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.

Signal pulse Medium under study Variably delayed Excitation Probe

pulses pulse Signal pulse energy Delay

The signal pulse energy (or change in energy) is plotted vs. delay.

The experimental temporal resolution is the pulse length. 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.

Unexcited medium Excited medium

Unexcited Excited medium medium absorbs absorbs heavily at weakly at wavelengths wavelengths corresponding corresponding to transitions to transitions from ground from ground state. state.

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. The simplest ultrafast spectroscopy method is the Excite-Probe technique.

Excite the sample with one pulse; probe it with Probe another a variable delay later; and measure pulse the change in the transmitted probe pulse energy or average E (t–) pr Sample power vs. delay. medium Detector

Variable Eex(t) delay,  Esig(t,) Excite The excite pulse pulse changes the sample absorption of the sample, temporarily. pulse energy

Change in probe 0 Delay,  The excite and probe pulses can be different colors. This technique is also called the Pump-Probe technique. Modeling excite-probe measurements

Let 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: ex 00L eL where L = sample length IIetransmitted incident

ex 00L eL  Ieeincident

0L ex assuming << Ieincident 1 0 eL 0 L 1

ex IeLtransmitted 01 0  Modeling excite-probe measurements (cont’d)

ex IItransmitted transmitted  01 0 eL

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

TItransmitted  I transmitted  0   TI0 transmitted  0

T   ex  0 eL T0  beam intensity Change in probe- 0 Delay,  Modeling excite-probe measurements (cont’d) 3 2 More complex decays occur if Excite Probe intermediate states are populated or if transition transition 1 the motion is complex. Imagine probing an intermediate transition, whose states temporarily fill with molecules on their 0 way back down to the ground state:

Excited molecules in state 2: stimulated emission of probe

Excited molecules in state 1: absorption 0 of probe Change in probe- beam transmitted intensity or power intensity 0 Delay,  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:

Chopper The excite pulse periodically Chopped excite changes the sample absorption pulse train seen by the probe pulse.

Lock-in detector

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

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! Excite-probe studies of bacterio-rhodopsin

Rhodopsin is the Native Artificial main molecule Probe at 460nm (increased involved in vision. absorption): After absorbing a photon, rhodopsin Zhong, et al., Ultrafast undergoes a many- Phenomena X, step process, p. 355 (1996). whose first three steps occur on fs Probe at 860 nm (stimulated or ps time scales emission): 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! Excite-probe measurements can reveal quantum beats

Since ultrashort pulses have broad bandwidths, they can excite two or more nearby states simultaneously.

Excitation- 2 pulse spectrum 1

Probe pulse Excite pulse 0

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

Electron transfer from a polymer to a buckyball is very fast.

It has applications to photo-voltaics, nonlinear optics, and artificial photosynthesis.

Brabec, et al., Ultrafast Phenomena XII, p. 589 (2000). The coherence spike in ultrafast spectroscopy

When the delay is zero, other nonlinear-optical processes occur,  involving coherent 4WM between the beams and generating additional 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. Excite Sample This spike pulse could be a very very fast event Probe that couldn’t pulse be resolved. Intensity fringes in Or it could sample when pulses be a arrive simultaneously coherence spike. Alternate picture: the pulses induce a grating in the absorption and/or refractive index, which diffracts light from each beam into the other. 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: Intensity Excite Sample fringes in pulse #1 sample due Excite pulse #2 to excitation pulses

Delay Probe Diffracted pulse pulse

This method is background-free, but the diffracted pulse energy goes as the square Diffracted

of the diffracted field and hence is weaker pulse energy than that in excite-probe measurements. 0 Delay,  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? Intensity Excite pulse fringes in #1 (acting as sample due the probe) to an Excite pulse #2 excitation pulse and the probe acting as an excitation pulse Delay Probe pulse (acting as an excite pulse)

A transient-grating experiment

with a coherence spike: Diffracted beam energy 0 Delay,  What the transient-grating technique measures

It measures the Pythagorean sum of the changes in the absorption and refractive index. The diffraction efficiency, , is given by:

 22   LnkL  42 Absorption Refractive index (amplitude) (phase) grating grating

This is in contrast to the excite-probe technique, which is only sensitive to the change in absorption and depends on it linearly.

