Measurement of orientation and alignment moment relaxation by polarization spectroscopy: Theory and experiment

Measurement of orientation and alignment moment relaxation by polarization

spectroscopy: Theory and experiment

Matthew L. Costen, Hilary J. Crichton, and Kenneth G. McKendrick

Citation: The Journal of Chemical Physics 120, 7910 (2004); doi: 10.1063/1.1691019

View online: http://dx.doi.org/10.1063/1.1691019

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/120/17?ver=pdfcov

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 120, NUMBER 17

1 MAY 2004

Measurement of orientation and alignment moment relaxation

by polarization spectroscopy: Theory and experiment

Matthew L. Costen,^a)Hilary J. Crichton, and Kenneth G. McKendrick

School of Engineering and Physical Sciences, William H. Perkin Building, Heriot-Watt University,

Edinburgh, EH14 4AS, United Kingdom

͑Received 23 October 2003; accepted 9 February 2004͒

A diagrammatic perturbation theory description of one-color polarization spectroscopy ͑PS͒ is

developed which emphasizes the signiﬁcance of orientation and alignment tensor moments of the

rotational angular momentum, and their collisional evolution. The inﬂuences of Doppler motion,

velocity-changing collisions, decay of population, orientation and alignment, and nuclear hyperﬁne

depolarization on the calculated PS signal are discussed. Illustrative simulations are presented of the

evolution of the PS signal as a function of pump–probe laser delay. These are generated by a Monte

Carlo integration of the derived equations for the signal electric ﬁeld over typical experimental

pump and probe laser temporal proﬁles and velocity distributions for a commonly studied system,

the OH A ²⌺^ϩ–X ²⌸ ͑0,0͒ band. These predictions are compared with a preliminary set of results

obtained in an experimental apparatus designed for one-color polarization spectroscopy using

independent pump and probe lasers. Measurements are presented using linearly polarized pump

light on the Q₁(2.5) transition of the OH A ²⌺^ϩ–X ²⌸ ͑0,0͒ band with He as the collision partner.

The decay of the experimental PS pump–probe signal is discussed with reference to inelastic

collisional population transfer rates in the literature. It is concluded that the collisional

depolarization of rotational alignment is rapid, with a rate approximately twice that of population

transfer. This is consistent with previous measurements in atmospheric pressure ﬂames. PS is shown

to be a viable novel spectroscopic method for determining rotational angular momentum orientation

and alignment relaxation rates, which are valuable quantities because they are sensitive probes of

͓DOI: 10.1063/1.1691019͔

Reichardt, Lucht, and co-workers,⁷used a numerical solution

to the time-dependent density matrices. These calculations

included collisional effects and Doppler broadening, and the

line shapes and saturation effects that result were discussed.

Further publications discussed the effects of multiaxial laser

modes⁸and short pulse ͑picosecond͒ lasers.⁹The results of

such calculations have been compared to measurements

made in ﬂames using saturated PS on known concentrations

of OH, with good agreement being found.¹⁰

The collisional properties of PS have been the subject of

fewer studies. Reichardt et al.¹¹have presented a diagram-

matic perturbative treatment of PS in which they describe the

laser pulses as exponentially decaying temporal functions.

They present analytical results for PS signal intensity as a

function of collision rate for the case where pump and probe

pulses are coincident in time. They show a decreased depen-

dence on collision rate for a short pulse ps-laser over the

longer pulses from a ns-pulsed laser. The effect of collisions

on PS is most clearly seen when the relative timings of the

pump and probe laser pulses are varied. The relaxation times

of PS signals from OH in atmospheric pressure ﬂames using

picosecond laser pulses have been measured.^12–15The re-

ported relaxation times are rapid, 120–700 ps, and depend on

J and pump polarization. The results were interpreted in a

perturbative theory approach, assuming that the probe pulse

only switched on after the pump pulse ﬁnished, and that the

I. INTRODUCTION

Polarization spectroscopy ͑PS͒ was ﬁrst developed by

1

¨

Hansch and co-workers as a variant of saturation spectros-

copy. In its simplest form, a relatively powerful polarized

pump laser beam is introduced to the sample, creating an

unequal m_Jdistribution in the ground and excited levels con-

nected by the transition. A weaker linearly polarized probe

laser beam, also resonant with the transition, is then co- or

counterpropagated with the pump beam. A signal beam is

generated which copropagates with the probe beam, but has

an orthogonal linear polarization. The signal beam is sepa-

rated from the probe beam using a linear polarizer. In the last

10 years, PS has seen increasing use in analytical measure-

ments in bright or collisionally perturbed environments

where the coherent signal beam properties are useful. These

have included a signiﬁcant number of studies of small radi-

cals in ﬂames, notably OH,^2,3NH,⁴NO,⁵and C₂.⁶This work

has mainly focused on the use of PS for relative and absolute

concentration measurements, along with the determination of

rotational temperatures.

There have been a number of recent papers describing a

numerical approach to the simulation of PS suitable for the

strong ﬁeld regime. Perhaps the most signiﬁcant of these, by

a͒

Author to whom correspondence should be addressed. Electronic mail:

M.L.Costen@hw.ac.uk

0021-9606/2004/120(17)/7910/17/$22.00

7910

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7911

II. THEORETICAL DESCRIPTION OF PS

A. Introduction

pump and probe pulses were short compared to the relax-

ation time scale.

We have been developing PS as a tool for studying in-

elastic energy transfer of small radicals.¹⁶We are interested

in the evolution of the orientation and alignment of rotational

angular momentum in the course of inelastic collisions.

These have been shown in both reactive and inelastic scat-

tering to be a sensitive probe of the forces involved in the

collision.¹⁷For example, in perhaps the most sophisticated of

the recent applications of related methods to nonreactive col-

lisions, measurements of the orientation and alignment of

NO (X ²⌸) created in collisions with Ar have been obtained

using crossed molecular beams with REMPI-TOF imaging

detection.^18,19Deviations from the simple predictions of the

kinematic apse model were seen, in particular a preferential

handedness of rotation as a function of scattering angle, il-

lustrating the depth of information that can be revealed.

We are particularly interested in electronically inelastic

collisions, and have recently been studying transfer between

the A ²⌬ and B ²⌺ states of CH.²⁰Those experiments in-

volved pulsed laser preparation of single rotational levels in

either of the electronic states, followed by dispersed ﬂuores-

cence resolution of the post collision state distribution at the

rotational level.²¹We have observed rapid transfer between

electronic states with a variety of colliders.²²Simple models,

that may be successful in explaining purely rotationally in-

elastic collisions, do not correctly predict the observed pro-

pensities for transfer or ﬁnal state distributions in these elec-

tronic state-changing collisions. Development of more

successful models would seem to require better information

on the forces acting in the collisions. Rotational angular mo-

mentum alignment provides just such a detailed window on

the forces present.

However, measurements of changes in orientation and

alignment in collisions of excited state species are very rare.

This is largely due to the lack of higher-lying excited states

suitable for conventional optical double-resonance experi-

ments in these small free radicals. We have therefore been

searching for a technique that will allow polarization-

sensitive detection without requiring such higher states. We

believe that polarization spectroscopy provides such a

method. In an earlier paper,¹⁶we demonstrated excellent sig-

nal to noise in one-color PS of the OH A ²⌺^ϩ–X ²⌸ transi-

tion, under controlled collider conditions in a sealed environ-

ment. In those experiments, a single ns-pulsed laser was used

to generate both pump and probe pulses, which were essen-

tially coincident in time. The effects of collisions on the PS

signal were explored by increasing the partial pressure of

collider gas. Very rapid decay rates for the PS signal were

found, which it was suggested might be the result of

velocity-changing collisions. Here we develop the theoretical

framework for the general analysis of one-color PS signals

with arbitrary pump–probe delay and demonstrate the appli-

cation of the method to an illustrative set of new measure-

ments for OH.

Polarization spectroscopy is a third order non-linear

spectroscopy, a variant of four wave mixing ͑FWM͒.²³FWM

in general arises from the interaction of three beams with a

polarizable medium, the nonlinear polarization induced in

the medium producing a fourth coherent signal beam. This

signal beam will propagate in a direction determined by the

phase matching conditions operating on the wave vectors of

the input beams. In PS there are only two independent input

beams ͑pump and probe͒ and the signal beam is detected in

the same propagation direction as the probe beam. The wave

vector for the signal beam is required to be the same as the

probe beam, k_signalϭk_probe, and hence two photons from the

pump beam must be used such that k_signalϭk_probeϩk_pump

Ϫk_pump. All calculations in this paper are made under an

assumption that the pump and probe beams copropagate. A

number of other possible third order processes may occur in

the sample under the inﬂuence of these laser ﬁelds, for ex-

ample, coherent anti-Stokes Raman spectroscopy ͑CARS͒

and degenerate four wave mixing ͑DWFM͒. These result in

various different signal beams, propagating in various pos-

sible directions and with distinct polarizations, deﬁned by the

phase matching conditions. However, by taking advantage of

the phase matching conditions and detecting a coherent sig-

nal beam with orthogonal polarization in the probe propaga-

tion direction we restrict our observation purely to PS. We

similarly aim to restrict our theoretical treatment to only

those interactions in the sample that give rise to this speciﬁc

signal beam.

We have performed experiments with low pulse energies

that correspond to the weak ﬁeld, perturbative limit. In this

publication, we correspondingly treat the problem theoreti-

cally in the perturbative limit, using a diagrammatic ap-

proach and angular momentum tensor theory. Williams

et al.^24,25have described a diagrammatic spherical tensor

treatment of DFWM in the weak ﬁeld limit, and Wasserman

et al.^26,27have extended this to include initial ground state

orientation and alignment, such as can arise from photodis-

sociation or reaction. These authors made several assump-

tions that are not valid in the experiments described in this

paper. As noted by Williams et al.,²⁴these monochromatic

frequency domain models for incident and generated ﬁelds

necessarily demand that the population relaxation and colli-

sional dephasing times must be much shorter than the tem-

poral duration ͑nominally inﬁnite͒ of the laser pulse. In sub-

sequent ﬁnite bandwidth treatments^11,28essentially the same

formalism has been applied in situations where the pulse

durations are much shorter than the decay rates. The func-

tional forms derived in these treatments consisted of the sum

over all interacting levels of a numerator, containing a four-

fold transition matrix element, and a denominator, containing

the resonant frequency and collisional dependence of the sig-

nal. The separation of the numerator and denominator was

achieved by assuming that the relaxation and dephasing rates

were independent of the m_Jsublevel. The numerator could

then be evaluated using the spherical tensor treatment. The

result is a separation of the geometrical factors introduced by

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7912

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

the different photon polarizations from the dynamical contri-

butions arising from population, orientation and alignment of

the sample. Phenomenological decay rates for these param-

eters were then introduced to the resonant denominator.

