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Periodically structured X-ray waveguides.

01 Sep 2013-Journal of Synchrotron Radiation (International Union of Crystallography)-Vol. 20, Iss: 5, pp 691-697
TL;DR: Both the simulated and the experimental diffraction patterns show in the far field that propagation takes place essentially only for low incidence angles, confirming the mode filtering properties of the structured X-ray waveguides.
Abstract: The properties of X-ray vacuum-gap waveguides (WGs) with additional periodic structure on one of the reflecting walls are studied. Theoretical considerations, numerical simulations and experimental results confirm that the periodic structure imposes additional conditions on efficient propagation of the electromagnetic field along the WGs. The transmission is maximum for guided modes that possess sufficient phase synchronism with the periodic structure (here called `super-resonances'). The field inside the WGs is essentially given at low incidence angle by the fundamental mode strongly coupled with the corresponding phased-matched mode. Both the simulated and the experimental diffraction patterns show in the far field that propagation takes place essentially only for low incidence angles, confirming the mode filtering properties of the structured X-ray waveguides.

Summary (3 min read)

1. Introduction

  • In particular, waveguides with additional periodicity have been extensively explored for visible light and microwaves for over half a century and find many applications in integrated optics (Yariv & Nakamura, 1977; Conwell, 1976).
  • On the other hand, X-ray waveguides, developed in the mid-1990s, are a relatively recent contribution to the field of optics.
  • Each resonance mode propagating inside the WG has a well defined wavefront (Bukreeva et al., 2010; Zwanenburg et al., 1999).
  • In a recent paper, Bukreeva et al. (2011) demonstrated theoretically that an additional symmetric periodic structure on the reflecting walls of a WG with a gap of a few hundred nanometers can filter out the asymmetric and the high-order modes.

2. Guided-mode propagation analysis in structured WGs

  • Since the transverse dimension of the guiding layer (the gap) is much smaller than the lateral one (along axis 0Y), the problem of X-ray propagation in the section has been treated two-dimensionally.
  • The authors present initially the basic equations for uniform (i.e. non-structured).
  • WGs, which are well known in the literature (Marcuse, 1974), and then the authors will extend the analysis to structured WGs.

2.1. X-ray propagation in a uniform waveguide

  • Within the coordinate system depicted in Fig. 1, the modes can be chosen inside the channel in the form (Ognev, 2010) mðxÞ ¼ sin qxmx ’mð Þ; ð3Þ where q2xm = k(2kzm) and qxm can be regarded as the projection of the wavevector k on the lateral axis 0X for the m mode.
  • The phase term in (4) takes into account the penetration of the electromagnetic field into the cladding material which causes effective broadening of the geometrical width d of the guiding layer.

2.2. X-ray propagation in structured waveguides

  • Mathematically the physical processes that occur in a periodic waveguide have been treated either with the guide modes as sums of Bloch–Floquet waves (Peng et al., 1974; Peng et al., 1975) or as a solution of the coupled-wave equations (Yariv & Nakamura, 1977; Conwell, 1976; Marcuse, 1969).
  • When the phase-matching condition, equation (10), is not fulfilled (i.e. when P 6¼ T), the propagation field is rapidly damped along the guiding layer.
  • In Fig. 2(a) the authors show the super-resonance effect, plotting the transmission calculated numerically as a function of the effective vacuum gap width deff, for the structured WG (solid line), compared with the transmission coefficient for a uniform waveguide (dashed line).
  • The structured waveguide transmits less energy compared with the uniform waveguide owing to the strong attenuation of modes which do not satisfy the phase-matching condition.
  • Periodically structured X-ray waveguides 693 equation (11).

3.1. WG fabrication

  • The other silicon slab, of the same dimensions as the first one, but without the Cr spacer, was positioned on the first one in such a way as to have a perfect superposition.
  • The two slabs were held firmly, one against the other, with the aid of a mechanical press (Fig. 5a) (Pelliccia et al., 2007).
  • Periodically structured X-ray waveguides J. Synchrotron Rad. (2013).
  • In the case of the structured WG, the periodic grating was fabricated directly on the silicon slab with the Cr shoulders, using electron beam lithography (Vistec EPBG5 High Resolution, acceleration voltage 100 keV) and silicon etching.
  • A 1.4 mm-thick layer of a positive-tone electronic resist, polymethyl methacrylate (PMMA), 600k molecular weight, was spun on the silicon top layer and baked at 442 K for 5 min on a hotplate, exposed with a dose of 800 mC cm 2, and developed with a methyl isobutyl ketone and isopropyl alcohol (1:1 solution) for 90 s.

