Page 1

Partially coherent nano-focused x-ray

radiation characterized by Talbot

interferometry

T.Salditt,1S.Kalbfleisch,1M.Osterhoff,1S.P.Kr¨ uger,1M.Bartels,1

K.Giewekemeyer,1H.Neubauer,1and M.Sprung1,2,∗

1Institut f¨ ur R¨ ontgenphysik, Universit¨ at G¨ ottingen, Friedrich-Hund-Platz 1,

37077 G¨ ottingen, Germany

2HASYLAB at DESY, Notkestr.85, 22607 Hamburg, Germany

*tsaldit@gwdg.de

Abstract:

nano-focused x-ray beam by grating (Talbot) interferometry in projection

geometry. The beam is focused by a fixed curvature mirror system op-

timized for high flux density under conditions of partial coherence. The

spatial coherence of the divergent exit wave emitted from the mirror focus

is measured by Talbot interferometry The results are compared to numerical

calculations of coherence propagation. In view of imaging applications,

the magnified in-line image of a test pattern formed under conditions of

partial coherence is analyzed quantitatively. Finally, additional coherence

filtering by use of x-ray waveguides is demonstrated. By insertion of x-ray

waveguides, the beam diameter can be reduced from typical values of 200

nm to values below 15 nm. In proportion to the reduction in the focal

spot size, the numerical aperture (NA) of the projection imaging system

is increased, as well as the coherence length, as quantified by grating

interferometry.

We have studied the spatial coherence properties of a

© 2011 Optical Society of America

OCIS codes: (340.7440) X-ray imaging; (340.7450) X-ray interferometry; (070.6760) Talbot

and self-imaging effects.

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1. Introduction

The high degree of spatial coherence required by modern lensless x-ray imaging techniques

calls for suitable methods to quantify wavefront distortion and spatial coherence [1–3]. The

mutual intensity function Γ or the complex degree of coherence needs to be quantified to control

the coherence of the wavefront. To this end, powerful methods based on interferometry have

been developed [4–7]. Most of these have been used almost exclusively on macroscopic scales,

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e.g. for collimated quasi-parallel beams. However, the most demanding high resolution imaging

experiments today require x-ray beams focused to nanoscale focal spot sizes. In this work, we

extend the methods of coherence characterization by the Talbot effect to nano-focused beams

with highly curved divergent wavefronts behind the focus. We extract the equal-time mutual

intensity function j(z,d) describing the phase correlations between two points separated by a

distanced inaplaneperpendiculartotheopticalaxisadistancez>0behindthefocus.Weshow

that by use of an x-ray waveguide the partial coherent wavefront of a Kirkpatrick-Baez (KB)

mirror system can be filtered to yield nearly fully coherent exit waves behind the waveguide at

the relevant sample distances.

Based on coherent x-ray beams, lensless x-ray imaging offers the potential to overcome

the resolution barrier associated with x-ray lens fabrication, in particular in the hard x-ray

range. Coherent x-ray imaging can be grouped into two classes: (A) Coherent diffraction imag-

ing (CDI) where the diffraction pattern is recorded in the (Fraunhofer) far-field [8], and (B)

propagation imaging in projection geometry, based on the measurement of the (Fresnel) near-

field [9–11]. For (A) quasi-plane wave illumination, and for (B) quasi-point source illumination

is desired, but for both cases, coherent wavefronts are essential. While the need for efficient

nano-focusing is obvious in projection propagation imaging, it is also essential for high reso-

lution CDI, requiring high photon flux densities in the sample plane, and hence in most cases

high gain focusing optics on the incidence side.

In the limiting case of full coherence, reconstruction of both the intensity and phase dis-

tribution of a nano-focused x-ray beam, in or around the focus position is achieved by iterative

reconstruction algorithms and the experimentally accessible far-field intensity distribution. Re-

construction algorithms can be based on support constraints from simple error reduction [12]

to more elaborate schemes [13,14], or on the overlap constraints defining the class of so-called

ptychographic algorithms [15]. In its advanced variants [16,17], ptychography is capable to re-

construct the unknown complex-valued illumination function probed by scanning along with an

generally also unknown object through the beam. This approach has been used to characterize

nanoscale wavefronts [18,19]. However, ptychographic wavefront reconstruction is based on

the assumption of a fully coherent beam, and thus fails or is compromised in the case of partial

coherence, at least in current implementations. We therefore turn to Talbot interferometry for

the quantification of spatial coherence.

We have performed the experiments with the KB mirror system and setup for propagation

imaging installed at the coherence beamline P10 of the new storage ring PETRA III at Hasy-

lab, DESY [20]. Two diagnostic tools have been used for beam characterization: (i) an x-ray

waveguide (WG), and (ii) a set of nanoscale gratings exhibiting the Talbot effect. Aside from

diagnostic purposes, insertion of the WG can also be used to reduce the beam diameter from

typical values of 200 nm (KB) to values below 15 nm [21]. The WG provides highly divergent

and coherent exit beams for phase contrast propagation imaging [11,22]. Importantly, the co-

herence properties and cross-section of the exit beam are decoupled from the primary source.

According to the reduction in the number of modes, the coherence of the exit wave is signifi-

cantly enhanced. In the first part of this work, before considering the coherence filtering effects,

we simply use the small diameter D of the WG guiding layer as an ’ultra-narrow slit’ to scan the

beam around the KB focus. In the last section, the characteristic mode structure of waveguides

is exploited, using the waveguide as a coherence filter. Waveguide mode structure and transmis-

sion is calculated using finite-difference (FD) simulations [23–25], based on the parabolic wave

equation [26]. The paper is organized as follows. After this introduction, the experimental setup

and parameters are presented, followed by a section on the focal intensity distribution. Section

4 contains the Talbot experiment and analysis. Section 5 studies the effect of partial coherence

on the propagation image of a simple test pattern, allowing for the comparison to an analytical

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z1

z2

focal

plane

grating

x

y

z

magnified

Talbot image

KB

mirrors

Fig. 1. Experimental setup: A parallel hard x-ray wave front is focused by the KB mirror

system. A Talbot grating is illuminated at a (defocus) distance z1behind the KB focal

plane, and the magnified projection Talbot image is recorded by an area detector positioned

in the far-field at a distance z1+z2. The schematic shows the example of a grating period

a = 1 μm with 500 nm lines and spaces (l&s) recorded by a pixel detector with 172 μm

(Pilatus, Dectris), while smaller grating periods corresponding to 200 nm l&s, and 50 nm

l&s required a CCD (LCX, RoperScientific) with smaller pixel size of 20μm.

expression. Section 6 presents a brief example of coherence filtering by a waveguide, before

the paper closes with a summary and conclusions.

