-
17. Ultrafast Laser Spectroscopy
How do we do ultrafast laser spectroscopy?
Generic ultrafast spectroscopy experiment
The excite-probe experiment
Lock-in detection
Transient-grating spectroscopy
Ultrafast polarization spectroscopy
Spectrally resolved excite-probe spectroscopy
Theory of ultrafast measurements: the Liouville equation
Iterative solution – example: photon echo
-
Ultrafast laser spectroscopy: How?
Ultrafast laser spectroscopy involves studying ultrafast events
that take place in a medium using ultrashort pulses and delays for
time resolution.
It usually involves exciting the medium with one (or more)
ultrashort laser pulse(s) and probing it a variable delay later
with another.
The signal pulse energy (or change in energy) is plotted vs.
delay.
The experimental temporal resolution is the pulse length. S
ignal puls
e e
nerg
y
DelayExcitation
pulses
Variably delayed Probe pulse
Signal pulse
Medium under study
-
What’s going on in spectroscopy measurements?
The excite pulse(s) excite(s) molecules into excited states,
which changes the medium’s absorption coefficient and refractive
index.
The excited states only live for a finite time (this lifetime is
often the quantity we’d like to find!), so the absorption and
refractive index return to their initial (before excitation) values
eventually.
Unexcited medium Excited medium
Unexcited
medium
absorbs
heavily at wavelengths
corresponding
to transitions
from ground
state.
Excited
medium
absorbs
weakly at wavelengths
corresponding
to transitions
from ground
state.
-
Epr(t–τ)
Variable delay, τ
Detector
Esig(t,τ)
Probe pulse
The simplest ultrafast spectroscopy method is the Excite-Probe
technique.
Excite the sample with one pulse; probe it with another a
variable delay later; and measure
the change in the transmitted probe pulse energy or average
power vs. delay.
The excite and probe pulses can be different colors.This
technique is also called the Pump-Probe technique.
Change in p
robe
puls
e e
nerg
y
Delay, τ0
The excite pulse changes the sample
absorption of the
sample, temporarily.
Excite pulse
Eex(t)
Samplemedium
-
Modeling excite-probe measurements
Let the unexcited medium have an absorption coefficient,
α0.Immediately after excitation, the absorption decreases by ∆α0.
Excited states usually decay exponentially:
∆α(τ) = ∆α0 exp(–τ /τex) for τ > 0
where τ is the delay after excitation, and τex is the
excited-state lifetime.
So the transmitted probe-beam intensity—and hence pulse energy
and average power—will depend on the delay, τ, and the lifetime,
τex:
where L = sample length( )0 0 exL e Ltransmitted incidentI I
e
τ τα α −− −∆ ⋅=
0 0exL e L
incidentI e eτ τα α −− ∆ ⋅=
( )0 01 exLincidentI e e Lα τ τα− −≈ + ∆ ⋅ assuming ∆α0 L
-
Modeling excite-probe measurements (cont’d)
The relative change in transmitted intensity vs. delay, τ,
is:
Change in p
robe-
beam
inte
nsity
Delay, τ0
⇒
( ) ( ) ( )( )0
0
0
transmitted transmitted
transmitted
T I I
T I
τ τ ττ
∆ − <=
<
( ) ( ) ( )00 1 extransmitted transmittedI I e Lτ ττ τ α −≈ <
+ ∆ ⋅
( )0
0
exT
e LT
τ ττ α −∆
≈ ∆ ⋅
-
Modeling excite-probe measurements (cont’d)
More complex decays occur if intermediate states are populated
or if the motion is complex. Imagine probing an intermediate
transition, whose states temporarily fill with molecules on their
way back down to the ground state:
Excite transition
Probe transition
0
1
2
3
Excited molecules in state 1: absorption of probe
Change in p
robe-
beam
tra
nsm
itte
d
inte
nsity o
r pow
er
Delay, τ0
0
Excited molecules in state 2: stimulated emission of probe
-
Lock-in Detection greatly increases the sensitivity in
excite-probe experiments.
This involves chopping the excite pulse at a given frequency and
detecting at that frequency with a lock-in detector:
Chopped excite
pulse train
Probe pulse
train
Lock-in detection automatically subtracts off the transmitted
power in the absence of the excite pulse. With high-rep-rate
lasers, it increases sensitivity by several orders of
magnitude!
The excite pulse periodicallychanges the sample absorption
seen by the probe pulse.
