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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...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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Page 1: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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

Page 2: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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

igna

l pul

se e

nerg

y

DelayExcitation

pulses

Variably delayed Probe pulse

Signal pulseMedium under study

Page 3: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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 mediumUnexcited medium absorbs heavily at wavelengths corresponding to transitions from ground state.

Excited medium absorbs

weakly at wavelengths

corresponding to transitions from ground

state.

Page 4: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Epr(t–)

Variable delay,

Detector

Esig(t,)

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.

Cha

nge

in p

robe

pu

lse

ener

gy

Delay, 0

The excite pulse changes the sample absorption of the sample, temporarily.

Excite pulse

Eex(t)

Samplemedium

Page 5: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Modeling excite-probe measurementsLet the unexcited medium have an absorption coefficient, 0.Immediately after excitation, the absorption decreases by 0. Excited states usually decay exponentially:

() = 0 exp(– /ex) for > 0

where is the delay after excitation, and ex is the excited-state lifetime.

So the transmitted probe-beam intensity—and hence pulse energy and average power—will depend on the delay, , and the lifetime, ex:

where L = sample length 0 0exL e L

transmitted incidentI I e

0 0exL e L

incidentI e e

001 exL

incidentI e e L assuming 0 L << 1

00 1 extransmittedI e L

Page 6: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Modeling excite-probe measurements (cont’d)

The relative change in transmitted intensity vs. delay, , is:

Cha

nge

in p

robe

-be

am in

tens

ity

Delay, 0

0

00

transmitted transmitted

transmitted

T I IT I

00 1 extransmitted transmittedI I e L

0

0

exT

e LT

Page 7: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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

23

Excited molecules in state 1: absorption of probe

Cha

nge

in p

robe

-be

am tr

ansm

itted

in

tens

ity o

r pow

er

Delay, 0

0

Excited molecules in state 2: stimulated emission of probe

Page 8: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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 pulsetrain

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-indetector

The lock-in detects only one frequency component of the detector voltage—chosen to be that of the chopper.

Page 9: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Excite-probe studies of bacterio-rhodopsin

Rhodopsin is the main molecule

involved in vision. After absorbing a

photon, rhodopsinundergoes a many-

step process, whose first three steps occur on fsor 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).

Page 10: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Excite-probe measurements can reveal quantum beats

Excitation-pulse spectrum

0

12

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.

Page 11: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Excite-probe measurements can reveal quantum beats: ExperimentHere, 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).

Page 12: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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.

Page 13: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

The coherence spike in ultrafast spectroscopyWhen the delay is zero, other nonlinear-optical processes occur, involving coherent 4WM between the beams and generatingadditional signal not described by the simple 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 veryfast event that couldn’t be resolved. Or it could be a coherence spike.

Page 14: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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.

Diff

ract

ed

puls

e en

ergy

Delay, 0

Delay

Excite pulse #1

SampleExcite pulse #2

Probe pulse

Diffracted pulse

Intensity fringes in

sample due to excitation

pulses

Page 15: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort 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

iffra

cted

be

am e

nerg

y

Delay, 0

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

Page 16: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

What the transient-grating technique measuresIt measures the Pythagorean sum of the changes in the absorption and refractive index. The diffraction efficiency, , 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.

Diff

ract

ed b

eam

in

tens

ity

Delay, 0

Tran

smitt

ed

inte

nsity

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(- /ex), then the TG decay will be exp(-2 /ex):

2 2

4 2L n kL

Page 17: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Time-resolved fluorescence is also useful.

Fluo

resc

ent

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

Page 18: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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).

Page 19: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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.

Page 20: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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.

Page 21: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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!

Page 22: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort 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 n mmn

Page 23: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

If the state of a single two-level atom is:

The density matrix, ij(t), is defined as:

Since excited state populations always eventually decay to ground state populations, ii generally depends on time, ii(t).

And coherence between two states usually decays even faster, so the off-diagonal elements also depends on time.

The density matrix

cc

* *

**

c c c cc ccc

ρ or ρ are the population densities of states and .

ρ and ρ are the degree of coherence between states and .

Page 24: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

For a many-atom system, the density matrix, ij(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

cc t

t c t c t c tt

t t

2

*

*

2

( ) ( ( )

( ) )

)

( ) (

c t

c t

c t c t

c c tt

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.

Page 25: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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.

Page 26: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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 [(t) or (t)]: T1 “relaxation time”

Coherence [(t) or (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 << T1.

Page 27: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

The Liouville equation for the density matrix is:

(in the interaction picture)

which can be formally integrated:

Nonlinear-Optical Perturbation Theory

,di Vdt

0

0

( ) ( ) 1/ ( '), ( ') 't

t t i V t t dt

t

This can be solved iteratively:

Note that i.e., a “time ordering.”

