microscopy with coherent x-rays - spring-8cheiron2014.spring8.or.jp/text/lec.18_c.schroer.pdfoptics...
TRANSCRIPT
![Page 1: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/1.jpg)
Christian G. Schroer DESY & Universität Hamburg
Microscopy with Coherent X-Rays
![Page 2: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/2.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
X-Ray Microscopy
2
KBroad field of applications:
>Physics, chemistry, biology, earth- and environmental science, nano-technology, ...
>Main advantage: large penetration depth !
>X-ray analytical contrasts: XRD, XAS, XRF, ...in-situ and in-operando studies
atomic scale resolution
Today: „mesoscopic gap“ catalystsCu(I)2O
M. Hanke, et al., APL 92, 193109 (2008)
quantum dots
real-space resolution: down to about 10 nmXRD and XAS: atomic scale
What happens on (mesoscopic) 1 - 10 nm scale?
![Page 3: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/3.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
X-Ray Microscopy
3
Today: „mesoscopic gap“
real-space resolution: down to about 10 nmXRD and XAS: atomic scale
What happens on (mesoscopic) 1 - 10 nm scale?
Many interesting physics and chemistry questions:
investigate local states:
> individual defects (0D): changes in electron density, charge ordering
> (structural) domain boundaries (2D), e. g., in multiferroics
> mesoscopic dynamics at (solid-state) phase transitions
> catalytic nanoparticles (activity in operando)
> ...
Mesoscale also very important for nanotechnology (e. g., defects in devices)!
![Page 4: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/4.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Current State of X-Ray Microscopy
4
Conventional x-ray microscopy
(typically: a few tens of nanometers)
optics are technology limited!Theoretical extrapolation of x-ray optical performance to the atomic level. [PRB 74, 033405 (2006); H. Yan, et al., PRB 76, 115438 (2007)]
Coherent x-ray imaging techniques (CXDI, ptychography)no imaging optic! limited by statistics of far-field diffraction patterns ...highest resolution: a few nanometers, focusing coherent beam [PRL 101, 090801 (2008); Y. Takahashi, et al., PRB 80, 054103 (2009); A. Schropp, et al., APL 100, 253112 (2012)]
optics limit spatial resolution: diffraction limit
Ernst Abbe
Airy disc
lens
↵
aperture
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Free Propagation of X-Rays
5
X-rays are electromagnetic radiation
propagation is described by homogeneous Maxwell equation.
Simplification: scalar wave field (small angles, paraxial)
r2'� 1
c2@2'
@t2= 0, ': scalar wavefield
Monochromatic radiation: Helmholtz equation
Polychromatic radiation is superposition of different frequencies ω
r2 !(~r)� k2 !(~r) = 0, '(~r, t) = !(~r) · ei!t, k =!
c
![Page 6: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/6.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Free Propagation of X-Rays
6
z
xy
0
source plane
��(⇥x, 0)
Helmholtz equation
can be solved with appropriate boundary conditions using Green’s theorem (Born & Wolf, pp. 417-425).
Boundary conditions:New nomenclature:
r2 !(~r)� k2 !(~r) = 0, '(~r, t) = !(~r) · ei!t, k =!
c
~r = (~x, z)
given in plane of source
other boundaries in half space z > 0 are far away and absorbing
![Page 7: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/7.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Free Propagation of X-Rays
7
x’
y’
x
y
z
0
θ
�(⇤x, ⇤x�) =�i cos �
2⇥r(�x, �x�) �x
�x�
�z(⇤x,⇥) =�
�0(⇤x�, ⇥) · eikr(�x,�x�)
r(⇤x, ⇤x�)· �(⇤x, ⇤x�)dx�dy�
Particular solution is the Fresnel-Kirchhoff integral:(Propagation mainly along the z direction)
(Fresnel integral, inclined incidence)
inclination factor:
propagation direction
target planeinitial plane
![Page 8: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/8.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Modeling Objects
8
d
n(x, y, z) = 1� ⇥(x, y, z) + i�(x, y, z)
X-rays interact with electrons in sample. Sample can be described by complex index of refraction:
Thin sample:No multiple scattering (1. Born approximation)
�z(~x,!) = O�z(~x) · 0(~x,!)