3.2. Experimental set-up

  • The experiment was carried out at the cSAXS beamline at the Swiss Light Source, Villigen, Switzerland, using the set-up shown in Fig. 5(b).
  • The authors used the X-ray undulator beam monochromated at 8 keV photon energy by a fixed-exit Si(111) monochromator.
  • The total distance between the source and the WG entrance was 34 m.
  • In addition, a third slit placed at 33.5 m from the source further reduced the beam size in the vertical direction to 0.1 mm.
  • The WG was mounted on a hexapod (http://www.hexapods. net/) allowing six degrees of freedom for careful orientation of the waveguide with respect to the incident beam.

3.3. Experimental results

  • The scope of the experiment was to measure the intensity distribution provided by the structured WG as a function of the incident angle, and to compare the experimental results with simulations based on the equations reported in x2.2.
  • Fig. 6(a) shows a representative far-field pattern, obtained for a small grazing incidence angle in (rotation around the y axis) corresponding to the maximum intensity at the detector.
  • The WG reflecting surfaces are in the yz plane (see Fig. 5b); consequently in the vertical direction the beam maintains its natural divergence, but in the horizontal direction the beam acquires a divergence owing to diffraction at the exit of the WG.
  • For comparison, Fig. 7(c) reports the same type of image I( in, det) calculated for a uniform WG with the same gap value of 241 nm.
  • With detectable intensity in the entire angular range up to the critical angle c.

4. Conclusions

  • The properties of X-ray vacuum-gap WGs with an additional periodic structure on the reflecting walls have been investigated.
  • When these conditions are not fulfilled, the attenuation of the electromagnetic wavefield along the structured WG is stronger.
  • Experimental results taken with synchrotron radiation confirm the theoretical findings with a good agreement between the experimental and simulated farfield diffraction patterns.
  • The authors thank Andreas Menzel for his comments on the manuscript and Xavier Donath for technical support during the measurements.
  • One of the authors (LO) was partially supported by grant No. 4361.2012.2 from the President of Russian Federation for Leading Scientific Schools.

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research papers
J. Synchrotron Rad. (2013). 20, 691–697 doi:10.1107/S0909049513018657 691
Journal of
Synchrotron
Radiation
ISSN 0909-0495
Received 18 December 2012
Accepted 5 July 2013
# 2013 International Union of Crystallography
Printed in Singapore all rights reserved
Periodically structured X-ray waveguides
Inna Bukreeva,
a
* Andrea Sorrent ino,
a,b
Alessia Cedola,
a
Ennio Giovine,
a
Ana Diaz,
c
Fernando Scarinci,
a
Werner Jark,
d
Leonid Ognev
e
and Stefano Lagomarsino
f,g
a
Institute for Photonics and Nanotechnologies, CNR, 00156 Rome, Italy,
b
ALBA Synchrotron Light
Source, Cerdanyola del Valles, 08290 Barcelona, Spain,
c
Paul Scherrer Institut, CH-5232 Villigen
PSI, Switzerland,
d
Elettra, Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy,
e
Kurchatov Institute,
NRC, Moscow 123182, Russian Federation,
f
Institute for Chemical-Physical Processes, CNR, 00185
Rome, Italy, and
g
Physics Department, Sapienza University, 00185 Rome, Italy.
E-mail: innabukreeva@yahoo.it
The properties of X-ray vacuum-gap waveguides (WGs) with additional
periodic structure on one of the reflecting walls are studied. Theoretical
considerations, numerical simulations and experimental results confirm that the
periodic structure imposes additional conditions on efficient propagation of the
electromagnetic field along the WGs. The transmission is maximum for guided
modes that possess sufficient phase synchronism with the periodic structure
(here called ‘super-resonances’). The field inside the WGs is essentially given at
low incidence angle by the fundamental mode strongly coupled with the
corresponding phased-matched mode. Both the simulated and the experimental
diffraction patterns show in the far field that propagation takes place essentially
only for low incidence angles, confirming the mode filtering properties of the
structured X-ray waveguides.
Keywords: X-ray beams and X-ray optics; synchrotron radiation instrumentation;
X-ray microscopes; wave propagation; transmission and absorption; interference.
1. Introduction
The propagation of modes in waveguides (WGs) with gratings
is a topic of considerable interest. In particular, waveguides
with additional periodicity have been extensively explored for
visible light and microwaves for over half a century and find
many applications in integrated optics (Yariv & Nakamura,
1977; Conwell, 1976). On the other hand, X-ray waveguides,
developed in the mid-1990s, are a relatively recent contribu-
tion to the field of optics. WGs are able to provide sub-
micrometer coherent X-ray beams in a large energy range,
both in one and two dimensions (Lagomarsino et al., 1996;
Feng et al., 1995; Pfeiffer et al., 2002). They have been applied
recently to X-ray microscopy, holography and coherent
diffraction imaging (Giewekemeyer et al., 2010; De Caro et al.,
2008; Pelliccia et al., 2010) at both synchrotron and laboratory
sources. Each resonance mode propagating inside the WG has
a well defined wavefront (Bukreeva et al., 2010; Zwanenburg
et al., 1999). However, in the front coupling geometry
(Zwanenburg et al., 1999; Pelliccia et al., 2007), several modes
are generally excited simultaneously, and the resulting wave-
front can thus be rather complex. Moreover, the degree of
coherence is conditioned by the coherence at the WG
entrance. Analysis of coherence properties and filtering of
X-ray beams in WGs can be performed by averaging of
radiation from several points (Osterhoff & Salditt, 2011).
Fabrication of single-mode X-ray WGs and reconstruction of
exit fields was discussed by Kru
¨
ger et al. (2012).
In a recent paper, Bukreeva et al. (2011) demonstrated
theoretically that an additional symmetric periodic structure
on the reflecting walls of a WG with a gap of a few hundred
nanometers can filter out the asymmetric and the high-order
modes. Then the WG can provide a highly coherent exit beam,
even with partially coherent illumination. A recent theoretical
and experimental investigation of the vertical multilayer B
4
C/
Al
2
O
3
periodic structure within WGs also demonstrated
resonant single-mode propagation of X-rays (Okamoto et al.,
2012).
In this paper we present both the oretical and experimental
studies of the properties of structured X-ray WGs, in parti-
cular of the mode filtering effect. This paper is organized as
follows: the theoretical analysis and computer simulations of
electromagnetic field propagation in structured WGs are given
in x2; expe rimental results, obtained at the cSAXS beamline at
the Swiss Light Source, are reported in x3; a general discussion
and the conclusions are provid ed in x4.
2. Guided-mode propagation analysis in structured
WGs
Let us consider a structured WG, schematically shown in Fig. 1,
in which the periodic structure (a reflecting grating with

