**4. Efficient imaging**

In a second step, the center of the hologram has to be found. This will be the center for the Fourier transform. At least one pixel precision should be aimed for to avoid strong wavy modulations of the reconstruction. However, by Fourier identities, a displacement in the hologram is equivalent to a phase shift in the reconstruction. Therefore, centering can also be performed

The third manipulation of the hologram is a numerical correction for the fact that specimen and reference typically have a finite relative shift in the propagation direction of the beam. In **Figure 1**, the reference exit wave is formed approximately 1 μm upstream of the specimen due to the finite thickness of mask, SiN membrane, and the specimen itself. Also, the reference itself acts as a focusing optics, introducing another in-line displacement between the smallest spot of the reference beam (i.e., the best resolution for the reconstruction) and the specimen [14]. The hologram contains the full wave field information (intensity and phase). Therefore, the reference focus can be numerically moved along the beam direction until both reference focus and specimen are in a common plane perpendicular to the beam [6]. Using this reference propagation algorithm often significantly improves the reconstruction quality and reso-

Strictly speaking, the hologram is the Fourier transform of the transmission function only if measured on a spherical detector. In practice, however, detectors are planar. For very close camera distances, an inverse gnomonic projection of the hologram is required to correct for this geometrical artifact [15]. In our measurements, such a correction has only been required

The reconstruction is in general complex valued. In theory, real and imaginary part of the reconstruction measure the refracting and absorptive part of the refractive index of the specimen, respectively [16]. However, such a quantitative correlation requires excellent centering, patching of the intensity blocked by the beam stop, a very low noise hologram, and a homogeneous phase across the reference beam. In most applications, it is therefore more practical to artificially shift all the relevant information into the real part by multiplying a constant phase

With an optimized sample design (see Section 4) and after following the previously described reconstruction steps, a high quality reconstruction of a magnetic sample as shown in **Figure 1** can be obtained from two camera accumulations (each with a maximum of 1500 photons per pixel), one for left-circular polarized light and one for right-circular polarized light. The total acquisition time, excluding camera readout and polarization change, can be as low as 1 s at a high intensity, high coherence beamline such as P04 at PETRA III in Hamburg, Germany.

In general, there are two ways of time-resolved imaging: The first way is single-shot imaging, where images are acquired on a time scale much shorter than the dynamics that is to be observed. Here, the dynamics is recorded as it is happening, without any need of

factor to the reconstruction. The real part can subsequently be displayed as an image.

lution and allows to obtain depth information of three dimensional specimen [4].

*x x*+*k y y*)

measure the center displacement in the hologram,

, where *x* and *y* are coordi-

in the reconstruction by multiplication with a plane wave *exp*(*i*(*<sup>k</sup>*

*x* and *k y*

both in pixels. With this method, sub-pixel centering can be achieved.

nates in the reconstruction and *k*

228 Holographic Materials and Optical Systems

if scattering angles exceeding 5° were recorded.

**3. Time-resolved measurements**

There are a great number of parameters that can be tweaked in X-ray holography in order to optimize the imaging for the specific scientific question. These parameters include the following: (i) object hole diameter; (ii) reference hole diameter; (iii) distance between object hole and reference hole; (iv) contrast of the specimen (e.g., thickness of a magnetic material); (v) camera-specimen distance; (vi) number of pixels of the camera and pixel size (changed by, e.g., binning and region of interest); (viii) camera gain or sensitivity; (ix) beam stop diameter; (x) an absorptive layer in or before the specimen (a parameter that is called here object linear transmission); and many more. Optimizing these parameters can yield orders of magnitude better contrast in the reconstructed image, and it is strongly recommended to use simulations to determine the optimum parameters.