However, the experiments described in this paper are per-

formed as a function of pump–probe delay rather than in the

frequency domain and measure decays on a time scale sig-

niﬁcantly longer than the laser pulsewidth. The measurement

of true orientation or alignment moments rather than phe-

nomenological decay rates is their primary purpose. The

main development that we describe here is a tensor moment

approach to pump–probe PS measurements that isolates true

orientation and alignment decay rates.

The diagrammatic approach to nonlinear optical calcula-

tions greatly simpliﬁes the calculation of the polarization

density matrix responsible for the generation of the third

order nonlinear signal.^29–31The complete density matrix can

be broken down into individual contributions from each pos-

sible time-ordered sequence of photon interactions, repre-

sented by a double-sided Feynman diagram. At each interac-

tion time, t_n, the quantum states participating are

represented above and below the interaction point. Between

the interaction times t₁Ͻt₂Ͻt₃Ͻt, corresponding to the

three photon interactions that generate the ﬁnal signal wave

at time, t, we have the respective delays ␶₁, ␶₂, and ␶₃.

During these delays, the system evolves ﬁeld free. Any re-

laxation during these delays is the property of the system that

we are interested in measuring.

Inspection of the diagrams shows clearly which interac-

tion sequences contribute to the measured signal, and the

relevant density matrix can then be calculated. When only

two isolated J states are involved, one in the ground elec-

tronic state and one in the excited electronic state, and only

the ground state has initial population, it is readily shown

that only eight diagrams can contribute to the signal. These

relevant diagrams are shown in Fig. 1, and can be broken

down into two groups. The ﬁrst group of four, labeled

‘‘pump–probe,’’ involve two interactions with the pump la-

ser, followed by the probe. These diagrams represent the

only processes that generate the signal when the probe pulse

is signiﬁcantly later than the pump pulse. The second four,

labeled ‘‘coherent,’’ are those where the probe interacts with

the sample laser either before, or in between, the pump in-

teractions. When the probe pulse is well before the pump

pulse only two of these diagrams will contribute to the sig-

nal, whereas all four will be signiﬁcant when the pump and

probe pulses overlap. The ‘‘pump–probe’’ diagrams also

contribute when the pump and probe pulses are coincident,

and hence all eight of the diagrams are important in this

situation.

FIG. 1. Feynman diagrams for the eight possible time-ordered sequences of

photon interactions giving rise to PS signals. First column, ‘‘pump–probe’’

diagrams IA–IVA, second column, ‘‘coherent’’ diagrams IB–IVB.

B. Perturbative derivation

The detailed derivation of the signal ﬁeld contribution

from diagram IA is given in the Appendix: here we discuss

the key features only. The measured signals are integrals

over the temporal characteristics of the experiment, princi-

pally the pulsewidths of the pump and probe lasers. The

relaxation time scales and laser pulsewidths may be of simi-

lar magnitude in our experiments, precluding any simpliﬁca-

tion of the integrals. The integration over the laser pulse-

widths is in general not a straightforward analytical task,

even if we were to assume an idealized functional form to

represent the complicated, not well deﬁned, experimental la-

ser pulse proﬁles.

Instead, we evaluate the signal ﬁeld for speciﬁc, unique

times and numerically integrate the results over suitable ex-

perimental parameters for the laser pulsewidths and detection

gate in a Monte Carlo fashion, a procedure discussed in Sec.

III. The calculation proceeds from an initially isotropic ͑and

hence diagonal͒ ground state density matrix. We assume that

at some time t₁it undergoes an interaction with the ﬁrst

photon. The resulting density matrix can be found by appli-

cation of the dipole operator. The result will be a linear su-

The Feynman diagrams may be used to direct the calcu-

lation of the signal electric ﬁeld generated by each of the

possible time-ordered sequences of interactions in the

sample. Because the temporal evolution of the electric ﬁeld

strength of our ns-pulsed lasers is much slower than the op-

tical cycle frequency, the resulting signal intensity that is

detected experimentally will be a cycle average of the square

modulus of the signal electric ﬁeld strength.

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7913

perposition of the ground and excited states, oscillating at the

transition frequency. With the aid of the Wigner–Eckart

theorem the result can be rearranged and expressed in terms

of an expansion in state multipoles and spherical tensors.³²

Provided the medium is isotropic, which is the case for our

experiments, the characterization of the relaxation of the

state multipoles that describe the expansion will be

straightforward.³³All projections, Q, of a state multipole will

relax at the same rate, but different ranks, K, will have dif-

ferent rates. The density matrix after this ﬁrst interaction is

described by a single multipole of rank Kϭ1, representing

coherence between the ground and excited states, or oscillat-

ing dipolar bulk polarization. The relaxation of this multipole

during the time interval ␶₁ϭt₂Ϫt₁is determined by a single

dephasing rate, ⌫_d.

At this point, the density matrix can be expanded once

more in terms of the state operators, the second photon in-

teraction is applied and the same procedure repeated. This

time, after some angular momentum manipulation, we ﬁnd

that the density matrix spans only the ground state, or only

the excited state, depending on which diagram is considered.

This corresponds physically to time-invariant, m_J-dependent

population transfer from the ground to excited states. The

hole burned in the ground state m_Jdistribution mirrors the

populations created in the excited state. The density matrix

may be deﬁned in terms of tensors of rank Kϭ0, 1, and 2.

These are population, orientation and alignment tensors, re-

spectively. Their evolution is the target of the experiments,

and is discussed in detail in Sec. II C. They evolve with

differing rates, ⌫_K, during the period ␶₂ϭt₃Ϫt₂. This pe-

2

riod may be long if the pump and probe pulses are widely

separated in time, as is possible with independent laser sys-

tems.

The third photon interaction is then applied in a similar

fashion, giving a density matrix that relaxes with rate ⌫_K

,

3

once again describing a superposition of the ground and ex-

cited states. The signal ﬁeld associated with this third-order

density matrix may be found by calculating the trace of the

product of the density matrix with the dipole operator. Some

straightforward angular momentum algebra shows that K₃

FIG. 2. ͑a͒ Orientation, Kϭ1, dependence of sum of squared 6-j symbols

from Eqs. ͑1͒ and ͑2͒ as a function of ground state rotational quantum

number J_gin half-integer increments. Filled triangles, P branch; ﬁlled

circles, Q branch; and ﬁlled squares, R branch. ͑b͒ Alignment, Kϭ2, depen-

dence, symbols as for ͑a͒.

ϭ1 only, and consequently the relaxation rate ⌫_Kduring the

3

time ␶₃ϭtϪt₃is the same as ⌫_dduring time ␶₁.

Similar calculations may be made for all eight diagrams,

and it is found that the signal electric ﬁeld generated by all of

the diagrams can be described in a simple fashion. The dia-

grams forming ground state orientation and alignments ͑IA,

IIA, IB, IIB͒ produce

2

1

J_eJ_g

͑3͒

Ϫ⌫ ͑␶ ϩ␶

͒

d 1 3

*

t͒"␧ ϭC⌽e

4

P

͑

ͭ

ͮ

͚

J_e

1

K₂

K

2

J

e

ϫF ␧ ␧ ␧ ␧ ;K ͒e^Ϫ⌫

.

͑2͒

␶

2

K

͑

2

1

2

3

4

2

These equations have a clear physical meaning. The ﬁrst

term, C, contains the dependence of the signal ﬁeld on the

number of molecules present, N, and the magnitudes of the

three incident electric ﬁelds, ␰_n. It also depends on the re-

duced matrix element carrying the transition line strength,

2

1

J_gJ_e

͑3͒

Ϫ⌫ ͑␶ ϩ␶

͒

d 1 3

*

t͒"␧ ϭC⌽e

4

P

͑

ͭ

ͮ

͚

J_g

1

K₂

K

2

͑1͒

4

N␰ ␰ ␰

͉

J

ʈ

͑

͗

e

␮͒

ʈ

J_g͉

͘

1

2

3

J

g

Cϭ

.

͑3͒

ϫF ␧ ␧ ␧ ␧ ;K ͒e^Ϫ⌫

.

͑1͒

␶

͑

8iប³2J ϩ1͒

͑

g

2

K

2

1

2

3

4

2

A coherent term, ⌽, deﬁnes the phase-matching resulting

from the time-ordered sequence of events and the ﬁeld-free

evolution of the system. The expression that holds for cases

I and III is

Those diagrams forming excited state orientation and align-

ments ͑IIIA, IVA, IIIB, IVB͒ result in

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7914

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

TABLE I. Polarization factors F(␧ ␧ ␧ ␧₄;K₂) for different tensor ranks

1

2 3

levels is required for orientation, and a minimum of three is

required for alignment. Hence, for example, alignment signal

cannot be generated on a Q(0.5) transition, as both ground

and excited states only possess two m_Jsubstates.

K₂and both left-hand circular and 45° linear pump polarizations. Values for

right-hand circular pump polarization are the same as for left circular other

than a change of sign for K₂ϭ1.

Left circular pump polarization

Linear pump polarization

The ﬁnal term F(␧ ␧ ␧ ␧₄;K₂) in Eqs. ͑1͒ and ͑2͒ con-

1

2 3

tains the geometrical dependence of the signal on the polar-

izations of the pump, probe, and signal photons. For cases I

and III this has the form

Diagram

K₂ϭ0

K₂ϭ1

K₂ϭ2

K₂ϭ0

K₂ϭ1

K₂ϭ2

IA

IIA

IIIA

IVA

IB

IIB

IIIB

IVB

0

i/2

i/4

0

1/2

0

1/2

͑1͒

ϪQ

Q

2

F ␧ ␧ ␧ ␧ ;K ͒ϭ

Ϫ1͒ 2K ϩ1͒ ␧ ͒

͑

͚

1

2

3

4

2

1

i/6

Ϫ5i/12

1/6

Ϫ1/4

1/12

Q

,Q ,Q

2 3

1

͑1͒

*

ϫ ␧ ͒

␧ ͒

3

␧ ͒

͑

2

Q ϪQ

4

Q

1

2

3

1

K₂

ϫ

ͩ

ͪ

Q₂ϪQ₁Q₁ϪQ₂

K₂

⌽ϭe^{Ϫi␻ tϩik "r}e^{i␶ ͑␻ Ϫ␻}

e

g

e^{i␶ ͑␻ Ϫ␻}e^{i␶ ͑␻ Ϫ␻}

.

