Mask-based Fourier transform holography is used to record images of biological objects with 2.2 nm X-ray wavelength. The holography mask and the object are decoupled from each other which allows us to move the field of view over a large area over the sample. Due to the separation of the mask and the sample on different X-ray windows, a gap between both windows in the micrometer range typically exists. Using standard Fourier transform holography, focussed images of the sample can directly be reconstructed only for gap distances within the setup’s depth of field. Here, we image diatoms as function of the gap distance and demonstrate the possibility to recover focussed images via a wavefield back-propagation technique. The limitations of our approach with respect to large separations are mainly associated with deviations from flat-field illumination of the object.
© 2010 Optical Society of America
Fourier transform holography (FTH) is a lensless, coherent imaging method capable of achieving high spatial resolution. The absence of optical elements makes this technique especially well suited for the application of (soft) X-rays. In FTH the hologram is formed by interference of a scattered object wave with a reference wave which encodes the relative phase information as intensity modulation directly in the diffraction pattern. Both waves originate from the object plane which is perpendicular to the optical axis. Due to this geometrical arrangement, the hologram is reconstructed by a single inverse Fourier transform . In recent years, the FTH geometry has successfully been established in the soft X-ray regime by an integrated mask-sample approach [2–7]. Here, the holography mask containing a transparent object region and a transversely offset reference region is rigidly coupled to the object, resulting in one fixed field of view (FOV) of the sample. The appeal of this approach is a stable FOV excluding any drift between mask and object, which allows long exposure times. Furthermore, the fixed FOV facilitates correlative microscopy with complementary types of measurements . Finally, the reference wave intensity, which is often a limiting factor , is maximized via suitable nanostructuring since the mask-sample structure can be made 100 % transmissive in the area of the reference aperture. The drawback of this approach is that the FOV cannot be changed during the X-ray holography experiment. While the FOV can be extended by a multiple object mask , a FOV versatility compared to standard microscopy is still not achieved.
A recently reported approach to overcome this limitation is the separation of the sample and the holography mask on different X-ray windows [10, 11]. The windows are supported on separate translation stages such that they can be transversely shifted with respect to each other. In this way, the FOV can be changed as long as the reference aperture is not obstructed by a strongly absorbing object feature. In particular, this is the case for sparse objects on a transparent membrane, a situation often encountered for studies of biological objects.
The location of mask and object on separate macroscopic support structures introduces a gap along the X-ray propagation direction between them. On one hand, this gap is desirable if the object is situated on the membrane side facing towards the holography mask and has to be protected from being touched. On the other hand, practical limitations may have to be considered. For typical Si3N4 membranes with 5 mm silicon support frames, an angular misalignment of 1° corresponds to a 40 µm separation at the sample position in the center. Similarly, contamination of the object and mask planes e.g. by dust particles can easily induce separations > 10 µm. From a practical point of view, it can thus be necessary to increase the working distance between mask and object, ideally without loss of image quality.
FTH image reconstruction via a single 2D Fourier transform will then reconstruct the wavefield in the plane of the reference source which in the actual holography experiment was offset from the object plane. The depth of field (DOF) for FTH imaging is given by DOF = λ /(2sin2 α) with λ being the wavelength and α being the maximum scattering angle of the object encoded in the hologram . We expect an increasing loss in resolution for increasing mask-object working distances, and significantly defocused images for gap distances larger than the DOF.
As holography records the complete wavefront information, it is possible to calculate a focused image based on this information even when the gap distance exceeds the DOF . This becomes essential for soft X-ray experiments with a high resolution below 50 nm and an associated large numerical aperture and small DOF. For example, a typical FTH X-ray setup used for high-resolution magnetic imaging of 3d transition metals consists of a 2 µm object hole, resonant illumination at a wavelength of λ = 1.59 nm and a numerical aperture of NA = sin α = 0.036. The corresponding DOF of only 0.6 µm makes it difficult to arrange the holography mask within the DOF of the sample.
In the present work, we carry out movable-mask-based FTH experiments as a function of the separation between mask and object. We demonstrate that the introduction of the wavefield back-propagation formalism to the FTH reconstruction process vastly improves the image quality, allowing to operate at mask-sample separations of hundreds of micrometers.
