Glover stanford nov 14 2011

X-ray/Optical wave mixing
Microprobing how light manipulates matter

Ernie Glover
Advanced Light Source Division, LBNL

New Opportunities in Nonlinear X-ray Scattering (X-ray Lasers LCLS)


Light scattering
scattering generally decreases with ω

scattering ~ polarization ~ displacement

δx decreases with ω
..
m x = eE cos(ωt)
(force acts for shorter time)


Nonlinear x-ray scattering to date

Spontaneous processes (PDC & Raman)

(large vacuum fields ~1019 W/cm2 at 1 Å)


• Understand which xNLO processes are feasible

• Understand similarities/differences in information obtainable


New capabilities

• Microprobe optical interactions (x/o sfg)
(directly measure induced charge, microfields)
(screening response)

• Determine Valence charge density (x/xuv sfg)
(backgnd free)
(flow of valence charge during dynamics)

• Pump and probe on microscopic level
(x-ray four wave mixing)

X-ray Four Wave Mixing : Exciton Dynamics
Tanaka & Mukamel PRL 89 043001 (2002) : polydiacetylene

How is exciton transported along a molecular chain ?

* delay kraman
A B

exciton dynamics

FWM Spectroscopy
(tunable source, multiple frequencies) creation detection

• valence exciton is created at site A (k1, k2) migration
• exciton migrates to site B

• time delayed detection at site B (k3,kraman) k1
~ 100 eV k2
k3 kraman

A B


New capabilities

• Microprobe optical interactions (x/o sfg)
(directly measure induced charge, microfields)

• Determine Valence charge density (x/xuv sfg)
(backgnd free)
(flow of valence charge during dynamics)

• Pump and probe on microscopic level
(x-ray four wave mixing)

Today : x/o mixing & light-matter interactions

Light-matter interactions are important

Vision

Photosynthesis

Photovoltaics

Light-matter interactions are important

Photonics / Optoelectronics

Quantum control over
how matter evolves

How does light catalyze dynamics ?

photochemistry Materials science
(isomerization) (phonons)

the Primary Light-Matter interaction is
microscopic rearrangement of valence charge

subsequent dynamics

Problem
We often lack a deep understanding of the microcopic details
of how light manipulates matter !
• theoretically complex
• tough to measure

Why is the optical response complex ?

Coupling between induced dipoles
Shine light on a material
electron
dipole field applied field
(over screening)

Screening response in a material
Apply light to a material. Generally don't know magnitude (or
even direction) of resulting force on charges in the system.

Emicroscopic = Eapplied + Epolarization

self-consistent internal field 'Local' Field in the material
(many body interactions) (to within self-field effects)

Local Field Effects
Apply light (Klight) to a crystal.

Emicroscopic(Klight+G)
Eapplied(Klight)
Lattice vector

Varies on scale of atoms

Constant on atomic
lengthscale




Local Field Effects

Eapplied(Klight)
Polarization varies on
scale of atoms
Constant on atomic
lengthscale




Local Field Effects

Eapplied(Klight)

Emicroscopic(Klight+G)

Constant on atomic
lengthscale

Local field effects refer to distinction between the macroscopic field
Eapplied(Klight) and the microscopic field Emicro(Klight+G)

Why is the optical response important ?

 Practical reasons (develop devices)

 Fundamental Materials Physics
(material properties)
(ground state charge distribution)

Analogy with screening of ionic cores in a material
Valence electron gas Ground state charge distribution

Screening
ions appear response

Microscopic details of Light-matter interactions ?

No methods to directly measure !

Optical probes average over
macroscopic (~µm) lengthscale

Atomic lengthscale
information is lost

“Seeing” matter on atomic lengthscales with X-rays

1935
Static Pictures

Why not simply use diffraction to 'see'
changes to valence charge density ?

X-rays in Theory & Experiment
(Compton & Allison)

X-ray Diffraction

Measures Qth Fourier component of the electronic charge density.

Problem

X-ray scattering dominated by scatter
x-ray from core charge
Q (Poor at probing valence charge !)

Valence charge is important !

(determines chemistry, charge conduction, etc)

A Solution ?