object
�z 0
�z
With the transmission function
O�z(~x) = eikRn dz = eik�z · e�ik
R� dz · e�k
R� dz
![Page 9: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/9.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Imaging
9
z
cohe
rent
ra
diat
ion
object
z = 0 z ⌧ d2
�z ⇡ d2
�z � d2
�
absorption near field Fresnel Fraunhofer
characteristic distance:
15 µm
d
object: Cu
d2/� = (30 µm)2/0.62 A = 14.5 m
thickness: >frame: 2 µm >thin stripes: 0.5 µm
!energy: 20 keV
![Page 10: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/10.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Imaging
10
z
cohe
rent
ra
diat
ion
object
z = 0 z ⌧ d2
�z ⇡ d2
�z � d2
�
absorption near field Fresnel Fraunhofer
characteristic distance:
15 µm
d
object: Cu
d2/� = (30 µm)2/0.62 A = 14.5 m
8% contrast
![Page 11: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/11.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Imaging
11
z
cohe
rent
ra
diat
ion
object
z = 0 z ⌧ d2
�z ⇡ d2
�z � d2
�
absorption near field Fresnel Fraunhofer
characteristic distance:
15 µm
d
object: Cu
d2/� = (30 µm)2/0.62 A = 14.5 m
22x contrast
z = 3 mm
![Page 12: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/12.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Imaging
12
z
cohe
rent
ra
diat
ion
object
z = 0 z ⌧ d2
�z ⇡ d2
�z � d2
�
absorption near field Fresnel Fraunhofer
characteristic distance:
15 µm
d
object: Cu
d2/� = (30 µm)2/0.62 A = 14.5 m
z = 14.5 mm
![Page 13: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/13.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Imaging
13
z
cohe
rent
ra
diat
ion
object
z = 0 z ⌧ d2
�z ⇡ d2
�z � d2
�
absorption near field Fresnel Fraunhofer
characteristic distance:
15 µm
d
object: Cu
d2/� = (30 µm)2/0.62 A = 14.5 m
z ! 1
![Page 14: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/14.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Fraunhofer Diffraction: Far Field
14
r0
rx
x’
z
⇥z(⌅x,⇤) = � ieikr0
2�r0
⇤⇥0(⌅x�, ⇤) · exp
��ik
xx� + yy�
r0
⇥dx�dy�
= � ieikr0
2�r0
⇤⇤0(⇧x�, ⌅) · exp
⌅�i⇧⇥⇧x�
⇧dx�dy�, ⇧⇥ =
k
r0
�x
y
⇥
r = r0 �x
r0x� � y
r0y�
r0 =�
z2 + x2 + y2
Extreme far field:
with
cos in inclination vanishes due to projection on sphere!
Far field is Fourier transform of wave field in initial plane!
![Page 15: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/15.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Fraunhofer Diffraction: Far Field
15
L2 = 3.1 m
far-field !
image
1 µm
Example: (E = 8 keV)Au (150 nm) on Si3N4:
(3 µm)2
1.55 A= 58 mm = z
![Page 16: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/16.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherence
16
probability to detect photon ~
Repeat experiment many times under identical conditions!!
So far, we considered the propagation of pure states only!
The ideal experiment:
>prepare probe (x-ray beam) in well defined initial state:
!(~x, 0) unimodal source
>propagate through fixed experimental setup
!(~x, det) =
ZP (~x, ~x00) !(~x
00, 0) d2x00
Also sample static and in well-defined state!
>detection:| !(~x, det)|2
![Page 17: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/17.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherence
17
�i
ensemble average under assumption of ergodicity!
Main problem:
We have no full control over the experiment:
>initial state not fully determined: ensemble of initial states that are realized statistically:
>experimental setup varies in time
time dependence of sample (dynamics)!
>detection:
from statistical ensemble with density
!(i,j)(~x, det) =
ZP (~x, ~x00
, tj) !i(~x00, 0) d2x00
dynamics of sample and setup described by j.
Idet(~x) =X
i,j
⇢(i,j) · | !(i,j)(~x, det)|2
!i(~x, 0)
![Page 18: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/18.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherence
18
Main problem:
What information can be extract from
Idet(~x) =X
i,j
⇢(i,j) · | !(i,j)(~x, det)|2
about the experiment! Can we deduce
!(i,j)(~x, det) and ⇢(i,j)
If the states are very different, e. g., when the initial state is badly prepared, this becomes more and more difficult!
