Experimental evaluation of signal-to-noise in spectro-holography via modified uniformly redundant arrays in the soft x-ray and extreme ultraviolet spectral regime [original]

Journal of Optics
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Experimental evaluation of signal-to-noise in
spectro-holography via modified uniformly
redundant arrays in the soft x-ray and extreme
ultraviolet spectral regime
To cite this article: Christian M Günther et al 2017 J. Opt. 19 064002

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Experimental evaluation of signal-to-noise in
spectro-holography via modi ﬁ ed uniformly
redundant arrays in the soft x-ray and
extreme ultraviolet spectral regime
Christian M Günther
1
, Erik Guehrs
1
, Michael Schneider
2
, Bastian Pfau
2
,
Clemens von Korff Schmising
2
, Jan Geilhufe
2 , 3
, Stefan Schaffert
1 , 4
and
Stefan Eisebitt
1 , 2
1
Technische Universität Berlin, D-10623 Berlin, Germany
2
Max-Born-Institut, D-12489, Berlin, Germany
E-mail: [email protected] ( Christian Günther ) and [email protected] ( Stefan Eisebitt )
Received 20 July 2016, revised 10 February 2017
Accepted for publication 28 February 2017
Published 8 May 2017
Abstract
We present dichroic x-ray lensless magnetic imaging by Fourier transform holography with an
extended reference scheme via a modi ﬁ ed uniformly redundant array ( mURA ) . Holographic
images of magnetic domains simultaneously generated by a single pinhole reference as well as
by a mURA reference are compared with respect to the signal-to-noise ratio ( SNR ) as a function
of exposure time. We apply this approach for spectro-holographic imaging of ferromagnetic
domain patterns in Co / Pt multilayer ﬁ lms. Soft x-rays with wavelengths of 1.59 nm ( Co L
3
absorption edge ) and 20.8 nm ( Co M
2,3
absorption edges ) are used for image formation and to
generate contrast via x-ray magnetic circular dichroism. For a given exposure time, the mURA-
based holography allows to decouple the reconstruction SNR from the spatial resolution. For
1.59 nm wavelength, the reconstruction via the extended reference scheme shows no signi ﬁ cant
loss of spatial resolution compared to the single pinhole reference. In contrast, at 20.8 nm
wavelength the single pinhole reveals some very intricate features which are lost in the image
generated by the mURA, although overall a high-quality image is generated. The SNR-
advantage of the mURA scheme is most notable when the hologram has to be encoded with few
photons, while errors associated with the increased complexity of the reconstruction process
reduce the advantage for high-photon-number experiments.
Keywords: magnetic circular dichroism, imaging and optical processing, holography, x-ray
imaging, diffraction ef ﬁ ciency, resolution and other hologram characteristics, magnetic domains
in thin ﬁ lms
( Some ﬁ gures may appear in colour only in the online journal )
1. Introduction
Fourier transform holography ( FTH ) is an imaging technique
that is based on the interference of the light scattered by an
object with a spherical reference wave originating in or close
to the object plane [ 1 ] . With the appreciable coherent photon
ﬂ ux from 3rd generation synchrotrons as well as the increased
availability of free-electron x-ray lasers on one hand and
Journal of Optics
J . Opt. 19 ( 2017 ) 064002 ( 11pp ) https: // doi.org / 10.1088 / 2040-8986 / aa6380
3
Present address: Canadian Light Source, SK S7N 2V3, Saskatoon, Canada.
4
Present address: Siemens Healthcare GmbH, 91052 Erlangen, Germany.
Original content from this work may be used under the terms
of the Creative Commons Attribution 3.0 licence . Any
further distribution of this work must maintain attribution to the author ( s ) and
the title of the work, journal citation and DOI.
2040-8978 / 17 / 064002 + 11$33.00 © 2017 IOP Publishing Ltd Printed in the UK 1

laser-driven high harmonic generation ( HHG ) sources on the
other hand, FTH has become possible over a large spectral
range, including the extreme ultraviolet ( XUV ) , the soft x-ray
regime and even hard x-rays [ 1 – 3 ] . With the advent of 4th
generation storage rings that will be diffraction-limited sour-
ces of x-rays well beyond 1 keV photon energy, interference-
based imaging can be expected to gain further importance,
similarly to its use at free-electron x-ray lasers [ 4 – 8 ] . For
laboratory sources, extending the wavelength range towards
such short wavelengths with coherent photon ﬂ uxes suitable
for high-resolution imaging is still a formidable challenge, but
progress in recent years has been fast [ 9 – 14 ] . Nevertheless,
while FTH with x-rays has become routine at 3rd generation
synchrotron sources [ 1 , 15 – 20 ] , when pushing the resolution
below the 10 nm region, investigating very weak contrast
objects, imaging with extreme temporal resolution or even via
a single femtosecond ﬂ ash exposure at short wavelengths, the
signal-to-noise ratio ( SNR ) achievable in the hologram at
high momentum transfer remains a challenge.