2. Experimental setup and parameters

Figure 1 shows the schematic of the experiment carried out at the holographic imaging end-

station of the coherence beamline P10 of PETRA III. The source in low β configuration con-

sists of a 5m long undulator with a period of 29mm, with a source size of 36μm×6μm (1σ,

horz.×vert.). The beamline was operated in monochromatic mode using a Si(111) double crys-

talmonochromator,positionedat35mbehindthesource.Atadistanceof87.7mthex-raybeam

is focused by two elliptically shaped mirrors (fixed shape) in Kirkpatrick-Baez (KB) geometry,

contained in a vaccuum vessel. The mirrors are configured to a fixed elliptical shape, corre-

sponding to an incidence angle of α = 4.00 mrad and α = 4.05 mrad, as well as a focal length

of 305mm and 200mm, for the vertically (v) focusing mirror (WinlightX, France) and the hor-

izontally (h) focusing mirror (JTEC Corporation, Japan), respectively. The mirrors consist of

Pd-covered silicon (v-mirror) and Pd-covered silica (h-mirror), respectively. For the h-mirror,

which was polished by elastic emission machining [27], the maximum deviation from the ideal

elliptical shape (uncoated) was independently measured to 3.1nm (peak-to-valley) [20]. The

entrance slits in front of the KB-system were set equal or larger than the geometric acceptance

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of the mirrors of 0.4 mm, i.e. the full mirror length was illuminated by the beam.

The KB near-field intensity distribution was characterized by scanning x-ray waveguides

through the KB focus, using a miniaturized fully motorized goniometer with optical encoders

(Attocube), with three translations in xyz, and two rotations along two directions orthogonal to

the optical axis. Alignment of the waveguide as well as the Talbot gratings was facilitated by

use of two on-axis optical microscopes whose focal planes coincide with the KB. The two mi-

croscopes are directed to the focus with a ’downstream’ and an ’upstream view’, respectively.

For x-ray beam path, the ’downstream view’ microscope (Optique Peter) has a hole in the de-

flecting mirror, while the ’upstream view’ microscope observing the sample at a fixed working

distance of 32mm, has a drilled objective (Bruker AXS), providing in-situ inspection. The sam-

ple stage is equipped with an air-bearing rotation (Micos) for ultra-high precision turns needed

for nano-tomography. On top of the rotation, a group of xyz piezos (Physik Instrumente) is used

for aligning the sample in the axis of rotation. Additional xyz stages (Micos) below the rotation

are used for aligning the rotation axis in the x-ray beam and for distance variation between the

waveguide and the sample. The detectors were placed at a distance of z1+z2= 5.29m behind

the sample, see Fig.1. Along with a direct-illumination CCD (LCX, Princeton Instruments)

used primarily for the data analyzed here, a scintillator CCD (SCX, Princeton Instruments)

and a single photon counting pixel detector (Pilatus 300K, Dectris) was available, and used to

cross-check the far-field signal and/or to determine the integral flux.

Waveguides: The beam is coupled into the waveguide by the so-called front-coupling

scheme [22]. Here we use two new waveguide types. (A) Lithographic channel waveguides

fabricated by wafer bonding (bonded-2DWG), yielding air filled channels [11,20], which can

be cut to a length according to the requirements of the photon energy E. An array of (well sepa-

rated) waveguide channels is first exposed into an e-beam resist layer on a Si wafer by electron

beam lithography. After development this resist layer acts as an etching mask for the reactive

ion etching (RIE) of the waveguide channels. After the removal of the etching mask, a second

(cap) wafer is bonded onto the first (structured) wafer, forming the waveguide chip. The en-

trances and exits on the cleaved edges of the waveguide chip are then polished by Focused Ion

Beam (FIB) polishing (FEI, Nova 600 Nanolab). The cross-section of the channels can be var-

ied,typicallyintherangebetween30nm×20nmandapprox.120nm×60nm,dependingonthe

specific application of the experiment. (B) For smaller waveguide dimensions, an arrangement

of crossed high-transmission planar x-ray waveguides is used similar to [28]. These waveguides

are made of a transmission optimized sequence of sputtered thin films, with amorphousC as the

guiding layer [21,29], even for very small layer thicknesses. For the specific guide used here, an

optical film layer sequence Ge/Mo[di=30 nm]/C[D=35 nm]/Mo[di=30 nm]/Ge was deposited

on a 3 mm thick Ge single crystal substrate (Incoatec GmbH, Germany). A second so-called

cap wafer (Ge, 440 μm thickness) was bonded onto the WG wafer by an alloying process to

block the beam areas not impinging onto the waveguide entrance. The resulting ‘sandwich’

sample was cut by a dicing saw to the desired lengths l = 200 μm, and cleaned by FIB.

Simulation of beam propagation and coherence: In order to simulate the focusing prop-

erties and near-field intensities of the KB-focus, wave-optical propagation was carried out nu-

merically. The undulator source at 7.9 keV with the nominal source size as given above was

discretized into a set of independent emitters, with Fresnel-Kirchhoff integrals used to calcu-

late the propagation from source to mirror, as well as from mirror to the focus. To this end,

the measured height profile of the mirror is taken into account. Partial coherence is modeled

by averaging stochastic realizations emitted by the virtual point-sources. The equal time com-

plex degree of coherence j1,2= ?u1u∗

complex-valued wave amplitudes u1,2and intensities I1,2was computed from a random phase

superposition u(x) = ∑nwncnun(x), with equally distributed) phases and (real valued) weight

2?/√I1I2between two points (1,2) with the associated

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factors wnadjusted to the Gaussian envelope of the source. For each basis field un, free space

propagation and reflection was carried out, and an ensemble average of 8000 stochastic realiza-

tions of random field distributions was evaluated in the ’detection’ plane of interest, modeling

the time-averaging measurement process. The reflection process was modeled by the Fresnel

equations,appliedlocallytoadiscretizedellipticallyshapedcurve(in1+1dimension)including

the measured surface deviation from the perfect ellipse, using 32000 data points for integration,

randomly distributed on the curve (surface). Details of this method will be described elsewhere

(Osterhoff and Salditt, in preparation). Results for the near-field intensity distribution around

the focal point are shown in Fig.2(e) for the case of the horizontally focusing mirror (JTEC)

on logarithmic scale, simulated for the experimental photon energy, horizontal source size, and

distances, as given above.

