1. Introduction

During the last decades, a tremendous development of X-ray sources has taken place, ranging from table-top sources to undulator-based sources at storage rings and X-ray free-electron lasers. X-ray radiation produced by synchrotron radiation sources is coherent to a certain degree [1,2]. There are several powerful X-ray techniques emerging along with the development of X-ray sources that require coherent X-ray radiation. These are, e.g., coherent diffractive imaging (CDI) [3–7], X-ray holographic microscopy (XHM) [8], Fourier transform holography (FTH) [9–11], X-ray ptychography [12–14] and X-ray photon correlation spectroscopy (XPCS) [15–17]. Consequently, the determination of the coherence properties of X-ray sources is of high interest since they are the essential prerequisites for these experiments.

There are several techniques existing to determine the coherence properties of X-ray radiation, e.g., Young’s interference experiments, measuring the fringe visibility as a function of pinhole or slit separation [18–21], or the more efficient use of spatially well-defined non-redundant arrays of apertures (NRA) [22,23] and uniformly redundant arrays of apertures (URA) [24,25]. In the latter cases the Fourier transform of the measured diffraction patterns is analyzed to extract the coherence properties of the illuminating radiation. Coherence measurements based on light scattering from random scatterers using laser light and light from a thermal source have been performed by Suzuki [26,27], Ross [28] and Asakura et al. [29]. They analyzed the Fourier transform of the observed speckle pattern to extract the coherence properties of the beam. Gutt et al. [30], Abernathy et al. [31] and Sandy et al. [32] performed coherence measurements using spatial intensity-correlation functions of speckle patterns from random scatterers. The obtained speckle contrast depends on the source, the optical imaging system, sample properties and the detector and is directly correlated to the coherence properties of the X-ray beam [33–35]. The speckle contrast can be extracted from the speckle patterns with low effort. The latter experiments, however, do not allow for a direct measurement of the transverse coherence length, in contrast to the method presented here.

We propose a method that allows determining the full two-dimensional spatial (transverse) mutual coherence function and hence the transverse coherence length of the X-ray radiation directly from a single magnetic speckle pattern. Basically it is the analysis of the Fourier transform of the magnetic speckle pattern obtained from magnetic maze pattern that is obtained using X-ray resonant magnetic scattering (XRMS). A ferromagnetic Co₃₅Pd₆₅ alloy film has been used as scattering medium utilizing a photon energy of 778 eV, which is the cobalt L₃ absorption edge. The method is based on a simple analysis and gives an easy access to the coherence properties of synchrotron radiation without using any special set-ups for coherence measurements, like double-pinhole or multi-pinhole apertures.

The Fourier analysis method is also applicable to speckle patterns from other sample systems that show an inherent high degree of structural size variation. Hence, the method is not restricted to magnetic samples.

2. Experimental

Figure 1 displays the layout of the X-ray resonant magnetic scattering (XRMS) and the Young’s double-pinhole experiment. The X-ray beam coming from the left passes through a pinhole with a diameter of 40 µm, which selects the coherent part of the beam and defines the illuminated area on the sample. The beam is scattered by the magnetic sample which is located 55 cm downstream of the pinhole. Optionally, an opaque mask with a double-pinhole structure can be placed in the sample plane to perform a Young’s interference experiment.

 

Fig. 1 A sketch of the XRMS and Young’s double pinhole experiment at P04. The beam passes through a pinhole with a diameter of 40 µm and is scattered at the magnetic sample. The scattering pattern is recorded by a CCD camera. The direct beam is blocked by a beam stop. Optionally, a dual aperture can be used instead of the magnetic sample to perform a Young’s double pinhole experiment.

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The scattered X-rays are detected by a Peltier-cooled 16 Mpx CCD camera (1100S, Spectral Instruments) with a pixel size of 15$\times$15 µm². The camera is protected from the high intensity zero-order beam by a central beam stop of 1 mm diameter.