Ronchi ruling with a period of about P = 200 mm, a duty cycle
of 1/2 and a depth of A =6mm) has been created on only one
of the two reflecting sidewall s of the WG. The material
constituting the walls is silicon with refraction index n
0
= "
1/2
=
1 + i. From Henke et al. (1993) we obtain = 7.6733
10
6
and = 1.7688 10
7
at the photon energy of 8 keV
considered here. In the coordinate system shown in Fig. 1 the
axis 0Z is parallel to the vacuum-slab boundary of the wave-
guide, and the axis 0X is along the WG gap. The waveguide,
whose channel length is 4 mm, is illuminated in front coupling
geometry by a plane wave. Since the transverse dimension of
the guiding layer (the gap) is much smaller than the lateral one
(along axis 0Y), the problem of X-ray propagation in the
section has been treated two-dimensionally.
We present initially the basic equations for uniform (i.e.
non-structured) WGs, which are well known in the literature
(Marcuse, 1974), and then we will extend the analysis to
structured WGs.
2.1. X-ray propagation in a uniform waveguide
Assuming the small va lue of reflection, diffraction and
scattering angles mainly contributing to the transmitted beam
[ <
c
=(2)
1/2
10
3
rad] and using the paraxial approx-
imation for the electric field E = (x, z) exp(ikz), the slowly
changing amplitude (x, z) is given as the solution of the
parabolic wave equations (PWEs),
2ik @=@z ¼
?
þ k
2
ð" 1Þ; ð1Þ
where "(x) is the permittivity of the material, which changes
abruptly at the vacuum–material boundaries. At a distance z
from the entrance of the multimodal waveguide the wave
amplitude with input profile (x, z =0) can be written as a
superposition of modes
m
(x),
ðx; zÞ¼
P
m
c
m
m
ðxÞ exp ik
zm
z
m
z