!(i,j)(~x, det)
Intensity washes outcoherence:
visibility of interference: how much does resemble the probability density ?
Idet(~x)
| !(i,j)(~x, det)|2
?
![Page 19: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/19.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherence
19
Visibility of interference:
Idet(~x) | !(i,j)(~x, det)|2how much does resemble the probability density ?
!(i,j)(~x, det)States must be sufficiently similar that the interference fringes
are not washed out!
Influence of the source
>transverse coherence
>longitudinal coherence
spatial differences of as function of i. !(i,j)(~x, det)
energetic differences of as function of i. !(i,j)(~x, det)
![Page 20: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/20.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Example: speckle size varies inversely with object size
20
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Transverse Coherence
21
�
d>
D
L1
�
d
D
L1
D
L1
z
sample detectorsource
d
Interference observable:
: angle between two speckles as seen from sample
: angle of source as seen from sample
![Page 22: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/22.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Transverse Coherence
22
lt =�L1
Dd < lt
D
L1
z
sample detectorsource
d
Transverse coherence length:
: interference visible
![Page 23: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/23.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Example: Transverse Coherence Length
23
lt
![Page 24: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/24.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Example: Transverse Coherence Length
24
lt
![Page 25: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/25.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Longitudinal Coherence: Monochromaticity
25
n =�
��
(n + 1)� = n(� + ��)
ll = ��
��, n� 1
Wave packet generated from superposition of monochromatic waves:
�
�+��
ll ll characteristic length of wave packet
![Page 26: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/26.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Longitudinal Coherence: Monochromaticity
26
objectsource
diffraction to large angles: path length difference large
Difference in length of different paths through sample must not exceed ll:
diffraction in forward direction: path length difference small
reduction of longitudinal coherence destroys visibility of interference at large momentum transfer!
![Page 27: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/27.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Example: Longitudinal Coherence Length
27
��
�= 0.12
Diffraction grating in lights of the bus to Kyoto
![Page 28: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/28.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Lensless Imaging: Coherent Diffraction Imaging
28
biomolecule
injection
x-ray pulse
diffraction pattern
the promise:> data up to large momentum transfer q
(large numerical aperture NA)
high resolution
A dream:
![Page 29: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/29.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Reconstruction by Phase Retrieval
29
Difficulty: no direct image
Diffraction pattern: intensity (phase of wave field lost)
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
What Information is Coded in Phase and Modulus?
30
Test object:
diffraction
![Page 31: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/31.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
What Information is Coded in Phase and Modulus?
31
FFT-1
Amplitudes: square root of diffraction pattern
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
What Information is Coded in Phase and Modulus?
32
Phase: modulus of all components set to 1 (shown: real part in gray scale)
FFT-1
phases are most important!
![Page 33: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/33.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Phase Problem: Information Lost
33
Support constraint!
Is diffraction pattern unique?
>NO! Half of the information is missing! (complex amplitudes vs. real diffraction pattern)
What additional information can be used to help solve the problem?>Knowledge of part of the object:
Number of unknowns reduced below the number of measured data!
>Object must be positive: positive electron density [small (real) object]
Example:>Object confined to certain area!
(rest of image empty)
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
How to Obtain a Large Support?
34
2k x 2k detector
detector further away
Large pixel and large field of view: number of unknowns too large!
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
How to Obtain a Large Support?
35
2k x 2k detectorSmall pixel and small field of view:
very few unknowns (a lot of redundant information)!
![Page 36: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/36.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
How to Obtain a Large Support?
36
2k x 2k detectorpixel half the size as in first example:
sufficiently few unknowns & high information content!
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Right Information Content: Oversampling
37
important result:
in order for the diffraction pattern to contain sufficient information to recover the real and imaginary part of the field amplitude, the speckles have to be nicely resolved:
appropriate sampling:
oversampling
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Is Solution Now Unique?
38
�r
e�i�q·�rFourier transform: additional phase: vanishes when taking absolute square of amplitude!
No!! Object can be arbitrarily shifted!
Both objects yield same diffraction pattern!
![Page 39: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/39.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Is Solution Now Unique?