The measurements were performed at the P04 beamline of the storage ring PETRA III. The beamline is equipped with an APPLE II undulator delivering 100% circularly polarized X-rays with energies ranging from 250 eV to 3000 eV [36]. A monochromator exit slit of 200 µm has been used for the experiments resulting in a resolving power of $\lambda /\Delta \lambda \approx 3\times 10³$ at the selected photon energy of 778 eV.

The vertical refocusing mirror unit (RMU) of the beamline is set to a focus distance of 2.5 m, and a vertical focal size of 70 µm, which is about one third of the monochromator exit-slit size. In the horizontal direction the beam is almost parallel (focal distance 16.9 m, focal size 100 µm). The latter asymmetric condition is due to the fact that the proper horizontal RMU was not yet installed at the time of the experiment.

The sample and double pinhole arrangement has been placed 18 cm downstream of the vertical focus. The detector was positioned 1.06 m behind the sample. Using the beam-defining 40 µm pinhole, a beam size at the sample position of ≈ 25 µm × 49 µm FWHM (h × v) was measured by scanning the beam with a 2 µm pinhole.

The ferromagnetic Co₃₅Pd₆₅ alloy film is fabricated using molecular beam epitaxy at room temperature [37]. The film is deposited on a 50 nm thick Si₃N₄ membrane of 100 × 100 µm² size. First, a seed layer of 2 nm Pd is grown. Onto this, 40 nm of Co₃₅Pd₆₅ are deposited and finally capped by a 2 nm Pd layer. The magnetic film system shows a perpendicular orientation of magnetization [38].

To fabricate the double-pinhole apertures, a Au_240nm(Pd_120nm/Au_240nm)₄ multilayer is sputter-coated onto a 100 nm thick Si₃N₄ membrane of 500 × 500 µm² size [8]. Double pinholes with a pinhole diameter of 300 nm at varied pinhole separations are milled into the multilayer using focused ion beam (FIB) [39]. The structuring process was performed with Ga⁺ ions of 30 keV (beam current 20 pA).

3. Coherent magnetic X-ray scattering

Out-of-plane magnetized Co₃₅Pd₆₅ alloy films display a maze-like domain configuration which creates an isotropic donut-shaped diffraction pattern in X-ray resonant magnetic scattering (XRMS) [40–43]. The diffraction pattern gives ensemble-averaged information on the magnetic domain structure in reciprocal space. Coherent or partially coherent illumination generates a magnetic speckle pattern whose speckle configuration is a fingerprint of the exact spatial arrangement of the magnetic domains within the probed area [33,41]. Furthermore, the size of the speckles is directly correlated to the beam size at the sample [19,33]. The scattered intensity as a function of momentum transfer q is given by

Within the framework of the Gaussian Schell model (GSM), which is widely used to describe synchrotron radiation from an undulator, the CDC and the intensity distribution of the X-ray beam are assumed to be Gaussian functions. In this case, a global degree of coherence can be introduced, which characterizes the transverse coherence properties of the beam by one number [13,21,22]

The autocorrelation function of the magnetic domain pattern or magnetic density ${\rho }_m\left({\text{r}}_{}\right)$ is given by [47]

The square of the modulus of the Fourier transformed domain pattern results in the simulated Fraunhofer diffraction pattern (Eq. (1)). A subsequent inverse Fourier transform of the diffraction pattern yields the autocorrelation (Patterson) function according to the autocorrelation theorem of the Fourier transform.

Figure 2 displays the modulus of the Patterson function for different standard deviations σ of the gamma distribution, i.e., for varying distributions of domain sizes, using a mean domain size of D = 100 nm. The graphs have been smoothed with a kernel of 3 µm width to suppress the strong high-frequency oscillations. The figure shows that if the domain pattern consists of domains with almost a single size (σ = 0.1 nm), the Patterson function is broad and triangular-shaped. With increasing variation of domain-sizes, the width of the central peak decreases until it can be described by almost a single narrow peak. Furthermore, with increasing σ, the decaying signal develops into perfectly flat side lobes. Similar results have been found byAsakura et al. [29] describing the autocorrelation of a diffuse plate as a function of mean-square phase variations, i.e. surface-height variations.