Grating and test structures: For coherence measurements by the Talbot effect, a high reso-

lution chart (NTT-AT, Japan, model # ATN/XRESO-50HC) consisting of a 500 nm thick nano-

structured tantalum layer on a Ru/SiC/SiN membrane was placed in the beam at a distance

z1downstream from the KB focus, as determined by the on-axis optical microscope. For this

study, we used three different gratings on this test structure, namely 500 nm lines and spaces,

200 nm lines and spaces, and 50 nm lines and spaces, respectively. Note that for example 500

nm lines and spaces (l&s), means 500 nm lines followed by 500 nm spaces, i.e. the given value

is half the grating period. At 7.9 keV, the calculated phase shift of a 500 nm Ta pattern is

φ = 0.830 rad, and the intensity transmission is T = 0.871. In addition to the grating measure-

ments, the hologram of a double slit test structure was analyzed, based on analytical forward

calculations and least square fitting. The double slit test structure was fabricated by FIB etching

(FEI, Nova 600 Nanolab). First, a thin Au layer was deposited by e-beam deposition on a 200

nm SiN foil (Silson), coated with a small 5 nm Ti adhesion layer. The Au film thickness was

determined to 35 nm by profilometry (Veeco Dektak 6M). The Au was then etched away in a

double slit pattern, with a slit length of 4 μm, a width w = 1.6 μm and a distance between the

two slits of 6 μm. After fabrication, the geometric layout was checked by e-beam microscopy.

3. Intensity distribution of the KB focus

Figures2(a) and 2(b) show the results of the x-ray waveguide scans through the focal plane

of the KB to characterize the near-field intensity distribution. By scanning one-dimensional

(planar) waveguides through the focal spot after careful angular alignment, they are used as a

direct probe for the beam width, see the schematic in (c). The thickness of the guiding layer

was D = 35nm in the horizontal and D = 30nm in the vertical direction. The lateral and ver-

tical profile of the focal intensity distribution can then be analyzed as a function of z at and

around the focal plane, to determine the spot sizes and the field of depth. In (a,b), the beam

cross section (FWHM) are plotted as a function of z after batch fitting of scans along the y

(vertical) and z (horizontal) directions for each z to a Gaussian peak profile. The over-estimated

error bar values drawn correspond to ±50 nm, and may considered an upper bounds for the

uncertainties, including systematic errors in parameter initialization during the batch fit. Single

profiles in particular in the focal plane have been analyzed by hand with individual initialization

of parameters, and fitting range, resulting in much smaller errors. Along with the experimen-

tal results (circles), the values determined from the Fresnel-Kirchhoff simulations (solid black

line) and an empirical fit to a Gaussian beam profile (solid red line) are shown. The simulations

of beam propagation take into account the geometrical parameters (undulator source size, dis-

tances) and measured height profiles of the mirrors, as shown in Fig.2(e), for the case of the

vertical direction. The resulting focal distribution is comparable to the experimentally measured

2D intensity distribution, as shown in (f), obtained from a series of waveguide scans (similar

to those shown in (a,b)), after normalization to the peak area for each scan. The simulation and

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the experimental results are in qualitative agreement, as far as beam width and depth of focus is

concerned, but show differences concerning the exact lineshapes and side maxima. The func-

tional form of the empirical fit is based on a convolution of a diffraction limited beam with a

Gaussian (FWHM value ζ) taking into account finite source size and all other spurious beam

broadening effects

?

The finite source size broadens the diffraction limited Gaussian beam with Rayleigh length z0,

butfromthez0valueofthefitsthedepthoffocuscanstillbequantified.Theverticalfocusseries

is fitted to z0= 92.1 μm and ζ = 194 nm (FWHM) with reduced χ2= 0.43. The horizontal

focus series is fitted to z0= 52.1 μm and ζ = 226 nm (FWHM) with reduced χ2= 1.46. The

Gaussian fits to the focus profiles in the focus plane as determined above yielded a spot size

of 203×221nm2(h×v, FWHM). In Fig.2(d), a vertical focus scan along with a Lorentzian

fit is shown yielding a FWHM of 186 nm. This indicates that depending on alignment and

lineshape used in fitting, individual scans also show FWHM values below 200 nm in the vertical

direction. Note that the exact fitting values also fluctuate slightly in the course of successive

scans, possibly also in response to mirror alignment after refilling of the storage ring and drift

of optical components. The largest source of errors, however, was due to the measurement

itself, i.e. the positioning system, which despite the use of encoders, resulted in stick-slip tilt

inaccuracies of the positioning, at least in the horizontal direction, as cross-checked by laser

interferometry. At a storage ring current of 60mA and a photon energy of E=7.9keV, the

measured flux in the 200 nm focal spot was 2.13·1011cps, as measured with a PIN diode

(Canberra PD300-500CB, 2.33·1012cps

pixel detector (Pilatus 300K). Note that this flux applies to the low bandpass Si(111) double

crystal monochromator setting, while a highest flux pink beam mode is also envisioned as an

option in future. Finally, the (far-field) divergence of the KB beam was measured to 1.15mrad×

1.97mrad, in the vertical and horizontal plane, respectively, in good agreement with the values

1.13mrad×2.0mrad expected from the mirror length and incidence angle.

4. Coherence measured by Talbot effect

FWHM =

2ln(2)zoλ

π

(1+z2/z2

0) +ζ2.

