nm science enabled by waveguides
Christian Fuhse, Christoph Ollinger and Tim Salditt
Institute of X-ray Physics, University of Göttingen (Germany)
Outline
Theoretical background● propagation of x-rays in waveguides● waveguide-based lensless imaging / holography
Imaging experiments● in-line phase-contrast imaging and in-line holography● off-axis holography
Outlook● Can we achieve 1 nm resolution by means of x-ray
waveguides?
dc≈2=
2e r e
2
X-ray waveguides
[Spiller & Segmüller, Appl. Phys.Lett. 24 (1974) 60]
θ n1
Re(n2) < Re (n
1)
n2
n2
Guided modes of a planar waveguide
standing waves → guided modes
n
n
n2
n2
z
xdd
(x)
(x)if
if
if
n1
n
n
n2
n2
n1
n
n
n2
n2
n1
Phase-contrast imaging
waveguide
sample
CCD camera
[Lagomarsino et al., Appl. Phys. Lett. 71 (1997) 2557]
12 µm
planar waveguide→ 140 nm spatial resolution in one direction
130 nm C
Cr
Cr
nylon fiber
≈ 500 µm
Two-dimensionally confining waveguides
[Pfeiffer et al., Science 297 (2002) 231]
„Resonant beam coupling“→ flux: 2 x 104 ph/s
beam dimensions 69 x 33 nm2
θr
PMMA Cr
waveguide
Probe
focusing device(Kirkpatrick-Baez mirrors, ID22)
1 – 3 µm
typical flux: 106-107 ph/s from 2D waveguides[Jarre, Fuhse, Ollinger, Seeger, Tucoulou, Salditt, Phys. Rev. Lett. 94 (2005), 074891]
synchrotron
beaml
Direct coupling of a focused beam
sample
CCD detector
30 – 100 nm
Direct coupling into 2D waveguides
E=8 keV[Fuhse & Salditt, Appl. Opt. (2006), in print]
0 0,5 1 z [mm]
x [n
m]
0 0,5 1 z [mm]
100
50
0
-50
-100
d = 40 nm
0 0,5 1 z [mm]
finite-difference field calculationsbased on the parabolic wave equation:
● strong absorption in the cladding● excitement of multiple guided modes● stronger damping of higher-order modes
→ field is dominated by the fundamental mode
d = 60 nm d = 100 nmx
[nm
]100
50
0
-50
-100
x [n
m]
100
50
0
-50
-100
0 2 4 I / I
0
PMMA
Si
d
d
Far-field pattern
● Far-field intensity (Fraunhofer approximation):
● guiding-core cross-sectional dimensions determine numerical aperture and
spatial resolution → spatial resolution ≈ diameter of the guiding core
2θ [deg]
2ω [deg]z
x
y
xy yxE(x,y)
Coherent imaging
point source
sampleplanewave
z1
z2 z
eff
sample
z
E1(x
1,y
1)
E2(x
2,y
2)
S1
S2
x x
y y
Kirchhoff's diffraction formula(Fresnel approximation):
zeff=z1 z2
z1 z2M =
z1 z2
z1
A PAbsorption Phase
edge enhancement
Imaging regimes
absorption & phase in “direct imaging”
A PAbsorption Phase
Imaging regimes
edge enhancement
absorption & phase in “direct imaging”
A PAbsorption Phase
hologram
Imaging regimes
edge enhancement
absorption & phase in “direct imaging”
“Direct” imaging of large objects
SEM
50 µm
gold structures (d=150 nm) → phase-shifting sample @E=10.4 keVtungsten wires ( ø 4.5 µm) → absorbing sample @E=10.4 keV
recorded image (E=10.4 keV)SEM
gold structures (d=150 nm) → phase contrast (edge enhancement)tungsten wires ( ø 4.5 µm) → absorption contrast
“Direct” imaging of large objects
50 µm
(SEM)
sample (Au, d=150 nm)hologram (11 individual exposures)
Imaging of smaller features
Imaging of smaller features
(SEM)
sample (Au, d=150 nm)hologram (11 individual exposures)
→ holographic reconstruction required
In-line holography
z z z
S
R
S S*
R
~ ~
reference beam signal wavephase-conjugate wave
”twin image“
recording: reconstruction:
sample
hologram hologram
Holographic reconstruction
phase of the reconstructed wave
(SEM)
sample (Au, d=150 nm)hologram (11 individual exposures)
Holographic reconstruction
resolution ≈ 360 nm→ but: d
WL < 100 nm
1,8
µm
phase of the reconstructed wave
(SEM)
sample (Au, d=150 nm)hologram (11 individual exposures)
sample
hologram
z z z
referencebeam
hologram
directimage
S S*
R
twinimage
Off-axis holography
→ direct image and twin image are spatially separated!