In mask-based FTH, a spherical reference wave is created
by a single reference aperture in the object plane [ 1 , 2 ] . The
image is generated by a two-dimensional Fourier transform,
where the achievable spatial resolution is limited by the lateral
dimension of this pinhole as well as the maximum momentum
transfer signal recorded. As a consequence, a tradeoff exists
between spatial resolution and contrast of the generated
image, as the intensity of the reference wave modulating the
hologram at high momentum transfer is typically the limiting
factor. This is the limit we consider in this work. A direct way
to improve the signal-to-noise ratio is reference multiplexing,
the generation of multiple independent images by multiple
references at suitable locations [ 4 , 21 ] . Here, the ‘ real estate ’
available for image reconstructions that can be added up is
connected to the size of the Fourier transformed hologram, the
so called Patterson map and, together with the pixel resolu-
tion, limited by the detector pixel array [ 1 ] . Extended refer-
ence approaches allow to overcome the tradeoff between
intensity and resolution by collecting more photons to con-
tribute to the reference signal, while at the same time pro-
viding a sharp point spread function and, thus, conserving the
high spatial resolution. In principle, very general structures
may be used to provide a reference signal [ 22 ] . For a known
shape of the reference aperture its contribution can be
removed from the cross-correlation term with the object,
yielding an image of the object. However, for certain special
reference schemes it is possible to obtain an image in a more
direct way without the necessity of an iterative reconstruction
algorithm. Representatives of such schemes with the goal to
increase reference wave intensity are uniformly redundant
arrays ( URAs ) [ 23 ] and Fresnel Zone plates [ 24 ] . The
reconstruction is obtained by operators based on convolution
or propagation.
In this work, we consider the SNR in magnetization
imaging by a modi ﬁ ed URA ( mURA ) [ 23 , 25 , 26 ] .A n
additional single pinhole is used to obtain an independent
FTH image in the exact same measurement, allowing to
compare the respective SNR. We image the nanoscale domain
pattern of perpendicular magnetic anisotropy samples via
x-ray magnetic circular dichroism ( XMCD ) [ 2 , 27 ] .W e
compare the magnetization contrast in images that are gen-
erated from the difference of two holograms recorded with
opposite helicities with images from a single helicity. The
latter case is especially important for destructive single-shot
measurements at XFEL sources. Finally, we demonstrate the
use of mURAs for magnetic imaging not only for one of the
widely used L -edges of 3d transition metal compounds, but
also for the M -edges in the XUV range, where magnetic
circular dichroism has recently been shown to be accessible
with HHG lab sources [ 28 ] .
2. Methods
2.1. Holography geometr y and magnetic sample composition
Resonant soft x-ray holography was carried out in Fourier
transform geometry, realized via a micro / nano-structured
mask integrated with the sample as introduced by Eisebitt
et al [ 2 ] . On a silicon nitride ( Si
3
N
4
) membrane an optical
mask is de ﬁ ned on one side and the specimen to be imaged is
rigidly attached on the opposite surface of the membrane [ 29 ] .
In our case the specimen to be imaged is the projected 2D
magnetization distribution within a thin metallic multilayer
ﬁ lm with perpendicular magnetic anisotropy. The parameters
of the three components — substrate, mask and magnetic layer
— are optimized for the scattering experiments at the very
different wavelengths in the XUV ( λ  =  20.8 nm, ħ ω  =
59.6 eV ) and soft x-ray ( λ  =  1.59 nm, ħ ω  =  778 eV ) regime.
The exact wavelengths used are determined by the ability to
generate magnetic contrast via XMCD at the corresponding
Co M
2,3
and L
3
absorption edges [ 30 , 31 ] . At both edges,
holograms with opposite circular helicity of the incoming soft
x-rays are recorded.
For L- edge imaging the thicknesses of the Si
3
N
4
mem-
brane and the Au layer fabricated by thermal evaporation are
100 nm and 1 μ m, respectively. The latter is used to de ﬁ ne the
holography geometry by focused-ion-beam milling. A cir-
cular object aperture of 2 μ m diameter is de ﬁ ned in the Au-
layer only. Laterally offset from the circular object mask, a
single pinhole and a mURA are de ﬁ ned as structures each
providing independent reference beams [ 21 ] , as shown in the
main panel of ﬁ gure 1 . These reference apertures penetrate the
complete sample structure, including the magnetic thin ﬁ lm.
Since the aspect ratio is a limiting factor for the focused-ion-
beam generation of such structures, the magnetic layer was
removed locally to ensure that pinholes with apex diameters
below 50 nm could be fabricated. The diameter of the pin-
holes again is a resolution-limiting factor in single-pinhole or
mURA FTH. The advantage of the mURA lies in faster data
acquisition as each individual pinhole generates an indepen-
dent signal that will add in the ﬁ nal decoded image. Upper
boundaries on the size of the mURA are set on one hand by
the associated required increase of the object-mURA distance,
which is limited by the coherence length of the light as well as
the q -sampling parameters of the detector. On the other hand,
a strong imbalance of the signal strengths originating from
2
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

object and reference results in reduced contrast in the ﬁ nal
image.
Here, the experimental realization of the mURA shown
in ﬁ gure 1 ( a ) consists of 60 transmissive pinholes on a square
grid with 300 nm pitch, similar to some of the structures used
by Marchesini et al [ 23 ] , but with individual apertures about a
factor of two smaller. The theoretical layout of the mURA
featuring 11  ×  11 elements and the corresponding decoding
pattern are visualized in panels ( b ) and ( c ) of ﬁ gure 1 [ 25 , 26 ] .
Note that the decoding pattern has a value range of [ − 1, + 1 ] ,
while the mURA is de ﬁ ned by a transmission function ran-
ging from zero to unity. Deviations from the theoretical pat-
tern, e.g. variations in position, shape and size of the
individual pinholes result in imperfect decoding and thus
contribute additional noise to the decoded image which is
discussed in the results section.
