Review Article Nonlinear Optical Spectroscopy of …...Review Article Nonlinear Optical Spectroscopy of Chiral Molecules PEER FISCHER1* and FRANC¸OIS HACHE2 1The Rowland Institute

Review Article

Nonlinear Optical Spectroscopy of Chiral MoleculesPEER FISCHER1* and FRANCOIS HACHE2

1The Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts2Laboratoire d’Optique et Biosciences, CNRS/INSERM/Ecole Polytechnique, Palaiseau, France

ABSTRACT We review nonlinear optical processes that are specific to chiralmolecules in solution and on surfaces. In contrast to conventional natural optical activityphenomena, which depend linearly on the electric field strength of the optical field, wediscuss how optical processes that are nonlinear (quadratic, cubic, and quartic)functions of the electromagnetic field strength may probe optically active centers andchiral vibrations. We show that nonlinear techniques open entirely new ways ofexploring chirality in chemical and biological systems: The cubic processes give rise tononlinear circular dichroism and nonlinear optical rotation and make it possible toobserve dynamic chiral processes at ultrafast time scales. The quadratic second-harmonic and sum-frequency-generation phenomena and the quartic processes mayarise entirely in the electric-dipole approximation and do not require the use of circularlypolarized light to detect chirality. They provide surface selectivity and their observablescan be relatively much larger than in linear optical activity. These processes also giverise to the generation of light at a new color, and in liquids this frequency conversiononly occurs if the solution is optically active. We survey recent chiral nonlinear opticalexperiments and give examples of their application to problems of biophysical interest.A 2005 Wiley-Liss, Inc. Chirality 17:421–437, 2005.

KEY WORDS: review; nonlinear optics; spectroscopy; molecular chirality; opticallyactive liquid; chiral monolayer; second-harmonic generation; SHG; sum-frequency generation; SFG; nonlinear circular dichroism; nonlinearoptical activity; BioCARS

Natural optical activity comprises optical rotation (circu-lar birefringence) and circular dichroism. Fresnel showedthat a medium gives rise to optical rotation if its refractiveindices for right- and left-circularly polarized light areunequal.1 A difference in the corresponding absorption in-dices causes plane-polarized light to become ellipticallypolarized (circular dichroism). In optical activity phenom-ena, neither the angle of rotation of a polarized light beamnor its ellipticity is expected to change with the intensity ofthe light. However, at typical peak powers of pulsed lasers,the optical field strength can become comparable to thefield that binds the valence electrons to the nucleus of anatom or a molecule, and under these conditions the opticalproperties will have contributions that depend nonlinearlyon the applied electromagnetic fields. This is the realm ofnonlinear optics.2 – 4 The usual linear refractive indicesobserved at low light intensities are then augmented byadditional contributions that depend on the intensity of thelight. Hence, nonlinear contributions to optical activity maybe observed in optically active media under appropriateexperimental conditions. We show that the cubic depen-dence on the electric field in nonlinear optical activityphenomena allows the observation of dynamic opticalactivity processes with unprecedented temporal resolution.

As we shall see, nonlinear optical phenomena may alsogive rise to the generation of light at a new wavelength.For instance, in sum-frequency generation, two laserbeams that are in the visible may ‘‘mix’’ in a sample andgenerate light in the ultraviolet (UV). Surprisingly, such aprocess is only symmetry-allowed in a liquid, if thesolution is optically active. In contrast to conventionaloptical activity, however, this nonlinear process can ariseentirely from induced electric dipoles (without magnetic orquadrupolar transitions) and is not circular differential.Hence, no polarization modulation is needed, and thegenerated photons themselves are a measure of thesolution’s chirality. Because an achiral solvent cannotcontribute to the signal, the technique is a sensitive,background-free probe of molecular chirality. In additionto the quadratic sum-frequency-generation process, which

Contract grant sponsor: Rowland Institute at Harvard*Correspondence to: Dr. Peer Fischer, The Rowland Institute at Harvard,100 Edwin H. Land Boulevard, Harvard University, Cambridge, MA 02142.E-mail: [email protected] for publication 15 February 2005; Accepted 4 April 2005

A 2005 Wiley-Liss, Inc.

CHIRALITY 17:421–437 (2005)

DOI: 10.1002/chir.20179Published online 4 August 2005 in Wiley InterScience(www.interscience.wiley.com).

has been observed, a higher-order (quartic) nonlinearRaman spectroscopy (BioCARS) has been predicted toexist with similar properties. These techniques make itpossible to access states of chiral molecules that lie in theUV as well as probe molecular vibrations with visible light.We discuss recent experiments and show how theapplication of an additional static electric field to sum-frequency generation allows the absolute configuration ofthe chiral solute to be determined.