The FTH geometry is realized as sketched in Fig. 1. A FTH mask is generated by depositing a 1 µm thick gold film on a Si3N4 membrane of 100 nm thickness. An object aperture with radius R = 5 µm and four reference holes with radii r = 100−125 nm are milled by focused ion beam lithography through the gold film and the silicon nitride membrane at a distance of d = 25 µm to the object aperture (center to center distance). Diatoms, constituting the object, are deposited on a separate silicon nitride membrane from aqueous suspension and subsequently air-dried. The object membrane can be moved with respect to the FTH mask by in-vacuum piezo-driven translation stages in x-,y- and z-direction with a step size of 50 nm. The travel range of the x- and y-actuators is larger than the transparent area of the object membrane (1 mm×1 mm). The distance Δ between sample and optics membrane can be changed from zero, i.e. both membranes are in contact, up to few mm. The entire assembly is steered into the X-ray beam by an UHV manipulator with a precision of 2.5 µm.
Experiments are performed with two slightly different parameter sets at the BESSY II UE52-SGM undulator beamline in the soft X-ray regime with λ = 2.21 nm (2.27 nm). The monochromator is set to a resolving power λ/Δλ ≈ 2000, resulting in a longitudinal coherence of 4.4 µm which exceeds our maximum path length difference of 0.52 µm at a maximum scattering angle of α = 1.0°. In order to ensure a transversely coherent illumination of the sample, we place the mask-object manipulator 300 mm downstream of the beamline focus, which is 17.4 µm FWHM in the horizontal and 120 µm fwhm in the vertical direction. The small angular acceptance of the transparent area of the mask in this geometry (8.7×10−10 sr for the object hole) in conjunction with the focal size allows us to treat the incident X-rays on the sample as plane-wave illumination simplifying FTH analysis and back-propagation. A charged-coupled device (CCD) camera (Princeton Instruments PI-MTE, 2048×2048 pixels, pixel size 13.5 µm) records the hologram 915 mm (795 mm) downstream of the sample. The distance between sample and CCD is chosen such that the camera can sample the shortest-period modulations in the hologram, generated by the interference of the waves emerging from the object aperture and the reference aperture. A beamstop blocks the central part of the hologram in order to use the limited dynamic range of the CCD for higher momentum transfer information. The center information is filled in by a second measurement without beamstop.
In FTH, the Fourier transform of the hologram H(kx,ky) reconstructs the wavefield of the object in the plane of the reference source via the reference-object cross-correlation term  (kx and ky represent the coordinate system in reciprocal space). In our case, the mask containing the reference source is located at z = −Δ, while the object is located at z = 0 and hence an out-of-focus image of the object is reconstructed in the reference plane. If Δ is smaller than the DOF no further operations are necessary as the reconstructed image of the sample is in focus. Otherwise wavefield propagation has to be applied to the holography procedure to obtain a focused image of the sample.
Free-space wavefield propagation is described in detail in optics literature [14,15]. The propagation of an arbitrary wavefront Ψz=−Δ at z = −Δ to z = 0 can be treated mathematically by decomposition into its plane wave components. The plane wave propagation through free-space over a distance Δ in direction of kz is accomplished by rotation of the phase by a factor of exp(iΔkz) only (kz is the z-component of the wave vector k). Afterwards the components are recomposed to the wavefront Ψz=0 at z = 0. The procedure is expressed using the free-space propagator:
where ℱ denotes the Fourier transform, ℱ−1 the inverse Fourier transform, k = 2π/λ the wave number, kx and ky the inverse coordinates in reciprocal space in the detector plane perpendicular to the z-axis.