; ð2Þ
where m =0,1,2,... is the mode number,
m
is the mode
damping coefficient, k
zm
is the ‘slow’ longitudinal wavevector
for the mth mode, and c
m
is the excitation coefficient for
mode m.
Within the coordinate system depicted in Fig. 1, the modes
can be chosen inside the channel in the form (Ognev, 2010)
m
ðxÞ¼sin q
xm
x
m
ðÞ; ð3Þ
where q
2
xm
= k(2k
zm
)andq
xm
can be regarded as the
projection of the wavevector k on the lateral axis 0X for the
m mode. The phase term
m
in (3) is given by
m
¼ arcsin q
xm
=k
c
ðÞþðm þ 1Þ; ð4Þ
where m is an integer and
c
= [Re(1 "
0
)]
1/2
is the Fresnel
critical angle for total external reflection.
The phase term in (4) takes into account the penetration of
the electromagnetic field into the cladding material which
causes effective broadening of the geometrical width d of
the guiding layer. For angles much less than the critical angle,
q
xm
/k
c
<< 1, the waveguide dispersion equation can be
expressed in a simple form,
q
xm
¼ ðm þ 1Þ=d
eff
; ð5Þ
where the effective width of the guiding layer is d
eff
d +
/
c
. For X-ray rad iation with photon energy 8 keV and a Si
substrate the effective guiding layer broadening is /
c
12.6 nm. Since
c
is proportional to , in this approxim ation
d
eff
is independent of the photon energy.
Taking into account (5), the longitudinal wavenumber k
zm
is
given by
k
zm
¼q
2
xm
=2k ¼2ðm þ 1Þ
2
=z
T
; ð6Þ
where z
T
is the self-imaging or Talbot distance for the wave-
guide (Bukreeva et al., 2011),
z
T
¼ 22d
eff
ðÞ
2
=: ð7Þ
2.2. X-ray propagation in structured waveguides
Mathematically the physical processes that occur in a
periodic waveguide have been treated either with the guide
modes as sums of Bloch–Floquet waves (Peng et al., 1974;
Peng et al., 1975) or as a solution of the coupled-wave equa-
tions (Yariv & Nakamura, 1977; Conwell, 1976; Marcuse,
1969). To solve coupled-wave equations one can usually make
some simplifying assumptions. A very common and usually
good assumption in optical WGs is that the interaction
between two given modes is particularly strong. In the
following we will demonstrate using computer simulations that
also in the X-ray region the propagation in a structured WG
can cause coupling between two strongly interacting modes.
We used two different computer codes based on the solution
of the parabolic wave equation. The first one is based on the
finite differences method (Kopylov et al., 1995); the second is
based on a splitting scheme (Ognev, 2002) with a successive
calculation of diffraction and the phase change at each step.
Both methods gave identical results. The X-ray optical prop-
erties for silicon were taken from Henke et al. (1993).
To study the propagation of the electromagnetic field in
structured WGs we started from the assumption that the
angular spectrum of the transmitted wavefield has to satisfy
the resonance conditions determined by two main factors. The
first is related to the periodicity of the wavefield in the
research papers
692 Inna Bukreeva et al.
Periodically structured X-ray waveguides J. Synchrotron Rad. (2013). 20, 691–697
Figure 1
Sketch of a structured X-ray waveguide with only one side structured
with a grating. The incoming X-rays impinge on the WG at an angle
in
. L
is the WG length, d is the gap value and P is the grating period.
det
are
the diffraction angles measured with respect to the incident beam.

waveguide; the second is connected to the periodicity of the
reflection grating. In general, the two periodicities do not
match each other.
The frequency spacing between two guided modes, taking
into account (6), can be written as
k
zl
k
zm
¼
q
2
xm
q
2
xl
2k
¼ 2
ðm þ 1Þ
2
ðl þ 1Þ
2
z
T
: ð8Þ
Therefore the lon gitudinal period of the wavefield modulation
or mode beating between the mode m and mode l in the
guiding layer is given by
T ¼
z
T
ðm þ 1Þ
2
ðl þ 1Þ
2
: ð9Þ
The modes will be coupled and propagate efficiently in the
WG only when the periodicity of the beating T [equation (9)]
matches the periodicity P imposed by the grating, k
zl
k
zm
=
(2/P)n, or in other words when
P ¼ nT ¼
z
T
ðm þ 1Þ
2
ðl þ 1Þ
2
n; ð10Þ
where n = 1, 2, ... are the Fourier harmonics of the
grating. Equation (10) is known as the Bragg condition or the
longitudinal phase matching for the guided modes which
hereinafter we will call ‘super-resonance’. Furthermore, we
restrict ourselves to the consideration of exclusively co-
directional interactions betw een two forward propagating
modes because, contrary to the optical case, the amplitude of
the backward running mode is negligibly small for X-ray
radiation. When the phase-matching condition, equation (10),
is not fulfilled (i.e. when P T), the propagation field is
rapidly damped along the guiding layer. Therefore, for an
efficient propagation according to (10), either the grating
periodicity has to be chosen appropriately, or the WG gap
needs to be properly adjusted.
With symmetric illu mination of the WG (zero grazing-
incidence angle
in
; see Fig. 1), the interference pattern in the
guiding layer is characterized by the beating of the funda-
mental mode l = 0 with the corresponding resonance mode m.
The lateral dimension of the vacuum gap which provides the
phase matching of the selected modes can be found from (10)
with n = 1 (Yariv & Nakamura, 1977),
d
eff
¼ð1=2Þ ðm þ 1Þ
2
1