͒

s

3

s

J J

2

1

2

1

J J

e g

1

͑4͒

ϫ

͑6͒

Q₂ϪQ₃Q₃ϪQ₂

The signal frequency ␻_sϭ␻₁Ϫ␻₂ϩ␻₃, with ␻ and ␻

1

2

from the pump laser and ␻ from the probe laser for cases IA

and for cases II and IV the result is

3

and IIIA. For cases IB and IIIB we must swap the sources of

␻ and ␻₃, so that the probe photon comes ﬁrst. The wave

1

K ϩQ

2

F ␧ ␧ ␧ ␧ ;K ͒ϭ

Ϫ1͒

2K ϩ1͒

͑

͚

1

2

3

4

2

vector deﬁnes the propagation direction of the signal beam,

k_sϭk₁Ϫk₂ϩk₃. The functional form of ⌽ for cases II and

IV is slightly different, namely,

Q

,Q ,Q

2 3

1

͑1͒

ϪQ

͑1͒

*

ϫ ␧ ͒

␧ ͒

3

͑

2

1

Q ϪQ

1

2

3

⌽ϭe^{Ϫi␻ tϩik}^"re^{i␶ ͑␻ Ϫ␻}

e

g

e^{i␶ ͑␻ Ϫ␻}e^{i␶ ͑␻}

e

.

͒

Ϫ␻

͒

J J 1

g

s

3

s

J J

2

1

K₂

͑1͒

͑5͒

*

ϫ ␧ ͒

ϫ

͑

ͩ

ͪ

4

Q

3

Q₂ϪQ₁Q₁ϪQ₂

Now ␻_sϭ␻₂Ϫ␻₁ϩ␻ with ␻ and ␻ from the pump laser

and ␻ from the probe for the ‘‘pump–probe,’’ case IIA and

IVA, whilst for the ‘‘coherent’’ case IIB and IVB we must

swap ␻ and ␻ so that the probe photon comes second. This

3

1

2

1

K₂

3

ϫ

.

͑7͒

ͩ

ͪ

Q₂ϪQ₃Q₃ϪQ₂

2

3

time the wave vector is given by k_sϭk₂Ϫk₁ϩk₃.

The experimental geometries used in polarization spectros-

copy deﬁne the probe and signal beams to have orthogonal

linear polarizations. The pump photons must, by deﬁnition,

both have the same polarization, which is either circular, or

linear ͑normally at 45° to the probe polarization, maximizing

the signal͒ at the choice of the experimenter. Using these

polarizations in the different cases to deﬁne ␧_n, and the rel-

evant Eqs. ͑6͒ or ͑7͒, the F(␧ ␧ ␧ ␧₄;K₂) can be found and

are listed in Table I. The ‘‘pump–probe’’ diagrams IA–IVA

are found, as might intuitively be expected, to be dependent

on a single tensor moment only, either K₂ϭ1 ͑orientation͒ if

the pump photons are circular, or K₂ϭ2 ͑alignment͒ if the

pump photons are linear. Crucially, the evolution of the sig-

nal when the pump and probe lasers are well separated in

time, when only these diagrams contribute, will therefore

enable the relaxation of a single tensor moment to be studied.

All of the cases are affected equally by the dephasing

rate, ⌫_d, during the combined times ␶₁ϩ␶₃. The population,

orientation and alignment sensitivity is contained in the sum

over K₂, which is restricted to K₂ϭ0, 1 or 2. The 6-j sym-

bol in Eqs. ͑1͒ and ͑2͒ reﬂects the transition sensitivity to

these different moments, depending on whether the moment

describes a population, orientation, or alignment in the

ground or excited state. The total signal electric ﬁeld will

result from the sum of the ﬁelds contributed by the different

diagrams. To illustrate the tensor moment dependence of the

signal electric ﬁeld on different transitions we evaluate the

sum of the squared 6-j symbols appearing in Eqs. ͑1͒ and

͑2͒. Figure 2͑a͒ shows the results for the orientation depen-

dence, K₂ϭ1, and Fig. 2͑b͒ for the alignment dependence,

K₂ϭ2. The sum is shown for P, Q, and R branch transitions,

as a function of the ground state rotational quantum number,

J_g, in half-integer steps. Strong dependencies on rotational

branch and quantum number are seen. Figure 2͑a͒ shows the

expected rapid decrease in orientation sensitivity via Q

branch transitions with increasing J_g, together with a more

gradual decline in P and R branch sensitivity tending towards

a constant value at high J_g. Figure 2͑b͒ shows that align-

ment sensitivity is higher on the Q branches at all except the

lowest J_g, with the P and R branch sensitivity declining to

one quarter of the Q branch sensitivity at high J_g. A mini-

mum of two m_Jsubstates in either the ground or excited

1

2 3

C. Orientation and alignment moments

The previous section showed how the signal is depen-

dent on the K₂tensor moments, which we have described as

orientation or alignment in the ground or excited states. Con-

sider the density matrix immediately after two interactions,

␳

⁽²⁾(t₂), for diagram IA ͓Eq. ͑A13͔͒. Collecting the complex

phase-dependent part and the square of the reduced matrix

element into a constant D this can be written

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7915

TABLE II. Alignment moments for Feynman diagrams IA or IIA in Fig. 1,

after interaction with two pump photons. The upper signs apply for left

circular polarization and the lower for right circular. ͑a͒ P branch, ͑b͒ Q

branch and ͑c͒ R branch.

that the experiments are directly sensitive to the orientation

and alignment of the rotational angular momentum of the

ground and excited states, and their evolution.

The results described in Sec. II A above show that for

Feynman diagram IA, only the decay of a single moment of

the distribution is measured, dependent on the polarization of

the pump light. It might be inferred that the population mo-

ment K₂ϭ0 is seemingly not measured. However, clearly the

signal must be sensitive to the population; as without a

sample present no signal can be measured! The resolution of

this apparent contradiction lies in recognizing that the relax-

ation rates for orientation and alignment as deﬁned are total

rates for the bulk loss of these properties, and include all

processes that remove population from the J state, as well as

the ⌬m_Jchanging events within that state.

Alignment moment

Circular pump

Linear pump

͑a͒

A⁽₀⁰⁾(J_g)

A⁽₀¹⁾(J_g)

1

0

1/2

1

J_gϩ1

ϯ

ͫ

ͬ

J_g

2)

A⁽₀²⁾(J_g)

1

2J_gϩ3

1

2J_gϩ3

ͫ

ͬ

ͫ

ͬ

10

J_g

10

J_g

(2)

2Ϫ

0

A

(J_g)

) 2J_gϩ3

ͫ

10

J_g

͑b͒

A⁽₀⁰⁾(J_g)

A⁽₀¹⁾(J_g)

1

0

D. Effect of Doppler motion

1/2

1

ϯ

ͫ

ͬ

The preceding sections have treated all molecules in the

sample as being at rest with respect to the beam propagation

direction. In practice the distribution of velocities in the

sample leads to different velocity subgroups ‘‘seeing’’ differ-

ent ﬁelds, and hence forming different third order density

matrices. This Doppler effect may be included in an estab-

lished fashion.³⁰We deﬁne the position of the molecule at

time t as r(t). At an earlier time, t_n, the position was r(t_n)

ϭr(t)Ϫ(tϪt_n)v, where v is the constant velocity of the

molecule. For example, at time t₁we can write the electric

ﬁeld incident in diagram IA as

J J ϩ1͒

͑

g

2)

g

A⁽₀²⁾(J_g)

1

2J Ϫ1͒ 2J ϩ3͒

1

2J Ϫ1͒ 2J ϩ3͒

͑ ͑

g g

͑

g

Ϫ

ͫ

ͬ

ͫ

ͬ

10

J_gJ ϩ1͒

10

J_gJ ϩ1͒

͑

g

͑

g

(2)

2Ϫ

0

A

(J_g)

)

2J Ϫ1͒ 2J ϩ3͒

͑

g g

͑

ͫ

ͬ

10

J_gJ ϩ1͒

͑

g

͑c͒

A⁽₀⁰⁾(J_g)

A⁽₀¹⁾(J_g)

1

0

1/2

1

J_gϩ1

Ϯ

ͫ

ͬ

J_g

2)

A⁽₀²⁾(J_g)

1

2J_gϪ1

1

2J_gϪ1

ͫ

ͬ

ͫ

ͬ

10 J_gϩ1

) 2J_gϪ1

(2)

2Ϫ

.r͑t͒Ϫik .v͑␶ ϩ␶ ϩ␶

͒

1

E t ͒ϭ␰ ␧ e^{Ϫi␻ t ϩik}

.

͑9͒

0

A

(J_g)

1 1

1

3

2

͑

1

1 1

ͫ

10 J_gϩ1

This Doppler effect can be applied to the incident electric

ﬁelds at all three times, and results in changes to the phase-

dependent term ⌽ in Eq. ͑1͒. The new ⌽ terms, assuming the

pump photons are identical, are listed in Table III. The dot

product k.v assures that only the projection of the velocity

onto the wave vectors is important, as required physically.

All of the wave vectors are assumed collinear in this treat-

ment, and we need only consider the one-dimensional pro-

jection of the Maxwell–Boltzmann velocity distribution,

which is a Gaussian distribution.

͑1͒

͑2͒

K ϩJ ϩJ

2 g e

*

␧ ͒

͑

ϪQ 2

1

␳

t ͒ϭD

2

Ϫ1͒

␧ ͒

1

͑

͚

Q ϪQ

1

2

K

,Q ,Q

2

1

2

1

J_gJ_e

1/2

ϫ 2K ϩ1͒

͑

ͭ

ͮ

2

J_g

1

K₂

The thermal motion manifests itself through two effects.

First, the lasers will come into resonance with molecular

subgroups at frequencies away from the natural transition

frequency. This gives rise to the expected Gaussian line

broadening. Second, the presence of the k.v term during de-

lays ␶ and ␶ results in the rapid loss of the signal. We

1

K₂

ϫ

T^KJ J ͒. ͑8͒

͑

g g

Q

2

ͩ

ͪ

Q₂ϪQ₁Q₁ϪQ₂

This can be expressed in terms of spherical tensor moments,

and those moments related to the alignment moments as con-

ventionally deﬁned.¹⁷In this derivation, the alignment mo-

ments are referenced to a Z-axis parallel to the direction of

propagation of the beams, regardless of whether circular or

linear pump light is used. Table II lists, as an example, the

renormalized alignment moments in this frame generated by

diagrams IA or IIA in the ground state for P, Q, and R

branches by a circular or 45° linear polarized pump beam.

This treatment shows that after two photons have inter-

acted with the sample it consists of an oriented or aligned

population in the ground state, and a complementary oriented

or aligned population in the excited state. The subsequent

polarization selection in the probe and detection steps con-

straints the sensitivity of PS to these moments further, as is

apparent from Table I. However, this analysis makes it clear

1

3

illustrate this effect using the example of the OH radical. The

OH A–X ͑0,0͒ band is at Ϸ308 nm, giving Ϸ2

͉k͉

ϫ10⁷m^Ϫ1. The Gaussian one-dimensional velocity distribu-

tion has a half width at half maximum equal to 381 m s^Ϫ1. A

phase shift of 2␲ then arises between molecules with this

velocity projection and those that have zero velocity projec-

tion in less than 1 ns. The Doppler motion of the molecules

thus gives rise to a wide range of phase angles within a short

delay time. The consequence of this is that the effective elec-

tric ﬁelds closely follow the temporal shapes of the laser

pulses, i.e., no signal is seen at negative pump–probe delays,

and no free induction decay is observed following the probe

pulse. During ␶ however, no dependence on the wave vector

2

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7916

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

TABLE III. Phase factors including the effect of Doppler motion, assuming pump photons are identical, for the different Feynman diagrams in Fig. 1.