X-ray / Optical wave mixing

Freund & Levine Phys. Rev. Lett. 25,1241 (1970)

Eisenberger & McCall Phys. Rev. A 3,1145 (1971)

X-rays : atomic lengthscales Optical : valence charge selectivity


X/O Sum Frequency Generation
x-ray + optical x-rays inelastically scatter from optically
driven charge density oscillations

optical hνx ± hνo
h νo
h νx

x-ray hνx
hνx+o
|V>

optical dipole

|G>

Directly microprobes optical interactions s p
Freund & Levine Phys. Rev. Lett. 25,1241 (1970)

Eisenberger & McCall Phys. Rev. A 3,1145 (1971)
Selective x-ray diffraction !
(preferential oscillation of valence charge)

ksum = kx + ko + GHKL momentum

ωsum = ωx + ωo energy

X-ray / Optical Sum Frequency Generation
What’s probed ?
x-ray diffraction measures charge densities
efficiency ~ ρ2Q

x-ray in x-ray out
momentum
Q
transfer
laser

Scattering regimes x-ray/optical SFG

optically induced redistribution
δρ = of valence charge
Scattering
Cross Section

(Lorentz
oscillator)

frequency

Rayleigh ~ 1/λ4 Resonant Thomson

What’s probed ?
efficiency ~ ρ2Q

x-ray in x-ray out
momentum
Q
transfer
laser

Scattering regimes x-ray/optical SFG

optically induced redistribution
δρ = of valence charge
Scattering
Cross Section

(Lorentz
oscillator)

frequency
hv

s p

What’s probed ?
efficiency ~ ρ2Q

x-ray in x-ray out
momentum
Q
transfer
laser

Scattering regimes x-ray/xuv SFG
δρ = full valence charge distribution

Scattering
Cross Section

(Lorentz
oscillator)

frequency

s p
All valence charge scatters as a
Thomson dipole.

What’s probed ?
efficiency ~ ρ2Q

x-ray in x-ray out
momentum
Q
transfer
laser

Scattering regimes x-ray/xuv SFG
δρ = full valence charge distribution

Scattering
Cross Section

(Lorentz
oscillator)

frequency

s p
I. Freund Chem. Phys. Lett. 12, 583
(1972)

X-ray / Optical Wave Mixing

Experiments tried in early 1970s failed
presumably due to weak xray sources

Experiment

optical wave mixing x-ray wave mixing

Experimental facility

4th Generation Light Source

X-ray Free Electron Laser

2-3 miles injector
to experiment

Experimental arrangement

Detector
Energy Filtering
Si 220

Monochromator
Si (111) Mixing Sample

apertures
x-ray o
X-rays ~15
1 eV, 2 µrad
Diamond 111
δE ~ 20 eV δθ ~ 2 µrad
optical

hνx
hνx + hνL
Reflectivity

Bragg angle

Experimental facility: X-ray Pump Probe Instrument

Slits, Be lenses, Intensity Monitors

800 nm, <10mJ, 50fs,

Diodes & 2MPixel array detector

Hutch 2

8 keV, 50fs, 20x250µm2,120 Hz

Hutch 3 Sample Mount
(rotation & translation )

Courtesy David Fritz

Experimental facility: X-ray Pump Probe Instrument

Experimental : Data Acquisition

Monochromator Detector
Si (111)

X-rays x-ray
1 eV, 2 µrad

δE ~ 20 eV δθ ~ 2 µrad Diamond 111

Diamond rocking curve

Experimental : Data Acquisition

Energy Filtering
Monochromator Si 220
Si (111)
Detector

X-rays x-ray
1 eV, 2 µrad

δE ~ 20 eV δθ ~ 2 µrad Diamond 111

Diamond rocking curve Si 220 calibration

Experimental : Space-time overlap

Translate to Bi (111) : laser perturbed diffraction for space-time overlap
(D.M. Fritz et al. Science 315, 633 (2007))