P=2

1=2
: ð11Þ
Taking the grating period P = 200 mm, one can obtain from
(11) the vacuum guiding layer widths which satisfy the long-
itudinal phase-matching conditions: d
eff
= 108 nm (m = 1),
d
eff
= 176 nm (m = 2), d
eff
= 241 nm (m = 3), d
eff
= 305 nm
(m = 4).
In Fig. 2(a) we show the super-resonance effect, plotting
the transmission calculated numerically as a function of the
effective vacuum gap width d
eff
, for the structured WG (solid
line), compared with the transmission coefficient for a uniform
waveguide (dashed line). For the structured WG, sharp super-
resonance maxima occur for the guiding layer dimensions
corresponding to equation (11). The structured waveguide
transmits less energy compared with the uniform waveguide
owing to the strong attenuation of modes which do not satisfy
the phase-matching condition. In Fig. 2(b) we show the field
attenuation along the WG for the gap d
eff
= 241 nm matching
the super-resonance conditions (black line) and for gaps just
below (d
eff
= 225 nm, dotted line) and just above (d
eff
=
280 nm, dashed line) the first one. As expected, the latter two
are damped more significantly along the guiding length. To
demonstrate that under super-resonance conditions mainly
two coupled modes propagate, we show in Figs. 3(a)–3(d) the
intensity distribution in the structured waveguide at
symmetric illumination in the case when the phase-matching
condition of (11) is satisfied for the modes with numbers m =1
(Fig. 3a), m =2(Fig.3b), m =3(Fig.3c) and m =4(Fig.3d).
Qualitative modal structure analysis in the WG is
performed with the Fourier transform of the field with respect
to the 0Z axis far from the WG entrance, where the modes out
of the super-resonance conditions are damped out. This
method was earlier applied to X-ray waveguides and electron
channelling (Fuhse & Salditt, 2005; Dabagov & Ognev, 1988).
The modulus of the Fourier transform is shown in Figs. 3(e)to
3(h) for m =1,m =2,m = 3 and m =4.
From Fig. 3 one can see that the structured WG transmits
mostly the fundamental mode and the mode m selected with
research papers
J. Synchrotron Rad. (2013). 20, 691–697 Inna Bukreeva et al.
Periodically structured X-ray waveguides 693
Figure 2
(a) Simulated transmission coefficient for the uniform (non-structured)
WG (dashed line) and for the structured WG (solid line) as a function
of the effective transverse dimension of the guiding layer. (b) Flux
attenuation along the structured waveguide with vacuum gap d
eff
=
241 nm (solid line) corresponding to super-resonance conditions d
eff
=
225 nm (dotted line), d
eff
= 280 nm (dashed line). In the calculation the
incident photon energy was E
inc
= 8 keV, the WG length L = 4 mm and
the grating period P = 200 mm.