(␻

Ϫ␻

Ϫk

(␻

Ϫ␻

pump

Ϫk

ϩk

"v)

⌽ϭe^Ϫi␻^{tϩik "r(t)}e^i␶

e

g

^"v)e^i␶

e

g

s

3

probe

J

probe

1

2

J

pump

IA and IIIA

IIA and IVA

IB and IIIB

IIB and IVB

Ϫ␻

pump

s

J

pump

e

g

Ϫ␻

ϩ͓k

Ϫk

(␻

Ϫ␻

probe

Ϫk

"v)

⌽ϭe^{Ϫi␻ tϩik "r(t)}e^i␶

^͔"v)e^i␶

s

probe

pump

probe

1

J

e g

probe

Ϫ␻

ϩ͓k

Ϫk

Ϫ␻

pump

⌽ϭe^Ϫi␻^{tϩik "r(t)}e^i␶

^͔"v)e^i␶

s

probe

J

e g

pump

is seen in Table III. Physically there is no spatial variation in

the population, orientation or alignment created. This is in

contrast to the spatially varying gratings generated in general

case of FWM experiments with noncollinear beams.²⁴

butions to the signal beam at time t. If instead, one of the

molecules undergoes a velocity-changing collision during

␶₁, then it will be at a different point in space to the other

molecule when it interacts with the second photon. This

dephasing destroys the PS signal. The same phenomenon

occurs during ␶₃. Both of these processes are operating dur-

ing delay periods deﬁned by the respective pump and probe

laser pulse-lengths. In an experiment where the pump–probe

delay is varied, they will reduce the absolute signal size by a

ﬁxed proportion, but will not cause decay as a function of the

pump–probe delay. In our previous experiments where the

pressure of collider gas was varied at a ﬁxed, coincident

pump-probe delay we saw extremely rapid decay of the PS

signal.¹⁶We have attributed this predominately to the effect

of these velocity-changing collisions.

E. Velocity-changing collisions

The previous section described how the velocity distri-

bution of the detected species was important in the evolution

of the PS signal. Clearly then, changes in the velocity distri-

bution during the overall pumping and probing process may

be expected to have a signiﬁcant effect on the signal. As far

as we are aware, however, velocity-changing collisions have

not been considered in the interpretation of any of the previ-

ous published studies of PS. However, there have been a

number of publications on their effect on other third order

nonlinear spectroscopic techniques, notably photon echo

spectroscopies.^34–36One approach to the inclusion of

velocity-changing collisions is to introduce a collision kernel

that gives the probability density per unit time that a colli-

We now consider collisions that occur during delay ␶₂.

In Table III we see that the premultiplier is not sensitive to

this delay at all, assuming that the pump photons are identi-

cal. Yet clearly if the molecule undergoes a collision during

this delay period, the velocity during ␶ will be different

sion changes the velocity from v to v . In principle, this

3

Ј

from that during ␶₁. What effect is expected from these col-

lisions? There is a straightforward physical interpretation.

That portion of the Doppler proﬁle that was in resonance

with the pump laser has been preferentially excited, with a

velocity width determined in practice by the bandwidth of

the laser. If we assume the probe laser has the same fre-

kernel can have almost any form, determined by the dynam-

ics of the interaction between molecule and collision partner.

We will therefore make some assumptions to simplify the

problem for illustrative purposes. It will be assumed that the

time scale of the collisions is negligible compared to the time

scale of the laser pulses. This impact approximation should

be valid for ns-pulsed lasers and ps collision time scales. It

will further be assumed that the collisions are hard, such that

the velocity distribution is completely thermalized after a

single collision. Finally, we assume that there is no correla-

tion between rotational angular momentum polarization and

velocity-changing collision cross sections.

quency as the pump laser then any collisions during ␶ that

2

change the velocity tend to move the sample out of reso-

nance with the probe laser. This will clearly result in a de-

crease in the observed signal. With single mode lasers, only a

small fraction of the Doppler proﬁle would be pumped and

probed at any laser center frequency. In that case, velocity-

changing collisions during this delay period would have a

signiﬁcant impact on the dependence of the signal on the

pump–probe delay. When using typical commercial ns-

pulsed dye lasers, however, the relatively broad spectral out-

put of the lasers will result in a signiﬁcant fraction of the

Doppler proﬁle being in resonance. In this case, these

velocity-changing collisions will be less important.

We will consider velocity-changing collisions in two dif-

ferent time regimes. The ﬁrst is during delay periods ␶ and

1

␶₃, when the system is in a coherent superposition of ground

and excited states. If the collisional interaction is different

for the two states, which will be the case in general, then the

collision kernel is no longer constrained to be positive deﬁ-

nite, but may be complex.³⁷The effect of the collisions, con-

sidering only the real part of the collision kernel, can be seen

by inspecting the relevant premultipliers in Table III. During

delays ␶ and ␶ the premultiplier depends on the velocity

We thus expect that the principal effect of velocity-

changing collisions on the observed PS signals using com-

mercial ns-pulsed lasers will be the reduction in absolute

magnitude of the signal, resulting from the dephasing inﬂu-

1

3

projection along the laser propagation direction. Changing

this velocity will result in a loss of signal, in a precisely

analogous fashion to the effect of the Doppler motion de-

scribed in Sec. II D. To illustrate this, consider two molecules

initially having the same velocity and same position. Without

any collisions, the molecules will stay exactly in phase, in-

teracting identically until ﬁnally generating in-phase contri-

ence of collisions during delays ␶ and ␶₃. We expect the

1

evolution of the signal as a function of pump–probe delay to

be dominated by collisional loss of population, orientation or

alignment, rather than by the modest effect of velocity-

changing collisions.

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7917

F. Nuclear hyperﬁne depolarization

symbol, ensuring that it is negligible when JӷI. In the simu-

lations described in Sec. III, we explore the effect of nuclear

hyperﬁne depolarization on PS signals for the OH

A ²⌺^ϩ–X ²⌸ transition.

Optical excitation in these small radicals creates orienta-

tion and alignment of the angular momentum, J, the result of

the coupling of the nuclear rotational and electronic spin and

orbital angular momenta. However, J may also be coupled to

the nuclear spin of the molecule. This effect has been de-

scribed in the study of angular momentum alignment result-

ing from photodissociation, reaction or optical

pumping.^17,32,33Although the inﬂuence of nuclear hyperﬁne

depolarization has been studied in FWM ͑Refs. 38–40͒ and

used to perform quantum beat spectroscopy, there has been

no explicit discussion of hyperﬁne interactions in PS. We

will simplify the problem by only considering the nuclear

hyperﬁne coupling during the delay period ␶₂. This is justi-

ﬁed on the following grounds. First, this is the only delay

period under our direct control, such that we may expect to

be able to probe any hyperﬁne effects as a modulation in the

observed signal. Second, the time scale of the nuclear hyper-

ﬁne coupling effects is determined by the strength of the

coupling between the nuclear spin I and angular momentum

J. If the different resultant states F from this coupling are to

be excited by the same laser pulse, then this coupling cannot

occur on a timescale much faster than the laser pulse. We do

not therefore expect signiﬁcant nuclear hyperﬁne depolariza-

III. SIMULATION OF PS SIGNALS

A. Simulation procedure

The preceding sections have detailed an approach to the

simulation of one-color PS signals. To simulate any experi-

mental signals we need to integrate the equations describing

the generated signal ﬁeld over the full range of possible

times, t_n, deﬁned by the laser pulsewidths and the signal

detection time, t, deﬁned by the timegate over which detect-

able signal is generated. This integration has been performed

numerically using a Monte Carlo ͑MC͒ approach, which we

now describe.

Three separate times, t_n, are MC selected from the two

laser pulsewidths; two from the pump pulse and one from the

probe pulse, both of which are modeled as Gaussian distri-

butions. The pump pulse is assumed to be centered at time

zero, and the probe pulse center time is stepped over the

desired range to reproduce that in the experiment; typically

Ϫ20 to ϩ100 ns. The pulsewidths are chosen to best reﬂect

the measured values of the experimental laser pulses. A

fourth time, t, is chosen as the detection time. This is MC

selected from a uniform distribution, which starts at the latest

of the three previously selected times and typically extends

for 5–10 ns. This represents a compromise between compu-

tational efﬁciency and simulation of the boxcar gate in the

experimental detection system. A target molecule velocity

along the laser propagation axis is MC selected from the

Gaussian distribution that is the one-dimensional projection

of the Maxwell–Boltzmann velocity distribution for that

temperature.

tion during the time scale of the laser pulse, and hence ␶

and ␶₃.

1

Making these assumptions, and recognizing that the sys-

tem during the delay ␶ may be described by an expansion in

2

orientation and alignment spherical tensors, we may use the

procedure described in the literature.^32,33The depolarization

of J can be described in the vector model in the following

fashion. J and the nuclear spin I couple to give F. J then

precesses about F so that the prepared direction of J in the

space-ﬁxed frame is altered. When I is of a similar magni-

tude to J the depolarization will be signiﬁcant. When J is

large then F and J will be nearly parallel and the depolariza-

tion will be small. The depolarization will display a periodic

nature, with J occasionally returning to its initial prepared

direction. For a single nuclear spin I coupling to J it may be

shown³²that the depolarization affects all spherical tensor

components of rank K equally, such that the time dependence

of a spherical tensor may be expressed as

The time ordering of the two pump and one probe pho-

tons selected is now used to ﬁnd which of the eight Feynman

diagrams IA–IVB are relevant. For example, when the probe

photon is the latest all four of the ‘‘pump–probe’’ diagrams,

IA–IVA, contribute to the signal. The signal ﬁeld that is

generated from these diagrams is then calculated using the

four selected times, using Eqs. ͑1͒–͑7͒. The effect of Doppler

motion is introduced using the selected velocity. The colli-

sional evolution of the signal is included by selected colli-

K

Q

K

␳

t͒ϭ␳ tϭ0͒G^Kt͒,

͑10͒

͑

Q

where the depolarization factor G^K(t) is given by

sional relaxation rates ⌫_dand ⌫_K. The effect of hyperﬁne

2

nuclear depolarization through the delay period ␶ is in-

2

cluded using Eq. ͑11͒. The signal ﬁelds generated by each

diagram are summed, for a large number, typically Ͼ10⁶, of

iterations of this procedure. The total signal ﬁeld generated is

the sum of the contributions of the individual diagrams. The

resulting signal intensity is the product of this signal ﬁeld

and its complex conjugate. This simulates the signal intensity

for a particular chosen delay between the pump and probe

pulsewidth centers. The probe pulse center time is then

moved and the process repeated. Software to simulate the PS

signals was written in FORTRAN, and the simulations per-

formed on a PC workstation ͑dual 2.8 GHz Intel Xeon͒.