AC o / Pt multilayer ﬁ lm of 100 nm thickness, with
composition Pt ( 30 Å ) / [ Co ( 12 Å ) / Pt ( 7Å ) ]
50
/ Pt ( 13 Å ) , was
fabricated by magnetron sputter deposition onto the Si
3
N
4
membrane. Pt ( 30 Å ) serves as growth buffer layer, while the
top Pt ( 13 Å ) layer prevents oxidation. The corresponding
hysteresis loop measured by polar magneto-optical Kerr effect
( MOKE ) indicates perpendicular magnetic anisotropy
( ﬁ gure 1 ( d )) which is typical for such multilayer systems [ 32 ] .
Demagnetization along the sample normal generates a
labyrinth domain pattern with alternatingly up and down
magnetized domains as observed in the magnetic force
microscopy ( MFM ) image in ﬁ gure 1 ( e ) .
For M -edge imaging the thickness of the Si
3
N
4
substrate
is reduced to 30 nm in order to minimize the substrate
absorption at 59.6 eV photon energy, where one attenuation
length is only 50 nm. At this photon energy, a [ Cr ( 5n m ) / Au
( 50 nm ) ]
5
multilayer with 300 nm overall thickness is suf ﬁ -
cient as an opaque mask to de ﬁ ne the holography geometry,
which laterally is similar to the one of the L -edge sample. The
diameter of the object aperture is again 2 μ m, the size of the
reference pinhole and the mURA pinholes was set to yield
apertures below 65 nm diameter. Due to an astigmatism of the
ion beam the ﬁ nal pinholes exhibit an oval shape with axes of
65 nm and 80 nm length. The composition of the magnetic
thin ﬁ lm is Ta ( 20 Å ) / Pt ( 30 Å ) / [ Co ( 8Å ) / Pt ( 14 Å ) ]
11
/ Pt ( 6Å )
resulting in a thin ﬁ lm of 30 nm overall thickness. The MOKE
hysteresis loop and a MFM image revealing the typical
labyrinth pattern of out-of-plane domains are shown in
ﬁ gures 1 ( d ) and ( f ) , respectively.
2.2. Experimental setup
The x-ray experiments were performed at the BESSY-II
synchrotron source in Berlin, Germany. A schematic of the
general FTH layout is shown in the main panel of ﬁ gure 1 .
The diffraction pattern generated upon coherent illumination
of the sample described above is recorded by a
27.6  ×  27.6 mm
2
( 2048 pixels  ×  2048 pixels ) in-vacuum,
back-illuminated charge-coupled device ( CCD ) camera
( Princeton Instruments PI-MTE ) .C o L - and M -edge imaging
Figure 1. General FTH setup ( a ) SEM image of the pinhole array representing the ( b ) theoretical mURA. ( c ) Corresponding decoding pattern
for ( b ) . Characterization of the magnetic properties of the thin ﬁ lms by ( d ) out-of-plane MOKE measurements and 5  ×  5 μ m
2
MFM-images
of the domain pattern for the samples used for ( e ) L - and ( f ) M -edge imaging.
3
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

was performed at the UE56 / 2-PGM-1 and UE112-PGM1
beamlines, respectively. The experimental parameters for
both beamlines are given in table 1 [ 33 ] . The maximum
XMCD contrast was determined by measuring the energy-
dependent transmission through a magnetically saturated
cobalt reference sample as function of the helicity of the
incident radiation. A circular beamstop of about 1 mm dia-
meter mounted in front of the CCD camera blocked the
intense direct beam to account for the limited dynamic range
of the CCD.
For L -edge imaging the strong spatial ﬁ ltering caused by
the high focus – sample distance was dictated by vacuum
chamber constraints. The coherence of the beamline was
determined by recording the scattering of a known aperture
array and comparing it to the pattern predicted by theory for
fully coherent illumination. Lower bounds for the horizontal
and vertical coherence lengths of 3.3 μ m and 4.2 μ m were
obtained, respectively [ 34 ] .
For measurements at the Co M -edge a 200 nm Al ﬁ lter
upstream of the sample was used to reduce the third harmonic
radiation residually present from the undulator. Given a
maximum scattering angle of 18 ° from center to edge of the
CCD chip, the small-angle approximation is not valid and the
recorded hologram needs to be corrected for Ewald sphere
curvature as well as for refraction effects [ 35 ] . An additional
guard aperture with approximately 1.5 mm diameter against
diffuse stray light bypassing the sample holder was placed
directly downstream of the sample.
3. Results
3.1. Hologram reconstr uction and mURA decoding
At the Co L -edge a set of 15 separate frames each with 60 s
exposure time was recorded for both helicities. After dark
ﬁ eld subtraction, each frame was normalized to denote the
number of detected photons. In ﬁ gure 2 ( a ) the decadic loga-
rithm of the sum of all 15 frames with positive helicity is
shown on a pseudocolor intensity scale. The central part close
to zero momentum transfer is blocked by the beamstop.
Adjacent, pronounced Airy fringes generated by the circular
object aperture are evident. In the momentum transfer range
of about q  =  25 μ m
− 1
to 50 μ m
− 1
the resonant scattering
caused by the presence of magnetic domains [ 36 ] is strongly
visible. These values translate to a domain periodicity of
125 – 250 nm. The prominent lattice of Bragg peaks is gener-
ated by the mURA. The brightest spots are separated by a Δ q
of 21 μ m
− 1
corresponding to the smallest distance between
neighboring elements of the mURA of 300 nm. Note that
these intense peaks — which will get more pronounced for
larger URAs — can present a problem if detectors with a
limited dynamic range such as our CCD are used. Once the
Table 1. Summary of the experimental parameters at the two
BESSY-II beamlines [ 33 ] .