z1
z2
synchrotron
beam
Kirkpatrick-Baez mirrors
DoutD
in
Din≈ 100 nm < transversal coherence length
Dout ≈ 5 µm
M =z1 z2
z1
.
magnified off-axis hologram
2 waveguides
CCD camera
sample
Waveguide-based off-axis holography
Off-axis holography
off-axishologram
(E=10.4 keV)
phase of the reconstructed wave
(from 25 holograms)
tungsten tip(SEM)
Quantitative analysis
→ Phase of the reconstructed wave in quantitative agreement with the simulation!
100 nm
Quantitative analysis
→ Phase of the reconstructed wave in quantitative agreement with the simulation!
Summary: State of the art
● off-axis holography with two waveguides– better image quality– better spatial resolution ( ≈ 100 nm)– quantitative determination of the phase
(→ projected electron density)
● lensless imaging with a single waveguide:– “direct” imaging of large objects
(phase & absorption)– holographic reconstruction of smaller features
resolution ≈ 360 nm but: severe artifacts (“twin image”)
1,8
µm
20 µm
Perspectives
● 3D imaging: tomography:small numerical aperture → large focal depthlarge penetration depth (in particular in the hard-X-ray regime)
ERL seems to be a well-suited source!
● temporal resolution: single-shot experiments with highly-brilliant pulsed sourcesXFEL seems well-suited
● increase spatial resolution:cross-sectional dimensions of the guiding core
fundamental limit: ~ 10 nm
depending on the electron density of the utilized materials[C. Bergemann,H. Keymeulen and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003)]
● waveguide core dimensions determine NA of the waveguide beamspatial resolution ≈ core dimensions
● higher spatial resolution requires the evaluation of radiation scattered outside the waveguide beam
● measure coherent scattering dataproblem: loss of phase information
Can we achieve 1 nm resolution?
sample
waveguide
● waveguide core dimensions determine NA of the waveguide beamspatial resolution ≈ core dimensions
● higher spatial resolution requires the evaluation of radiation scattered outside the waveguide beam
● measure coherent scattering dataproblem: loss of phase information
use an additional bent waveguide beam as a referenceto measure the phase!→ solution of the phase problem ?→ How far can we bend the waveguides ?
Can we achieve 1 nm resolution?
waveguide
sample
Can we achieve 1 nm resolution?
NA = 2° → spatial resolution ≈ λ / 2NA ≈ 1.5 nm
bent to 1°losses: a factor of about 3
bent to 2°losses: a factor of about 15
beams from bent waveguides (Si/calixarene, d ≈ 60 nm, E=12 keV)(ROBL beamline, ESRF)
Acknowledgments
● Ansgar Jarre, Jens Seeger, Sebastian Kalbfleisch,(Insitute of X-ray Physics, Göttingen)
● ESRF:- ID22 (Remi Tucoulou) → imaging experiments- ROBL (Norbert Schell) → curved waveguides
● Institute of Material Physics, Göttingen:Talaat Al-Kassab (tungsten tips)
● German Federal Ministry of Education and Research (BMBF, grant no. 05 KS4MGA/9)
Comparison off-axis / in-line
Comparison of phase and intensity
intensity phase
far-fieldwithout sample
hologram ofthe sample
reconstruction(intensity)
reconstruction(phase)
Off-axis holography
Beam damage
Fabrication of x-ray waveguides
substrate
resist
e-beamexposure
development
evaporation of thetop cladding layer
lead tapesilver dag
incidentbeam
200 nm
beam
incident
4 mm