2

F

J

K

I

Ј

2Fϩ1͒ 2F ϩ1͒

͑

Ј

G^Kt͒ϭ

͑

ͭ

ͮ

͚

2Iϩ1͒

͑

J

F,F

Ј

E

ϪE_F͒t

͑

F

Ј

ϫcos

.

͑11͒

ͫ

ͬ

ប

The hyperﬁne nuclear depolarization is thus shown to be

oscillatory in nature, with periods deﬁned by the energy

splittings, E_FϪE_F, between the different nuclear hyperﬁne

Ј

states. The magnitude of the effect is determined by the 6-j

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7918

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

2

⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

2

FIG. 3. Simulation of PS of OH A

⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

FIG. 4. Simulation of PS of OH A

of pump–probe delay for different translational temperatures. Simulations

assumed no collisional decay processes and laser pulsewidths of 5 ns for

both pump and probe. Solid line, 1ϫ10^Ϫ8K; dashed line, 1 K intensity

scaled by a factor of 3; dotted line, 298 K scaled by a factor of 500.

of pump–probe delay for different, assumed equal, pump and probe laser

pulsewidths. Simulations assumed no decay processes and a translational

temperature of 1ϫ10^Ϫ8K. Solid line, 1 ns pulsewidths; dashed line, 5 ns;

dotted line, 10 ns.

lation in the ground and excited states, with the maximum

being reached at the end of the pump pulse.

B. Illustrative simulations

The simulations in Fig. 4 are the result of varying the

translational temperature from 1ϫ10^Ϫ8K through 1 K to

298 K, but maintaining the assumption of no other decay

processes. Each of these simulations assumed a 5 ns pulse-

width and involved 1ϫ10⁶MC iterations per pump–probe

delay time step. Two temperature dependent effects stand

out. First, both the 1 and 298 K simulations show effectively

no signal before the pump pulse occurs. Second, the 1 K

signal is approximately three times, and the 298 K signal

approximately 500 times, smaller than the 1ϫ10^Ϫ8K signal.

Both of these effects are a result of the Doppler motion of the

sample. The loss of the signal at negative pump–probe de-

lays is the result of the velocity-induced dephasing process

described in Sec. II C above. As the temperature increases, a

smaller proportion of the velocity distribution is in resonance

with the lasers, decreasing the signal size.

Simulations were performed for illustrative purposes on

the Q₁(2.5) transition of the OH A ²⌺^ϩ–X ²⌸(0,0) band,

assuming linearly polarized pump light. Pump and probe la-

sers frequencies were centered at the transition frequency,

and the lasers were assumed to have identical pulsewidths. In

all the simulations, except that exploring the effect of

velocity-changing collisions, the laser outputs were assumed

to be of a single common central frequency. Typical ns-

pulsed dye lasers do not produce a pure single mode output.

It might be expected that using multimode lasers would pro-

duce a change in the absolute signal size, but no variation in

the signal as a function of pump–probe delay. We have per-

formed exploratory calculations on the effect of multimode

lasers, and have indeed found no changes in the relative

variation of the simulated signal with pump–probe delay. We

therefore do not discuss further the effects of multimode la-

sers in this paper.

We ﬁrst demonstrate the dependence of the PS signal on

the pump–probe delay on the purely hypothetical assump-

tions that there are no decay processes operating, and that

Doppler motion is insigniﬁcant ͑translational temperature 1

ϫ10^Ϫ8K). Figure 3 shows the resulting simulations for la-

ser pulsewidths of 1, 5, and 10 ns. All three simulations show

signal at negative pump–probe delays, which arises from the

processes shown in Feynman diagrams IB and IIIB. The sig-

nal rises sharply at the pump ﬁring time and then continues

at a constant value determined by the four ‘‘pump–probe’’

Feynman diagrams. The risetime of the signal is clearly de-

pendent on the pulsewidth. However, the signal peak is not

centered at zero delay, but is instead displaced to positive

pump–probe delay, increasing with pulsewidth. This is a re-

sult of the pump pulse building up oriented or aligned popu-

Figure 5 shows the effect of including the radiative de-

cay of the excited state. This simulation assumes a 5 ns laser

pulsewidth, 298 K thermal velocity distribution, and uses the

LIFBASE ͑Ref. 41͒ total radiative rate for the A ^ϩ⌺ (

v

ϭ0,Jϭ2.5,F₁) level probed by the Q₁(2.5) transition, k_rad

ϭ1.45ϫ10⁶s^Ϫ1. The resulting simulation shows a decay

due to the radiative process as signal resulting from the ex-

cited state is lost, followed at long delays by a constant sig-

nal from the remaining ground state. Note the large change

of time scale from Figs. 3 and 4. This remnant signal is a

quarter of the magnitude of the initial ⌬tϭ0 signal. The

initial signal was determined by the four pump–probe dia-

grams, which for this transition and polarization each con-

tribute the same amount to the overall signal electric ﬁeld. At

long times only two of these diagrams remain, IA and IIA.

Consequently, the signal ﬁeld is halved in strength and the

signal intensity drops to a quarter of the initial value. This

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7919

2

FIG. 5. Simulation of PS of OH A

⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

_radϭ1.45

^Ϫ1, of the excited state. Simulations assumed pump and probe laser

FIG. 7. Simulation of PS of OH A ⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

of pump–probe delay showing the inﬂuence of nuclear hyperﬁne depolar-

ization for different pump and probe laser pulsewidths. Simulations included

radiative decay of k_radϭ1.45ϫ10⁶s^Ϫ1but no collisional decay processes.

The translation temperature was 298 K. Solid line, laser pulsewidth 0.5 ns;

dashed line, laser pulsewidth 5 ns.

of pump–probe delay showing effect of radiative decay,

k

ϫ10⁶

s

pulsewidths of 5 ns, no other collisional decay processes and a translational

temperature of 298 K.

shows how the signal may not decay as a single exponential,

as it depends on the sum of different exponential decays with

possibly different weighting factors.

Figure 6 shows simulations with increasing collisional

loss rates, introducing conditions more typical of those in

experiments. The pulsewidths are again 5 ns, the temperature

298 K, the radiative decay rate is k_radϭ1.45ϫ10⁶s^Ϫ1, and

the collider number density is 3.24ϫ10¹⁶cm^Ϫ3. The colli-

sional rate constants for ground and excited states are as-

sumed to be identical, and only the K₂ϭ2 alignment rate

constants are signiﬁcant for this case of linear pump light.

The decay rate constants are ͑a͒ 1ϫ10^Ϫ10cm³s^Ϫ1, ͑b͒ 5

ϫ10^Ϫ10cm³s^Ϫ1, and ͑c͒ 1ϫ10^Ϫ9cm³s^Ϫ1. The resulting

curves are essentially single exponential decays, with rates

twice that implied by the rate constants. The single exponen-

tial nature of the decay is a consequence of using the same

decay constants for ground and excited states. These curves

show that at the typical pressures, of order 1 Torr, used in our

own collisional energy transfer studies, and physically rea-

sonable collisional decay constants, the PS signal is expected

to disappear for pump–probe delays of the order of 100 ns.

Nuclear hyperﬁne depolarization effects are introduced

in Fig. 7. In the simulation shown by the solid line, the laser

pulsewidths are reduced to 0.5 ns, the time step to 0.2 ns, and

2ϫ10⁶iterations of the MC procedure were calculated at

each time step. Collisional relaxation processes have not

been included, although the radiative decay of the excited

state is included, with the same value as before. Nuclear

hyperﬁne depolarization is simulated for the Q₁(2.5) transi-

tion. The hyperﬁne energy levels are calculated^42,43for the

relevant rotational, spin–orbit and ⌳-doublet levels of the

X ²⌸ and A ²⌺^ϩstates using literature coupling

constants.^44,45The resulting simulation shows clear periodic

oscillations. These oscillations show two separate frequen-

cies. The higher of these arises from the A ²⌺^ϩstate and has

a period of Ϸ2.4 ns, the lower from the X ²⌸ state having a

period of Ϸ8.6 ns. These different time scales simply reﬂect

the different hyperﬁne coupling constants in these states.

Each of these oscillations is a sine function, resulting from

the two nuclear hyperﬁne levels in each of the ground and

excited states.

2

FIG. 6. Simulation of PS of OH A

⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

of pump–probe delay for different collisional decay rate constants at a num-

ber density of 3.24ϫ10¹⁶cm^Ϫ3. Simulations assumed pump and probe laser

pulsewidths of 5 ns, a radiative decay rate k_radϭ1.45ϫ10^{6 Ϫ1}, and a trans-

s

lational temperature of 298 K. Collisional rate constants were identical for

ground and excited states. Solid line, 1ϫ10^Ϫ10cm³s^Ϫ1; dashed line, 5

The second simulation shown in Fig. 7, given by the

dashed line, is identical to the ﬁrst except for the laser pulse-

ϫ10^Ϫ10cm³s^Ϫ1; dotted line, 1ϫ10^Ϫ9cm³s^Ϫ1

.

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7920

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

and assumed ␶ the probability of at least one collision oc-

2

curring is used in the usual MC fashion. If a collision does

occurs, then a new velocity is MC selected from a 1D

Maxwell–Boltzmann distribution, and is used in the subse-

quent calculation of the phase factor arising from delay ␶₃.

Laser bandwidths are introduced by MC selecting the pump

and probe frequencies from separate Gaussian distributions

centered at the transition frequency.

These additional dimensions in the MC integration re-

quired an increased number of iterations per simulation time

step. We have chosen 1ϫ10⁹as a reasonable compromise

between statistical noise and computational time. Simula-

tions for three pump laser bandwidths, 0.06 cm^Ϫ1, 0.1 cm^Ϫ1

,

and 0.2 cm^Ϫ1, are shown in Fig. 8. In each case, the probe

laser bandwidth is assumed to be 0.06 cm^Ϫ1. A clear decrease

in the PS signal as a function of pump–probe delay is seen

for a pump bandwidth of 0.06 cm^Ϫ1, with a slower but dis-

cernible decrease for 0.1 cm^Ϫ1, and effectively no decrease

on the experimental time scale for the 0.2 cm^Ϫ1pump band-

width. This reﬂects the increasing proportion of the velocity

distribution that is being excited as the pump laser bandwidth

is increased. At 298 K the thermal OH velocity distribution

gives a transition linewidth of Ϸ0.1 cm^Ϫ1. Thus, a pump

bandwidth of 0.2 cm^Ϫ1pumps almost the entire velocity dis-

tribution.