UE56 / 2-
PGM-1 UE112-PGM1
energy ( eV ) 778 59.6
energy resolution ( eV ) 0.880 0.027
ﬂ ux ( ph / ( s 0.1% bandwidth )) 4  ×  10
13
4  ×  10
13
divergence ( horz. × vert. )( mrad ) 10  ×  10 1.4  ×  0.6
focus – sample distance ( cm ) 28 11
sample – detector distance ( cm ) 41 4.1
detectable momentum transfer
q
max
( μ m
− 1
)
266 204
diffraction-limited resolu-
tion ( nm )
48 65
Figure 2. ( a ) Hologram recorded with positive helicity accumulated for 15  ×  60 s. The pseudocolor intensity scale corresponds to the
logarithm of the number of detected photons. ( b ) Difference of two holograms ( each 15 x 60 s ) recorded with opposite helicities on linear
intensity scale. In both panels the full detector image ( 2048  ×  2048 pixels ) is shown.
4
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

Bragg peak intensities reach values limiting a weaker sample
scattering being detected due to the given dynamic range, one
would like to attenuate these beams similar to the q  =  0
beam, which is obviously a dif ﬁ cult task. Figure 2 ( b ) shows
the difference of two holograms recorded with positive and
negative helicities ( 15 frames each ) on linear intensity scale.
Here the non-magnetic charge part of the scattering cancels
out to ﬁ rst order in the subtraction and, thus, the magnetic part
of the scattering signal is enhanced. As a consequence, one
can directly observe weak magnetic scattering up to
q  =  100 μ m
− 1
.
In ﬁ gure 3 the central part of the Patterson map obtained
by a two-dimensional Fourier transform of the positive heli-
city hologram  visualized in ﬁ gure 2 ( a ) is presented. Numer-
ical treatment of the hologram included q  =  0 centering,
cosmic ray removal and smoothing the abrupt intensity
changes at the beamstop edges to reduce FFT artifacts as well
as an interpolation of the ﬁ nal Patterson map to half the
pixel size.
Beams through all transmissive areas of the holography
mask are able to interfere and generate respective cross-terms
in the Patterson map. The center contains the autocorrelation
of all transmissive areas and it is surrounded by strong ringing
artifacts due to the beamstop, which damp out towards the
cross-correlation areas of the Patterson map. The three cross-
correlation pair ‘ images ’ of object, mURA and single pinhole
aperture form at locations de ﬁ ned by the connecting vector of
the speci ﬁ c two apertures [ 1 ] . In detail these are the cross-
correlations of: pinhole & object forming above and below
the center at position marked ‘ A ’ . This term corresponds to
standard mask-based FTH and directly provides an image of
the magnetic domain state within the object aperture de ﬁ ning
the ﬁ eld of view ( FOV ) . Pinhole & mURA cross-correlations
are located in the lower left and upper right corner, labeled
‘ B ’ . Here, the single pinhole directly images the transmission
of the mURA pinhole array. In that way, the actual trans-
mission through the individual mURA pinholes can be
directly measured in the very same experiment used to image
the object. Note that this at-wavelength measurement of the
transmission and exact location of each source point within
the mURA is much superior to SEM images of the reference
object and can be used to correct mURA imperfections to
obtain higher resolution, as is generally true with knowledge
of extended references [ 22 ] . One could thus use a single high-
resolution aperture to characterize a larger reference structure
such as the mURA to combine highest spatial resolution with
optimized reconstruction intensity. While this would require a
two-step, but non-iterative analysis, the data could be
acquired in a single exposure. This analysis is beyond the
scope of this paper; here we restrict ourselves to determining
the exact mURA orientation ( 2.3 ° ) and pixel-pitch ( 25.1
pixel ) from this measurement, as these parameters are
required for decoding the cross-correlation of the object and
the mURA reference. The diameter of single spots in this
image of the mURA is 50 nm. Note that this value results
from the cross-correlation of two ( nominal ) identical pin-
holes. Thus, assuming a Gaussian shape for the transmission
of the individual apertures, their transmission FWHM is a
factor of 2 smaller. Finally, the mURA & object cross-cor-
relation term is clearly visible in the Patterson map in ﬁ gure 3
at ‘ C ’ . This cross-correlation term constitutes the super-
position of the 60 images each generated by one of the 60
open elements of the mURA, analogous to the isolated pin-
hole, and as previously demonstrated by Marchesini et al
Figure 3. Center region ( 45%  ×  30% ) of the real part of the 2D-Fourier transform of the hologram visualized in ﬁ gure 2 ( a ) . The linear
intensity scale was adjusted for good visibility of the three cross-correlation images generated between pinhole, mURA and object.
5
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

[ 23 ] . Since the intermediate distance between the mURA
pinholes is smaller than the object size, these images are
overlapping. Note that for larger mURAs containing more
pinhole apertures the mURA & object cross-correlation term
grows accordingly and requires an increased distance between
both apertures to avoid overlap. The maximum distance
between the apertures is limited by the coherence length and
the pixel number of the detector [ 5 ] .
The FTH image ( A ) and the mURA-object cross-corre-
lation ( C ) are cropped for further processing. For both, the
contrast is maximized into the real part by complex rotation as
the analysis of the spatial domain distribution rather than an
absorption versus phase contrast analysis is the typical ima-
ging goal for this kind of sample.