 

Fig. 2 Fig. 2. Normalized Patterson function of a simulated one-dimensional magnetic domain pattern with gamma-distributed domain sizes for different values of the standard deviation σ of the distribution function. The side lobes vary from 0.38 to 0.003. The average domain size has been set to 100 nm.

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The same analysis can be performed in two dimensions using a Kerr microscopy image of a magnetic maze domain pattern (Fig. 3(a)). The square of the modulus of the Fourier transformed Kerr image yields the simulated two dimensional diffraction pattern (Fig. 3(b)). The subsequent inverse Fourier transform of the latter provides the two-dimensional Patterson function (Fig. 3(c)). A distinct high-intensity peak structure can be seen in the center together with slight intensity fluctuations spread over the entire range. A slight asymmetry of the central peak is observable resulting from a slightly asymmetric simulated magnetic diffraction pattern, indicating a preferred domain alignment. One-dimensional profiles have been extracted by radially averaging the Patterson function around the center using circle segments with an angular width of 10 degree in horizontal and vertical directions. They reveal a narrow peak at the center and perfectly flat side lobes, which are independent of direction in the slightly asymmetric scattering pattern. The findings of the two-dimensional Patterson function are consistent with the ones obtained in one dimension using a large variation of domain sizes within the domain pattern. Thus, the Patterson function of a maze-like magnetic domain pattern, which is discussed in the following, can be expressed by a distinct narrow central peak and perfectly flat side lobes.

 

Fig. 3 a) Kerr microscopy image of a magnetic maze domain pattern with an average domain size of 481 nm. b) Central area of the modulus square of the Fourier transformed maze pattern showing a simulated donut-like diffraction pattern. c) Modulus of the Fourier transformed diffraction pattern showing the Patterson map of the maze pattern. d) Plot of the Patterson function in vertical and horizontal directions (blue and green solid line) obtained from radial averaging around the center utilizing circle segments. They reveal the high non-constant contribution of the Patterson function at the center position and flat side lobes (black solid line).

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4. Fourier analysis of the magnetic speckle pattern

The scattering intensity, i.e. the magnetic speckle pattern, can be described by Eq. (2). A Fourier transformation of the recorded speckle pattern yields [29]

In the statistically stationary model, which is the limit of the Schell model where the illuminating field components are uniform and planar [2], the MCF depends only on the separation x of the coordinates ($\Gamma \left(r_1,r_1-x\right)={{\rm I}}_0\gamma \left(x\right)$). The model is frequently described and used in the literature [2,23,30–32,51]. The MCF can then be rewritten as$\Gamma \left(r_1,r_1-x\right)=\Gamma \left(x\right)$and Eq. (7) gives

The CDCs extracted from the Fourier transformed speckle pattern in the framework of both models can be used to determine the transverse coherence length of the X-ray beam as it is described in the following.

5. Determination of the two-dimensional spatial (transverse) coherence

An XRMS experiment was performed at a photon energy of 778 eV using the Co₃₅Pd₆₅ alloy sample. Figure 4(a) displays a magnetic diffraction pattern averaged over fifty successively recorded images, each with an exposure time of 0.02 s. The ring structure indicates scattering from a magnetic maze pattern. The speckle structure within the annulus proves a coherent illumination of the magnetic sample. Figure 4(b) shows the Fourier transform of the speckle pattern. The high-intensity fringe-like structure in the center of the image is the non-constant contribution of the Patterson function. It is only visible in the image center and has a width of 500 nm in total, resulting from the large variation of domain sizes (see above).

 

Fig. 4 a) Magnetic diffraction pattern recorded at 778 eV photon energy from a Co₃₅Pd₆₅ alloy film. The inset shows a small section of the annulus revealing its speckled structure. b) FFT of the magnetic diffraction pattern showing the product of the modulus of the mutual coherence function with the modulus of the Patterson function of the magnetic diffraction pattern (logarithmic scale). The inset displays the center position of the image where the variation of the Patterson function is dominant. The small red shaded areas represent the angular ranges used to determine the MCF and the transverse coherence length in the horizontal and vertical directions.