equation (11). Under these conditions the interference pattern
is characterized by the beating of the fundamental mode with
the corresponding resonance mode m =1atd
eff
= 108 nm
[Figs. 3(a) and 3(e)], m =2atd
eff
= 176 nm [Figs. 3(b) and
3(f )], m =3atd
eff
= 242 nm [Figs. 3( c) and 3(g)] and m =4at
d
eff
= 305 nm [Figs. 3(d) and 3( h)], for the given period P =
200 mm of the grating.
Different from the uniform waveguide, which at symmetric
illumination (
in
= 0) propagates only symmetric modes, the
structured waveguide can propagate both symmetric and
asymmetric modes.
The filtering properti es of the structured WG of high-order
modes are shown in Fig. 4, where we report the X-ray flux
attenuation in the structured (a) and uniform (b) waveguide
with resonance vacuum gap d
eff
= 241 nm at different angles
in
of the waveguide illumination. It follows from the figure
that the transmission in the structured waveguide is essentially
limited to sma ll incident angles, and that effective mode
filtering takes place with respect to uniform WGs. A slight
increase of X-ray beam transmission at the negative angle
in
2.2 mrad in Fig. 4(a) corresponds to the phase matching
of high-order modes with l = 6 and m = 7 [see equation (10)].
However, one can see from the figure that the structured
waveguide effectively suppresses these high-order coupled
modes excited at angles different from zero.
Simulation for structured WGs with different groove depths
has shown that the transmission of X-rays does not depend on
groove de pth if this exceeds the gap value.
3. Experiment
3.1. WG fabrication
The WG was made from two silicon slabs, 8 mm 4 mm; in
one of them an empty channel was obtained by depositing a Cr
layer with thickness equal to the desired WG gap (240 nm in
our case) on the entire surface, except for a central channel,
approximately 1 mm wide. The other silicon slab, of the same
dimensions as the first one, but without the Cr spacer, was
positioned on the first one in such a way as to have a perfect
superposition . The two slabs were held firmly, one against the
other, with the aid of a mechanical press (Fig. 5a) (Pelliccia et
al., 2007).
research papers
694 Inna Bukreeva et al.
Periodically structured X-ray waveguides J. Synchrotron Rad. (2013). 20, 691–697
Figure 4
Variation along the WG of the intensity integrated over the channel
width, normalized to the incident integrated intensity, for different
incident angles
in
,in(a) the structured WG, (b) the uniform WG. In both
cases the vacuum gap was d
eff
= 241 nm and the incident energy was
8 keV. The period P in the structured WG was 200 mm. The figure clearly
shows selective transmission for the structured WG.
Figure 3
Top: interference pattern given by modes propagating in the guiding layer
with lateral dimensions d
eff
equal to (a) 108 nm, (b) 176 nm, (c) 241 nm,
(d) 305 nm. The color bar indicates intensity in a.u. Bottom: modal
structure of the field propagated in the WG found as a Fourier transform
of the field with respect to the optical axis of the waveguide (axis 0Z) for
the same vacuum gaps: (e) 108 nm, ( f ) 176 nm, (g) 241 nm, (h) 305 nm.
As in the previous figure the incident photon energy was E
inc
= 8 keV, the
WG length L = 4 mm and the grating period P = 200 mm