(1)

mA), which was calibrated with a single photon counting

Following [1,30], the normalized mutual coherence function also termed complex degree of

coherence describes the correlation between two wave field u probed at two points (1) and (2),

and at times t and t +τ, respectively, and can be written for stationary fields as

γ1,2(τ) =?u1(t)u∗

2(t +τ)?

??I1??I2?

,

(2)

where Ii= ?uiu∗

notational clarity, the mutual coherence function is denoted by capital Γ = ?u1(t)u∗

(before normalization), and the cross-spectral density as the Fourier transform of Γ by W1,2=

?

Again, before normalization we define J1,2= Γ1,2(0). In the following, the complex degree of

coherence will always implicitly denote the equal-time function j, which is relevant here. If not

stated otherwise, point (1) will be located on the optical axis, point (2) in a plane normal to

the optical axis, at a distance d from point (1). The degree of coherence will then be written as

a function j(d), for fixed distance z along the optical axis, or as a function of two arguments

j(z,d), if its evolution as a function of defocus distance is considered. z1can be the defocus

distance in the divergent beam (laboratory) coordinate system, or zef f= z1z2/(z1+z2), if the

i? is the intensity at the two points, and < ··· > denotes temporal average. For

2(t +τ)?

dτ Γ exp(iωτ). In the case of stationary and quasi-monochromatic fields, it is sufficient to

consider the mutual intensity function or equal time complex degree of coherence j1,2=γ1,2(0).

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- 1200- 1000 - 800 - 600 - 400 - 2000 200 400600 800 1000 1200

100

200

300

400

500

600

700

800

900

1000

f whm [ nm ]

z [ µm ]

- 1400 - 1200 - 1000 - 800 - 600 - 400 - 2000 200400600

200

400

600

800

1000

1200

1400

1600

exp. FW HM ver t .

exp. FW HM hor z.

FW HM f i t t o:

( 2l n( 2) z0λ/π ( 1+z

m i r r or si m ul at i on

2/ z0

2) +σ

2)

0. 5

f whm [ nm ]

z[ µ m]

x [ nm ]

- 1000 - 5000 5001000

- 1500

- 1000

- 500

0

500

1000

- 1. 5

- 1

- 0. 5

0

0. 5

1

1. 5

z [ µ m ]

x [ nm ]

- 1000 - 5000 5001000

- 1000

- 500

0

500

1000

- 2. 5

- 2

- 1. 5

- 1

- 0. 5

- 1600- 80008001600

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

nor m . I nt ensi t y

z [ nm ]

KB f ocus scan

Lor ent zi an f i t

FW HM = 186 nm

KB- beam

z

x, y

wavegui de

( a)

( b)

( c)

( d)

( e)( f )

Fig. 2. Beam width (FWHM) along the (a) horizontal (x) and (b) vertical (y) direction, as

a function of z along the optical axis, as measured by scanning the waveguide through the

focus, along with the Fresnel-Kirchhoff simulations (solid black line) and a fit to a Gaussian

beam profile (solid red line), convolved with a residual width (see text). (c) Schematic of

the waveguide scan. (d) Focus intensity profile along the vertical direction in the focal

plane after iterative alignment, along with a Lorentzian fit (solid red line). Side minima

and maxima are observed around the central focus. Two-dimensional (e) simulated and

(f) experimental intensity distribution in the focal region (vertical plane), after logarithmic

color encoding (see colorbar).

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parallel beam equivalent coordinate system is used. According to the so-called Fresnel scaling

theorem for paraxial beams, free space propagation and the associated contrast formation is

equivalent in the cone beam and parallel beam case, up to a simple coordinate transformation of

thedetector pixel sizeby thegeometric magnification M =(z2+z1)/z1and thedefocus distance

z→zef f[2]. Apart from this simple rescaling and the associated changes in spatial distances, in

particular of the Talbot replication distance, the experiment and analysis essentially follows the

parallel beam case, as described in [4]. The idea is simple: according to wave optics a periodic

object (in an infinite) beam creates periodic self-images downstream of itself. This so-called

Talbot effect with perfect replication occurs for |j| = 1, while partial coherence leads to a loss

of contrast. The contrast in the detector plane evaluated for the first order Fourier coefficient

m = 1 of a lattice with period a, reflects the degree of coherence between two points separated

by λzef f/a. Thus, the diffraction angle of the m = 1 lattice reflection and the distance between

object and detector (the defocus distance) determine the argument d of the measured function

|j(d)|.

Before turning to the experimental procedure, let us briefly consider the expected spatial

coherence properties based on the simulations. Figure 3 shows the results for the relevant field

behind the KB focus in the horizontal plane, i.e. (a) a two-dimensional plot of the |j(z,y)| ,

as well as selected curves |j(y)| for z const., with the corresponding Gaussian fits. Finally,

(c) shows the linear scaling of the full width at half maximum (FWHM) of the curves |j(y)|

as a function of z, in other words the lateral coherence length ξFWHM. The linear scaling of

the coherence length and the Gaussian functional form, support the use of a Gaussian Shell-

model (GSM) for the coherence properties, as put forward in [3]. In the simulation, the actual

undulator source was assumed to be completely incoherent, but a finite coherence ‘builds up by

free space propagation. According to the GSM, the cross-spectral density W (or for the quasi-

monochromatic the mutual intensity J) for any two points in a plane orthogonal to the optical

axis separated by a distance d =

x2

?

?

1−x2

2can be written as

J1,2(x1,x2) =

I1(x1)I2(x2) exp(−(x1−x2)2/(2ξ2)) ,

(3)

where ξ or the FWHM equivalent ξFWHM= 2√2ln2ξ can be defined as the lateral coherence

length. A characteristic property of the GSM is the fact that the ratio between the beam size and

the coherence length is a constant parameter q. In the horizontal plane, q?0.37 is thus obtained

from the constant opening angle of the beam (2 mrad) and the opening angle subtended by the

lateral coherence length, which is ξFWHM/z ? 0.733 mrad. In the vertical plane, simulations

predict full coherence q ≥ 1.