2

FIG. 8. Simulation of PS of OH A

⌺

^ϩ–X ²⌸ ͑0,0͒ Q₁(2.5) as a function

of pump–probe delay showing inﬂuence of velocity changing collisions for

different pump laser bandwidths. He collider number density of 3.24

ϫ10¹⁶cm^Ϫ3, velocity-changing collision cross section 100 Å². The simula-

tions assumed no other decay processes and a translational temperature of

298 K. The probe laser bandwidth was 0.06 cm^Ϫ1in all simulations. Solid

line, pump laser bandwidth 0.2 cm^Ϫ1; dashed line, pump laser bandwidth 0.1

cm^Ϫ1; dotted line, pump laser bandwidth 0.06 cm^Ϫ1

.

IV. COMPARISON TO EXPERIMENT

width, which is increased to 5 ns. Instead of the previously

clear oscillations at two different frequencies, this simulation

shows only a slight modulation at the lower frequency. The

laser pulsewidth is now too broad to observe the high fre-

quency oscillation from the A ²⌺^ϩstate. The width of the

laser pulse has also largely averaged out the lower frequency

oscillations from the X ²⌸ state. These simulations show that

with typical ns-pulsed dye lasers, hyperﬁne nuclear depolar-

ization will not generally be observable for OH

A ²⌺^ϩ–X ²⌸ transitions in the form of oscillations as a

function of pump–probe delay. They will however reduce

the total signal level. Other species with different hyperﬁne

coupling constants, and hence hyperﬁne depolarization ti-

mescales may produce measurable oscillations. For example,

quantum beats from nuclear hyperﬁne interactions have been

observed using time-resolved two-color resonant four-wave

mixing in the ground X ²⌸ and excited A ²⌺^ϩstates of

NO.³⁹In a one-color PS experiment on NO using ns-pulsed

lasers we would expect to observe equivalent oscillations,

with a combination of the frequencies observed for ground

and excited states.

In this section, we compare simulations produced using

the procedure described in the preceding sections to repre-

sentative experimentally measured PS signals. These pre-

liminary measurements were made on the OH radical using

the A ²⌺^ϩ–X ²⌸(0,0) transition. A more comprehensive se-

ries of measurements of OH collisional polarization proper-

ties will be reported in a later publication.

A. Experimental procedure

The experimental apparatus is practically the same as

that described in a previous publication,¹⁶with the signiﬁ-

cant addition of another Nd:YAG pumped pulsed dye laser

system. Only the essential details and differences are re-

ported here.

OH X ²⌸ radicals were generated by the 266 nm photo-

dissociation of H₂O₂. The H₂O₂͑50% by weight with H₂O)

was picked up by a ﬂow of the collider gas, He. A total

pressure of Ϸ600 mTorr was maintained in the cell by vary-

ing the ﬂow rate, as monitored by a capacitance manometer

͑MKS͒. Three independent laser systems were used in the

experiment, the relative timings of which were controlled by

two digital delay generators ͓Stanford Research Systems

͑SRS͒ DG535͔. The ﬁrst laser, an Nd:YAG ͑Continuum

Surelite II-10͒, generated 266 nm which was used to photo-

dissociate the H₂O₂. The second laser system was a

Nd:YAG pumped dye laser, ͑Lumonics Hyperdye/R640 laser

dye͒, which produced doubled output at Ϸ308 nm suitable

for the OH A ²⌸–X ²⌺^ϩ͑0,0͒ transition. This acted as the

pump laser system. The ﬁnal system was another Nd:YAG

laser ͑Continuum Surelite II-10/Sirah Cobrastretch/R640 la-

ser dye͒ which also generated doubled output at Ϸ308 nm.

This system acted as the probe laser. The pump laser ﬁred 8

Finally, the effect of velocity-changing collisions is in-

troduced in Fig. 8. These simulations assume no collisional

or radiative relaxation, and ignore the inﬂuence of hyperﬁne

depolarization. Velocity-changing collisions during the delay

␶ are introduced in the following fashion. A collision cross

2

section is assumed, for these simulations 100 Å², and the

resulting collision rate calculated for He as a collider at a

number density of 3.24ϫ10¹⁶cm^Ϫ3and temperature of 298

K. The probability densities of integer numbers of velocity-

changing collisions are assumed to be distributed according

to Poisson statistics. For each iteration of the MC procedure

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7921

thermal 298 K OH velocities, a 13 ns pulsewidth for the

pump laser system, and a 5 ns pulsewidth for the probe laser,

consistent with manufacturers’ data. The assumed radiative

decay rate of the excited state was k_radϭ1.45ϫ10⁶s^Ϫ1

.

Velocity-changing collisions were not included, for the rea-

sons discussed in Sec. III B above. The simulations assume

that the collisional rate constants for the ground and excited

states are equal, at 0.9ϫ10^Ϫ9cm³s^Ϫ1, 1.1ϫ10^Ϫ9cm³s^Ϫ1

,

and 1.3ϫ10^Ϫ9cm³s^Ϫ1

,

respectively. The simulations

clearly ﬁt the data well, reproducing both the growth and

decline of the experimental signal. As discussed in Sec. III B,

the growth of the signal is deﬁned by the pulsewidths of the

pump and probe lasers. The decay of the signal is determined

by the collisional loss of the population and the alignment

created by the pump laser, as discussed in Sec. II C. The

simulation assuming 1.1ϫ10^Ϫ9cm³s^Ϫ1ﬁts the experimen-

tal data best, with the other simulations giving an indication

of the statistical, but not necessarily systematic, uncertainty

in this measurement of about 1ϫ10^Ϫ10cm³s^Ϫ1

.

2

FIG. 9. Experimental data and simulation of PS of OH A

⌺

^ϩ–X ²⌸ ͑0,0͒

Q₁(2.5) as a function of pump–probe delay. Linear pump laser polarization,

total pressure of H₂O₂/H₂O precursor, and He collider, 600 mTorr. Simula-

tions assumed a 13 ns pump laser pulsewidth and a 5 ns probe laser pulse-

Rate constants for the total loss processes occurring in

the ground and excited states can be calculated from litera-

ture measurements of the rotational energy transfer ͑RET͒

and electronic quenching rate constants. The total pressure of

collider and precursor gases in the experiment was 600

mTorr, of which we deduce that approximately 100 mTorr is

the precursor H₂O₂/H₂O mixture ͑majority H₂O) from an

analysis of the LIF radiative lifetime. A total collisional loss

rate of 6.7ϫ10⁶s^Ϫ1is predicted for the gas mixture for the

width, a radiative decay rate k_radϭ1.45ϫ10^{6 Ϫ1}, no velocity-changing col-

s

lisions and a translational temperature of 298 K. Open circles, experimental

data; dashed line, simulation assuming kϭ0.9ϫ10^Ϫ9cm³s^Ϫ1; solid line,

simulation assuming kϭ1.1ϫ10^Ϫ9cm³s^Ϫ1; dotted line, simulation assum-

ing kϭ1.3ϫ10^Ϫ9cm³s^Ϫ1

.

␮s after the photolysis laser, sufﬁcient time for the nascent

OH translational distribution to relax to thermal equilibrium

and for the nascent rotational distribution to be largely ther-

malized. The probe laser also ﬁred at Ϸ8 ␮s after the pho-

tolysis laser, with the exact timing determining the pump–

probe delay. This was scanned using the digital delay

generator, typically from Ϫ20 ns to ϩ100 ns.

2

excited ⌺^ϩstate including both total RET rates out of the

probed level^46,47and electronic quenching rates.⁴⁸This may

be converted to an apparent rate constant, in which we as-

sume that all 600 mTorr of the collider was the same gas, of

k

_appϭ3.4ϫ10^Ϫ10cm³s^Ϫ1. A similar prediction of the total

49,50

2

collisional loss rate through RET in the ground ⌸ state

may be made, and is found to give a total rate of 6.2

ϫ10⁶s^Ϫ1and an apparent rate constant,

_appϭ3.2

ϫ10^Ϫ10cm³s^Ϫ1

PS was performed in a copropagating geometry with lin-

ear polarized pump light. The pump laser ﬂuence was attenu-

ated to ensure operation in the perturbative regime, as shown

in a previous publication.¹⁶Stress induced birefringence in

the cell windows alters the polarization of the probe beam

and generates a wavelength-independent background signal.

This was minimized before each experiment by careful ap-

plication of a compensating stress to the cell through its

mounting clamps. The pump and probe laser ﬂuences were

monitored using energy meters ͑Molectron͒. LIF was moni-

tored through the cell side window and used to ensure that

each of the pump and probe lasers was resonant with the

desired transition. Both PS and LIF signals, and the output of

the two energy monitors were passed to boxcar integrators

͑SRS SR250͒ and via a computer interface ͑SRS SR245͒ to a

PC. The laser timing and probe laser wavelength, together

with the data acquisition, were all controlled via LABVIEW

software.

k

.

These apparent rate constants calculated for the total

population loss processes in the ground and excited states are

only approximately 1/3 of the magnitude of the bulk polar-

ization decay rate constant required to correctly simulate the

experimental data. Two possible processes that may account

for this are velocity-changing collisions, and angular mo-

mentum realignment. We do not believe that velocity-

changing collisions are likely to be a major loss process in

these experiments. The bandwidths of the pump and probe

lasers are a signiﬁcant fraction of the total Doppler width,

being Ϸ0.1 cm^Ϫ1and Ϸ0.06 cm^Ϫ1FWHM, respectively. The

simulations presented in Sec. III suggest that under these

circumstances, even when assuming complete thermalization

of velocities on every collision and a large cross section, no

signiﬁcant signal decay will occur on the experimental time

scale. We are therefore led to the important conclusion that

the observed rapid signal decay is the result of realignment

of the rotational angular momentum and the consequent de-

polarization of the sample, with a second order rate constant,

B. Comparison of simulations to experimental results

Figure 9 shows the variation with pump–probe delay of

the experimentally measured PS signals from the OH

Q₁(2.5) transition with linearly polarized pump light. It is

compared to three simulations. The simulations assumed

k

_alignϷ7ϫ10^Ϫ10cm³s^Ϫ1

There have been a number of previous measurements of

depolarization of rotational angular momentum, using

.

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7922

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

optical-optical double resonance techniques. Studies prior to

1986 are summarized in the review article by McCaffery

et al.⁵¹In some cases, depolarization has been found to be

considerably slower than RET, for example self-collisions of

duced to be 17% H₂O and 83% He. Our measurements there-

fore appear to conﬁrm the ﬁndings of Williams et al. that

elastic reorientating collisions are favored over RET colli-

sions at low J for OH with He as a collider.

N ( ϭ1),⁵²of CO,⁵³and C₂H₂.⁵⁴Other systems have pro-

v

In a future paper, we will present further measurements

of OH collisions using one-color PS, exploring the inﬂuence

of different collision partners and varying rotational quantum

number. One limitation of one-color PS is that both the

ground and excited states contribute to the signal together, as

does the decay of the prepared polarization in both levels.