An image from the mURA & object cross-correlation is
obtained by convolution with the 2  ×  2 tiled decoding pattern
( ﬁ gure 1 ( c )) matched to the pixel distance of the mURA in the
reconstruction. Note that for the scaled decoding pattern the
individual 2  ×  2  ×  11
2
elements represented by single pixels
with values of ± 1 are separated by large areas of zeros which
are omitted in ﬁ gure 1 ( c ) . For clarity, in ﬁ gure 3 we have
demonstrated the contributions to the Patterson map using a
single-helicity measurement. It is clear that with suf ﬁ cient
SNR such as obtained from the hologram in ﬁ gure 2 ( a )
domain images can easily be obtained with one helicity only
[ 4 , 35 ] . This is of particular relevance for destructive single-
shot experiments, e.g., at free-electron x-ray lasers. Now we
will continue the discussion with standard helicity difference
holograms, which allow to suppress non-magnetic contribu-
tions [ 1 , 16 , 17 ] .
3.2. Ev aluation of the signal-to-noise ratio
The hologram reconstruction process as described in the
previous section produces two independent images of the
sample ’ s nanoscale magnetic domain pattern, one originating
from the single pinhole reference and the other from the
mURA. These two images generated from the Co L -edge
helicity difference hologram shown in ﬁ gure 2 ( b ) are com-
pared in ﬁ gures 4 ( a ) and ( b ) , respectively. Both, the single
pinhole and the mURA provide almost identical reconstruc-
tions allowing to unambiguously identify the arrangement of
the up / down oriented ferromagnetic domains in the multi-
layer ﬁ lm. Only very slight differences can be found, for
example looking at the bubble-like domain indicated by the
green arrow, illustrating the high level of con ﬁ dence in the
structure obtained. A reason for these minute deviations might
be imperfect matching of the decoding pattern to the exper-
imental data. Line pro ﬁ les ( ﬁ gure 4 ( e )) crossing domain
borders at the indicated locations con ﬁ rm a spatial resolution
of approximately 50 nm for both images, as de ﬁ ned by the
10% – 90% criterion. Note that this is a conservative estimate
as the domain wall itself can have a sizeable width, depending
on multilayer composition and thickness [ 37 ] . That value ﬁ ts
well with the recorded momentum transfer of q  =  266 μ m
− 1
which translates to a spatial resolution of 48 nm and is cor-
roborated by the diameter of 50 nm of single spots in the
pinhole / mURA cross-correlation term. As the hologram
leading to the reconstructions in ﬁ gure 4 is recorded with
1800 s overall exposure time and correspondingly high SNR,
a signi ﬁ cant advantage of the mURA hologram over the
standard single pinhole reference version is not to be
expected.
Our samples would in an ideal case result in a bimodal
intensity histogram due to the dominance of magnetic
domains either being fully magnetized up or down, i.e.,
appearing black or white, with only smaller fractions of the
FOV being in intermediate states at the domain walls and thus
represented as shades of gray [ 37 ] . We therefore quantify the
SNR in the reconstructed images by analysis of the image
contrast. In ﬁ gures 4 ( c ) and ( d ) histograms featuring 40 bins
generated from panels ( a ) and ( b ) visualize the contrast dis-
tribution within the circular FOV ( signal ) and the surrounding
area ( noise ) .
The latter region should in an ideal case be at a constant
intermediate gray level, as no signal from this masked area
contributes to the hologram. Intensity ﬂ uctuations in this
region of the reconstruction are thus indicative of systematic
errors, and the ratio of the two variance values within and
Figure 4. The magnetic domain patterns reconstructed from the
helicity-difference hologram in ﬁ gure 2 ( b ) generated by ( a ) the FTH
pinhole and ( b ) the mURA. The contrast is scaled to 1% – 99% and
the circular FOV is 2 μ m in diameter. Histograms of the FOV
( signal ) and the surrounding area ( noise ) are visualized in ( c ) for the
FTH and ( d ) the mURA image. Bins correspond to the intensity
values in the reconstructions. ( e ) Normalized line pro ﬁ les as
indicated in panels ( a ) and ( b ) .
6
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

outside the FOV can be used to quantify how much stronger
real variations in the image contrast are detectable over arti-
ﬁ cial ﬂ uctuations which are seen outside the FOV and can be
expected to exist within the FOV at the same level. For the
pinhole- and mURA-generated images the signal exhibits two
peaks distributed symmetrically around zero generated by the
black and white domains. The noise follows a Gaussian dis-
tribution around zero. We approximate the widths of these
histogram curves, which is proportional to the contrast, with
the square root of the variance, i.e., the standard deviation, of
the contributing image part. Finally, we denote the ratio of
these two standard deviations inside and outside of the
FOV as SNR of the image. For ﬁ gures 4 ( a ) and ( b ) we
obtain values of SNR
pinhole
 =  3.9 and SNR
mURA
 =  4.8,
respectively.
To investigate the SNR advantage of a mURA hologram
when an image has to be formed with less incident photons,
the separately recorded frames each with 60 s exposure time
of the hologram are used to illustrate how the contrast builds
up in the image series. The reconstructions shown in
ﬁ gures 5 ( a ) – ( d ) consider only the scattered photons detected
up to frame numbers of 1 to 4, 10 and 15. Photons detected in
the immediate vicinity of the beamstop shadow were exclu-
ded up to the same momentum transfer boundary for each
hologram, in order to avoid any systematic errors associated
with a potential shift of the beamstop relative to the
diffraction pattern. In panels ( a ) and ( b ) helicity difference
reconstructions are shown, while panels ( c ) and ( d ) corre-
spond to reconstructions from single-helicity holograms
alone. For both cases the images generated by the single
pinhole (( a ) and ( c )) and the mURA (( b ) and ( d )) are
presented.