We now turn to the Talbot experiment. Figure 4 gives illustrative examples of the Talbot ef-

fect in the magnification setup and the data analysis scheme for a grating with 200 nm lines

and spaces (l&s), aligned vertically, i.e. in the case of probing the coherence properties in the

horizontal plane. As the sample is scanned in z1, the visibility of the fringes oscillates between

weak or no contrast as shown in (a) to high contrast as shown in (b). After vertical averaging

over a region of interest (ROI), the spectral density of the profile is computed. The integrated

intensity of the first order (|fm=1|) is then evaluated as a function of z1. Since the magnification

M also changes with z1, the position of the m = 1 moves linearly in the PSD curve. An auto-

mated peak search function was used to single out the m = 1 peaks. When z1corresponds to

a Talbot minimum, this procedure did not work well for the 200 nm l&s grating. In a refined

procedure, the movement of the peak was analyzed by linear regression to predict the peak

position, followed by intensity readout, see also Fig. 5(b), comparing the two procedures of

simple peak search (open symbols) and of peak search based on regression (solid symbols).

According to Talbot theory (Eq. (3) in [4]) the first order Fourier coefficient of the intensity is

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0.0 2.55.0 7.5 10.0

0.0

0.2

0.4

0.6

0.8

1.0

z=15mm

z=10mm

z= 5mm

z= 2mm

Gaussian fit

degree of coherence

lateral distance x [mm]

degree of coherence

defocus z [mm]

x [ µm]

02468 101214 16

0

2

4

6

8

10

12

FWHM [µm]

defocus z [mm]

FWHM

-0.20(5) + 0.733(5) z

(a)

(b)

(c)

Fig. 3. Simulation results. (a) Degree of coherence |j(x,z)| in the horizontal (xz) plane,

evaluated for point (1) on the optical axis (0,z) , and point (2) at (x,z). (b) The degree of

coherence |j| as a function of x for selected constant values of z, along with a fit (red solid

line) to a Gaussian. (c) The full width at half maximum ξFWHMof the Gaussian fits as a

function of z shows linear scaling (solid red line), defining the (full) opening angle of the

coherent beam cone ξFWHM/z ? 0.733 mrad, which is smaller than the 2 mrad (FWHM)

angle of the full beam in the horizontal direction. In the vertical plane, simulations predict

full coherence, as a result of the significantly smaller vertical undulator source size.

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−0.1

0

0.1

signal

00.05 0.10.15 0.2

0.01

0.02

0.03

spatial frequency (periods/pixel)

psd

0.5

1

1.5

pixel

norm. intensity

Fig. 4. (a) Intensity distribution (linear gray scale) of the 200nm lines and spaces (l&s) at

z1= 8.4 mm recorded with the CCD detector, after empty beam correction (scale bar: 0.8

mm in the detector coordinates, corresponding to 1.3μm in the sample plane. (b) same as

in (a) but for z1=8.5 mm, where high contrast is observed. (c) Intensity profile obtained by

vertical averaging of (b) and subtraction of the mean. (d) Modulus of the Fourier transform

of the profile (spectral density, PSD) as a function of inverse pixel, showing the strong

m = 1 order.

fm=1= j(λzef f/a)1

a

?a

0

dx exp[−i2πx/a] g(x)g∗(x+λzef f/a) ,

(4)

where a is the periodicity of the lattice and g(x) is the complex-valued function describing the

transmittance of the lattice. For the particular case of a binary phase grating, the integral can be

solved analytically, see Eqs. (4) and (5) in [4]. For the rectangular lattices used here with the

parameters a, φ and T given above, as well as the duty cycle 0.5, the integral can be evaluated

numerically.

The experimental values of the m = 1 Fourier coefficient extracted from the Talbot scans are

fitted to the equation

|fm=1| = c |j(λzef f/a) sin(πzef f/ZT)| ,

where ZT= a2/λ is the Talbot (replication) distance, and zef f= z1z2/(z1+z2) is the effective

defocus distance. c = 0.440 is a prefactor which is determined numerically, by solving Eq. (3)

of [4] for the parameters of the grating. Thus, it is the amplitude of the sinosoidal Talbot curves

which carries the information on |j|.

Figure 5(a) and 5(b) show the measured Talbot curves (symbols) along with the least-square

fits(solidlines),forthe200nml&s/a=400nmgrating,andforthe500nml&s/a=1μmgrat-

ing. The magnitude of the first order (m = 1) maximum in the power spectral density (PSD), as

computed by Fourier analysis of the recorded Fresnel diffraction pattern, is plotted as a function

of the defocus distance z1. The fitted Talbot distances for the two gratings of Z200= 1.020 μm,

(5)

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Fig. 5. (a) The magnitude of the first (m=1) order Fourier component |fm=1| of the 200 nm

l&s grating (solid symbols), as a function of the defocus distance z1, computed by Fourier

analysis of the recorded Fresnel patterns, along with a least-square fit to Eq. (5) (solid line).

The intensity oscillates periodically as the grating position is scanned, with a period of the

expected Talbot distance Z. (b) The same for the 500 nm l&s grating, with (solid symbols)

and without (open symbols) a refined image analysis procedure (see text). As expected, the

amplitude of the |fm=1| oscillation is higher for the 500 nm l&s grating than for the 200

nm l&s grating.

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and Z500= 6.306 μm correspond well with the expected Talbot distances a2/λ, within the ex-

perimental uncertainties. A small error δz1of the nominal defocus z1was allowed as a free

parameter in the least-square fit. The resulting value δz1= −0.62 mm and δz1= −0.44 mm

for the two gratings, respectively, show that the position of the test pattern as determined by

the optical microscope was probably not exact. The focal plane of the optical microscope had

initially been adjusted to the KB focus, but due to realignment after injection the focal plane

of the KB can vary slightly from one ring filling to the next. Finally, the third and last free

parameter was the fitted |j|-value associated with the amplitudes of the Talbot curves, which

were evaluated to j200= 0.415±0.003, and j500= 0.927±0.007, respectively. This measured

coefficient |j(z1,d)| corresponds to the degree of coherence in the sample plane z1for any two

points separated by a lateral distance d = λzef f/a. During the scan, z1and d both increase

linearly. For z1? z2, z1? zef f, and a constant value |j| = const. is measured along the line

forming an angle λ/a with respect to the optical axis. Note that the data sets could of course be

fitted with models parameterizing a variation of j along the scan, but the amplitude of the Tal-

bot effect proves constant on visual inspection, and the rather good agreement of the data with

the three parameters motivates the conclusion that |j(z1,λz1/a)| is constant. This conclusion is

plausible, since the coherence length ξ(z1) increases linearly with z1due to propagation. The

reduced mean square sum is χ2= 4.0 and χ2= 0.51, for the two fits. The overall agreement is

quite satisfactory of a three parameter fit, and precludes models with more parameters.