The use of Eq. (1) requires amplitude and phase information of a wavefield. Both are encoded within the hologram. In the limit of a transversely small reference source approximating a delta function δ (x−x 0,y−y 0), the object wavefront in the plane z = −Δ is recovered without loss in spatial resolution. Note that the reference source size affects the spatial resolution of the reconstruction in the same way as in the ideal situation of an FTH experiment with Δ = 0, i.e. its influence is independent of the propagation procedure. A Fourier transform hologram is reconstructed by an inverse Fourier transform :
The first two terms describe the auto-correlations of the object wave and the reference wave. We identify Ψ(x+x 0,y+y 0)z=−Δ with the defocused virtual image at z = −Δ of the object which is actually located at z = 0. In the reconstruction matrix the image is offset from auto-correlations by the vector (x 0,y 0) which is defined by the distance between the object and reference holes on the holography mask, i.e. d = ∣(x 0,y 0)∣. Ψ*(−x+x 0,y+y 0)z=−Δ is the socalled twin image. We are only interested in the object wave Ψ(x+x 0,y+y 0), which is usually well separated from the auto-correlations and the twin image. We ignore these terms and write Eq. (2) for simplicity:
A focused image is thus obtained by:
i.e. by adaption of the free-space propagator to the hologram prior to inverse Fourier transformation. Note, that the autocorrelation is more extended for increasing gap distances after usage of Eq. (4), which means that for very high gap distances it may not be completely separated from the object wave. This effect does not occur in our experiments as the object reference spacing is sufficiently large.
3. Results and Discussion
To illustrate the potential application of the method for lensless imaging, we use diatoms dispersed on a membrane. First we demonstrate the possibility to stepwise move the FOV to obtain a larger stitched image analogously to [10, 11]. Our main focus is then on the application of wavefront back-propagation for high-resolution imaging at variable mask-sample separations.
In Fig. 2 we present a typical hologram corresponding to a single FOV of Fig. 3 (bottom left). Five holograms of diatoms are taken and the overlap between the single real-space images is used to patch the images to a larger FOV (Fig. 3). The fine structure of the diatoms is clearly visible. The reconstructions are calculated without the use of wavefield back-propagation as the gap between the holography mask and the object was smaller than the corresponding DOF in this case. We estimate the lateral resolution with a linecut at several positions (10 % − 90 % criterion) to be 110 nm. As the maximum detected momentum transfer corresponds to a lateral resolution to 89 nm, our resolution is limited by the size of the reference source.
Usually, the pixelation of the detector limits the lateral resolution as a single large FOV requires a large sample-detector distance which decreases the maximum measured momemtum transfer. Additionally, a single large FOV requires a large reference aperture to ensure a sufficient intensity ratio between the reference and the object wave which is directly connected to the contrast of the reconstructed image. Anyhow, large effective FOVs can be built up without the loss of spatial resolution if a larger FOV is holographically recorded with the same number of detector pixels n × n in a multi-object approach, where a number of fixed FOVs on a holographic mask are imaged in one experiment. As a result the ratio between highest and lowest spatial frequencies for FTH corresponds to n/4 for the entire lateral object-reference extent or equivalently n/6 for the FOV alone . A continous large area can be imaged holograpically with a high lateral resolution, if this approach is used. Compared to other recent experiments using FTH without a separated holography mask to image biological samples, we achieve a four times better resolution with a similar total FOV . Note, for sparse objects on an open support membrane, the scanning mask approach is straight forward. Typically the reference beam is not blocked by the sample, in contrast to the situation in , where a trench was fabricated lithographically through the object membrane which limits the flexibility of the holography mask.
Next, we explore the influence of the mask-object separation along the X-ray propagation direction on the imaging performance. A series of measurements for increasing distances between mask and object is shown in Fig. 4. The quality of the directly reconstructed images drastically deteriorates for increasing distances. At distances larger than 150 µm the image is defocused so strongly that a conclusion on the object structure becomes impossible. In all reconstructions, the object hole boundary remains in focus as it is located in the reference hole plane.
