divergent LG helical wave. Performing an inverse Fourier transform on the CCD image (corresponding to a holographic film being reconstructed using a plane wave), the 1^st diffracted order just after the hologram will be the spectrum of the object multiplied by a LG radial Hilbert filter. The inverse Fourier transform will thus result in an edge-enhanced image of the object.

 

Fig. 1. (a) Schematic view of a binary Laguerre-Gaussian zone plate (LGZP). (b) The proposed edge-enhancement Fourier hologram x-ray microscopy setup: the object (O, grey shaded area) is placed in a pinhole at the back focal plane of the LGZP illuminated by a collimated x-ray beam; a CCD is placed at a distance z from the object, where z is large enough to satisfy the condition for Fraunhofer diffraction. The CCD image is inverse Fourier transformed to obtain the edge-enhanced image of O.

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To verify our design, we performed numerical simulations for the setup on Fig. 1(b). The following parameters were used: The pinhole has a diameter of 15μm, so that the maximum distance between an object point and the reference doughnut-spot is s = 7.5μm. A soft x-ray with λ = 4.4nm was considered [13]. This choice for illumination wavelength was motivated by the fact that carbon compounds have a strong sharp absorption line at ~4.35nm, which causes anomalous dispersion. Although the compounds are completely transparent, their index markedly changes around 4.4nm. Thus this wavelength is thus suitable for detecting phase jumps in carbon compounds, including biological molecules. The CCD distance satisfying the Fraunhofer condition is z > s²/λ = 1.3cm. We chose z = 6cm, so that the ~17μm pixel size of commercial x-ray CCD cameras [14] is capable of resolving the densest grating created by the helical reference beam and the extreme object points. The hologram on the CCD was calculated at 128×128 pixels, yielding an effective CCD area of 2mm×2mm, and a numerical aperture (NA) of 0.018 for the system. The grating period at the edge of the LGZP is thus λ/NA=240nm. We chose a LGZP with a focal length of f = 1cm, having ~380 “zones” (i.e. near-circular spiral lines), and a lateral size of ~0.37mm. The condition for efficient interference at the CCD is that the maximum pathlength difference between extreme rays Δz_max ≈ s ∙ NA is much smaller than the temporal coherence length L = λ²/Δλ [8]. Assuming an x-ray source with a monochromaticity of Δλ/λ = 2.5∙10^-4 [8], this condition is satisfied for the proposed setup.

 

Fig. 2. (a) Amplitude object (axes are labeled in μm); calculated reconstructed image using (b) the Laguerre-Gaussian Hilbert filter setup of Fig. 1(b), and (c) a conventional Fourier transform x-ray microscope setup; (d), (e) and (f) show intensity cross-sections of (a), (b) and (c), respectively, along the vertical direction (denoted by the broken vertical lines) at the center of the field.

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2.1 Amplitude objects

Figure 2(a) shows an amplitude object having curved and straight contours, and a transmittance of 1 (top part), 0.4 (bottom part), and 0 (outer regions). The axes are labeled in μm, and show the object position inside the pinhole. For the calculation, the object was assumed to consist of point sources at a distance of 60nm from each other, corresponding to a pixel size of 60nm in the object field. First the CCD hologram pattern was calculated using numerical scalar ray-tracing to produce the interference between light coming from the coherently illuminated by the object and a diverging LG spherical wave. For simplicity, we assumed that the peak intensity of the LG spherical wave at the CCD plane equals the intensity of the wave coming from the object. Considering the geometry of our setup and the fact that the LG spherical wave and the object wave originate from the 1^st and 0^th order diffraction of the LGZP, respectively, our choice for relative intensities corresponds to a diffraction efficiency for the LGZP which is well below the theoretical limit for both phase and amplitude zone plates. As can be seen from the geometry in Fig. 1(b), the center of the hologram is overexposed, due to the 0^th diffraction order wave that passes directly through the part of the pinhole not occupied by the object. As a rough estimate, assuming the entire 15μm diameter pinhole causing the overexposed part, its diameter on the CCD (taking diffraction into account) is ~50μm, corresponding to 2-3 pixels. We neglected this effect and calculated the hologram as if the pinhole regions outside the object and the focused reference doughnut-spot were dark.