Next, the same gratings with l&s oriented horizontally were used in analogous scans to de-

termine the |j| values in the vertical (yz)-plane (data not shown). Least-square analysis resulted

in j200=0.473±0.009, and j500=0.948±0.007, for the 200 nm l&s and the 500 nm l&s grat-

ing, respectively. The |j| values in the vertical plane are therefore only slightly larger than in the

horizontal plane. One would have expected a higher degree of coherence based on the smaller

vertical (primary) source size, which leads to a nominally almost fully coherent illumination of

the vertically focusing mirror. However, the presence of the double crystal monochromator in

the vertical plane was observed to introduce vibrations, leading to a decease in coherence. This

effect does not seem to be as systematic as the finite horizontal source size. Thus, apart from the

reduced amplitude of the Talbot scans, which already disprove full coherence, the Talbot self

image quality seems somewhat inferior and more irregular. In the Talbot scans, |fm=1| never

decreased to zero, and the quality of the fits was worse than in the horizontal direction. In any

case, the vertical coherence length ξx(z1) was thus also smaller than the beam diameter.

Figure 6 compiles the results on |j(z1,d)| in the xz-plane and illustrates the geometry of the

probing scans. (a) shows the two experimental j values corresponding to the two gratings, as

a function of normalized lateral distance X = (x/z). A one-parameter fit to a Gaussian of unit

amplitudeexp(−X2/2Σ2)yieldsanormalizedcoherencelengthΣ=0.30±0.02,orcorrespond-

ingly a Full Width at Half Maximum (FWHM)ξFWHM=2?2ln(2)Σ z=0.71·10−3z. The cor-

spectively. Along with the experimental values, the resulting |j(X)| curve of the coherence sim-

ulations is shown (thin black line), as calculated for the plane z1=10 mm, scaled to the X =x/z

coordinate, showing good agreement. In (b), The function |j(z1,x)| = exp(−(x/z1)2/2Σ2) is

shown as a contour plot, with selected contour levels. The two oblique lines (white dotted lines

on both sides of the optical axis ) in the (xz)-plane indicate the measurement points probed

by the two defocus scans. The solid white lines indicate the divergence of the KB-beam in the

vertical (xz) plane. Finally, the lateral coherence length ξFWHM(z1) is plotted in (c), for the

experimental values (open circles) with errors corresponding to the Σ fit, along with the numer-

ical coherence simulations (solid red squares), and the analytical results for an equivalent fully

incoherent source at distance z1. The only parameter in this expression is the ’effective’ hori-

zontal source size, which best approximates the experimental results when set to s = 194.6 nm

responding values for the vertical yz plane are Σ = 0.33±0.02, and ξFWHM= 0.78·10−3z, re-

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Fig. 6. (a) Degree of coherence |j(z1,d)| in the xz-plane as a function of normalized lat-

eral distance X = (x/z), with the two experimental j values corresponding to the two

gratings (open squares), along with a one-parameter fit to a Gaussian of unit amplitude

exp(−X2/(2Σ2)), yielding a normalized coherence length Σ = 0.30±0.02, or correspond-

ingly a FWHM value o ξFWHM= 2?2ln(2)Σ z = 0.707·10−3z. Along with the experi-

as evaluated for the plane z1= 10 mm, scaled to the X = x/z coordinate. The j values

corresponding to the vertical direction (filled triangles) are also included for compari-

son. A corresponding Gaussian fit yields Σ = 0.33±0.02 (not shown). (b) The function

|j(z1,x)| = exp(−(x/z1)2/(2Σ2)) as determined for the horizontal plane, shown as a con-

tour plot, with selected contour levels, along with the two oblique lines (white dotted lines)

in the (xz)-plane probed by the two defocus scans, and the edge of the beam (solid white

lines) indicating the divergence of the KB-beam, here for the horizontal (xz) plane. (c) The

lateral coherence length ξFWHM(z1) corresponding to the experimental values (open cir-

cles) (with errors from the Σ fit), and the numerical simulations (solid red squares), as well

as the coherence length corresponding to a fully incoherent source of size s, observed at

distance z1. The fitted slope corresponds to s = 194.6 nm (FWHM), see text.

mental values, the degree of coherence |j| from the simulations is shown (thin black line),

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(FWHM). In other words, all observations at the respective distances probed experimentally

are equivalent to the coherence properties of an incoherent source with the above cross section.

The corresponding treatment for the vertical plane yields an effective vertical source size of

s=179.0 nm (FWHM). In conlcusion, the coherent opening angle of the beam was determined

to 0.71 mrad in the horizontal and 0.78 mrad in the vertical plane, which divided by the total

beam opening angle, 1.97 mrad and 1.15 mrad, fixes the coherence parameter q = 0.36, and

q = 0.68, for the two planes, respectively. Throughout this treatment we have assumed that

the focusing, propagation and coherence properties factorize in the two orthogonal directions.

The results are in excellent agreement in the horizontal plane, but deviate from expectation in

the vertical plane, for which q ≥ 1 was expected, since the vertical mirror should have been

illuminated coherently due to the smaller vertical undulator source.

5. Holographic imaging of a double slit

Next, we will deduce information on the lateral coherence by analyzing the hologram of a

well-defined test structure. The idea is to model the measured intensity profile by an analytical

expression derived for full coherence, followed by Gaussian convolution to account for partial

coherence. The convolution width should then be related to the coherence length ξ. The use

of a simple double slit (DS) pattern was motivated by the fact that one can easily derive an

analyticalfunctionfortheFresneldiffractionpattern,forarbitrarycomplex-valued transmission

functions. For the present photon energy and film thickness, the sample primarily interacts by

phase contrast. E = 7.9keV, a structure in 35 nm thick Au layer primarily interacts by phase

contrast with an expected phase shift of φ = 0.069 rad and a transmission of T = 0.986. As

described above, the DS had a width w = 1.6μm and a distance between the two slits of 2l =

6μm, adapted to the expected coherence length. To avoid the complications associated with

adaptation of the phase reconstruction algorithms to partially coherent wave fields, we here

take a forward-scattering based approach, i.e. we analyze the hologram, based on a simple

analytical equation which can easily be derived for the DS pattern as a function of the detector

coordinate xD,

?