Chopper
Lock-in
detector
The lock-in detects only one frequency
component of the detector voltage—
chosen to be that of the chopper.
-
Excite-probe studies of bacterio-rhodopsin
Rhodopsin is the main molecule
involved in vision. After absorbing a
photon, rhodopsin undergoes a many-
step process, whose first three steps occur on fs or ps time
scales
and are poorly understood.
Excitation populates a new state, which absorbs at 460nm and
emits at 860nm. It is thought that this state involves motion of
the carbon atoms (12, 13, 14). An artificial version of rhodopsin,
with those atoms held in place, reveals this change on a much
slower time scale, confirming this theory!
Native ArtificialProbe at 460nm
(increased
absorption):
Probe at 860 nm
(stimulated
emission):
Zhong, et al., Ultrafast Phenomena X, p. 355 (1996).
-
Excite-probe measurements can reveal quantum beats
Excitation-
pulse
spectrum
0
1
2
Excite pulse
Probe pulse
Since ultrashort pulses have broad bandwidths, they can excite
two or more nearby states simultaneously.
Probing the 1-2 superposition of states can yield quantum beats
in the excite-probe data.
-
Excite-probe measurements can reveal quantum beats:
Experiment
Here, two nearby vibrational states in molecular iodine
interfere.
These beats also indicate the motion of the molecular wave
packet on its potential surface. A small fraction of the I2
molecules dissociate every period.
Zadoyan, et al., Ultrafast Phenomena X, p. 194 (1996).
-
Time-frequency-domain absorption spectroscopy of
Buckminster-fullerene
Brabec, et al., Ultrafast Phenomena XII, p. 589 (2000).
Electron transfer from a polymer
to a buckyball is very fast.
It has applications to photo-voltaics,
nonlinear optics, and artificial
photosynthesis.
-
The coherence spike in ultrafast spectroscopy
When the delay is zero, other nonlinear-optical processes occur,
αinvolving coherent 4WM between the beams and generatingadditional
signal not described by the simple ∆α model. As in autocorrelation,
it’s called the coherence spike or coherent artifact. Sometimes you
see it; sometimes you don’t.
Alternate picture: the pulses induce a grating in the absorption
and/or refractive index,
which diffracts light from each beam into the other.
Intensity fringes in sample when pulses arrive
simultaneously
Probe
pulse
SampleExcite
pulseThis spike could be a
very very
fast event
that couldn’t
be resolved.
Or it could
be a
coherence
spike.
-
Taking advantage of the induced grating: the Transient-Grating
Technique.
Two simultaneous excitation pulses induce a weak diffraction
grating, followed, a variable delay later, by a probe pulse.
Measure the diffracted pulse energy vs. delay:
This method is background-free, but the diffracted pulse energy
goes as the square of the diffracted field and hence is weaker than
that in excite-probe measurements.
Diffr
acte
d
puls
e e
nerg
y
Delay, τ0
Delay
Excite
pulse #1Sample
Excite pulse #2
Probe pulse
Diffracted pulse
Intensity fringes in
sample due to excitation
pulses
-
A transient-grating measurement may still have a coherence
spike!
When all the pulses overlap in time, who’s to say which are the
excitation pulses and which is the probe pulse?
A transient-grating experiment with a coherence spike: D
iffr
acte
d
beam
energ
y
Delay, τ0
Delay
Excite pulse #1 (acting as the probe)
Excite pulse #2
Probe pulse (acting as an
excite pulse)
Intensity fringes in
sample due to an
excitation pulse and the probe
acting as an excitation
pulse
-
What the transient-grating technique measures
It measures the Pythagorean sum of the changes in the absorption
and refractive index. The diffraction efficiency, η(τ), is given
by:
This is in contrast to the excite-probe technique, which is only
sensitive to the change in absorption and depends on it
linearly.
Dif
fracte
d b
eam
in
ten
sit
y
Delay, τ0
Tra
nsm
itte
d
inte
nsit
y
Absorption
(amplitude)
grating
Refractive index
(phase) grating
H. Eichler, Laser-Induced Dynamic
Gratings, Springer-Verlag, 1986.
If the absorption grating dominates
and the excite-probe decay is exp(-τ /τex), then the TG decay
will be exp(-2τ /τex):
( ) ( ) ( )2 2
4 2
L n kLα τ τη τ
∆ ∆ ≈ +
-
Time-resolved fluorescence is also useful.
