**2. How to obtain a real space image**

A typical and convenient realization of X-ray holography is in a lensless off-axis geometry as depicted in **Figure 1**. Here, the entire optics of the imaging process constitutes of a mask in the X-ray beam with two circular apertures, the so-called object and reference holes. The object hole is typically approximately 1 μm in diameter and the reference hole is roughly 50 nm in diameter. The two holes are separated by 2–5 μm (at least three times the object hole radius) in the direction transverse to the incident beam. The specimen that is to be imaged is placed behind the object hole, ideally rigidly attached to the mask to exclude drift of the reconstructed image. The beam is transmitted through both holes and the specimen and is diffracted due to absorption and phase modulation during the transmission process. The hologram, that is, the interference pattern of the two diffracted beams, is recorded using a camera. For details regarding the wave propagation, see Refs. [9, 13]. Here, we consider the typical application where the camera is placed in the far field of the scattering process. Therefore, the scattering amplitude is conveniently described by the Fourier transform of the total transmission function of sample and mask.

The camera detects the intensity of the scattered light. Mathematically, the intensity is the absolute square of the scattering amplitude. An inverse Fourier transform of this intensity pattern yields the so-called reconstruction, which is the autocorrelation of the original total transmission function. The reconstruction can be written as a sum of the autocorrelations of object hole transmission function and reference hole transmission function plus the crosscorrelations between these two transmission functions. When pictured, the autocorrelation is located in the center of the reconstruction and the cross-correlation terms are displaced by the vector from object hole to reference and vice versa, see **Figure 1**. Hence, there is no  spatial overlap between these terms. Since the reference is small, the cross-correlation yields an image of the transmission function of the specimen with a resolution determined mainly by the reference hole diameter [1, 13].

imaging (a merit that is derived from Fourier space imaging because drift in Fourier space

Using holography for soft X-ray imaging, first demonstrated in a lensless setup in 2004 [1], is a rather novel approach that is still under heavy development and significant improvements have been made in the past few years [2–12]. Still, permanent holography end stations with user support, as common for more established techniques such as (scanning) transmission X-ray microscopy (STXM) or photo emission electron microscopy (PEEM), are not yet available. Therefore, the following chapter discusses the key ingredients and solutions for the main challenges of X-ray holography from an end user perspective. Specifically, the content of this chapter is organized as follows: The basic theory of how to obtain a hologram and how to reconstruct the real space information is presented in Section 2; the tricks of time resolved measurements and how to measure time zero in Section 3; some of the most important considerations for efficient imaging in Section 4; a suggested end station for magnetic imaging in Section 5; steps of how to fabricate suitable samples in Section 6; and finally, an outlook of anticipated future developments in Section 7. For those interested in more fundamental and technical aspects of

translates to phase shifts in real space) and in situ sample manipulations.

X-ray holography, I suggest the other specialized literature, for example [9, 13].

A typical and convenient realization of X-ray holography is in a lensless off-axis geometry as depicted in **Figure 1**. Here, the entire optics of the imaging process constitutes of a mask in the X-ray beam with two circular apertures, the so-called object and reference holes. The object hole is typically approximately 1 μm in diameter and the reference hole is roughly 50 nm in diameter. The two holes are separated by 2–5 μm (at least three times the object hole radius) in the direction transverse to the incident beam. The specimen that is to be imaged is placed behind the object hole, ideally rigidly attached to the mask to exclude drift of the reconstructed image. The beam is transmitted through both holes and the specimen and is diffracted due to absorption and phase modulation during the transmission process. The hologram, that is, the interference pattern of the two diffracted beams, is recorded using a camera. For details regarding the wave propagation, see Refs. [9, 13]. Here, we consider the typical application where the camera is placed in the far field of the scattering process. Therefore, the scattering amplitude is conveniently described by the Fourier transform of the total transmis-

The camera detects the intensity of the scattered light. Mathematically, the intensity is the absolute square of the scattering amplitude. An inverse Fourier transform of this intensity pattern yields the so-called reconstruction, which is the autocorrelation of the original total transmission function. The reconstruction can be written as a sum of the autocorrelations of object hole transmission function and reference hole transmission function plus the crosscorrelations between these two transmission functions. When pictured, the autocorrelation is located in the center of the reconstruction and the cross-correlation terms are displaced by the vector from object hole to reference and vice versa, see **Figure 1**. Hence, there is no

**2. How to obtain a real space image**

226 Holographic Materials and Optical Systems

sion function of sample and mask.

In theory, reconstructing the real space image of a specimen from a hologram is just a simple Fourier transform. However, in practice, a number of post-processing steps are required to obtain a high quality image, as briefly discussed subsequently. First, a hologram is seldom recorded without artifacts. For instance, cosmic rays, stray light, pixel defects, and particles on the camera chip lead to overly bright or dark pixels in the hologram and to wave-like noise in the reconstruction. Such artifacts can be eliminated by statistical analysis (e.g., identifying artifacts as pixels deviating from the local average intensity by many standard deviations and by much more than a single photon count) and by subtracting a dark image. Other artificial artifacts, such as a central beam stop and the wire holding it, can be smoothed to avoid high intensity noise in the reconstruction due to sharp edges in the hologram. Best results are obtained when using high quality camera chips (typically 2048 × 2048 pixels) with a low noise readout electronics and by carefully eliminating sources of stray light.

**Figure 1.** Schematic illustration of X-ray holography. The sample (labeled mask) is illuminated with a coherent plane wave from the left. The sample consists of a transparent SiN membrane. On the back side is an opaque Au layer with two holes, a larger one (the object hole, typically 1 μm in diameter and a smaller one (the reference hole, typically 50 nm in diameter). These holes are visible as dark shadows in the scanning electron micrograph overlaid on the mask. On the front side of the SiN membrane is the specimen that is to be imaged: In the present example, a magnetic film patterned into a wire. The incident beam is transmitted through the two holes and gets scattered. In the far field, the interference of the two scattered beams forms the hologram, which is recorded with a camera. The direct beam is blocked via a circular beam stop (typically made by a drop of glue on a thin wire, both of which are visible as a black circle and a black line in the hologram) so that the dynamic range of the camera can be optimized to detect the diffracted beam. The hologram can be digitally reconstructed via an inverse Fourier transform FFT <sup>−</sup><sup>1</sup> to obtain the local transmission function of the sample, here showing the out-of-plane magnetization of the sample as black and white contrast. The image of the sample appears twice, a phenomenon known as twin image formation.

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 in the reconstruction by multiplication with a plane wave *exp*(*i*(*<sup>k</sup> x x*+*k y y*) , where *x* and *y* are coordinates in the reconstruction and *k x* and *k y* measure the center displacement in the hologram, both in pixels. With this method, sub-pixel centering can be achieved.

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

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 if scattering angles exceeding 5° were recorded.

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

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.