If the absorption grating dominates

and the excite-probe decay is exp(- /ex), then the TG decay will be exp(-2 /ex): intensity intensity Transmitted H. Eichler, Laser-Induced Dynamic Diffracted beam Delay,  Gratings, Springer-Verlag, 1986. 0 Time-resolved fluorescence is also useful.

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 Excite with a probe pulse in pulse Fluorescence a SFG crystal. SFG Sample crystal

Slow detector Probe pulse Lens

Delay Fluorescent beam power Delay Time-resolved fluorescence decay Normal tissue

When different tissues look alike (i.e., have similar absorption spectra), looking at the time- resolved fluorescence can help distinguish them. Malignant tumor Here, a malignant tumor can be distinguished from normal tissue due to Svanberg, its longer decay Ultrafast Phenomena IX, time. 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

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. 45° polarized excite 90° polarizer pulse HWP Sample

Probe pulse 0° polarizer Delay 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. 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! Semiclassical Nonlinear-Optical Perturbation Theory

• 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: mn  m n For any operator A, the mean value is given by: A  TraceA

• Effects that are not included in this approach: saturation, population of other states by spontaneous emission, photon statistics. The density matrix   If the state of a single two-level atom is:  

 c  c  ρ or ρ are the population densities of The density matrix, ij(t), is defined as: states  and .  * *    cc cc       * *   c c cc      

ρ and ρ are the degree of  coherence between states  and . 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 for a many-atom system

For a many-atom system, the density matrix, ij(t), is defined as:  (t) (t)  ctct()* () ct ()ct* ()       * *   ()t  ()t ct( )()c ()t ctct( )     where the sums are over all atoms or molecules  in the system.

2 * Simplifying:  ct() ct ()ct()         * 2  ct()c ()t ct( ) 

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?

A macroscopic coherence is the sum over all the atoms in the medium.

Atom #1 The collisions Atom #2 "dephase" the emission, causing Atom #3 cancellation of the total emitted light, typically exponentially. Sum: 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”

Collisions cause dephasing but not necessarily de-excitation; therefore, it is generally true that T2 << T1.

The measurement of these times is often the goal of nonlinear spectroscopy! Nonlinear-Optical Perturbation Theory

The Liouville equation for the density matrix is: d  iV  ,  (in the interaction picture) dt

which can be formally integrated: t ()tt  ()  1/ iVttdt  ('),(') ' 0  t0  This can be solved iteratively: ()tt   ()n () n0

t t1 tn-1 n ()n (t )  1/ i dt dt ... dt Vt ( ), Vt ( ), ... Vt ( ), ( t ) ...  12 nn 1 2 0  tt00 t 0

Note that ttt 0-11  nn  ... tt i.e., a “time ordering.” Perturbation Theory (continued)

Expand the commutators in the integrand:

Vt( ), Vt ( ), ... Vt ( ), ( t ) ... 12  n 0 

Consider, for example, n = 2:

Vt(),(),()120 Vt t  Vt (),() 12002 Vt () t ()() t Vt 

 Vt()120 Vt() () t Vt () 102 () t Vt ()

 Vt()201 () t Vt ()  ()() t 021 Vt Vt ()

Thus,  () n contains 2n terms. Density matrix & Hamiltonian

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

2 • population of state j = jj   0   1 • polarization operator p is:     0  and so the macroscopic polarization P is: P  N p  N Tr p   11 12  0    N Tr    0    21  22       N 12  21

Hamiltonian H = H0 + Hint:

E1 0  0 0   0  E H   Hint  p  E   0      E* 0   0 E2  0    (we have defined the zero of energy as the energy of state 1) Diagonal elements of   Liouville equation for the diagonal element:       eq i 22  2 H, 2  i 22 22 t T1  eq  0 H12  11 12   11 12  0 H12  22  22  2        2  i H 21  21 22   21 22 H 21  T1

eq 22  22  H2112  H1221  i T1

…and similarly for 11.

