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)