. Coherence of light and matter:
from basic concepts to modern applications
Part III
Vorlesung im GrK 1355
SS 2011
A. Hemmerich & G. Grübel
Location:
SemRm 052, Gebäude 69, Bahrenfeld
Tuesdays 10.15 – 11.45
G.Grübel (GR), A.Hemmerich (HE), M.Yurkov (YU)
1
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Coherence of light and matter:
from basic concepts to modern applications
Introduction
12.4.
(HE)
19.4+3.5. Coherence of classical light , basic concepts and examples
(HE)
10.+17.5. Coherence of quantized light, basic concepts and examples
(HE)
24.5.
Coherence of matter waves
31.5.
Coherence properties of the radiation from
7.6.
(HE)
x-ray free electron lasers
(YU)
Coherence based X-ray techniques: Introduction
(GR)
21+28.6. Imaging techniques
(GR)
5.+12.7. X-ray Photon Correlation Spectroscopy
(GR)
2
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
Literature
Basic concepts: The quantum theory of light
Rodney Loudon, Oxford University Press (1990)
Quantum Optics
Marlan O. Scully, M. Suhal Zubairy, Cambridge
University Press (1997)
Dynamic Light Scattering with Applications
B.J. Berne and R. Pecora, John Wiley&Sons (1976)
Elements of Modern X-Ray Physics
J. A. Nielsen and D. McMorrow, J. Wiley&Sons (2001)
Matter Waves:
Bose-Einstein Condensation in Dilute Gases
C. J. Pethick and H. Smith, Cambridge University Press (2002)
Lecture Notes
Part I:
Part II+III:
3
http://www.photon.physnet.uni-hamburg.de/ilp/
hemmerich/teaching.html
http://hasylab.desy.de/science/studentsteaching/lectures/ss11/
graduiertenkolleg/……
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Coherence of light and matter:
from basic concepts to modern applications
Part III: G. Grübel
Script 1
Coherence based X-ray techniques
Overview, Introduction to X-ray Scattering, Sources of Coherent X-rays,
Speckle pattern and their analysis
Imaging techniques
Phase Retrieval, Sampling Theory, Reconstruction of Oversampled Data,
Fourier Transform Holography, Applications
X-ray Photon Correlation Spectroscopy (XPCS)
Introduction, Equilibrium Dynamics (Brownian Motion), Surface Dynamics,
Non-Equilibrium Dynamics
4
Imaging and XPCS at FEL Sources
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
Coherence based X-ray techniques:
Introduction
5
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Introduction:
Experimental Set-Up
50
source (visible light, x-rays,..)
100
150
source parameters: source
size, λ, λ/λ, ...
200
250
300
coherence properties:
(incoherent, partially coherent,
coherent)
50
100
200
detector
sample
interacts with radiation
(e.g. x-rays)
L
6
150
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
250
300
.
Introduction: Scattering with coherent X-rays
If coherent light is scattered from a disordered system it gives rise to a
random (grainy) diffraction pattern, known as “speckle”. A speckle pattern
is an interference pattern and related to the exact spatial arrangement of
the scatterers in the disordered system.
I(Q,t) ~ Sc(Q,t) ~ |
e iQRj(t) |2
j in coherence volume c= ξ t2ξ l
Incoherent Light:
S(Q,t) = < Sc(Q,t)>V>>c
7
ensemble
average
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Introduction: Speckle Pattern
A speckle pattern contains information on both, the source and sample
that produced it.
If the source is fully coherent and the scattering amplitudes and phases of
the scattering are statistically independent and distributed over 2π one
finds for the probability amplitude of the intensities:
P(I) = (1/<I>) exp (- I/<I>)
Mean:
Std.Dev. ζ:
Contrast:
8
<I>
sqrt (<I2> -<I>2) = <I>
ß = ζ2/<I>2 = 1
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. The Linac Coherent Light Source (LCLS)
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Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. The Linac Coherent Light Source (LCLS)
Single pulse hard X-ray speckle
pattern captured from nano-particles
in a colloidal liquid (photon wavelength = 1.37 Å
10
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
Introduction: Scattering with coherent X-rays
If coherent light is scattered from a disordered system it gives rise to a
random (grainy) diffraction pattern, known as “speckle”. A speckle pattern
is an interference pattern and related to the exact spatial arrangement of
the scatterers in the disordered system.