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

   0 i  2H2112  H1221  i t T1 Off-diagonal elements of  Liouville equation for the off-diagonal element:   i 12  1 H, 2  i 21 t T2   i   12   i  H21     21 t  T2  Now use the known form for the perturbation:

* it * it H12  H21  E(t)e  E (t)e 

 2i * it it   0  21  12 E e  Ee  t  T1  i  1  21 it * it    Ee  E e   i  21 t   T2  Rotating wave approximation

(n) int 21(t)  21 e n (t)   (m)eimt m Insert into the previous equations, and match terms of like frequency: (n) (n) (n)  2i (n1) * *(n1)   0  21 E  21 E  t  T1 (n) i  1  21 (n) (n1) A    i     A21  E   t   T2  which can be integrated to yield: t  2i  t't  (n)  dt' (n1)E* (t')  *(n1)E(t') exp    21 21        T1  t  i   t  (n)  dt' E(t')(n1) exp Adt" 21           t'  Iterative method for solving perturbatively (n)  F(n1)  We have a system of equations of the form: 21 (n) (n1) 21  G

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

(1) (0) (2) (1) (0) 21  G    F21  FG 

(3) (2) (0)  21  G  GFG 

(3) 21 term looks like: 3 t t    t 1 2 t  t t  (3)  2i dt dt dt exp 2 1  Adt'  21 0    1  2  3       T1     t1 

t t   2   2  E(t )E* (t )E(t )exp Adt'  E(t )E(t )E* (t )exp A*dt'  1 2 3   1 2 3    t3  t3 

Must be a (3) process! Multiple pulses

3 t t    t 1 2 t  t t  (3)  2i dt dt dt exp 2 1  Adt'  21 0    1  2  3       T1     t1 

t t   2   2  E(t )E* (t )E(t )exp Adt'  E(t )E(t )E* (t )exp A*dt'  1 2 3   1 2 3    t3  t3  Suppose there are two pulses, both short compared to all relevant time scales:

k2

 ik1r ik2r E(t) = E1 (t) e + E2 (t ) e

k1

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 Example: application to the two-pulse echo

k2

Choose the 2k1 -k2 direction.

Then only two terms (of 16 in  (3)) contribute: k1 21

3 t t t t 2k1  k2    1 2 t  t  (3)  2i E2E dt dt dt exp 2 1  Adt'  21 0 1 2    1  2  3       T1     t1 

t t   2   2  (t )(t  )(t )exp Adt'  (t )(t )(t  )exp A*dt'  1 2 3   1 2 3    t3  t3 

First term: must have t2 ≥ 0 AND t2 =  ≥ 0 signal only must have t ≥  AND t = 0 ≤ 0} for  = 0 1 1

"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 k 2 3     t 0  t > 0 (3)  2i E2E   exp Adt'  A*dt' 21 0 1 2     < 0     0  

k1 Inhomogeneous broadening 2k1 -k2 We must integrate over the Homogeneous broadening inhomogeneous distribution g():  P(3)  d g  (3) Polarization = N 21 ~ exp[(t + )/T2']  21  ~ et T2 d ei  t g measured signal S():  t   2  T' (3) 2 e 2 for   0 P(3) is non-zero only at t =  S   P dt      0 otherwise 2 S()   P(3) dt  e2 T2  dt e2t T2 t   1 1 1 0 0 where = + 4 '  ' T T T e T2 for   0 2 1 2 S    0 otherwise Echo vs. FID: can we tell?

Homogeneous case: phase-matched free-induction decay

pulse #1 pulse #2 free-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: how to tell

One way to distinguish:

k2

k1

2k1 -k2

FWM upconversion using a third optical pulse

e.g., M. Mycek et al., Appl. Phys. Lett., 1992 Photon echo: what’s going on?

The pseudo-vector: z component denote population state xy components denote polarization state z z z

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)