Flu
ore
scent
beam
pow
er
Delay
Exciting a sample with an ultrashort pulse and then observing
the fluorescence vs. time also yields sample dynamics. This can
be done by directly observing the fluorescence or, if it’s too
fast, by time-gating it
with a probe pulse in a SFG crystal.
Delay
Slow detector
Excite pulse
Sample
LensProbe pulse
SFG crystal
Fluorescence
-
Time-resolved fluorescence decay
When different tissues look alike (i.e., have similar absorption
spectra), looking at the time-resolved fluorescence can help
distinguish them.
Here, a malignant
tumor can be
distinguished from normal tissue due to
its longer decay time.
Normal tissue
Malignant tumor
Svanberg, Ultrafast Phenomena IX, p. 34 (1994).
-
Temporally and spectrally resolving the fluorescence of an
excited molecule
Exciting a molecule and watching its fluorescence reveals much
about its potential surfaces. Ideally, one would measure the
time-resolved spectrum, equivalent to its intensity and phase vs.
time (or frequency).
Here, excitation occurs to a predissociative state, but other
situations are just as interesting. Analogous studies can be
performed in absorption.
-
Ultrafast Polarization Spectroscopy
It’s also possible to change the absorption coefficient
differently for the two
polarizations. This is called induced dichroism. It also rotates
the probe polarization and can also be used to study orientational
relaxation.
Delay
45°polarized excite pulse Sample
Probe pulse
HWP
0° polarizer
90° polarizer
A 45º-polarized excite pulse induces birefringence in an
ordinarily isotropic sample via the Kerr effect. A variably delayed
probe pulse
between crossed polarizers watches the birefringence decay,
revealing the sample orientational
relaxation.
-
Other ultrafast spectroscopic techniques
Photon Echo
Transient Coherent Raman Spectroscopy
Transient Coherent Anti-Stokes Raman Spectroscopy
Transient Surface SHG Spectroscopy
Transient Photo-electron Spectroscopy
Almost any physical effect that can be induced by ultrashort
light
pulses!
-
• Treat the medium quantum-mechanically and the light
classically.
• Assume negligible transfer of population due to the light.
• Assume that collisions are very frequent, but very weak: they
yield exponential decay of any coherence
• Use the density matrix to describe the system. The density
matrix is defined according to:
For any operator A, the mean value is given by:
• Effects that are not included in this approach: saturation,
population of other states by spontaneousemission, photon
statistics.
Semiclassical Nonlinear-OpticalPerturbation Theory
( )ATraceA ρ=n mmn =ρ
-
If the state of a single two-level atom is:
The density matrix, ρij(t), is defined as:
Since excited state populations always eventually decay to
ground state populations, ρii generally depends on time,
ρii(t).
And coherence between two states usually decays even faster, so
the off-diagonal elements also depends on time.
The density matrix
cc βα α βψ = +
* *
**
c c c c
c ccc
β β
β ββ β
αα α α α α
α α β βρρρ
ρ
=
α
β
ω
ραα or ρββ are the population densities of
states α and β.
ραβ and ρβα are the degree of coherence between states α and
β.
-
For a many-atom system, the density matrix, ρij(t), is defined
as:
where the sums are over all atoms or molecules in the
system.
The density matrix for a many-atom system
*
*
*
*
( )
( ) (
( () ( ) ( ) ( )
) ( )
)
( ) (( ) )
t c t c t c t
c
c t
t c t c t c t
t
t t
αα α α α α
α α
β β
β ββ β β β
ρ ρρρ
=
∑ ∑∑ ∑
2
*
*
2
( ) ( ( )
( ) )
)
( ) (
c t
c t
c t c t
c c tt
β
β β
α α
α
=
∑ ∑∑ ∑
Simplifying:
The diagonal elements (gratings) are always positive, while the
off-diagonal elements (coherences) can be negative or even
complex.
So cancellations can occur in coherences.
-
Why do coherences decay?
Atom #1
Atom #2
Atom #3
Sum:
A macroscopic coherence is the sum over all the atoms in the
medium.
The collisions "dephase"the emission, causingcancellation of the
total emitted light, typically exponentially.
-
Grating and coherence decay: T1
and T2
A grating or coherence decays as excited states decay back to
ground.
A coherence can also cancel out if collisions have randomized
the phase of each oscillating atomic dipole.
The time-scales for these decays to occur are always written
as:
Grating [ραα(t) or ρββ(t)]: T1 “relaxation time”
Coherence [ραβ(t) or ρβα(t)]: T2 “dephasing time”
The measurement of these times is often the goal of nonlinear
spectroscopy!