I(Q,t) ~ Sc(Q,t) ~ |
e iQRj(t) |2
j in coherence volume c= ξ t2ξ l
Incoherent Light:
S(Q,t) = < Sc(Q,t)>V>>c
11
ensemble
average
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
Introduction: Speckle Reconstruction
Reconstruction (phasing) of a speckle pattern: “oversampling” technique
gold dots on SiN membrane
(0.1 m diameter, 80 nm thick)
=17Å coherent beam at X1A
(NSLS), 1.3.109 ph/s 10 m pinhole
24 m x 24 m pixel CCD
reconstruction
“oversampling” technique
Miao, Charalambous, Kirz, Sayre, Nature, 400, July 1999
12
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Reconstruction of “oversampled” data
Model structure in 20 nm SiN
membrane
Incident FEL pulse:
25 fs, 32 nm,
4 x 1014 W cm-2 (1012
ph/pulse)
13
Speckle pattern recorded
with a single (25 fs) pulse
Reconstructed image
H. Chapman et al.,
Nature Physics,
2,839 (2006)
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
Introduction: Scattering with coherent X-rays
If coherent light is scattered from a disordered system it gives rise to a
random (grainy) diffraction pattern, known as “speckle”. A speckle pattern
is an interference pattern and related to the exact spatial arrangement of
the scatterers in the disordered system.
I(Q,t) ~ Sc(Q,t) ~ |
e iQRj(t) |2
j in coherence volume c= ξ t2ξ l
Incoherent Light:
S(Q,t) = < Sc(Q,t)>V>>c
14
ensemble
average
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Introduction:
X-ray Photon Correlation
Spectroscopy (XPCS)
colloidal silica particles
undergoing Brownian
motion in high viscosity
glycerol
V. Trappe and A. Robert
15
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
quantify dynamics in terms of the intensity correlation function g2(Q,t):
I(Q,t) =|E(Q,t)|2 = |
bn(Q) exp[iQ rn(t)|2
Note: E(Q,t) = dr’ ρ(r’) exp [iQ r’(t)]
g2(Q,t)
ρ(r’): charge density
= <I(Q,0) I(Q,t)> / <I(Q)>2
if E(Q,t) is a zero mean, complex gaussian variable:
g2(Q,t)
= 1 + ß(Q) <E(Q,0)E*(Q,t)>2 / <I(Q)>2
g2(Q,t)
= 1 + ß(Q) |f(Q,t)|2
F(Q,t) =
16
<> ensemble av.; ß(Q) contrast
with f(Q,t) = F(Q,t) / F(Q,0)
F(Q,0): static structure factor
N: number of scatterers
[1/N{b2(Q)}
|
N
m=1
N
n=1
<bn(Q)bm(Q) exp{iQ[rn(0)-rm(t)]}>
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
Coherence based X-ray techniques:
An X-ray Scattering Primer
17
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
Experimental Set-Up for Scattering Experiments
50
source (visible light, x-rays,..)
100
150
source parameters: source
size, λ, λ/λ, ...
200
250
300
coherence properties:
(incoherent, partially coherent,
coherent)
50
100
150
200
250
detector
k’
k
sample
scattering angle 2
interacts with radiation
(e.g. x-rays)
L
18
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
300
. Scattering of X-rays: A primer
consider a monochromatic plane (electromagnetic) wave with
wavevector k:
E(r,t) = ε Eo exp{i(kr-ωt)}
with |k|=2π/λ, λ[Å]=hc/E, ω=2π/
k’
elastic scattering:
Q
ħ k’ =ħ k+ħQ
k
Scattering by a single electron:
Erad(R,t)/Ein =
-(e2/4πεomc2)exp(ikR)/R cosψ
spherical wave
thomson scattering length ro
(=2.82*10-5 Å)
19
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
scattered intensity:
IS/Io = |Erad|2 R2
/ |EIn|2 Ao
R2 : solid angle seen by detector
Ao incident beam size
Is = (dζ/d ) (Io/Ao)
with the differential cross section (for Thomson scattering)
(dζ/d ) = ro2 P
note: ζtotal =
20
1
P = cos2ψ
½(1+cos2ψ)
vertical
horizontal
unpolarized
(dζ/d ) = (8π/3) ro2
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
scattering by a single atom:
phase factor
scattering amplitude by
an ensemble of electrons
-rofo(Q) = –ro
(atomic) formfactor