**Figure 2** illustrates how the contrast of a magnetic stripe domain image depends on some of the aforementioned parameters. The reconstructions were obtained from simulations of a purely two dimensional sample with a circular reference and a Co-containing magnetic specimen in a stripe domain phase. Contrast is generated exploiting the X-ray magnetic circular dichroism of the Co magnetic domains, that is, in absorption lengths of 13.7 nm for up domains and 23.4 nm for down domains [18]. For the simulations, the specimen is modeled as a magnetic layer with a total Co thickness of 2 nm, a hexagonal lattice of 125-nm diameter bubble domains, and a object linear transmission of 25%. The object hole has a diameter of 1.5 μm, the reference hole a diameter of 50 nm, and the distance between the two is fixed to 4 μm. The camera is a chip of 2048 × 2048 pixels of 13.5 × 13.5 μm2 size at a distance of 24 cm to the sample. The sensitivity is assumed to be 50 counts per photon and the readout noise is 3 counts per pixel in average. The camera saturates at 64,000 counts. The model includes a beam stop of 1 mm diameter, a coherence of 80.00% and sample vibrations relative to the camera of 300 nm.

The simulations in **Figure 2** demonstrate how significantly the contrast is determined by the intensity ratio of object and reference, the overall coherence, and the dynamic range of the camera. Within a reasonable range and assuming that the intensity of the incoming flux is not a limiting factor, more absorption of the object and a smaller object hole improve the intensity ratio of reference to object and yield better contrast. That is, the image quality can be improved by coating the specimen with some absorptive layer.

**Figure 2.** Contrast of the reconstruction as a function of imaging parameters. Top row: Object linear transmission, from 0.125 (left) to 1.0 (right) in steps of 0.125. Second row: Beamstop diameter, from 0.4 mm (left) to 3.2 mm (right) in steps of 0.4 mm. Third row: Vibrations (sigma of a Gaussian distribution), from 0 (left) to 28 μm (right) in steps of 4 m. Bottom row: Object hole size, from 700 nm (left) to 2100 nm (right) in steps of 200 nm.

The presence of a beam stop allows for using the dynamic range of the camera to detect signals at high scattering angles, which generally increases the contrast. However, if the beam stop becomes too large, the missing small scattering angle signals (corresponding to constant or low frequency modulations in the hologram) offset the intensity level of the reconstruction. The effect can be seen in **Figure 2** for a beam stop diameter of 2.4 mm. The white domains have now a color comparable to the background. For an even larger beam stop, the effect of ringing can be observed, as illustrated for the 3.2 mm beam stop. The optimum size of the beam stop depends on most of the other imaging parameters and has to be determined from simulations.

Finally, vibrations between the sample and the camera have a similar effect as a reduced coherence of the incoming beam [19]: Speckles and interference modulations become blurred and the image contrast reduces. Similar to the beam stop size, the question of at which magnitude the vibrations negatively effect the contrast can be answered only through simulations. In the present case, vibrations of as much as 8 μm have almost no effect on the image quality. However, for other geometries, the threshold may be as low as 300 nm.

Experimentally, the following parameters have been verified to yield full black-white contrast of magnetic domains in a single camera acquisition per helicity and a spatial resolution of better than 20 nm: (almost) fully coherent photons with energy at the Co L3 edge (778 eV); an object hole with 800 nm field of view; a 50 nm reference that has its most constricted part on the camera side of the mask (considering that most references are conical); a centerto-center distance of 3 μm between object and reference holes; a specimen with a total of 9 nm of Co and a nonmagnetic absorption of 75%; a camera with 2048 × 2048 pixels of 13.5 μm size at a distance of 17 cm to the sample operated using 1 MHz low noise readout; a beam stop diameter of 0.75 mm diameter blocking just the central part of the object hole's airy disk; and a rigid mounting of the sample and the camera. The relatively small size of the object hole and the absorptive layer lead to a favorable total intensity ratio of object and reference beams, which manifests in a clear interference modulation of the entire hologram, including the center, as visible in **Figure 1**. Fine speckle patterns can be distinguished in the hologram due to the good coherence and the reduction in relative transverse vibrations between camera and sample. The excellent resolution is obtained due to a focus formed by a cylindrical reference in a thick mask material [14] and the large numerical aperture associated with a close camera distance.