In the case of the structured WG, the periodic gra ting was
fabricated directly on the silicon slab with the Cr shoulders,
using electron beam lithography (Vistec EPBG5 High Reso-
lution, acceleration voltage 100 keV) and silicon etching. A
1.4 mm-th ick layer of a positive-tone electronic resist, poly-
methyl methacrylate (PMMA), 600k molecular weight, was
spun on the silicon top layer and baked at 442 K for 5 min on a
hotplate, exposed with a dose of 800 mCcm
2
, and developed
with a methyl isobutyl ketone and isopropyl alcohol (1:1
solution) for 90 s. The pattern was then transferred into the
substrate by means of an inductively coupled plasma (ICP )
system, using a two-step process for Si etching. In the first step
a plasma containing C
4
F
8
(30 sccm; sccm = standard cubic
centimeters per minute) and Ar (187 sccm) gases, at 70 mtorr
pressure and 600 W RF power (t = 2 s), was used to passivate
the walls of the trench; in the second step the etching was
performed by Ar (100 sccm) and SF
6
(130 sccm) gases, at
30 mtor r pressure and 500 W RF power for t = 10 s. The silicon
grating height was 6 mm, obtained in six steps of the ICP
process.
3.2. Experimental set-up
The experiment was carried out at the cSAXS beamline at
the Swiss Light Source, Villigen, Switzerland, using the set-up
shown in Fig. 5(b).
We used the X-ray undulator beam monochromated at
8 keV photon energy by a fixed-exit Si(111) monochromator.
The total distance between the source and the WG entrance
was 34 m. The beam size was firstly defined by a pair of slits at
26 m from the source to 0.5 mm in the vertical (y) direction
and 0.6 mm in the horizontal (x) direction. In addition, a third
slit placed at 33.5 m from the source further reduced the beam
size in the vertical direc tion to 0.1 mm. This slits setting gives a
final divergence on the WG entrance plane of 18 mrad on the
horizontal and 3 mrad on the vertical direction. No pre-
focusing optics was used. The photon flux was estimated to be
1.6 10
11
photons s
1
spread over a 0.6 mm 0.1 mm area,
corresponding to a flux density of 2.7 10
12
photons mm
2
.
The WG was mounted on a hexapod (http://www.hexapods.
net/) allowing six degrees of freedom for careful orientation of
the waveguide with respect to the incident beam. The WG
entrance was first centered on the beam using the x and y
translations taking care that its reflecting surface is in the yz
plane, and then the different incident angles were selected
rotating the WG around the y axis. A PILATUS 2M detector
(Kraft et al., 2009) with pixel size of 172 mm was placed
downstream of the WG at a distance of 7.27 m. A He-filled
flight tube between the WG and the detector reduced air
scattering and absorption.
3.3. Experimental results
The scope of the experiment was to measure the intensity
distribution provided by the structured WG as a function of
the incident angle, and to compare the experimental results
with simulations based on the equations reported in x2.2. To
this purpose the far-field diffraction pattern has been recorded
for a given range of incident angles. The WG was rotated in
steps of 0.0025
over a total range of 0.6
, slightly exceeding
double the critical angle
c
, and at each step an image was
recorded at the detector with an exposure time of 1 s for a
total scanning time of 4 min. Fig. 6(a) shows a representative
far-field pattern, obtained for a small grazing incidence angle
in
(rotation around the y axis) corresponding to the maximum
intensity at the detector. The WG reflecting surfaces are in the
yz plane (see Fig. 5b); consequently in the vertical direction
the beam maintains its natural divergence, but in the hori-
zontal direction the beam acquires a divergence owing to
diffraction at the exit of the WG. Fig. 6(b) shows the beam
profile along the x direction, obtained by integrating the
intensity along 21 pixels in the y direction. The same proce-
dure was repeated for all the images, each one corresponding
to a given incidence angle, and the two-dimensional intensity
distribution I(
in
,
det
) shown in Fig. 7(a) was eventually
obtained, where the abscissa
in
is the grazing-incidence angle
of the incoming beam an d the ordinate
det
is the exit angle of
radiation with respect to the incident direction (see Fig. 1).
Fig. 7(b) shows the corresponding computer simulation
research papers
J. Synchrotron Rad. (2013). 20, 691–697 Inna Bukreeva et al.
Periodically structured X-ray waveguides 695
Figure 5
(a) Schematic picture of the WG with air gap. Two 4 mm 8 mm Si slabs
were firmly pressed one against the other. One slab has two Cr shoulders,
about 200 nm thick, which leave one free channel with one nanometric
dimension d corresponding to the WG gap, and one macroscopic
dimension w (in this case 1 mm); one slab had in addition a periodic
structure with period P = 200 mm along the WG length L (see Fig. 1).
(b) Experimental set-up assembled at the cSAXS beamline for testing the
structured WG. The beam was defined by two slit systems, defining a
beam at the WG entrance of about 0.6 mm (H) 0.1 mm (V), with a
divergence of about 18 mrad (H) 3 mrad (V).

Citations
More filters
Journal ArticleDOI
L. I. Ognev1
TL;DR: In this paper, an approximate solution to the dispersion equation for the dynamics of an X-ray beam trapped in a plane waveguide with smooth walls is used to study the dependence of extinction depth on the optical density of wall material and Xray energy.
Abstract: An approximate solution to the dispersion equation for the dynamics of an X-ray beam trapped in a plane waveguide with smooth walls is used to study the dependence of extinction depth on the optical density of wall material and X-ray energy. For angles of incidence of the X-ray beam that are no greater than the Fresnel angle, the extinction depth in nonabsorbing medium depends only on difference N e1–N e1 of electron densities in the wall material and waveguide layer as (πr 0(N e1–N e2))–1/2, where r 0 ~ 2.81794 × 10–13 cm is the classical electron radius. The results are in good agreement with the penetration depth of X-ray radiation into surface that is obtained using the Fresnel formulas in the approximation of plane waves at grazing angles that range from zero to one half of the critical angle of total external reflection.

1 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, the atomic scattering factors for all angles of coherent scattering and at the higher photon energies are obtained from these tabulated forward-scattering values by adding a simple angle-dependent form-factor correction.

5,470 citations


"Periodically structured X-ray waveg..." refers background in this paper

  • ...From Henke et al. (1993) we obtain = 7.6733 10 6 and = 1.7688 10 7 at the photon energy of 8 keV considered here....

    [...]

  • ...The X-ray optical properties for silicon were taken from Henke et al. (1993)....

    [...]

Book
01 Jan 1974
TL;DR: The asymmetric slab waveguide weakly guiding optical fibers coupled mode theory applications of the coupled power theory theory of the directional coupler grating-assisted direction couplers approximate and numerical methods nonlinear effects as discussed by the authors.
Abstract: The asymmetric slab waveguide weakly guiding optical fibers coupled mode theory applications of the coupled mode theory coupled power theory theory of the directional coupler grating-assisted direction couplers approximate and numerical methods nonlinear effects.