Once the CCD pattern was calculated from numerical ray-tracing (corresponding to the experimental recording of a digital hologram on the CCD), an inverse fast-Fourier-transform (iFFT) was applied to the CCD pattern. Figure 2(b) shows the reconstructed image of the object obtained by iFFT. As seen, the LGZP setup of Fig. 1(b) indeed produces an edge-enhanced image of the object, with the brightness of the edges corresponding to the intensity steps of the original object. For comparison, we also calculated the reconstructed image of the same amplitude object produced by a normal Fourier transform x-ray microscope setup [7, 8]. For this calculation, we simply replaced the LGZP of Fig. 1(b) with a FZP. The result is presented in Fig. 2(c). As expected, the FZP setup produces an ordinary intensity image of the object, rather than an edge-enhanced contour image. Figures 2(d), (e) and (f) show intensity cross-sections of Figs 2(a), (b) and (c), respectively, along the vertical direction denoted by the broken lines in Figs 2(a), (b) and (c). For amplitude objects the edge-enhancement technique is advantageous over conventional microscopy if the contrast of the object, (I_max - I _min)/(I_max + I_min), is small, where I_max and I_min are the maximum and minimum intensity values of the object field, respectively. In such cases conventional microscopy produces poor contrast images, whereas the contrast of edge-enhancement microscopy is – neglecting noise – inherently 100%.

2.2 Phase objects

Next we illustrate the edge-enhancement property of our setup for purely phase objects. Figure 3(a) depicts a phase object divided into two areas that produce different phase shifts on the incoming light (displayed as different grey-levels in Fig. 3(a)), while keeping the amplitude constant. The difference in phase shifts is Φ. Figures 3(b)-(d) show calculated reconstructed images of Fig. 3(a) using the LGZP setup, for Φ values of 2π/2, 2/10 and 2π/50, respectively. Since the object of Fig. 3(a) is surrounded by a rectangular amplitude-window, a rectangular contour also appears on the reconstructed image. We suppressed this rectangular contour by displaying only the central 90%×90% of the reconstructed images in Figs 3(b)-(d). As illustrated, the proposed setup produces sharp, isotropic edge-enhanced images of objects having a phase step. It is capable of detecting even very small phase steps (corresponding, e.g. in the case of Fig. 3(d), to an optical path difference of λ/50) with high-contrast. Figures 3(e), (f) and (g) show intensity cross-sections of Figs 3(b), (c) and (d), respectively, along the vertical direction denoted by the broken lines in Figs 3(b), (c) and (d). The faint periodic structure observed in Fig. 3(d) is probably caused by the discrete grid-like nature of the CCD hologram, and becomes pronounced as the phase step is decreased. Note that for the case of Φ = 2π/2 (Fig. 3(b)) the two areas on the object interfere destructively, hence the 0^th order diffraction produced by the object is very weak (and exactly zero, if the two areas are equal). Thus for this case the Zernike phase contrast method would produce very poor results [6]. On the other hand, since our setup is based on the radial Laguerre-Gaussian Hilbert transform – which detects even small values of Φ with high sensitivity –, the image contrast is high for large as well as small phase jumps. We assume that the performance of the method is fundamentally limited by issues such as the signal-to-noise ratio in the system.

 

Fig. 3. (a) Phase object with two areas having a difference Φ in phase shift (axes are labeled in μm); calculated edge-enhanced reconstructed images using the setup of Fig. 1(b), for (b) Φ = 2π/2, (c) Φ = 2π/10, and (d) Φ = 2π/50. Figures 3(e), (f) and (g) show intensity cross-sections of (b), (c) and (d), respectively, along the vertical direction denoted by the broken vertical lines.

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3. Concluding remarks

Important biological phenomena often occur on the surface of thin cell membranes and minute fibers, of a size of several 10nm. While conventional microscopy is inefficient for analyzing such phenomena, we believe that the proposed microscope system, by being able to detect small phase jumps in such systems, may become a powerful tool for the analysis of certain life phenomena.

Finally we note that similar calculations were performed using a SZP instead of a LGZP in the setup. Our results confirmed those presented for a visible light setup in ref. [9]. The SZP produces similar images as Figs 2(b), and 3(b)-(d), but the reconstructed images obtained by the LGZP – which efficiently suppresses the sidelobes in the point-spread-function for the Hilbert transform [9] – have sharper contours and are more sensitive to small phase-jumps. However, as noted earlier, the local gratings on a SZP have constant duty cycle, and this may be a practical advantage over the LGZP in the fabrication process for an experimental setup.