Collisions cause dephasing but not necessarily de-excitation;
therefore, it is generally true that T2
-
The Liouville equation for the density matrix is:
(in the interaction picture)
which can be formally integrated:
Nonlinear-Optical Perturbation Theory
[ ],di Vdt
ρ ρ=ℏ
( ) [ ]00
( ) ( ) 1/ ( '), ( ') '
t
t t i V t t dt
t
ρ ρ ρ= + ∫ℏ
This can be solved iteratively:
Note that i.e., a “time ordering.”
( )
0
( ) ( )n
n
t tρ ρ∞
=
=∑
0 -1 1 ... n nt t t t t≤ ≤ ≤ ≤ ≤ ←
( ) [ ]-11
0 0 0
( )
1 2 1 2 0( ) 1/ ... ( ), ( ), ... ( ), ( ) ...
nttt
nn
n n
t t t
t i dt dt dt V t V t V t tρ ρ = ∫ ∫ ∫ℏ
-
Expand the commutators in the integrand:
Consider, for example, n = 2:
Thus, contains 2n terms.
Perturbation Theory (continued)
[ ] [ ]1 2 0 1 2 0 0 2( ), ( ), ( ) ( ), ( ) ( ) ( ) ( )V t V t
t V t V t t t V tρ ρ ρ = −
2 0 1 0 2 1( ) ( ) ( ) ( ) ( ) ( )V t t V t t V t V tρ ρ− +
( ) nρ
[ ]1 2 0( ), ( ), ... ( ), ( ) ...nV t V t V t tρ
1 2 0 1 0 2( ) ( ) ( ) ( ) ( ) ( )V t V t t V t t V tρ ρ= −
-
• population of state j = ρjj
• polarization operator p is:
µ−µ−
0
0
and so the macroscopic polarization P is:
( )
( )21122221
1211
0
0
ρρµµ
µρρρρ
ρ
+−=
−−
⋅=
⋅==
N
TrN
pTrNpNP
Hamiltonian H = H0 + Hint:
−−
=⋅=0
0*int
E
EEpH
µµ
For an ensemble of 2-level systems in the presence of a laser
field:
1
2
Ωℏ
Density matrix & Hamiltonian
Ω=
=
ℏ0
00
E0
0EH
2
1
0
(we have defined the zero of
energy as the energy of state 1)
-
Liouville equation for the diagonal element:
[ ]1
eq
222222
Ti2,H2
ti
ρ−ρ−ρ=∂ρ∂
ℏℏ
1
eq
222221121221
TiHH
ρ−ρ−ρ−ρ= ℏ
…and similarly for ρ11.
1
2222
21
12
2221
1211
2221
1211
21
122
002
Ti
H
H
H
H eqρρρρρρ
ρρρρ −−
Ω
−
Ω= ℏ
ℏℏ
( )1
021121221
TiHH2
ti
ρ∆−ρ∆−ρ−ρ=∂
ρ∆∂ℏℏ
Derive from these an equation for the population difference ∆ρ =
ρ22 - ρ11
Diagonal elements of ρ
-
Liouville equation for the off-diagonal element:
[ ]2
2112
Ti2,H1
ti
ρ−ρ=∂ρ∂
ℏℏ
21
2
2112
T
iH
ti ρ
−Ω+ρ∆−=
∂ρ∂ ℏ
ℏℏ
Now use the known form for the perturbation:
( )ti*ti*2112 e)t(Ee)t(EHH ωω− −µ−==
( )( )1
0titi*
1221T
EeeEi2
t
ρ∆−ρ∆−−ρ+ρµ=∂
ρ∆∂ ω−ωℏ
( ) 212
ti*ti21
T
1ieEEe
i
tρ
+Ω−ρ∆−µ=
∂ρ∂ ωω−
ℏ
Off-diagonal elements of ρ
-
tim
m
)m(
tin
n
)n(
2121
e)t(
e)t(
ω−
ω−
∑
∑
ρ∆=ρ∆
ρ=ρ
Insert into the previous equations, and match terms of like
frequency:
( )1
)n(
0
)n()1n*(
21
*)1n(
21
)n(
TEE
i2
t
ρ∆−ρ∆−ρ−ρµ=∂ρ∆∂ −−
ℏ
)1n()n(
21
)n(
21 Ei
At
−ρ∆µ+ρ=∂ρ∂
ℏ( )
ω−Ω−−= i
T
1A
2
which can be integrated to yield:
ρ∆
µ=ρ ∫∫−
∞−
t
't
)1n(
t
)n(
21 "Adtexp)'t(Ei
'dtℏ
( )
−ρ−ρ
µ=ρ∆ −−∞−∫
1
)1n*(
21
*)1n(