2,290 citations

Journal ArticleDOI
TL;DR: In this paper, the propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated as a rigorous and exact boundary-value problem, and the characteristic field solutions are shown to be of the surface-wave or leaky-wave type, depending on the ratio of periodicity to wavelength (d/lambda).
Abstract: The propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated here as a rigorous and exact boundary-value problem. The characteristic field solutions are shown to be of the surface-wave or leaky-wave type, depending on the ratio of periodicity to wavelength (d/lambda). The dispersion curves and the space-harmonic amplitudes of these fields are examined for both TE and TM modes. Specific numerical examples are given for the cases of holographic layers and for rectangularly corrugated gratings; these show the detailed behavior of the principal field components and the dependence of waveguiding and leakage characteristics on the physical parameters of the periodic configuration.

452 citations


"Periodically structured X-ray waveg..." refers methods in this paper

  • ...Mathematically the physical processes that occur in a periodic waveguide have been treated either with the guide modes as sums of Bloch–Floquet waves (Peng et al., 1974; Peng et al., 1975) or as a solution of the coupled-wave equations (Yariv & Nakamura, 1977; Conwell, 1976; Marcuse, 1969)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a perturbation theory which is applicable to the scattering losses suffered by guided modes of a dielectric slab waveguide as a consequence of imperfections of the waveguide wall is presented.
Abstract: This paper contains a perturbation theory which is applicable to the scattering losses suffered by guided modes of a dielectric slab waveguide as a consequence of imperfections of the waveguide wall. The development of the theory occupies the bulk of the paper. Numerical results appear in Sections VI and VIII to which a reader less interested in the theory is referred. The theory allows us to conclude that random deviations of the waveguide wall in the order of 1 percent, for guides designed to guide an optical wave of Λ o = 1μ wavelength, can cause scattering losses of 10 percent per centimeter or 0.46 dB per centimeter. A systematic sinusoidal deviation of the waveguide wall can cause total exchange of energy from the lowest order to the first order guided mode in a distance of approximately 1 cm if the amplitude of the sinusoidal deviation from perfect straightness is only 0.5 percent of the thickness of the guide. An rms deviation of one of the waveguide walls of 9A causes a radiation loss of 10 dB per kilometer (index difference 1 percent, guide width 2.5μ).

378 citations


"Periodically structured X-ray waveg..." refers methods in this paper

  • ...Mathematically the physical processes that occur in a periodic waveguide have been treated either with the guide modes as sums of Bloch–Floquet waves (Peng et al., 1974; Peng et al., 1975) or as a solution of the coupled-wave equations (Yariv & Nakamura, 1977; Conwell, 1976; Marcuse, 1969)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the theory and device applications of periodic thin-film waveguides are discussed, including mode solutions, optical filters, distributed feedback lasers, distributed Bragg reflector (DBR) lasers, grating couplers, and phase matching in nonlinear interactions.
Abstract: This paper deals with the theory and device applications of periodic thin-film waveguides. Topics treated include mode solutions, optical filters, distributed feedback lasers (DFB), distributed Bragg reflector (DBR) lasers, grating couplers, and phase matching in nonlinear interactions.

370 citations


"Periodically structured X-ray waveg..." refers background or methods in this paper

  • ...In particular, waveguides with additional periodicity have been extensively explored for visible light and microwaves for over half a century and find many applications in integrated optics (Yariv & Nakamura, 1977; Conwell, 1976)....

    [...]

  • ...The lateral dimension of the vacuum gap which provides the phase matching of the selected modes can be found from (10) with n = 1 (Yariv & Nakamura, 1977), deff ¼ ð1=2Þ ðmþ 1Þ 2 1 P=2 1=2 : ð11Þ Taking the grating period P = 200 mm, one can obtain from (11) the vacuum guiding layer widths which…...

    [...]

  • ...Mathematically the physical processes that occur in a periodic waveguide have been treated either with the guide modes as sums of Bloch–Floquet waves (Peng et al., 1974; Peng et al., 1975) or as a solution of the coupled-wave equations (Yariv & Nakamura, 1977; Conwell, 1976; Marcuse, 1969)....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions in "Periodically structured x-ray waveguides" ?

In this paper, the authors present both theoretical and experimental studies of the properties of structured X-ray waveguides with gratings, in particular of the mode filtering effect.