Detector
Energy Filtering
Si 220

Monochromator
Si (111) Bi (111) Sample

apertures
x-ray o
X-rays ~15
1 eV, 2 µrad

δE ~ 20 eV δθ ~ 2 µrad
optical

x/o delay

Experimental : SFG Data

X-ray / optical cross-correlation
SFG signal vs x-ray / optical delay


X-ray / optical cross-correlation

vary x-ray / optical time delay

x-ray

Diamond

optical


Rotate sample angle

No laser Yes laser Yes laser

x/o SFG peak

rotate sample angle

x-ray

Diamond

optical

rotate analyzer angle

x-ray analyzer

Diamond

optical

rotate optical polarization

x-ray

Diamond

optical


vary optical intensity

x-ray

Diamond

optical

Experimental : Measured Efficiency

x-ray

Diamond

optical
Ioptical ~ 1010 W/cm2

Absolute efficiency Relative efficiency

efficiency relative
SFG power / input x-ray power
to ‘regular’ diffraction

2.4 x 10-7 1.7 x 10-6

estimated uncertainty ~ factor of 2

Wave Equation Model for x/o SFG

Wave equation

∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2

dPNL/dt = JNL (ωx + ωo)

X-rays see free electrons

.
mv=F

d v/dt = ∂v/∂t + ( v . ) v = (q/m) (E + vxB/c)
∆

JNL (ωx + ωo) = ρ(0)v(2) + ρ(1)v(1)


Wave equation

∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2



.
mv=F

∆

JNL (ωx + ωo) = i(e/2m) Dωx ρo(1) Ex - Doppler
(ρ(u)/2ωsum) (e2/2m2) Dωo Dωx (Eo. kx)E + Displacement

(ρ(u)/2ωsum) (e2/m2) (Dωo/ωx) Eox(kx x Ex) Lorentz

Dωj ≡ ωj /(ωb2- ωj2)


Wave equation

∆ 2 1 d2 E 2β d E 4π d2 PNL
E =
c 2
dt 2 c dt c2 dt2



.
mv=F

∆

JNL (ωx + ωo) = i(e/2m) Dωx ρo(1) Ex

Eoe ωot
-i
Exe-iωxt
x/o SFG : Optical Doppler term Dominates !

Wave Equation Model : SFG power vs angle & energy

1 um crystal δE ~ 720 meV δθ ~ 14 urads

δ angle
δ ω



Wave Equation Model : Predicted Efficiency
Efficiency vs Crystal thicknesss Induced Charge/Microfields

absorption induced charge is the single
no absorption free parameter

JNL = ρ(111) vx

charge & microfield related
by Gauss’ law

4π ρ
∆ . E =

i G111 . E111 = 4π ρ111

Crystal thickness (m)
ρ(111) ~ 7.3x10-5 e/Å3
Efficiency = SFG power / X-ray power in
E111 / Emacro ~1/6
input beam properties
δΕx-ray ~ 1 eV δλoptical ~ 35 nm Reproduce measured efficiency

δθx-ray ~ 2 ur δθoptical ~ 4 mr
δτx-ray ~ 80 fs δτoptical ~ 2 ps

Models for microscopic optical response

• Bond Charge Model
(semi-empirical)

• Molecular Orbital Calculation (1974)

• Pseudopotential Calculation (1972)

• Density Functional Calculation
('first principles')

Diamond unit cell and primitive cell

Unit cell FCC with two atom basis

Two types of bonding orientation
Primitive cell

8 atoms (16 bonds) in Unit Cell

2 atoms (4 bonds) in Primitive Cell

Covalent Bond Formation

isolated atoms Molecular Orbital View

Diamond bond : sp3 orbitals

covalently bonded atoms

Covalent Bond Formation

isolated atoms Molecular Orbital View

Diamond bond : sp3 orbitals

o
109.5

covalently bonded atoms

Covalent Bond in Diamond

Valence electron gas Pseudopotential Strategy

Determine how a free (valence)
electron gas responds to the sudden
appearance of the ions.

Replace ionic cores (nucleus & tightly
bound electrons) with an effective
(pseudo) potential.


Ionic cores appear Pseudopotential Strategy


(pseudo) potential.

Ions (pseudopotential) polarize the
valence electrons leading to a self-
consistent valence charge distribution



Screening response to lowest order Pseudopotential Strategy

overlap charge density Determine how a free (valence)

(pseudo) potential.


Co-ordinate and nonspherical (screening response)
charge described beyond lowest
order (nonlinear screening)


Nonlinear screening central to
covalent bond formation
Pseudopotential Strategy


(pseudo) potential.