( )

0

( ) ( )n

n

t t

0 -1 1 ... n nt t t t t

-11

0 0 0

( )1 2 1 2 0( ) 1/ ... ( ), ( ), ... ( ), ( ) ...

ntttnn

n n

t t t

t i dt dt dt V t V t V t t

Page 28: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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 t

2 0 1 0 2 1( ) ( ) ( ) ( ) ( ) ( )V t t V t t V t V t

( ) n

1 2 0( ), ( ), ... ( ), ( ) ...nV t V t V t t

1 2 0 1 0 2( ) ( ) ( ) ( ) ( ) ( )V t V t t V t t V t

Page 29: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

• population of state j = jj

• polarization operator p is:

0

0

and so the macroscopic polarization P is:

2112

2221

1211

00

N

TrN

pTrNpNP

Hamiltonian H = H0 + Hint:

00

*int EE

EpH

For an ensemble of 2-level systems in the presence of a laser field:

1

2

Density matrix & Hamiltonian

000

E00E

H2

10

(we have defined the zero of energy as the energy of state 1)

Page 30: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Liouville equation for the diagonal element:

1

eq222222

Ti2,H2

ti

1

eq2222

21121221 TiHH

…and similarly for 11.

1

2222

21

12

2221

1211

2221

1211

21

12 200

2T

iH

HH

H eq

1

021121221 T

iHH2t

i

Derive from these an equation for the population difference = 22 - 11

Diagonal elements of

Page 31: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Liouville equation for the off-diagonal element:

2

2112

Ti2,H1

ti

212

2112

TiH

ti

Now use the known form for the perturbation:

ti*ti*2112 e)t(Ee)t(EHH

1

0titi*1221 T

EeeEi2t

212

ti*ti21

T1ieEEei

t

Off-diagonal elements of

Page 32: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

tim

m

)m(

tin

n

)n(2121

e)t(

e)t(

Insert into the previous equations, and match terms of like frequency:

1

)n(0

)n()1n*(

21*)1n(

21

)n(

TEEi2

t

)1n()n(21

)n(21 EiAt

i

T1A

2

which can be integrated to yield:

t

't

)1n(t

)n(21 "Adtexp)'t(Ei'dt

1

)1n*(21

*)1n(21

t)n(

Tt'texp)'t(E)'t(Ei2'dt

Rotating wave approximation

Page 33: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

We have a system of equations of the form: )1n()n(

21

)1n(21

)n(

G

F

Start with 21(0) = 0 and (0) = 0 and iterate:

)0()1(21

)2()0()1(21 GFFG

)0()2()3(21 GFGG

21(3) term looks like:

2

3

2

3

1

21

t

t

*3

*21

t

t32

*1

t

t1

12t

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

Must be a (3) process!

Iterative method for solving perturbatively

Page 34: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

2

3

2

3

1

21

t

t

*3

*21

t

t32

*1

t

t1

12t

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

Suppose there are two pulses, both short compared to all relevant time scales:

k1

k2

E(t) = E1 (t) eik1r + E2 (t ) 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

Page 35: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

k1

k2

2k1 k2

Choose the 2k1 - k2 direction.

Then only two terms (of 16 in 21(3)) contribute:

2

3

2

3

1

21

t

t

*321

t

t321

t

t1

12t

3

t

2

t

1

3

2210

)3(21

'dtAexp)t()t()t('Adtexp)t()t()t(

'AdtT

ttexpdtdtdtEEi2

First term: must have t2 ≥ 0 AND t2 =

must have t1 ≥ AND t1 = 0

≥ 0≤ 0} signal only

for = 0

"coherent spike"

Second term: must have < 0 and t > 0 (no other constraints);it gives rise to signal at values of other than merely = 0

Example: application to the two-pulse echo

Page 36: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

k1

k2

2k1 - k2

0*

t

0

3

2210

)3(21 'dtA'AdtexpEEi2

t > 0 < 0

Homogeneous broadening

Polarization = N 21 ~ exp[(t + )/T2']

measured signal S():

otherwise

0 for

0edtPS

'2T

22)3(

where21

'2 T

1T1

T1 +=

P(3) is non-zero only at t =

otherwise

0 for

0eS

'2T

4

Example: application to the two-pulse echo

t

Inhomogeneous broadeningWe must integrate over the inhomogeneous distribution g():

gede

gdPtiTt

)3(21

)3(

2~

tedtedtP)(S 22 Tt2

0

T2

0

2)3(

Page 37: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Homogeneous case: phase-matched free-induction decay

pulse #1 pulse #2free-induction decay: 2k1 - k2

Inhomogeneous case: photon echo

pulse #1 pulse #2

echo

It can be difficult to distinguish between these two cases experimentally!

Echo vs. FID: can we tell?

Page 38: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

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

Page 39: Ultrafast Laser Spectroscopy...Ultrafast laser spectroscopy: How? Ultrafast laser spectroscopy involves studying ultrafast events that take place in a medium using ultrashort pulses

Photon echo: what’s going on?The pseudo-vector: z component denote population state

xy components denote polarization statez 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)