21
t
)n(
T
t'texp)'t(E)'t(E
i2'dtℏ
Rotating wave approximation
-
We have a system of equations of the form:( )
( ))1n()n(21)1n(
21
)n(
G
F
−
−
ρ∆=ρ
ρ=ρ∆
Start with ρ21(0) = 0 and ∆ρ(0) = ρ0 and iterate:
( ) ( ) ( )( ))0()1(21)2()0()1(21 GFFG ρ∆=ρ=ρ∆→ρ∆=ρ( ) ( )( )(
))0()2()3(21 GFGG ρ∆=ρ∆=ρ→
ρ21(3) term looks like:
+
×
+−
µρ−=ρ
∫∫
∫∫∫∫∞−∞−∞−
2
3
2
3
1
21
t
t
*
3
*
21
t
t
32
*
1
t
t1
12
t
3
t
2
t
1
3
0
)3(
21
'dtAexp)t(E)t(E)t(E'Adtexp)t(E)t(E)t(E
'AdtT
ttexpdtdtdti2
ℏ
Must be a χ(3) process!
Iterative method for solving perturbatively
-
+
×
+−
µρ−=ρ
∫∫
∫∫∫∫∞−∞−∞−
2
3
2
3
1
21
t
t
*
3
*
21
t
t
32
*
1
t
t1
12
t
3
t
2
t
1
3
0
)3(
21
'dtAexp)t(E)t(E)t(E'Adtexp)t(E)t(E)t(E
'AdtT
ttexpdtdtdti2
ℏ
Suppose there are two pulses, both short compared to all
relevant time scales:
τ
k1
k2
E(t) = E1 δ(t) eik1r + E2 δ(t − τ) eik2r
A product of three of these E(t) fields gives 8 terms, each with
one of these four wave vectors:
k1+k2-k2 k1+k2-k1 k1+k1-k2 k2+k2-k1
phase matching, of a sort: pick the direction you care about
Multiple pulses
-
k1
k2
2k1 − k2
Choose the 2k1 - k2 direction.
Then only two terms (of 16 in ρ21(3)) contribute:
τ−δδδ+
δτ−δδ
×
+−
µρ−=ρ
∫∫
∫∫∫∫∞−∞−∞−
2
3
2
3
1
21
t
t
*
321
t
t
321
t
t1
12
t
3
t
2
t
1
3
2
2
10
)3(
21
'dtAexp)t()t()t('Adtexp)t()t()t(
'AdtT
ttexpdtdtdtEEi2
ℏ
First term: must have t2 ≥ 0 AND t2 = τ
must have t1 ≥ τ AND t1 = 0
τ ≥ 0
τ ≤ 0} signal only for τ = 0"coherent spike"
Second term: must have τ < 0 and t > 0 (no other
constraints);it gives rise to signal at values of τ other than
merely τ = 0
Example: application to the two-pulse echo
-
k1
k2
2k1 - k2
+
µρ−=ρ ∫∫τ
0
*
t
0
3
2
2
10
)3(
21 'dtA'AdtexpEEi2ℏ
t > 0τ < 0
Homogeneous broadening
Polarization = N µ ρ21 ~ exp[(−t + τ)/T2']
measured signal S(τ):
( ) otherwise
0 for
-
Homogeneous case: phase-matched free-induction decay
τ
pulse #1 pulse #2free-induction decay: 2k1 - k2
Inhomogeneous case: photon echo
τ
pulse #1 pulse #2
τ
echo
It can be difficult to distinguish between these two cases
experimentally!
Echo vs. FID: can we tell?
-
k1
k2
2k1 - k2
One way to distinguish:
FWM upconversion using a third optical pulse
e.g., M. Mycek et al., Appl. Phys. Lett., 1992
Echo vs. FID: how to tell
-
Photon echo: what’s going on?
The pseudo-vector: z component denote population statexy
components denote polarization state
z zz
The photon echo is physically equivalent to the “spin echo” in
NMR spectroscopy, except for the extra complication of wave-vector
phase matching (2k2 – k1)