39
Diffraction pattern nearly point symmetric!
transmission function:
>real: pure absorptionFT symmetric
>imaginary: pure phase objectFT anti-symmetric
In both cases:
Diffraction pattern has point symmetry!
![Page 40: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/40.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Is Solution Now Unique?
40
No!!
Both objects yield same diffraction pattern!
Object and its point inversion are both solutions
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Is Solution Now Unique?
41
No!!Both objects yield same diffraction pattern!Object and its point inversion are both solutions
(except for central beam usually covered by beam stop.)
Babinet’s theorem!!
![Page 42: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/42.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Iterative Phase Reconstruction
42
R.W.Gerchberg & W.O. Saxton, Optic (1972) 35, 237 J.R. Fienup, Appl Opt. (1982). 21, 2758 R.P. Millane & W.J. Stroud, J. Opt. Soc. Am. (1997) A14, 568
> support > positivity > ...
FFT
FFT-1
real-space constraint Fourier-space constraint
real-space constraint: Fourier-space constraint:
i(~x, obj)
0i(~x, obj)
0i(~q, obj) :=
0i(~x, obj)
| 0i(~q, obj)|
·pIdet(~q)
0i(~q, obj)
i(~q, obj)
![Page 43: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/43.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Example
43
FT
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Reconstruction
44
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Reconstruction: Stagnation
45
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Repeat Reconstruction
46
stagnated
converged
Deviation of model from diffraction pattern:
![Page 47: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/47.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Coherent Diffraction Imaging with Soft X-Rays
47
J. Miao, et. al., Nature 400, 342(1999)
Object: Gold on Si3N4 light micrograph:
reconstruction:diffraction pattern:
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
1 µm
XFEL: Image and Destroy
48
single-pulse diffraction pattern
reconstruction
structure etched into membrane
H. Chapman et al., Nature Phys. 2, 839 (2006)
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
XFEL: Image and Destroy
49
H. Chapman et al., Nature Phys. 2, 839 (2006)
second pulse: sample destroyed
first pulse
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
CXDI of Biological Samples: Mimivirus
50
reconstruction
M. Seibert, et al., Nature 469, 78 (2011)
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Spatial Resolution
51
~ q-4
increase fluence
increase resolution
{
{
Diffraction pattern of a yeast cell:
D. Shapiro, et al., PNAS 102, 15343 (2005).
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Nanofocusing Optics
52
(a) (b)
source
focuspair of mirrorsin KB-geometryreflection:
diffraction:
refraction:
>Fresnel zone plates (~20 nm) >multilayer mirrors (7 nm) >multilayer Laue lenses (16 nm) >bent crystals
>mirrors (25 nm) >capillaries >wave guides (~10 nm)
>lenses (43 nm, 18 nm)
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Refractive X-Ray Lenses
53
100 µm
> first realized in 1996 (Snigirev et al.) > a variety of refractive lenses have been developed since > applied in full-field imaging and scanning microscopy > most important to achieve optimal performance:
parabolic lens shape
Lengeler, RXOptics
nanofocusing lenses
![Page 54: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/54.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Nanofocus
100 µm
54
De↵
focus
aper
ture
NA
f
De↵ = 4
sf�
µ/
pf
NA =De↵
2f/ 1p
f
transition to nanofocusing lenses (NFLs)
Large focal length f: aperture limited by absorption
(⇒ minimize focal length f)
minimize µ/δ (⇒ small atomic number Z)
![Page 55: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/55.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Nanofocusing Lenses (NFLs) Made of Silicon
55
single lens
optical axis
APL 82, 1485 (2003).
Made at IHM at TU Dresden together with SAW Components.
high accuracy, reproducibility
3136 NFLs on wafer! about 600000 single lenses!
![Page 56: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/56.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
point focus
sample & scanner
slits
synchrotron radiation beam
horizontal lens
vertical lens
Nanofocusing Lenses (NFLs)
56
point focus requires two lenses in crossed geometry
horiz. focus. lensvert. focus. lens
ESRF ID13
![Page 57: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/57.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Nanoprobe: Coherent X-Ray Diffraction Imaging
57
E = 15.25 keVλ = 0.813 Åbeam size:< 150 x 150 nm2
(amplitude)
gain: 104
sample
beamstop
Nanoprobe at ID13 of ESRF
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Imaging of Small Objects
58
1 µm
200 nm
4 gold particles:
diameter ~ 40 nm
7.5·106 atoms
fluence:1·105 ph/nm2
dose:
1·1010 ph
![Page 59: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/59.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Reconstruction
59
100 nm
resolution: ~ 10 nm
HIO
shrink wrap
100nm
record resolution: 5 nm (1.5·106 ph/nm2)diffraction at single gold particlePRL 101, 090801 (2008).