(pseudo) potential.



The self consistent field / charge
distribution develops




(pseudo) potential.



Covalent bonds stabalize lattice against
shear distortion

Bond Charge Model
Covalent bond charge is the polarizable J.C. Phillips PRL 1967
charge in the system Dielectric properties of covalent
semiconductors dominated by bond charge

B.F. Levine PRL 1969
The
Optical polarizability confined
to bond charge
Optical
pulse  Nonlinear optical susceptibilities χ(2), χ(3)

 Raman scattering

Semi-empirical model
(Freund & Levine, PRL 25, 1241 (1970) "Optically Modulated X-ray Diffraction")

Basic idea
Goal : compute optically induced charge density
{magnitude of (average) optical response & microscopic spatial distribution}

• Magnitude : from macroscopic optical measurements

• Spatial distribution : make a guess
Induced charge density identical to (measured) bond charge density.
(rigid bond model)

Bond Charge Model

The
to bond charge
Optical



Equations
Nonlocal response
δρ(r) = - .P
∆

Polarization at ith bond
P(r) = Ncell ∫cell αmicro(r,r’) Emicro(r’) d3r’
influenced by field from
other polarized bonds
P(q,G) = Ncell ∑G’αmicro(q,G,G’) Emicro(q,G’)

Bond Charge Model

The
to bond charge
Optical



Equations
Local response approximation
δρ(r) = - .P
∆

P(r) = Ncell αmicro(r) Emacro(r)
determined onlyith bond
Polarization at by
macroscoic field by
determined only
macroscoic field P(q,G) = Ncell αmicro(q,G) Emacro(q)

Bond Charge Model

The
to bond charge
Optical



Equations
Local response approximation P(q,G) = χG Emacro(q)

FHKL χmacroscopic
structure factor
Polarization at by
macroscoic field by
determined only (for bond charge)
Emacro = {3/(2+ε)} Eoptical, vacuum
macroscoic field
ε = 1+ 4πχmacro

Bond Charge Model

The
to bond charge
Optical



Equations

P(q,G) = χG {3/(2+ε)} Eoptical, vacuum
Polarization at by
Overestimates measurement by ~ x2
macroscoic field by
determined only
macroscoic field ( 1 Fourier component )

Two models for induced valence charge

Pseudopotential

Molecular Orbital

Two 1970s calculations of microscopic fields
How precisely does light exert its force ? Two predictions

δρ laser via Molecular Orbital δρ laser via Pseudopotential
Arya & Jha Phys. Rev. B 10,4485 (1974) Van Vechten & Martin Phys. Rev. Lett 28,446 (1972)

agrees with semi-empirical δρ more delocalized
(spreads beyond bond charge)
bond charge model
Mixing efficiency ~ x100 lower

Presupposes a localized response Calculation decides degree of localization

Two 1970s calculations of microscopic fields
How precisely does light exert its force ? Two predictions

δρ laser via Molecular Orbital δρ laser via Pseudopotential
Arya & Jha Phys. Rev. B 10,4485 (1974) Van Vechten & Martin Phys. Rev. Lett 28,446 (1972)

agrees with semi-empirical δρ more delocalized
(spreads beyond bond charge)
bond charge model
Mixing efficiency ~ x100 lower

Presupposes a localized response Calculation decides degree of localization
Overestimates measurement by ~ x2 Underestimates measurement by ~ x6

Density Functional Theory
Calculation of induced charge in diamond

covalent bond
Ground state valence charge density

0.25

0.20

0.15
Charge density

0.10

0.05

0.00
0.0 0.2 0.4 0.6 0.8
Position

atoms

Calculation of induced charge in diamond

Induced (valence) charge density

Charge
density


Overlay Induced over ground state charge density
ground

induced
x1000

Atoms

Optical activity
pretty well
confined to the
bond charge

Compare model prediction to measurement

Induced charge density (e/Å3) Absolute efficiency

x/o SFG measurement 7.3 x 10-5 (x or / √2) 2.4 x 10-7 (x or / 2)

DFT prediction 1.1 x 10-4 5.4 x 10-7

BC (MO) prediction 1.3 x 10-4 7.6 x 10-7

VVM pseudopotential ~ ρBC / 10 ~ 7.6 x 10-9

Density Functional Calculation & Bond Charge Model
agree with data to within error bars

optical How does light interact with Diamond ?