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The modulus of the MCF can be extracted for all axial direction in the two-dimensional plane using radial averaging of small circle segments around the center position (see Fig. 4(b)). In the framework of the statistically stationary model (see above), the CDC as a function of separation Δx is obtained by normalizing the MCF with its value at zero separation. A Gaussian fit$\exp (-\Delta x^2/2{\xi }_{\text{T}}^2)$yields an upper estimate for the transverse coherence length ξ_T, i.e., the standard deviation of the Gaussian [21,45,46]. Figure 5(a) shows the obtained ξ_T values along different axial directions. Circle segments with an angular width of 10 degree are used. The transverse coherence length is found to be constant for all axial directions and thus shows no asymmetry with respect to the horizontal and vertical direction. In this case, an average MCF and CDC can be extracted using radial averaging of a full circle, which are shown in Fig. 5(b), yielding an average transverse coherence length of ${\xi }_{\text{T,avg}}=\left(15.6\pm \text{0}\text{.5}\right)$µm. The small constant offset of the modulus of the MCF cannot be explained by the theoretical description of the inverse Fourier transform of the speckle pattern, as the mutual coherence function is, by its definition, converging to zero at large separation.

 

Fig. 5 a) Polar diagram showing the transverse coherence length in all axial directions determined in the frame of the statistically stationary model (black circles) and in the frame of the Gaussian Schell model (blue circles). The red dashed lines denote the general shape along the axial directions and the green solid line represents the shape of the incident beam, characterized through its (rms) width${\sigma }_{\text{B}}$in horizontal and vertical directions. b) Modulus of the average MCF obtained by radial averaging around the center position of the Fourier transformed magnetic diffraction pattern (black circles). Within the statistically stationary model, the MCF is normalized with its maximum value close to zero separation resulting in the modulus of the CDC. Using a Gaussian fit (red line) an average transverse coherence length of${\xi }_{\text{T,avg}}=\left(15.6\pm \text{ 0}\text{.5}\right)\text{ µm}$is obtained.

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We found that the offset originates from the read-out noise of the CCD detector, which gives a constant background in the modulus of the Fourier transform even after appropriate dark image correction. The latter results in an additive contribution to the Fourier-transformed speckle pattern due to the linearity property of the Fourier transform.

The global degree of coherence $\zeta$ can be determined using Eq. (5), the transverse coherence lengths${\xi }_{\text{T,v}}=\left(15.3\pm \text{0}\text{.5}\right)$µm and ${\xi }_{\text{T,h}}=\left(15.6\pm \text{0}\text{.5}\right)$µm the rms widths ${\sigma }_{\text{B,v}}\approx 20.8$µm and ${\sigma }_{\text{B,h}}\approx 10.6$µm of the illuminated X-ray beam. The obtained values for the vertical and horizontal direction are${\zeta }_{\text{v}}\approx 0.35$and${\zeta }_{\text{h}}\approx 0.59$. Thus, the total degree of coherence results in$\zeta ={\zeta }_{\text{v}}{\zeta }_{\text{h}}\approx 0.21$. From these results, it is obvious that the horizontal beam size is significantly smaller than the transverse coherence length in the same direction. Thus, the prerequisites for applying the statistically stationary model in the horizontal direction are questionable. In the following, we will therefore reanalyze the same set of data accounting for the respective beam diameter in each direction.