Focused reconstructions obtained by applying wavefield back-propagation according to Eq. (4) are presented in the second and fourth rows of Fig. 4. Zero distance is coarsely determined by observed contact between mask and object. Δ is then treated as a free parameter in the back-propagation procedure. For all distances we are able to obtain a focused reconstruction of the object with a resolution of 110 nm, without prior exact knowledge of Δ. The resolution is again limited by the size of the reference aperture as the maximum measured momentum transfer corresponds to 80 nm. If the mask-object separation is larger than 60 µm, concentric rings become visible in the reconstruction. These rings originate from the illumination of the diatom object with the Fresnel diffraction generated by the aperture in the holography mask. The Fresnel number is F = R 2/(Δλ) < 184 for separations Δ exceeding 60 µm in our experiment. Usually, this diffraction structure in the illumination function is not desirable and is thus limiting the maximum mask-object separation for imaging applications. However, we would like to emphasize that this effect is not an artifact of the wavefield propagation and the presented formalism works free from aberrations even for the largest distances. Generally, the reconstructed image displays the focused object superimposed with the illumination function. In principle this could be corrected by normalizing with a calculated or measured illumination function. In practice this approach is limited and prone to errors as the information content in low illumination regions is correspondingly low and small errors in the illumination function used for normalization will create significant artifacts.
Obviously, the deviation from a flat-field illumination becomes stronger for increasing gap distance Δ. In accordance with Fresnel diffraction theory, deviations from flat field illumination are most noticeable at the fringe and become stronger towards the FOV center for increasing Δ. The numerical calculation of the Fresnel diffracted intensity is presented in Fig. 5. It was used to determine the deviations in the illumination function from flat-field illumination.
While the tolerable variation in illumination depends on the specific sample features and contrast, the following illumination situation may be considered as sufficient for many applications: the illumination within the inner 90 % of the FOV diameter can be considered sufficiently flat, if the intensity variation within this area amounts to less than 10 %. This is the case for Δ≤cR 2/λ with c = 0.0056 taken from the calculations in Fig. 5 and results in a Fresnel number F = λz/R 2 ≥ 179. For our experimental parameters, this situation corresponds to Δ ≤ 62 µm which is in agreement with our imaging results. As obvious from Fig. 4 and 5, good images addressing a specific imaging problem may also be recorded for larger Δ. Contrary to the estimate of a 2 mm maximum working distance in Ref.  based on the longitudinal coherence length and optical path length difference, our results lead us to the conclusion that the limitation due to non-flat illumination of the sample discussed here is a more severe limitation on Δ in order to obtain high quality images. We note that this limitation is not present in an optically inverted situation where the reference wave is generated by a small scattering object rather than an aperture. In this case, no opaque mask is required in the reference plane and the object illumination is defined directly by the incident X-ray beam.
We want to draw attention to a potential undesired disturbance of the reference wave by the object. The scattered wavefield of the reference aperture is also passing through the object plane. The distance where the minimum of the Airy disc of the reference wave starts to reach the object is Δ = dr/(1.22λ) where d is the distance between reference and object hole on the holography mask and r is the radius of the reference hole. In our experiments this situation is reached for Δ ≥ 542 µm, which is much larger than all distances Δ in the experiments presented here. For a practical purpose this effect can thus be neglected for sparse samples on a transparent substrate.
Finally, we note that in coherent diffractive imaging e.g. via “keyhole diffractive imaging” or “ptychography”, where no reference beam is present, a movable field of view has also been realized [17, 18]. In the latter case, the overlap of illuminated regions is required to provide additional information on the system in order to aid the phase retrieval process. In contrast, in the FTH approach presented here, each image is retrieved independently by direct Fourier inversion.
We demonstrate the possibility to decouple the X-ray holography mask from the sample to image biological objects which exceed the size of the object aperture. This approach is particularly useful to image biological samples as they can typically easily be prepared sparsely on a membrane.
We investigate the influence of the gap size between the X-ray holography mask and the object membrane as it impairs the image quality if it is larger than the DOF. We demonstrate a combination of FTH and wavefield back-propagation to obtain focused high-resolution image reconstructions. Although we are able to reconstruct a sharp image also for large gap distances (> 60 µm), the technique is limited by the influence of the object hole Fresnel diffraction generating deviations from flat-field illumination. Particularly for experiments with high lateral resolution and corresponding low DOF our approach is mandatory in order to to extract high-quality images from holographic data with a moveable field of view.
We thank the H. Zabel and his group for the possibility to carry out the experiments in the ALICE scattering chamber and associated support. The diatoms were provided by P. Guttmann. We are thankful for the possibility to carry out focused ion beam lithography at the Centre for Free-Electron Laser Science, Hamburg.
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