Nonlinear spectroscopies can also be applied to orientedassemblies of (chiral) molecules. Such is the case with sum-frequency generation (SFG) and the bulk-forbidden butsurface-specific second-harmonic generation process(SHG). Linear-polarized as well as circularly polarizedlight can probe the nonlinear optical response from a chiralmolecular monolayer. Remarkably, the nonlinear intensitydifferentials in SHG can be much larger than for linearoptical activity. For instance, the chiral SHG observable in amolecular film of a Troger’s base can be 6 orders ofmagnitude larger than the corresponding linear opticalrotation. We give examples of SHG experiments on chiralsurfaces of biological interest, including studies on proteins.

The aim of this review is to introduce the reader to thebasic principles of chiral nonlinear optical spectroscopiesand to give an overview of some recent experimental work.Although the use of chiral molecules as nonlinear opticalmaterials is a major focus of current research, space doesnot permit us to discuss these in detail. The reader mayconsult the literature for further details.5 – 7

We first outline some basic theoretical concepts thatrender certain nonlinear optical processes specific tochiral molecules.

THEORETICAL BACKGROUND

In this section, we consider the linear and nonlinearoptical properties of molecules. We discuss the theoreticalbasis of linear2,8 – 11 and nonlinear optical activity as well asfrequency conversion in chiral media.

Linear Optical Activity

Optical fields that are incident upon a molecule inducetime-varying molecular moments which themselves radi-ate. Most linear optical phenomena such as refraction (andabsorption) of light, and hence the refractive index (andthe absorption index), as well as Rayleigh scattering canbe interpreted through a molecule’s oscillating electric-dipole moment A

!ind linear in the electric field, E

!,a

A!

ind ¼ a$E!þ:::; ð1Þ

where a$ is the polarizability of the molecule. In general, a$

relates the three components of the electric field vectorto those of the dipole moment and is therefore a tensorwith nine components. Molecular symmetry may reduce

the number of independent components. The induceddipole moments in an ensemble of molecules give rise toan average macroscopic polarization P

!in the medium

P!¼ NhA!indi; ð2Þ

where the angular brackets denote an orientationalaverage over a region of space containing N moleculesper unit volume. In analogy to eq. (1), we can write

P!¼ "0m

$ð1ÞE!

þ:::; ð3Þ

where q0 is the permittivity of the vacuum. The linearsusceptibility m$ð1Þ is the macroscopic analogue of themolecular polarizability. It relates the electric field vectorto the macroscopic polarization vector. In a liquid, thelinear susceptibility is a scalar and is related to themolecular polarizability via

mð1Þ ¼ N

"0

1

3ðaxx þ ayy þ azzÞ �

N

"0a: ð4Þ

The linear refractive index n0 of an isotropic, non-magnetic dielectric is a function of the susceptibility,

n0 ¼ ð1 þ mð1ÞÞ1=2 �ffiffiffi"

p: ð5Þ

To describe the refractive index of an optically activeliquid, it becomes necessary to go beyond the electric-dipole approximation in eq. (1), and to include the inducedmagnetic dipole moment m

!ind (and for oriented samples

also the induced electric-quadrupole moment Q$

ind).12

In analogy to eq. (2), the macroscopic magnetization andthe macroscopic quadrupole density are respectivelyM!

¼ Nhm!indi and Q$ ¼ NhQ$ indi. For an optically active liq-

uid, we need to consider8,13,14

A!

ind ¼ a$E!þN�1 G

$0B!.þ:::;

m!

ind ¼ �N�1G$0

E!.þ:::; ð6Þ

where the dot denotes a derivative with respect to time,and where G

$V is the optical rotation tensor,b which is afunction of the rotational strength.

We now consider a monochromatic circularly polar-ized plane wave traveling along the z direction. In an op-tically active medium, the electric field E

!� with amplitudeE0, and the magnetic field B

!� can (at a point z = 0) bewritten as

E!� ¼ E0ðx

*cosðNtÞ � y

*sinðNtÞÞ;

B!� ¼ E0

n0

cð� x

*sinðNtÞ þ y

*cosðNtÞÞ: ð7Þ

The upper sign corresponds to right circularly polar-ized and the lower sign to left circularly polarized light; x*

aIn condensed media, such as liquids, the field at the molecule will be dif-ferent from the applied optical electric field, E

!, due to induced dipole –

dipole interactions of the surrounding molecules. E!

in microscopic ex-pressions, such as eq. (1), should be replaced with a ‘‘local field,’’ whichis approximated in the Lorentz model by ð"þ 2Þ=3ð ÞE!.

bNote that some authors use h$

for the optical rotation tensor. However, inthis article h

$is used to describe a nonlinear optical property tensor, the

first hyperpolarizability.