Within the frame of the Gaussian Schell model, the intensity distribution of the incident X-ray beam is included as a normalization factor for the MCF. In this case, the Fourier transformed speckle pattern has to be normalized by$K\left(\text{x}\right)$, the autocorrelation function of square root of the beam intensity distribution to obtain the CDC (see Eq. (9). For simplicity, an elliptical Gaussian beam profile has been assumed here, using the experimentally obtained FWHM values. This means dividing by a Gaussian profile with twice the beam width. In principle, the autocorrelation of the square root of any experimentally obtained two-dimensional beam profile can be used, as long as it shows no variation on the length scale of the magnetic domain pattern. The CDC for all axial directions has been extracted from the normalized Fourier transformed speckle pattern with the same procedure described above and the ξ_T values are determined using Gaussian fits (see Fig. 5(a)). The resulting ξ_T values show a pronounced asymmetry with respect to the vertical${\xi }_{\text{T,v}}=\left(16.2\pm \text{0}\text{.5}\right)$µm and horizontal${\xi }_{\text{T,h}}=\left(24.6\pm \text{1}\text{.5}\right)$µm directions (see Fig. 6). Due to the narrow width of the incident beam in the horizontal direction, the first order peak of the Airy pattern of the beam emerging from the 40 µm pinhole appears at a radius of around 45 µm in the window of investigation.

 

Fig. 6 Modulus of the CDC in horizontal and vertical directions (black circles). The profiles are extracted from the normalized Fourier transformed speckle pattern using radial averaging of small circle segments with an angular width of 10°. Using Gaussian fits (red lines) transverse coherence lengths of${\xi }_{\text{T,h}}=\left(24.6\pm \text{ 1}\text{.5}\right)\text{ µm}$in horizontal and${\xi }_{\text{T,v}}=\left(16.2\pm \text{ 0}\text{.5}\right)\text{ µm}$in vertical direction are obtained. The green triangles represent the CDC values obtained from Young´s double pinhole experiment from averaged line profiles (see Fig.7 (a)) at different pinhole separations Δx. Using Gaussian fits (green dashed lines), we determined a transverse coherence length of ${\xi }_{\text{T,h}}=\left(22.6\pm \text{ 0}\text{.4}\right)\text{ µm}$and ${\xi }_{\text{T,v}}=\left(16.1\pm \text{ 0}\text{.4}\right)\text{ µm}$in vertical and horizontal direction, respectively.

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The presence of this feature confirms the validity of our normalization, as it results in deviations from the assumed Gaussian profile. It has an impact on the extracted CDC at large separations and on the determined transverse coherence length ξ_T, which results in an increased error of our results in horizontal direction. The global degree of coherence yields${\zeta }_{\text{v}}\approx 0.36$, ${\zeta }_{\text{h}}\approx 0.76$for the vertical and horizontal direction and a total degree of coherence of$\zeta ={\zeta }_{\text{v}}{\zeta }_{\text{h}}\approx 0.27$.

The findings of both models reveal that if the coherent fraction of the beam is small compared to the beam size as it is in the vertical direction, the statistically stationary model describes the coherence properties of the beam sufficiently well (see Fig. 5(a)). In this case, the CDC is the dominant contribution and Eq. (9) and Eq. (10) give comparable results. On the contrary, if the coherent fraction of the beam is large compared to the beam size as it is in the horizontal direction, the contribution of the beam intensity distribution is the dominant contribution and has to be taken into account for a correct description of the coherence properties of the beam. In this case, the statistically stationary model underestimates the coherence properties.

The detectable field of view$x_{FOV}=\lambda L/s=112$µm (see Fig. 4(b)) is determined by the wavelength λ, the sample-detector distance L and the pixel size s of the detector, where the number of pixels N defines the resolution in the space domain$x_{res}=\lambda L/Ns=27.4\text{nm}\text{.}$ The MCF can be mapped out up to a separation of $x=112\text{µm}/2=56\text{µm}$due to its centro-symmetry. Hence, it can be easily seen that a sufficiently large sample-detector distance and a small pixel size are prerequisites for the detection of the full two-dimensional MCF.

6. Spatial coherence measurements using Young’s double pinhole experiment

To corroborate the above-described result, a Young’s double-pinhole experiment at a photon energy of 778 eV has been performed. A set of double pinholes is positioned in the sample plane to measure the transverse coherence lengths of the X-ray beam in horizontal and vertical direction.