4 gold particles:
4⇥
10 � 1.8
>fluence 10 x smaller >resolution reduced by
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
CXDI of Silver Nanocube
60
Y. Takahashi, et al., Phys. Rev. B, 80, 054103 (2009).
SPring-8:11.8 keV
total fluence:16·106 ph/nm2
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Hard X-ray Scanning Microscopy at PETRA III
61
98.2m43.3m38.5m
nanoprobemicroprobe
prefocusingmirrormono.
X-ray energy: 10 - 50 keV
different contrasts:
> fluorescence > diffraction (SAXS, WAXS) > absorption (XAS) > ptychography & CXDI
spatial resolution: down to < 50 nm down to < 5 nm (CXDI)
Microscope:
~98 m from source
Nucl. Instrum. Meth. A 616 (2-3), 93 (2010).
![Page 62: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/62.jpg)
Christian G. Schroer | Coherence 2014 | 28. Aug. 2014 | 62
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Scanning Coherent Diffraction Imaging: Ptychography
63
J. Rodenburg, H. Faulkner. Appl. Phys. Lett. 85, 4795 (2004), P. Thibault, et al., Science 321, 379 (2008), A. Schropp, et al., Appl. Phys. Lett. 96, 091102 (2010), M. Dierolf, et al., Nature 467, 436 (2010).
>Sample is raster scanned through confined beam >At each position of scan: diffraction pattern is recorded >Overlap in illumination between adjacent points
far-field diffraction pattern
sample
![Page 64: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/64.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Ptychographic Microscopy
64
SEM image (I. Snigireva)
Experiment at P06:detector:
Pilatus 300k (172µm pixel size)
sample-detector distance: 2080 mm
exposure time: 1.5 s per point
Sample: NTT AT test pattern
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Ptychography: Reconstruction
65
O
P
Ii�(⇥r) = O(⇥r) · P (⇥r � ⇥ri)
�0(⇥r)
propagate
cycle many times
scan position
0 =p
Ii · ei·phase( )
Maiden & Rodenburg, Ultramicroscopy 109, 1256 (2009).
generate transmitted wavefield
update amplitudes
back- propagate
update object &
illumination
![Page 66: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/66.jpg)
Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Scanning Microscopy: Fluorescence Imaging
66
500 nm
50 nm lines and spaces
Ta Lα fluorescence
E = 15.25 keV 50 x 50 steps of 40 x 40 nm2
2 x 2 µm2 FOV exposure: 1.5 s per point
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Scanning Microscopy: Ptychography
67
im
re
0.0
-0.1
-0.2
-0.3
-0.4
[rad]
500 nm
50 nm lines and spaces
A. Schropp, et al., APL 96, 091102 (2010), S. Hönig, et al., Opt. Exp. 19, 16325 (2011).
A. Schropp, et al., APL 100, 253112 (2012).
E = 15.25 keV 50 x 50 steps of 40 x 40 nm2
2 x 2 µm2 FOV exposure: 1.5 s per point detected fluence: 2.75·104 ph/nm2
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
500 nm
Scanning Microscopy: Ptychography
68
1
2
3
4
pixel size: 3.84 nm
Different features have different resolution!!
A. Schropp, et al., APL 100, 253112 (2012).
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
500 nm
Scanning Microscopy: Ptychography
69
1
2
3
4
A. Schropp, et al., APL 100, 253112 (2012).
local diffraction pattern
qmax
qmax
Resolution of a feature depends on its structure factor!
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Contrast in Coherent Diffraction Imaging
70
M1C2M2
C1
CSCG
silane nitridetitanium/titanium nitridesilicon oxidevoidsaluminiumtungstensilicongate stack
M0
200 nm
M. Seibert, et al., Nature 469, 78 (2011).
Mimivirus (TEM) Microchip
Only tungsten! Where is the Al, Si, ... ?