422 FISCHER AND HACHE

and y* are unit vectors; and c is the speed of light in

a vacuum.Substitution of the fields into A!ind in eq. (6) and use of

eq. (3) yield

P$� ¼ "0

Na

"0� NG

0n0

"0c

!E!� ð8Þ

for a liquid, where the isotropic component GV u (G Vxx + G VyyG Vzz)/3. The induced magnetic dipole term is best in-cluded by considering an effective polarization that con-tains the contributions due to the magnetization and thequadrupole density15,16

P!

eff ¼ P!þ i

Nr!�M

!�r

!�Q$; ð9Þ

such that

P$�

eff ¼ "0Na

"0� 2NG

0n0

"0c

!E!� ð10Þ

The effective polarization is of the form

P!

eff¼ "0meff E

!; ð11Þ

and since the refractive index of a dielectric is, in general,given by n = (1 + �eff)1/2, we obtain the refractive index ofan optically active liquid for right (+) and left (�) circularlypolarized light:

nð�Þ � n0 � g0; where g0 ¼ NG0

"0c: ð12Þ

The optical rotation in radians developed over a pathlength l is a function of the wavelength l and the circularbirefringence and is given by:8

# ¼ kl

Enð�Þ � nðþÞ� �

: ð13Þ

We thus obtain the familiar Rosenfeld equation for op-tical rotation:

# � � 2kl

Eg0: ð14Þ

Nonlinear Optical Activity

In order to describe nonlinear optical contributions tothe refractive index we need to consider the inducedmoments expanded in powers of the electromagnetic field.The induced electric-dipole moment of eq. (1) oscillatingat x becomes

A!

ind ¼ a$E!þ g

$E!E!E!þ:::: ð15Þ

For an electric field E!

/ E!

0 cosðNtÞ , the term that iscubic in the electric field has a time dependence cos3(Nt).

Because cos3(Nt) / 3 cos(Nt) + cos(3Nt), there is a termthat varies as cos(Nt) and therefore contributes to therefractive index at N. The term that is quadratic in theelectric field h

$E!E!

has been omitted from eq. (15) asit has no term that oscillates at the input angular fre-quency N. h

$and g

$ are known as the first and secondhyperpolarizabilities, respectively. The first hyperpolariz-ability h

$gives rise to second-harmonic generation and

sum-frequency generation (vide infra). It is seen thatseveral nonlinear optical processes may occur simulta-neously and that the respective hyperpolarizabilities aredistinguished by their frequency argument.

The refractive index that includes the nonlinear(intensity-dependent) electric-dipolar contribution imme-diately follows from eqs. (2) and (15). We definemð3Þ � Ng="0 and write

P!¼ "0 mð1Þ þ mð3Þ

E!

0

�� 2� �E!; ð16Þ

such that (cf. eq. (11)) n � n0 þ mð3Þ E!

0

�� 2=ð2n0Þ.c We canwrite the refractive index in terms of the intensity,I / E

!0

�� 2, and use a more compact notation by introducingthe nonlinear index of refraction, n2:4

n � n0 þ n2I : ð17Þ

By considering the cubic field terms in eq. (6), theintensity-dependent refractive index of an optically activeliquid can be derived in a similar fashion. Inspection ofeqs. (12) and (17) suggests its general form:

nð�Þ � ðn0 þ n2IÞ � ðg0 þ g2I Þ; ð18Þ

where we have introduced a ‘‘nonlinear optical activityindex,’’ g2. Hence, the circular birefringence of an opticallyactive liquid is modified in the presence of intense light. Inanalogy to the Rosenfeld equation, we can write a gen-eralized equation that describes linear and nonlinear op-tical rotation:

# � � 2kl

Eðg0 þ g2IÞ: ð19Þ

The corresponding absorption indices describe linear andnonlinear circular dichroism.