To good approximation, optical
activity confined to bond charge !

Summary

• Observation of x-ray/optical sum frequency generation

• Measurement and ab initio simulations suggest simple bond
charge model accurate down to microscopic length scales

• New opportunities in nonlinear x-ray scattering created by
x-ray FELs
X-ray/optical wave mixing particularly important sub-field of
nonlinear x-ray scattering due to relatively high efficiency

X/O Collaboration

Jerry Hastings Steve Harris Jan Feldkamp

David Fritz Sharon Schwartz Diling Zhu
Marco Cammarata David Reis Sinisa Coh

Henrik Lemke Ryan Coffee Tom Allison

X-ray Mixing Options : Relative Strengths

Nonlinear Current at ω1 + ω2

JDoppler (ω1+ ω2) = i(e/2m) { ω1 /(ωB2- ω12)} ρ(E2)induced E1

JDisplacement (ω1+ ω2) = (-e2/4m2) (ρ(0)/ωsum) { ω1 /(ωb2- ω12)} { ω2 /(ωB2- ω22)} (E1 . k2)E2

JLorentz (ω1+ ω2) = (e2/2m2) (ρ(0)/ωsum) { ω1 /(ωB2- ω12)}(1/ω 2) E1x(k2 x E2)

For free electrons (ωb=0)
ρ(Ei)induced = ρ(0){G.Ei/e}αfree(ωi) ~ ρ(0) G.Ei / ωi2)

Relative strength of JNL (ω1+ ω2)
JDoppler (ω1+ ω2) ~ 1/ { ω1 ω22}
x-ray = 8 keV xuv = 100 eV optical = 800 nm
JDisplacement (ω1+ ω2) ~ 1/ { ω1 ωsum }
JDoppler {x/x, x/xuv, x/o} ~ {1, 6400, 3x107}
JDisplacement (ω1+ ω2) ~ 1/ { ω1 ωsum }
JDisp/Lorentz {x/x, x/xuv, x/o} ~ {1, 160, 104}

Must account for phasematching, different charge densities, absorption …

Still efficiency x/o >> efficiency x/x or x/xuv !


Now: Ultrafast x-ray diffraction Next: Non-linear Ultrafast diffraction
follows motion of atom cores x-ray wave mixing selectively
microprobes valence charge

optical excitation (x-ray + ) optical excitation

x-ray probe x-ray + VUV probe

δt
δt

X-ray scattering dominated by atom cores View chemical bond dynamics on
natural length (Å) & time (fs) scales !
(poor at probing valence charge density)
(see breaking/formation of bonds)

X-ray Wave Mixing
New scientific opportunity to microprobe valence charge

A Bragg scattering experiment

X-rays
(atomic spatial resolution) hνx
Upconverted X-rays
hνx ± hνL (scattered solely from valence charge)
Optical or VUV
hνL
(modulates VALENCE charge)

How does light manipulate matter ? Probe full valence charge density
(X-ray/optical wave mixing) (X-ray/VUV wave mixing)
hvVUV > valence binding energy

Micro-probes how visible light distorts the Micro-probes full valence charge density. All
optically active valence charge (chemical bonds) valence electrons respond as Thomson dipoles

Wave Equation Model: Efficiency vs (δE,δθ) of input beams

δΕ x-ray ~ 2.6 meV δλ laser ~ 140 nm

δθ x-ray ~ 4 ur δθ laser ~ 90 mr

Wave Equation Model : SFG beam properties

δΕx-ray ~ 130 meV

δθx-ray ~ 6 ur
δτx-ray ~ 0.9 ps

δΕx-ray ~ 1 eV δλoptical ~ 35 nm

Wave Equation Model : Predicted Efficiency

absorption absorption
no absorption no absorption

Crystal thickness (m) Crystal thickness (m)
Loss due to imperfect collimation &
Efficiency = SFG power / X-ray power in
monochromaticity of input beams

δΕx-ray ~ 1 eV δλoptical ~ 35 nm

Glover stanford nov 14 2011

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