A. Schropp, et al., J. Microscopy, 241(1), 9 (2011).
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Christian G. Schroer | DLSR Workshop | 5. Sept. 2014 |
Resolution and Contrast in Ptychography
71
coherent fluence on feature:
500 nm 500 nm
-0.4 -0.2 0.0[rad]
-0.3 -0.1 0.1[rad]
(a) (b)6.7·103 ph/nm2 3.9·102 ph/nm2
Schropp & Schroer, NJP 12, 035016 (2010).
Signal = fluence * scattering cross section
~ structure factor of feature:
For a given feature:
![Page 72: Microscopy with Coherent X-Rays - SPring-8cheiron2014.spring8.or.jp/text/Lec.18_C.Schroer.pdfoptics are technology limited! Theoretical extrapolation of x-ray optical performance to](https://reader034.vdocuments.us/reader034/viewer/2022050514/5f9e4d205132675a0f1ced1f/html5/thumbnails/72.jpg)
Christian Schroer | SNI 2014 | 23. September 2014 | Seite
Imaging at the X-Ray Free-Electron Laser
74
Linac Coherent Light Source (LCLS at SLAC)> short, dense electron bunches> long undulator
ultra short, very intense x-ray pulses (10 - 100 fs)
Investigating the dynamics of matter on nanoscale.
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
>increase fluence for imaging !!
>increase intensity on sample
!>increase energy deposited in
sample
>...
Focusing Hard X-Ray FEL Pulses
75
L1
L2
g
b
Focusing needed:
resolution determined by fluence on sample
induce non-linear optical effects
create matter in extreme conditions
source
lens
Diffraction limited imaging of source
B. Lengeler, RXOptics
X-ray optics:
beryllium parabolic refractive x-ray lenses
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Experiment at MEC at LCLS
76
undulatorz = -464.0 m
Be-CRLsz = -0.25 m
samplez = 0 m
CCDz = 4.14 m
monochromatorz = -376.4 m
f = 250 mm dt = 115 nm}
Attenuation:
E = 8.2 keV, λ = 1.51 Å
monochrom. beam: �E
E⇡ 10�4 ⇡ 10�3
>N = 20 >R = 50 µm >300 µm aperture
Focusing by CRLs:
test object (KTH)
2 µm
A. Schropp, et al., Sci. Rep. 3, 1633 (2013).
unattenuated beam
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Experiment at MEC at LCLS
77
undulatorz = -464.0 m
Be-CRLsz = -0.25 m
samplez = 0 m
CCDz = 4.14 m
monochromatorz = -376.4 m
E = 8.2 keV, λ = 1.51 Å
A. Schropp, et al., Sci. Rep. 3, 1633 (2013).
Ptychogram:
20 x 20 steps 100 x 100 nm2 step size single shot exposure: 2 exposures per point 384 out of 400 (800) images for reconstruction
512 x 512 pixel 54 x 54 µm2 pixel size
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Reconstruction
78
0.0
-0.3
-0.6
-0.9
-1.2
-1.8
-1.5
2 ѥm
re
im2 ѥm
1.0
0.8
0.6
0.4
0.2
0.0re
lativ
e in
tens
ity
-1000 -500 0 500 1000horizontal position [nm]
average position 1 position 2
125 nm
object phase [rad]
illumination
A. Schropp, et al., Sci. Rep. 3, 1633 (2013).
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
Nanofocused LCLS Beam Profile
79
1 2z
5 mm
5 μm
5 μm
re
im
identify aberrations of lensfeed into fabrication
A. Schropp, et al., Sci. Rep. 3, 1633 (2013).
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
High Resolution X-Ray Imaging at XFEL Sources
80
Method: magnifying phase contrast imaging
>imaging with secondary point source: efficient use of fluence >phase contrast: for quantitative results, phase retrieval required
Alternatively:Time-resolved diffraction in focus of nanobeam
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Christian G. Schroer | Cheiron School 2014 | 29. September 2014 | page
An X-Ray Microscopists Dream
82
access to all length scales(in principle) from Å to millimeters
Quantitive in-situ measurement of physical properties of matter
>on all relevant length scales >on all relevant time scales
Key technology: brilliant, coherent x-rays with time structure
Fusion of real and reciprocal space:
Requirement: a lot of coherent light
X-ray free-electron lasers diffraction limited storage rings