This may be overcome using two-color PS, probing on a

different transition to that used in the pump stage. Two-color

PS may be analyzed in a precisely analogous fashion to the

one-color PS derivation presented in this paper, and will be

the subject of a future publication.⁵⁶

2

duced signiﬁcantly faster depolarization rates, notably

H₂(B ¹⌺^ϩ_u) with He, where depolarization of Jϭ1, m_Jϭ0

was found to have a cross section of 30 Å².⁵⁵These varia-

tions may have several explanations. Depolarization is gen-

erally found to be signiﬁcantly faster for lower J. This is not

surprising, as a given absolute change in the magnitude of

͉

m_J͉ will result in a much larger proportional change in de-

gree of orientation or alignment at low J. Second, and of

more signiﬁcance in dynamics, the shape of the potential will

have a strong inﬂuence. For example, the H₂(B ¹⌺_u^ϩ)–He

potential is strongly anisotropic, being repulsive in a linear

geometry, but attractive in the ‘‘T-shape’’ geometry. The

large reorientation cross section observed is attributed to this

anisotropy, in contrast to H₂(X ¹⌺_g^ϩ)–He, where the poten-

tial is effectively isotropic and little reorientation is seen.

The most directly relevant studies to compare with our

current data are FWM and PS measurements of OH, in at-

mospheric pressure ﬂames. Williams et al.²⁵measured FWM

line shapes of OH A ²⌺^ϩ–X ²⌸ in H₂/O₂/He ﬂames using

different polarizations and derived line-broadening param-

eters for population and alignment relaxation rates. A notable

ﬁnding for the two low J transitions studied, Q₂₁(1.5) and

Q₂₁(3.5), was that the polarization relaxation rate was al-

most twice the measured population relaxation rate. The

ﬂame conditions are clearly very different ͑atmospheric pres-

sure and 1380 K͒ from our room temperature, low pressure

experiments. However, neglecting any dependence of the

collision cross sections on temperature we can convert these

line broadening parameters to rate constants for the ﬂame

conditions, and then to equivalent rate constants at room

temperature. The resulting predicted rate constants are, for

population decay, k_popϭ3.3ϫ10^Ϫ10cm³s^Ϫ1, and for align-

V. CONCLUSIONS

The motivation behind this work has been the desire to

measure rotational angular momentum relaxation and trans-

fer in inelastic collisions of electronically excited small free

radicals. This paper has described an angular momentum ten-

sor moment analysis of the signals arising in one-color PS, in

which the process of keeping track of the time-ordered se-

quence of interactions is simpliﬁed by the use of double-

sided Feynman diagrams. Unlike previous related work on

FWM, we do not assume that collisional decay rates are

rapid with respect to the laser pulsewidth, nor do we assume

that decay rates are independent of the m_Jlevel. The result

shows that when the pump and probe lasers are well sepa-

rated in time, the pump laser prepares either orientation or

alignment of the rotational angular momentum in the reso-

nant rotational levels, depending on the pump polarization.

The signal is generated by the interaction of the probe laser

with this oriented or aligned sample, and is sensitive to the

decay of both the populations of the probed levels and the

anisotropy of the rotational angular momentum. If the colli-

sional population transfer rates are known from other mea-

surements, then the rate of loss of angular momentum orien-

tation or alignment can be inferred directly.

ment decay, k_alignϭ6.4ϫ10^Ϫ10cm³s^Ϫ1

.

The decay of FWM and PS signals has also been studied

independently by Drier and co-workers using picosecond la-

sers on the OH A ²⌺^ϩ–X ²⌸ transition in CH₄/air and

H₂/O₂atmospheric pressure ﬂames. Population, orientation

and alignment decay were measured for a range of J.^12–15

The general trends found were that orientation relaxes slower

than alignment, and for both, relaxation slows with increas-

ing J. FWM measurements on the Q₁(1.5) transition yielded

both population and orientation decay rates, and the orienta-

tion was seen to decay faster than the population, in agree-

ment with the observations of Williams et al. A conversion

of these decay rates to equivalent second order collision rates

at room temperature assuming equivalent collision cross sec-

tions and a H₂/O₂ﬂame at a temperature of 1600 K, yields

a population decay rate, k_popϭ4.4ϫ10^Ϫ10cm³s^Ϫ1, and an

We have also analyzed the inﬂuence of other processes,

including the Doppler motion of the probed molecules, the

effect of velocity-changing collisions, and of nuclear hyper-

ﬁne depolarization. The Doppler motion of the sample intro-

duces a dephasing process during delays following an odd

number of photon interactions. As a result, the growth of the

PS signal closely follows the overlap of the pump and probe

lasers. For the same reason, the signal pulse follows closely

the temporal proﬁle of the probe laser. Nuclear hyperﬁne

depolarization during the pump–probe delay was introduced

to the analysis using the standard literature description of

angular momentum depolarization. The resulting oscillations

of the prepared orientation and alignment prove to be too

rapid to be measured using commercial ns-pulsed lasers in

the current example of the OH A ²⌺^ϩ–X ²⌸ transition.

Velocity-changing collisions are predicted to reduce the ab-

solute magnitude of the measured signals when they occur

during the ␶ and ␶ periods, by introducing an additional

orientation decay rate, k_orϭ6.5ϫ10^Ϫ10cm³s^Ϫ1

.

The correspondence between our new measured values

and these estimates from both Williams et al. and Suvernev

et al. is striking. Under Williams et al.’s conditions the ﬂame

composition is estimated to be 3% O₂, 24% H₂O, and 73%

He, not markedly different from our gas composition, de-

1

3

dephasing process. The effect of velocity-changing collisions

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J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7923

during the second delay period is distinct and less pro-

nounced. Assuming the laser bandwidth is less than the Dop-

pler width, a velocity subgroup is pumped, and subsequently

probed, subject to resonance conditions with the pump and

probe lasers. Only collisions that change the velocity sufﬁ-

ciently to destroy this resonance condition result in signal

decay. With common commercial ns-pulsed dye laser sys-

tems the typical laser bandwidth will cover most of the ther-

mal Doppler width, and such collisional effects will be rela-

tively unimportant.

The ﬁrst perturbation Hamiltonian, ͑A1͒, is now applied to

give the ﬁrst order density matrix. The diagram shows that

this acts from the left, on the ket space, and accordingly the

ﬁrst order density matrix can be written as

"r

1

Ϫ␰₁e^{Ϫi␻ t ϩik}

1 1

͑1͒

␳

t ͒ϭ

1

͑

2iប 2J ϩ1͒

͑

g

ϫ

͉

,m

e

J_em_eJ m

͗͘

͉

␮"␧

͉

J_gm_gJ m ͉.

͗͘

g

͚

e

1

g

m

g

Preliminary measurements of a representative PS signal

from OH in collision with He were made on the Q₁(2.5)

transition of the A ²⌺^ϩ–X ²⌸ ͑0,0͒ band. These measure-

ments were compared to simulations, and the rate constants

required to reproduce the experimental data deduced. The

results were discussed in terms of literature RET and

quenching rates. The striking ﬁnding is that a realignment

rate of approximately twice the RET and quenching rates is

required to accurately model the data. This is in satisfactory

agreement with previous ps-pulsed PS and FWM measure-

ments in ﬂames^13–15and with FWM line shape measure-

ments in He-rich ﬂames.²⁵These measurements thus support

the conclusion that polarization decay is faster than RET for

low J OH.

͑A5͒

The tensor dot product ͑␮"␧͒ can be expressed in the spheri-

cal basis, and the Wigner–Eckart theorem applied to the re-

sulting matrix element.³²This gives the following expanded

form:

.r

1

͑1͒

Ϫ␰₁e^{Ϫi␻ t ϩik}

J

ʈ

␮ ͒

ʈ

J_g

͘

1 1

͑

͗

e

1

͑1͒

␳

t ͒ϭ

1

͑

2iប 2J ϩ1͒

͑

g

͑1͒

P ϩJ Ϫm

e

1

e

ϫ

Ϫ1͒

␧ ͒

͑

1

ϪP

͑

͚

1

m

,m ,P

g

e

1

J_e

1

J_g

͉

J m J m ͉.

͗͘

g

͑A6͒

ͩ

ͪ

e

g

Ϫm_eP₁m_g

The density matrix operators can be expressed in a spherical

basis in terms of the spherical tensor operators T^K_Q¹(J_eJ_g) as

deﬁned by Blum³³

ACKNOWLEDGMENTS

We thank the EPSRC for a research grant including a

studentship for H.J.C. M.L.C. thanks the Royal Society of

Edinburgh for a BP Research Fellowship.

J Ϫm

1/2

e

͉

J_em_eJ m

͗͘

͉

ϭ

Ϫ1͒

2K ϩ1͒

͑

͚

g

1

K

,Q

1

J_e

J_g

K₁

ϫ

T^KJ J ͒.

1

͑

ͩ

ͪ

e

g

Q

APPENDIX: PERTURBATIVE DERIVATION

OF PS SIGNAL

1

m_eϪm_gϪQ₁

͑A7͒

In this Appendix the derivation of the functional form

for the electric ﬁeld arising from Feynman diagram IA of

Fig. 1 is outlined in detail. The rules for determining the

operators for a particular interaction in the Feynman diagram

are given in Ref. 29. Equations ͑A1͒–͑A3͒ list the perturba-

tion Hamiltonians for the three separate photon interactions

shown in Fig. 1, diagram IA,

Substituting Eq. ͑A7͒ into Eq. ͑A6͒ and using the orthogo-

nality of the 3-j symbols we ﬁnd

.r

1

͑1͒

Ϫ␰₁e^{Ϫi␻ t ϩik}

J

ʈ

␮ ͒

1

ʈ

J_g

1 1

͑

͗

͘

e

͑1͒

␳

t ͒ϭ

1

͑

2iប 2J ϩ1͒

͑

g

T^K_QJ J ͒␦

␦

K ,1 Q

1 1

͑1͒

ϪP

P

1

Ϫ1͒ ␧ ͒

͑

1

e

g

,P

1

͑1͒

ϫ

.

H

t ͒

1

͚

͑

1/2

1

"r

␮"␧ ͒␰ e^{Ϫi␻ t ϩik}

,

͑A1͒

͑A2͒

͑A3͒

2K ϩ1͒

͑

K

,Q ,P

1 1

1

ϭϪ

͑

1

iប

2iប

͑A8͒

͑2͒

t ͒

2

1

͑

ϩi␻ t Ϫik "r

2 2

2

*

␮"␧ ͒␰ e

,

The Kroenecker delta functions in Eq. ͑A8͒ indicate that the

expression is nonzero only when K₁ϭ1 and Q₁ϭP₁.

We now consider the time evolution of the density ma-

2

iប

2iប

͑3͒

t ͒

3

1

͑

"r

3

␮"␧ ͒␰ e^{Ϫi␻ t ϩik}

.