Inspection of the noise and contrast in the reconstructed
images clearly shows that for lower total exposure time better
images are obtained for the mURA reference compared to the
pinhole reference, and that helicity difference images are
superior to single helicity reconstructions. As can be expec-
ted, for very high total exposure resulting in high SNR in the
holograms, these differences become less pronounced. As
detailed before, we calculate the SNRs for each of the 15
frames within the exposure series. The corresponding plots of
the SNR are shown in ﬁ gure 5 ( e ) . The SNR values are plotted
over the square root of the detected photon number ( rather
than over the square root of the exposure time ) to compensate
for varying sample illumination caused by a slow drift of the
x-ray beam that was observed during the recording of the
exposure series. Note that scaling the x -axis with the square
root of detected photons introduces a non-linearity for the
growth of the detector ’ s readout noise which increases line-
arly with the square root of the number of exposures.
In in-line holography, only photons that have passed the
ﬁ rst pinhole aperture contribute to the scattering from the
Figure 5. Domain patterns generated from helicity-difference holograms ( compare ﬁ gure 2 ( b )) with exposure times of 120 s per frame
number generated by ( a ) the single pinhole and ( b ) the mURA. ( c ) Single pinhole and ( d ) mURA generated images from positive helicity
alone ( compare ﬁ gure 2 ( a )) . Here, the exposure time is 60 s per frame number. ( e ) SNR derived from panels ( a ) – ( d ) , ﬁ lled markers
correspond to images shown. ( f ) SNR
mURA
/ SNR
pinhole
for images recorded from helicity differences and positive helicity only.
7
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

object to be imaged. The intensities of object and reference
beam are thus not entirely independent. In this case the SNR
was found to scale with the square of the number of photons
that are incident on the object [ 38 ] , in accordance with what
one may expect from Poisson photon statistics. Similarly, for
a given intensity balance between reference and object wave
( which is of course subject to change for imaging the same
object with different reference structures such as a single
pinhole and a mURA ) , the same quadratic dependence is
expected in FTH [ 23 ] .
In th e SNR es tima te pl ot s in ﬁ gu re 5 ( e ) we ob se rv e the
tre nd th at fo r sing le -p in ho le -b as ed F TH the SNR gr ow s with
the sq ua re root of the nu mb er of ac cu m ul at ed ph ot on s, wi th
he li ci ty -d if fere nc e ho l ogra m s pr ov id in g subs ta nt ia ll y in cr ea se d
SN R for a give n phot on nu mb er . A smal l de viat io n from th is
lin ear be havi or is ob serv ed, ca used by th e dr if t of th e ill um i-
na ti on disc usse d abov e. In co nt rast , the SNR fo r th e mURA
de vi at es fr om this Poi sson st ati stic s be havi or, with co mpara -
tiv ely lo we r SNR incr ease at hi gher ex posu res. We at trib ute
thi s to syst emat ic co nt ribu tion s to the ba ckgr ound no ise in the
ima ges ge nera ted vi a the mU RA re fere nc e. Not e th e diff eren t
ch ar ac te r of in te ns ity ﬂ uc tuat ions ou tsid e the F OV in the
re cons tr uc ti on s i n ﬁ gu re s 4 ( a ) , ( b ) for th e di ff eren t refe renc e
sch emes . In thi s area , whi ch shou ld appe ar ﬂ at , an incr ease of
ﬂ uc tuat ions wi th a spat ial corr elat ion leng th simi lar to th e in-
pl ane doma in corr elat io n len gt h is obse rved in t he mUR A
re cons tr uc ti on i n ( b ) , bu t not i n th e sing le pi nhol e reco ns tru c-
tion i n ( a ) . We susp ect th e caus e to be im perf ect m atch ing of
the m UR A deco di ng pa tt ern to th e real sp at ia l dist ribu tio n of
the mURA t rans mis si on fu ncti on. As a re sult , ther e is a
co mpo nent in th e back grou nd no ise th at scal es wit h the si gnal
fro m the ob ject . Con sequ entl y, th e SN R for th e mURA t ends to
gr ow sl ower than the s quar e root of th e ex po su re fo r high
ex po su re ti mes. On the ot her ha nd , th e subs tan ti al SNR
ad va nt ag e of th e mURA re cons truc tion s ov er th e sin gl e pin-
ho le FTH fo r low er phot on numb ers is evid ent from th e data
co mpi led in ﬁ gu re 5 .