rj
exp(iQ.rj)
position of scatterers
fo(Q0) = Z, fo(Q ) = 0
form factor of an atom:
f(Q,ħω) = fo(Q) +f ’ (ħω) + i f ’’ (ħω)
dispersion corrections:
scattering intensity:
21
level structure
absorption effects
Is = ro2 f(Q) f *(Q) P
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
scattering by a crystal:
rj’ = Rn + rj
lattice vector + atomic position in lattice
Fcrystal (Q) =
rj fj(Q)exp(iQrj)
unit cell structure factor
Rnexp(iQRn)
lattice sum
Is = ro2 F(Q) F*(Q) P
lattice sum
phase factor of order unity or N (number of unit cells) if
Q Rn =2π x integer ($)
22
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
evaluation of lattice sums:
construct reciprocal space such that:
ai
aj* = 2π δij
with ai defining a
reciprocal lattice such that
G = h a1* + k a2* + l a3*
and G fullfills ($) for Q = G (Laue condition)
k + Q = k’
Ewald sphere
sin (θ/2) = (Q/2) / k
Laue condition
lattice sum:
|
Rn
exp(iQRn) |2
Bragg’s law
 N vc* δ (Q-G)
N number of unit cells; vc* unit cell volume in reciprocal space
23
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
24
construction of reciprocal space:
(real space lattice constants a1, a2, a3);
vc = a1
a1* = 2π/vc (a2 x a3)
a3* = 2π/vc (a1 x a2)
a2* = 2π/vc (a3 x a1)
(a2 x a3)
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. The unit cell structure factor
Fuc(Q) =
rj
z
Fjmol(Q) exp(iQrj)
r2
y
example: fcc lattice (use conventional cubic unit cell)
r4
r1 = 0, r2 = ½ a (y + z), r3 = ½ a (z + x), r4 = ½ a(x + y)
G = ha1* + ka2* + la3*
a1* = 2π/vc (a2xa3) = 2π/a3 [ay x az] = 2π/a [y x z] = 2π/a x
a2* = 2π/vc (a2xa3) = 2π/a3 [az x ax] = 2π/a [z x x] = 2π/a y
a3* = 2π/vc (a2xa3) = 2π/a3 [ax x ay] = 2π/a [x x y] = 2π/a z
vc = a1 (a2 x a3)
G
G
G
G
25
r1 = 2π/a ( hx + ky + lz)
r2 = 2π/a ( hx + ky + lz)
r3 = 2π/a ( hx + ky + lz)
r4 = 2π/a ( hx + ky + lz)
0
1/2a(y + z)
1/2a(z + x)
1/2a(x + y)
=0
= π (k+l)
= π (h+l)
= π (h+k)
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
x
. The unit cell structure factor for a fcc lattice
Fhklfcc(Q) =
j=1- 4
f(Q) exp(iQrj) = f(Q) [ exp(iGr1) + … exp(iGr4)]
Fhkl fcc(Q)= f(Q) [ 1 + exp(iπ(k+l)) + exp(iπ(h+l)) + exp(iπ(h+k))]
4
if h,k,l are all even or odd
0
otherwise
=
Ihkl fcc (Q) = F(Q) F*(Q)
Reflections:
100 forbidden
111 allowed
200 allowed
26
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
.
From a measurement of a (large) set of crystal reflections |Fhkl|2 it is
possible to deduce the positions of the atoms in the unit cell.
Limitations:
phaseproblem:
| F(Q )| = | F(-Q) |
| F(Q) | = | F(Q)e iΦ |
27
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Coherence
Longitudinal coherence:
Two waves are in phase at point P. How far
can one proceed until the two waves have
a phase difference of π:
l
= (λ/2) (λ/ λ)
Transverse coherence:
Two waves are in phase at P. How far does
one have to proceed along A to produce a
phase difference of π:
2
t
28
t
θ =λ
= (λ/2) (R/D)
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Fraunhofer Diffraction
Fraunhofer diffraction of a
rectangular aperture 8 x 7 mm2,
taken with mercury light λ=579nm
(from Born&Wolf, chap. 8)
29
Fraunhofer diffraction of a
circular aperture, taken with
mercury light λ=579nm
(from Born&Wolf, chap. 8)
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Fraunhofer Diffraction (λ=0.1nm)
30
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel
. Speckle pattern
random arrangement of
apertures: speckle
31
regular arrangement of
apertures
Coherence of light and matter: from basic concepts to modern applications, Vorlesung im GrK 1355, SS 2011 A. Hemmerich & G. Gruebel