Diffraction patterns from the double pinholes were recorded using pinhole separations of Δx = 4 µm, 8 µm, 16 µm and 20 µm along the horizontal direction and of Δx = 4 µm, 8 µm,12 µm and 16 µm along the vertical direction. Averaged line profiles were extracted from each diffraction pattern as presented in Fig. 7(a) and 7(b). The intensity distribution of a double pinhole fringe pattern is given by [19–21]

 

Fig. 7 a) Fringe pattern of a double-pinhole arrangement at Δx = 4 µm separation in vertical direction (logarithmic scale). The inset shows a SEM micrograph of the double pinhole structure. The tilt of the diffraction pattern is due to a tilted CCD chip with respect to the vertical axis. The analyzed region is indicated by a red rectangle revealing the averaged line profile (black dots) shown in panel b). The blue line is a fit of the model (see text). The inset shows an enlarged region of the curve at the center.

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Figure 6 shows the obtained values of the modulus of the effective CDC as a function of pinhole separation $\Delta x$. Using a Gaussian fit [21,22], the root-mean-square (rms) transverse coherence length in vertical and horizontal direction can be determined. We obtain${\xi }_{\text{T,v}}=\left(16.1\pm \text{0}\text{.4}\right)\text{µm}$and ${\xi }_{\text{T,h}}=\left(22.6\pm \text{0}\text{.4}\right)\text{µm}$for the vertical and horizontal direction.

The global degree of coherence yields${\zeta }_{\text{v}}\approx 0.36$, ${\zeta }_{\text{h}}\approx 0.73$for the vertical and horizontal direction and a total degree of coherence of$\zeta ={\zeta }_{\text{v}}{\zeta }_{\text{h}}\approx 0.26$.

7. Conclusions

We have deployed a Fourier analysis to the magnetic speckle pattern from a Co₃₅Pd₆₅ alloy sample illuminated by a partially coherent beam at a photon energy of 778 eV in order to extract the two-dimensional spatial (transverse) coherence function of the illuminating X-ray beam. We have found that magnetic samples with maze-domain pattern are perfect objects for this Fourier analysis method as they provide a Patterson function which does not distort the MCF of the Fourier transformed speckle pattern. The coherence properties of the beam in any axial direction have been analyzed within the Gaussian Schell model and the statistically stationary model. The transverse coherence length determined in the framework of the statistically stationary model show no asymmetry with respect to the vertical ${\xi }_{\text{T,v}}=\left(15.3\pm \text{0}\text{.5}\right)$µm and horizontal ${\xi }_{\text{T,h}}=\left(15.6\pm \text{0}\text{.5}\right)$µm directions. The vertical direction is in a good agreement with the one obtained from Young’s double pinhole experiment${\xi }_{\text{T,v}}=\left(16.1\pm \text{0}\text{.4}\right)\text{µm}$, but we find a significant deviation in the horizontal direction${\xi }_{\text{T,h}}=\left(22.6\pm \text{0}\text{.3}\right)$µm. In the Gaussian Schell model and taking the intensity distribution of the beam into account, the extracted transverse coherence length in vertical ${\xi }_{\text{T,v}}=\left(16.2\pm \text{0}\text{.5}\right)$µm and horizontal ${\xi }_{\text{T,h}}=\left(24.6\pm \text{1}\text{.5}\right)$µm direction show both a very good agreement with the ones from Young’s double pinhole experiment. We found that if the coherent fraction of the beam is small compared to the beam size, the statistically stationary and the Gaussian Schell model show equal results as well as a good agreement with Young’s double pinhole experiment. On the contrary, if the coherent fraction of the beam is large compared with the beam size, the intensity distribution of the beam has a strong impact on the extraction of the CDC and the transverse coherence length. In this case, a normalization using the beam profile is required to describe the coherence properties of the beam correctly.

We conclude that by using the Fourier transform of a magnetic speckle pattern the full two-dimensional complex degree of coherence can be extracted without utilizing any apertures such as double pinholes or non-redundant arrays of apertures. Thus, the Fourier analysis method presented here can be used for an online check of the coherence properties in experiments which are very sensitive to the degree of coherence. Furthermore the method is not restricted to magnetic samples and can be applied to various other sample systems with a high degree of structural size variations along with an emerging speckle pattern.