To directly compare the SNRs of mURA and pinhole,
their ratio SNR
mURA
/ SNR
pinhole
is plotted in ﬁ gure 5 ( f ) . For
low photon numbers, the mURA provides a 1.5 times higher
SNR than the pinhole for a single difference image and the
factor rises to 2.5 for a magnetization map obtained from a
helicity difference hologram. According to Marchesini et al
[ 23 ] for an URA with n open elements the SNR scales with:
n
S

NR SNR
22
n
1
=
resulting in SNR
60
/ SNR
1
 =  2.7 which is in reasonable
agreement with our observations for difference-helicity ima-
ging in the low-photon limit, where the counting statistics of
photons detected in the hologram is expected to be the
dominant limiting factor. We suspect that the reduced SNR
gain in the case of single-helicity holography is due to arti-
facts that originate from non-magnetic contributions. These
include ringing of the Fourier transform at steep intensity
edges and affect the background SNR . Such contributions are
usually eliminated in reconstructions from difference holo-
grams. A detailed understanding of this ﬁ nding will have to
await numerical modeling, which is beyond the scope of this
article. The less important photon counting statistics does
become a dominant source of noise, the more the SNR ratio
decreases. This is the case for increasingly higher exposure
times for both single and difference helicity reconstructions
for the reasons discussed above. Under high-dose conditions
where the incident coherent photon ﬂ ux is not a limiting
factor, the ef ﬁ ciency gain of holographic encoding via more
complex high-intensity references such as a URAs or FZPs
[ 24 ] is thus limited, while the complexity of experiment and
the potential for systematic errors introduced, e.g., via
imperfect knowledge of the URA, is increased. Vice versa,
the ef ﬁ ciency advantage is obviously relevant when the
available coherent photon ﬂ ux is limited, as will be encoun-
tered for many experiments in single-shot experiments or
when using table-top sources.
We would like to point out that the analysis presented
here, based on a multiplexed experiment containing a single
pinhole and a mURA reference at the same time, neglects any
cross-talk effects of the two holograms recorded on the same
detector within the same exposures. These depend on several
experimental factors including the coherence of the illumi-
nation and the dynamic range of the detector.
3.3. mURA spectro-holography at the Co M-edge
While XMCD is a very strong contrast mechanism at the 3d
transition metal L -edges with asymmetries, e.g., exceeding
20% for iron and cobalt [ 31 ] , much weaker contrast on the
order of only 1% is available at the respective M -edges. At the
same time, the formation of a high-resolution image is more
challenging given that the wavelength of the radiation
employed is more than a factor of ten larger compared to the
L
3
resonances of these elements. Nevertheless, spectro-holo-
graphic imaging at these wavelength is of large interest for
sub-picosecond time-resolved studies, as this spectral range
can currently be reached by the soft-x-ray free-electron lasers
FLASH [ 5 , 7 , 39 ] and FERMI [ 8 ] and has become increas-
ingly accessible with suf ﬁ cient coherent photon ﬂ ux via the
rapidly developing laser-based laboratory sources exploiting
HHG [ 13 , 14 , 28 ] . Imaging with XMCD contrast in the XUV
spectral range is thus an interesting test case for mURA-based
spectro-holography.
With a suitably designed sample as described above, we
have carried out a FTH experiment again based on a multi-
plexed measurement containing both a single aperture and a
mURA reference structure. A single dataset consisting of two
holograms with opposite helicity and an exposure time of
150 s each was recorded. The helicity-difference hologram
and the resulting reconstructions are presented in ﬁ gure 6 .
Focusing on the Bragg peaks in ﬁ gure 6 ( a ) generated by
the mURA it becomes evident that their regular spacing is
mapped with increasing distortion for increasing momentum
transfer. This is a direct consequence of the large scattering
angle up to 18 ° and the resulting mapping of the strongly
curved Ewald sphere on the ﬂ at 2D detector. The appropriate
corrections to obtain high-quality images have been discussed
elsewhere [ 35 ] and are applied here, together with a
8
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

Bla ckma nn-H arri s win do w func tion to re du ce FF T arti fa ct s
or igi nat in g from a guar d aper ture be hi nd th e samp le crea ting a
sha rp cut- off at the hi gh scat te ri ng an gl es . Agai n, t he mUR A
pi tch and ro ta ti on ca n be dete rmin ed from th e at-w avel engt h
tra nsm is sion of t he mURA as ma pped vi a the cr oss- corr elat ion
wit h the sing le pinh ole re fere nce. T he di am et er of si ngle sp ots
in th is cr oss- co rrel atio n corr espo nds t o 90 nm .
The resulting pinhole and mURA reconstructions are
shown in ﬁ gures 6 ( b ) and ( c ) , respectively. In these images,
after zeropadding, one pixel corresponds to 19 nm. As in the
case of L -edge holography, in general the domain patterns
seen in the reconstructions agree very well. A line pro ﬁ le
crossing domain walls indicates a spatial resolution of about
90 nm, as de ﬁ ned by the 10% – 90% criterion. Again, note that
this is a conservative estimate due to the existence of a
magnetization transition region in the domain wall [ 37 ] .
However, the pinhole image reveals small domains which are
lost in the mURA reconstruction, examples are marked by the
green arrows. These small features appear ‘ smeared over ’
with intermediate contrast corresponding to unphysical
‘ intermediate magnetization ’ at positions in the labyrinth
domain pattern where a full remanent magnetization can be
expected. Apparently, the overall spatial resolution of the
single pinhole is thus higher compared to the mURA. We
attribute this ﬁ nding to inhomogeneities of the 60 pinholes
forming the mURA and consequently a mismatch of the
decoding pattern. In fact, SEM images recorded after the
experiment revealed a strong contamination of the illuminated
areas, i.e., the object and reference apertures, probably with
carbon which is a typical deposit on optical elements for high-
intensity x-ray illumination due to cracking of residual gas
contaminants. While such contaminations play a minor role at
the Co L -edge, where the transmission through carbon is high,
its effect is much stronger for photons of lower energy at the
Co M -edges. Such a contamination may increase the resolu-
tion of a single pinhole by reducing its diameter. On the other
hand, for each individual pinhole of an URA the con-
tamination may affect the size and shape somewhat differ-
ently, distorting the homogeneity of the array and, thus,
resulting in an overall adverse effect. In that case the single
reference image would pro ﬁ t from the contamination whereas
the mURA image would not. Again, a possible way to
increase the spatial resolution for the mURA image in the
future might be to use the measured pinhole image of the
mURA to quantify the at-wavelength transmission of each
single aperture and use this information as input for further
iterative re ﬁ nement.