3 3

trix in the interval ␶ between t₁and t₂. The matrix can be

3

1

iប

2iប

described at any time by an expansion in the spherical tensor

operators and state multipoles. The time evolution of the

density matrix is contained within the state multipoles. If the

collisional environment is isotropic then all projections, Q,

of a given multipole moment rank, K, evolve with the same

rate.³³Including the energy propagator, which is independent

of the multipole rank, the density matrix at time t₂before the

interaction with the second photon can be written as

The initial sample is assumed to be thermalized, with a uni-

form m_Jstate population distribution in the ground state, and

no population in the excited state. The resulting initial den-

sity matrix, at time t₁, is nonzero only along the diagonal,

͉

J_gm_gJ m ͉

͗͘

g

͑0͒

␳

t ͒ϭ

1

.

͑A4͒

͑

͚

2J ϩ1͒

͑

m

g

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7924

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Costen, Crichton, and McKendrick

␶

Ϫ⌫ ␶

d 1

͑1͒

Ϫ␰₁e^{Ϫi␻ t ϩik}^"re^Ϫi␻

e

g

e

J

ʈ

␮ ͒

1

ʈ

J_g

͘

1 1

1

J J

1

͑

͗

K

Q

e

t ͒ϭe^Ϫi␻

e

g

␳

J J ͒T^KJ J ͒e^Ϫ⌫

. ͑A9͒

͑1͒

␶

K 1

1

͑1͒

J J

1

␳

͑

e

Q

1

␳

t ͒ϭ

2

͑

͚

2

e

g

2iប 2J ϩ1͒

͑

K

,Q

1

g

1

͑1͒

T¹_QJ J ͒

͑

e g

1

Q

1

Ϫ1͒ ␧ ͒

͑

It was noted above that the only nonzero result is for K₁

ϭ1, assured by the delta function in Eq. ͑A8͒. Hence only

one state multipole is nonzero. There is only a single decay

rate to consider, involving the off-diagonal elements of a

density matrix between ground and excited states. We hence-

forth denote this dephasing rate constant as ⌫_d, and the re-

sulting ﬁrst order density matrix at time t₂is

1

ϪQ

1

ϫ

.

͑A10͒

͚

Q

)

1

We can now proceed with the second optical interaction.

We expand the spherical tensor operator, as deﬁned by

Blum,³³once again

J

k

Ј

T^k_qJ J͒ϭ

Ϫ1͒

2kϩ1͒

͑

͉

J m J,m

͉

,

͑A11͒

J Ϫm

Ј

1/2

Ј

͑

Ј

Ј Ј

͗͘

ͩ

ͪ

͚

m

Ϫm Ϫq

Ј

m ,m

Ј

then apply the second photon interaction Hamiltonian to yield

Ϫ␰ ␰₂e^{Ϫi͑␻ t ϩ␻ t}e^{i͑k Ϫk ͒•r}e^Ϫi␻

e

g

e

J

ʈ

␮ ͒

1

ʈ

J_g

͒

␶

Ϫ⌫ ␶

d 1

͑1͒

1 1

2 2

1

2

J J

1

͑

͗

͘

1

e

͑2͒

␳

t ͒ϭ

2

͑

4ប²2J ϩ1͒

͑

g

J_e

J_g

1

͑1͒

ϪQ

Q

J Ϫm

e e

1

*

ϫ

Ϫ1͒ ␧ ͒

Ϫ1͒

͉

J m

g

J m

g

͉ ␮"␧ ͒

͑

g

Ј

͉

J_em_eJ m ͉.

͗͘

g

͑A12͒

͑

͗͘

ͩ

ͪ

͚

Ј

1

g

Ј

1

m_eϪm_gϪQ₁

Q

m ,m ,m

e g g

1

The further steps applied to the ﬁrst order density matrix can now be applied again. After some rearrangement of the 3-j

symbols to simplify the result we ﬁnd

Ϫ␰ ␰₂e^{Ϫi͑␻ t ϩ␻ t}e^{i͑k Ϫk ͒"r}e^Ϫi␻

e

g

e

J

ʈ

␮ ͒

1

ʈ

J_g

J

ʈ

␮ ͒

͑

ʈ

J_e

͒

␶

Ϫ⌫ ␶

d 1

͑1͒

1 1

2 2

1

2

J J

1

͑

͗

͗͘

͘

1

e

g

2

͑2͒

␳

t ͒ϭ

2

͑

4ប²2J ϩ1͒

͑

g

1

J_gJ_e

1

K₂

T^KJ J ͒.

͑A13͒

͑1͒

ϪQ

͑1͒

K ϩ2J

1/2

2

g

*

ϫ

Ϫ1͒

␧ ͒

1

␧ ͒

2K ϩ1͒

͑

Q ϪQ

2

͑

g

Q

2

ͩ

ͪ

ͭ

ͮ

͚

2

g

2

1

J_g

1

K₂Q₂ϪQ₁Q₁ϪQ₂

K

,Q ,Q

1 2

2

Here we have succeeded in separating the geometrical and dynamical factors determining the signal. The 3- j symbol and light

polarization tensors restrict K₂to 0, 1 or 2, i.e., population, orientation or alignment, respectively. The 6-j symbol contains the

sensitivity to spectroscopic branch. Equation ͑A13͒ could equally well be written in terms of angular momentum alignment

operators, as discussed in Sec. II C.

The second order density matrix evolves during the delay ␶₂ϭt₃Ϫt₂. Crucially, because the states involved are degenerate

the only term in the evolution arises from collisional processes,

J

g

͑2͒

͑t Ϫt

͒

2

␳

t ͒ϭ␳ t ͒e^Ϫ⌫

.

͑A14͒

3

K

͑

2

3

We include this decay rate, dependent on the spherical tensor rank K₂, and apply the third photon interaction at time t₃.

Exactly the same procedure as with the previous two interaction Hamiltonians followed by a similar angular momentum

rearrangement leads to

␰ ␰ ␰₃e^{Ϫi͑␻ t ϩ␻ t Ϫ␻ t}e^{i͑k Ϫk ϩk ͒"r}e^Ϫi␻

e

g

e

J

ʈ

␮ ͒

1

ʈ

J_g

J

ʈ

␮ ͒

2

ʈ

J_e

J ␮ ͒

ʈ

͑

e

ʈ

J_g

͘

͒

␶

Ϫ⌫ ␶

d 1

͑1͒

1 1

2 2

3 3

1

2

3

J J

1

͑

͗

͗͘

1

2

e

g

3

͑3͒

␳

t ͒ϭ

3

͑

8iប³2J ϩ1͒

͑

g

͑1͒

1ϩK ϩ3J ϩJ

e

2

g

*

ϫ

Ϫ1͒

␧ ͒

3

͑

͚

1

ϪQ

2

Q ϪQ

1

2

3

K

,K ,Q ,Q ,Q

3 1 2 3

2

1

J_gJ_e

1

K₂

J

g

e^Ϫ⌫

2K ϩ1͒ 2K ϩ1͒

͑

2 3

͑t Ϫt

͒

2

1/2

3

K

͑

ͩ

ͪ

ͭ

ͮ

2

J_g

1

K₂Q₂ϪQ₁Q₁ϪQ₂

K₂J_gJ_gK₃

K₂

1

T^KJ J ͒.

͑A15͒

3

͑

ͩ

ͪ

ͮ

e

g

Q

3

J_eK₃

1

Q₃ϪQ₂Q₂ϪQ₃

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140.254.87.149 On: Mon, 22 Dec 2014 11:02:20

J. Chem. Phys., Vol. 120, No. 17, 1 May 2004

Polarization spectroscopy

7925

The ﬁnal time evolution of the system is during delay ␶

ϭtϪt₃. This introduces a ﬁnal decay rate giving the density

matrix

3

K

Q

͑3͒

J Ϫm

1/2

3

e

Ј

*

P

t͒•␧ ϭN

␳

t͒ Ϫ1͒

͑

2K ϩ1͒

͑

3

͑

͚

4

K

,Q ,m ,m

3 e g

Ј

3

J_e

J_g

K₃

͑3͒

␶

Ϫi␻

␶

J J 3

g

K

3

␳

t͒ϭ␳ t ͒e^Ϫ⌫

3

e

.

͑A16͒

͑

3

ϫ

ͩ

ͪ

m_eϪm_gϪQ₃

Ј

The observed signal depends on the electric ﬁeld generated

by the third-order nonlinear polarization represented by this

density matrix. This electric ﬁeld can be found as the expec-

tation of the dipole operator acting on the density matrix,²³

*

ϫ J m

͉

␮"␧

͉

J_em_eJ m

͉

J_gm_g.

͘

͑A18͒

͗

͗͘

g

4

Ј

The dipole operator can once again be expressed in the

spherical basis and the Wigner–Eckart theorem applied. The

orthogonality of the 3-j symbols restricts the nonzero value

of K₃to unity and the signal photon polarization spherical

tensor projection to Q₃. The relaxation during delay ␶ is

physically the same process as during delay ␶ and thus

͑3͒

*

͖

P

t͒•␧ ϭN Tr ␳ t͒• ␮"␧ ͒ ,

͑

͕

4

3

͑3͒

*

P

t͒•␧ ϭN

J m

͉␳

g

t͒• ␮"␧ ͒

͉

J_gm_g, ͑A17͒

͑

͗

͘

͚

g

4

1

m

g

⌫

₃ϭ⌫_d. Hence, the electric ﬁeld associated with the time-

K

where N is the total number of molecules present. Writing

the density matrix as an expansion in state multipoles gives

ordered sequence of photon interactions described by Feyn-

man diagram IA is ﬁnally

N␰ ␰ ␰₃e^{Ϫi͑␻ t Ϫ␻ t ϩ␻ t}e^{i͑k Ϫk ϩk ͒"r}e^Ϫi␻

g

e^{Ϫ⌫ ͑␶ ϩ␶}

͒

͑␶ ϩ␶

͒

3

1 1

2 2

3 3

1

2

3

J J

e

1

3

d

1

2

͑3͒

*

t͒"␧ ϭ

4

P

͑

8iប³2J ϩ1͒

͑

g

2

1

J_gJ_e

J

g

͑1͒

4

␶

ϫ

͉

J

ʈ

␮͒

ʈ

J_g

͉

͘

e^Ϫ⌫

K

2

͑

͗

ͭ

ͮ

͚

2

e

J_g

1

K₂

K

2

͑1͒

ϪQ

͑1͒

Q

2

*

␧ ͒

4

Ϫ1͒ 2K ϩ1͒ ␧ ͒

␧ ͒

3

͑

͚

2

1

2

Q ϪQ

Q

1

2

3

Q

,Q ,Q

1

2

3

1

K₂

1

K₂

ϫ

.

͑A19͒

ͩ

ͪͩ

ͪ

Q₂ϪQ₁Q₁ϪQ₂Q₂ϪQ₃Q₃ϪQ₂

The phase factor in Eq. ͑A19͒ may be rewritten in terms of

delays, ␶_n, resulting in the functional forms given in Eqs.

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Measurement of orientation and alignment moment relaxation by polarization spectroscopy: Theory and experiment

DOI: 10.1063/1.1691019

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