+Erf[

4λz(w+2(l−xD))]+Erf[

I/I0(xD) = |T +1

2(eiφ−T)(Erf[

π

4λz(w−2(l+xD))]+Erf[

?

π

4λz(w−2(l−xD))])|2.

π

4λz(w+2(l+xD))]

?

π

?

(6)

Figure 7(a) shows the hologram (region of interest) measured at z1= 20 mm with the CCD

(LCX,RoperScientific), after correction of dark current and empty beam (flat field) shows the

two slots accompanied by characteristic intensity oscillations. These oscillations as well as the

overall contrast contain the information about spatial coherence. For the forward calculation,

the projection geometry was mapped onto parallel beam propagation as described above, see

also [31]. Given the distance z1= 20 mm between source and sample and z2= 5290mm−z1

between sample and detector, parallel beam propagation (and reconstruction) by Fresnel prop-

agators can be applied using the effective defocus (propagation) zef f= z1z2/(z1+z2), and

geometrically de-magnified pixel sizes with M = (z1+z2)/z1, just as for the Talbot images

discussed above. The expression was then fitted to a horizontal cross section through the DS

pattern. To account for an arbitrary horizontal zero position, as well as small errors in the

normalization, constant (sub-pixel) horizontal and vertical (intensity) shifts were also allowed

in the fit. All geometric (w,l) and experimental parameters (M,zef f) were kept constant. Fig-

ure 7(b) shows the horizontal Fresnel oscillatory intensity trace, after vertical integration over

91 pixels for optimized signal to noise, with errors determined by evaluating the variance of a

corresponding flat image region, along with a one-dimensional least-square fit (χ2= 3.15) to

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50

100

150

200

250

0.8

0.9

1

1.1

1.2

1.3

−1000100

0.9

0.95

1

1.05

detector position [pixel]

normalized intensity

(b)

(a)

Fig. 7. (a) Near-field propagation image (linear gray scale) of a double slit pattern with

predominant phase contrast after correction of dark current and empty beam (flat field).

The DS is illuminated by the partially coherent and divergent KB beam. Scale bar denotes

0.8 mm in the detector plane, corresponding to 3.025μm in the sample plane. (b) Hor-

izontal intensity trace showing the characteristic Fresnel oscillations (dark circles) with

least-square fit (red line), see text.

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the expression given above, followed by convolution with a Gaussian with FWHM ζ. Aside

from the FWHM parameter ζ = 4.03 pixel, governing the Gaussian convolution, the resulting

fitting parameters were φ = 0.0864 and T = 0.987. The higher value for φ is an indication that

the thickness measurement of the film by the profilometer may have been off by a factor 1.2.

How to understand the convolution in simple terms, based on partial coherence correspond-

ingtothefinitesourcesize?Takingasimplegeometricrelationfortheexpected width(FWHM)

of the convolution function ζ = Δs z2/z1[32], and the source size Δs ? 200 nm (FWHM), a

FWHM value of 2.64 pixel is expected, substantially smaller then the fit result of 4.03 pixel.

One may be tempted to readily attribute the discrepancy to all other smearing effects aside

from coherence, such as vibrations between beam and sample and/or detector point spread

(PSF) function. However, the PSF of the direct illumination CCD was measured, confirming

essentially the 1 pixel value expected for a CCD with negligible cross-talk. Vibration ampli-

tudes on the order of 200 nm are also unlikely. Instead, investigating the error returned for each

parameter by the least-square fit, we see that the lower bound of the confidence interval defined

by the increase of χ2→ χ2+1 is ζ ?1.4, and that the expected value would thus still be within

the confidence interval. Accordingly, the smearing of the Fresnel diffraction pattern is not in

disagreement with the experimental coherence length or source size. At the same time, we also

see that imaging a test structure at fixed defocus is not sensitive enough to determine |j|. At

the same time, we can conclude that imaging under given parameters and conditions of partial

coherence is well described by the ideal propagation contrast followed by an semi-empirical

Gaussian convolution, and that imaging by the KB beam alone without further coherence filter-

ing already gives quantitatively tractable results.

6. Coherence filtering by a waveguide

Next, the Talbot scans were continued, but now with a waveguide inserted in the KB focal

plane for further reduction of the beam size by an order of magnitude, down to below 20 nm,

in order to demonstrate the associated gain in spatial coherence. The sputtered thin film sys-

tem Ge/Mo[30 nm]/C[35 nm]/Mo[30 nm]/Ge with a waveguide length of 200μm was used as

described above. Simulation and previous experiments have shown that very small near-field

beam cross sections (FWHM) in the range of 10-15 nm can be obtained in theses waveguides by

multi-modal interference and damping of higher orders [21]. Figure 8(a) shows the schematic

of the setup, and (b) a numerical calculation of the field propagation in and directly behind the

waveguide, based on finite difference equations [23]. Since the grating is a one-dimensional

structure, one-dimensional beam confinement was sufficient and thus the horizontal WG (for

vertical beam confinement) was used in the Talbot scan. The grating defocus distance was first

calibrated by the optical microscope. At its reference position, the fully motorized microscope

has a focal spot coinciding with the KB mirror. Next, the z1value with respect to the waveguide

exit plane was determined in the x-ray beam by vertical translations of the grating, yielding

a magnification of M = 8997 and correspondingly a defocus of z1= 0.5880 mm, in perfect

agreement with the thickness of the waveguide. Figure 8(c) shows the far-field intensity pattern

of the 50 nm l&s grating at 1 mm defocus with respect to the KB focus. The intensity ma-

trix was divided first by the empty waveguide far-field matrix, and then divided by the average

horizontal (vector) intensity profile (mean over all rows). Finally, (d) shows the result of the