The SNR ratio of mURA and pinhole is 1.2 for the one
accumulation time recorded, i.e., a slight advantage for
mURA imaging over use of a single pinhole as a reference
structure is present in this high photon dose regime. We note
that in comparison to the use of XMCD at the Co L -edge, the
magnetic contrast within the object in the M -edge case is
much weaker. Furthermore, the overall transmission through
the entire object FOV is different as determined by the
effective optical constants of such multilayer ﬁ lms. Especially
in the XUV spectral range, the latter are not well known and
further detailed analysis of the impact of the object contrast as
Figure 6. ( a ) Unprojected helicity difference hologram with z -scale ranging from − 300 to 300 analog-to-digital units ( ADUs ) . The distance
between two neighboring Bragg peaks corresponds to 21 μ m
− 1
. ( b ) Pinhole and ( c ) mURA reconstruction with contrast scaled to 1% to 99%
obtained from ( a ) plus their corresponding normalized line pro ﬁ les shown in ( d ) . The FOV corresponds to 2 μ m.
9
J . Opt. 19 ( 2017 ) 064002 C M Günther et al

well as of the total intensity balance between object and
reference wave and its impact on SNR in the reconstructions
will require additional investigations.
4. Conclusion
In c on clu si on, w e hav e dem ons tra ted th e ﬁ r s t use of a mo di ﬁ ed
U RA f or r eso na nt sp ec tro- ho logr ap hic i ma gi n g at X UV and
so ft x- ray w av el eng ths . We ar e abl e to im age f err oma gne tic
domain p attern s at b oth the L
3
and M
2, 3
abs o rp t ion ed ges of C o
and directly comp are them with a respective hologram reco rded
un de r id ent ica l con dit ion s vi a a sin gl e pin hol e ref ere nce. A t the
Co L -e d ge th e do mai n str uct ure is rep rod uce d wit hin the s pat ial
resolut ion of 50 nm. At the Co M - e dg es we e sti mat e a spa ti al
re sol uti on of ab out 9 0 nm f rom t he pi nho le / mURA c ross cor -
relation . I n c ontrast, th e s ingle p inho le can resolve s maller
do mai n fea tur es of ab out 65 nm , in agr ee ment wi th t he no min al
di am et er of th e ref ere nc e ape rtu re. T he lo ss of s pat ia l res olu ti on
is attr ibuted to inho mogen eities of th e mURA, pos sibly ca used
by c on tam in ati on. W e ana lyz e the g ain i n sign a l- to -n oi s e in t he
re co nst ruc ted r eal -sp ace ma gn et iz ati on ma p ob tai ned b y us ing
th e mU RA ve rsu s si ngl e pi n ho le re fer ence as a fu nc ti on of
in cid ent p ho ton d os e an d com par ing s ing le hel ic ity and h eli ci ty
di ffe ren ce ho log ram s. In th e lo w- exp os ur e li mit , a 2. 5-f old
in cre ase i n th e S NR f or a gi ven ex pos ur e is ob ser ved fo r d if-
ferenc e helicity holograms, whic h is in line with theoretical
ex pe ct a ti ons . For si ngl e hel ic ity h olo gra ms , we se e a 1. 5-f old
S NR in cre as e . We s pe cul ate t hat t his d iff ere nce i s du e to t he
cancelatio n of s ystematic a rtifacts sc aling wit h th e d ose in th e
differe nce helicity case, but in -depth simulation s will be
re qu ir ed to c lar ify t his p oin t. Fo r in cr eas ing e xpo sur e tim e of
the holog rams, the relative SNR ad vantage of the mURA ove r a
si ngl e pin ho le dec rea s es . Ar ti fac ts from i mpe rfe ct mU RA
manufac turing a nd, thus, impe rfect matchi ng of the de coding
pa tte rn ca n be ex pec ted a s the or ig in of t his d ecr eas ing be ne ﬁ t.
We have used the at-wavelength tran smission of the mURA
w hi c h is en co ded v ia th e addi ti ona l sing le pi n ho le re f er en ce in
th e mul tipl exe d ho log ram s o far o nly t o det er min e mU RA pi tch
and o rienta tion; the a dditional us e of th e mURA tra nsmission in
ea ch of it s ap e rt u re s wi ll all ow hi ghe r or de r co rre cti ons in
mU RA- bas ed im ag ing in the fu tur e. Th e sign al -to- n oi s e ben e ﬁ t
of mURA based ( so ft ) x- ray h olo gr aph y w il l o bv iou sly be
de cis ive in lim it ed p ho ton nu mbe r si tua tion s, su c h as en cou n-
te red fo r s ing le- sho t im agi ng, e .g. , at fre e- ele ctr on x-r ay las er
so urc es an d at la bo rat ory so urc es su ch a s when p ush ing
to wa rd s eith er sh ort er w ave len gth o r sub -f s te mpo ral r eso lu ti o n.
Acknowledgments
Support from BMBF under contract 05K13KT3, from the
Leibniz Graduate School DinL and the Helmholtz Virtual
Institute VH-VI-419 is gratefully acknowledged.
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