Talbot scan (solid circles) with a least-square fit to Eq. (5) (solid line). Given the significantly

reduced Talbot replication period ZT= a2/λ, the curve is not sampled finely enough, but a

least-square fit is nevertheless possible, if the ZTis kept constant, or fixed within a small range

of values. The resulting amplitude of the Talbot curve is significantly increased with respect to

the results without waveguide. The amplitude fit parameter of 0.445 directly illustrates the gain

in coherence, since within error it equals the prefactor in Eq. (5), with c = 0.440 calculated

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Fig. 8. (a) Schematic of the waveguide Talbot setup. (b) Numerical simulation of beam

propagation in the waveguide, assuming a plane incidence wave. Transmission, beam con-

finement and the coupling of the field into and out of the waveguide can be studied. Due

to damping of the higher order modes, the field in the exit plane is highly confined, with

a width (FWHM) smaller than the guiding layer D, resulting in a broad diffracted emis-

sion cone used to illuminate the grating. (c) Illustrative example of the Talbot image as

measured with the detector at the same position as above, with the 50 nm l&s grating at a

defocus value z1= 0.5880 mm behind the exit plane of the waveguide. The vertical scale

bar represents 1.8 mm in the detector plane, corresponding to 200 nm in the sample plane.

(d) Talbot scan yielding the magnitude of the first (m = 1) order Fourier component of the

50 nm l&s grating (circles), as determined from Fourier analysis of the recorded Fresnel

patterns, along with a least-square fit to Eq. (5) (solid line), as a function of the defocus

distance (here with respect to KB focus).

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Page 19

for the given optical constants of the grating and photon energy. Accordingly, this indicates a

degree of coherence |j| ? 1 over the measured range, in line with the coherence simulations,

which predict a fully coherent waveguide exit beam.

7.Summary and conclusion

The spatial coherence of an x-ray beam emanating from a nano-focused partially coherent sec-

ondary source of 200 nm cross section has been characterized. We have first mapped out the

focal intensity distribution by scanning x-ray waveguides through the beam. At a storage ring

current of 60mA and a photon energy of E=7.9keV, the monochromatic flux (Si(111) dou-

ble crystal monochromator) in the focal spot was 2.13·1011cps. To characterize the lateral

coherence in particular of the defocused beam behind the mirror focus, grating interferometry

measurements as well as numerical wave propagation were employed. Defocus values have

been chosen in view of propagation imaging. This study is part of a long term goal to relax

coherence requirements in imaging applications, i.e. to develop advanced reconstruction algo-

rithms which take a (measured) finite degree of coherence into account [1], and enable imaging

at optimized flux for tomography and dynamical imaging. To this end, the magnified propa-

gation image of a test pattern was analyzed, and fitted by an analytical expression convolved

with a Gaussian smearing function which takes into account the effects of partial coherence.

This example shows that imaging under partial coherence can in this case be well described

by the ideal propagation contrast followed by a semi-empirical Gaussian convolution, and that

imaging by the KB beam alone without further coherence filtering already gives quantitatively

tractable results.

A more complete treatment of the lateral coherence in two orthogonal directions to the beam

was achieved by grating interferometry using gratings of 500 nm, 200 nm, and 50 nm lines and

spaces, respectively, at defocus distances in the range of 1 mm to 15 mm, relevant for magni-

fied projection imaging. In this range the coherence length ξx,ywas essentially indistinguishable

from that of an incoherent secondary source of the same size and distance. In other words the

effect of partial coherence simply changes the ratio of beam diameter and coherence length,

but not the coherence length in absolute value. This is true for defocus distances larger than

the Rayleigh length, in line with simple wave optical coherence propagation, both in analyti-

cal approximation as in a more complete numerical treatment. The agreement in the horizontal

directions with the numerical results was excellent. In the vertical direction, the degree of co-

herence was very similar to the horizontal values, in contrast to expectation based on the smaller

vertical (primary) source size. The small source size (σ = 6μm) in the vertical direction should

have led to a nearly fully coherent vertical beam, over the acceptance of the KB mirror. This

was, however, not the case, and the discrepancy can be attributed to the double nitrogen cooled

crystal monochromator which is known to introduce vibrations, and to compromise coherence.

Possible improvements are now under close investigation at PETRA III.

In conclusion, as the coherence of the x-ray optical system is an essential prerequisite for

coherent imaging, one needs suitable diagnostic tools and procedures for the case of propa-

gation imaging and for coherent diffraction imaging alike. Extending previous studies carried

out in unfocused or moderately focused synchrotron beams, the present work has addressed the

case of highly focused radiation by Talbot interferometry, which was previously treated only

in the limit of full coherence, e.g. by ptychographic wavefield reconstruction. We expect that

for future applications of full field propagation imaging, the coherence requirements can be re-

laxed, if properly measured. If this is achieved, a much wider range of partially coherent optical

systems can be used, to the advantage of higher flux and lower accumulation times. However,

for applications needing more stringent coherence requirements and high aperture divergent

beams, further optical coherence (and spatial) filtering elements are needed. To this end, we

#143927 - $15.00 USD

(C) 2011 OSA

Received 9 Mar 2011; revised 24 Apr 2011; accepted 25 Apr 2011; published 3 May 2011

9 May 2011 / Vol. 19, No. 10 / OPTICS EXPRESS 9674

Page 20

have inserted an x-ray waveguides in the KB focus, leading to an increase in spatial coherence,

as evidenced by the increased amplitude of the oscillatory grating visibility, in the Talbot scans.

Acknowledgments

We thank the HASYLAB/DESY for support and commissioning of the low emittance radiation

storage ring PETRA III. We acknowledge financial support by Deutsche Forschungsgemein-

schaft through SFB755 Nanoscale Photonic Imaging and the German Ministry of Education

and Research under Grants No. 05KS7MGA and 05K10MGA .

#143927 - $15.00 USD

(C) 2011 OSA

Received 9 Mar 2011; revised 24 Apr 2011; accepted 25 Apr 2011; published 3 May 2011

9 May 2011 / Vol. 19, No. 10 / OPTICS EXPRESS 9675