Nonlinear optical phenomena may not only modify therefractive index of an optically active liquid, but they canalso generate light at wavelengths different from those ofthe incident laser. We now discuss such frequency-conversion processes that are specific to chiral moleculesin liquids and on surfaces.

cOther definitions of the fields and polarizations can be found in thenonlinear optics literature. The numerical factors that enter theexpression for the refractive index and that accompany the susceptibilities(here and elsewhere in this paper) may therefore differ from those usedin other conventions.

423NONLINEAR OPTICAL SPECTROSCOPY OF CHIRAL MOLECULES

Coherent Frequency Conversion Processesin Chiral Media

The induced dipole quadratic in the electric field,

A!

ind ¼ h$E!E!; ð20Þ

with E!

/ E!

0 cosðNtÞ , has a static term and a term thatoscillates at twice the applied frequency, since cos2(Nt) /1 + cos(2Nt). The induced dipole oscillating at 2N radiatesand is the source of second-harmonic generation. Shouldthe incident optical field contain two frequencies N1 andN2, then radiation at the sum-frequency (N1 + N2) and thedifference-frequency (N1 � N2) may be generated. Onlysum (and difference) frequencies can be generated in aliquid, and only if the liquid is optically active.

Sum-frequency generation in liquids. The polariza-tion for a sum-frequency-generation process is in generaldescribed byd

P!ðN1 þ N2Þ ¼ "0m

$ð2ÞðN1 þ N2ÞE!ðN1ÞE

!ðN2Þ; ð21Þ

where the second-order susceptibility tensor m$ð2Þ is written

with its frequency argument.The intensity of the sum frequency is proportional to

jP!

ðN1 þ N2Þj2 . In a liquid, one needs to consider anisotropic average in eq. (21) with the result that thepolarization is given by the vector cross product of theincident fields

P!ðN1 þ N2Þ ¼ "0mð2ÞðN1 þ N2Þ E

!ðN1Þ � E

!ðN2Þ

� �; ð22Þ

and the susceptibility (for one enantiomer) is a scalar ofthe form

mð2Þ ¼ N

"0

1

6ðhxyz � hxzy þ hyzx � hyxz þ hzxy � hzyxÞ

� N

"0h: ð23Þ

The term in parentheses vanishes for any molecule thatpossesses reflection planes, a center of inversion, orrotation–reflection axes, and h is thus only nonzero fora chiral molecule.e It is of opposite sign for the enantio-mers of a chiral molecule. h is therefore a pseudoscalar (ascalar that changes sign under parity), as are the chirality

observables GV, g0, and g2. However, in contrast to linearand nonlinear optical activity phenomena, b arises entirelyin the electric-dipole approximation (no magnetic orelectric-quadrupolar contributions). m(2) is zero in a race-mic mixture, as here the number densities for the R- andS-enantiomers are equal and their contributions to eq. (23)cancel. Sum-frequency generation in a liquid is thus in itselfa measure of a solution’s chirality.

Second-harmonic and sum-frequency generation atsurfaces. Second-harmonic generation cannot occur in aliquid, even if the solution is optically active.f In SHG N1 =N2 and the hyperpolarizability components in eq. (23)become symmetric in the corresponding indices, e.g., bxyz =bxzy etc., such that their antisymmetric sum vanishes.However, both sum-frequency generation (SFG) andsecond-harmonic generation (SHG) are allowed at a sur-face provided the molecules at the interface adopt apreferred orientation. This is, for instance, the case for athin layer of dipolar molecules that lie between twoisotropic media with different optical properties, such asthe liquid/air interface. Generally, one assumes that sucha surface is symmetric about its normal, giving Cl

symmetry for chiral molecules and Clv for achiralmolecules. The interface is then characterized by oneaxis, the surface normal, around which it is rotationallyinvariant and which is parallel to the direction of theaverage dipole moment. In addition, the arrangement ofthe molecules at the interface can often be characterizedby an average tilt angle u of the molecular axis (takento be along ‘‘z’’) with respect to the surface normal (along‘‘Z ’’). The nonvanishing components of the second-ordersurface susceptibility tensor, evaluated in the frame ofthe laboratory that contains the optical fields, are thenrelated to the first hyperpolarizability tensor, which isexpressed in molecular coordinates, by the followingmatrix equations:17,18

mð2ÞZZZ

mð2ÞZXX

mð2ÞXZX

mð2ÞXXZ

0BBBB@

1CCCCA ¼ Nj

"0

a 2b 2b 2b2b c �b �b2b �b c �b2b �b �b c

2664

3775

�

hzzz

hzxx þ hzyy

hxzx þ hyzy

hxxz þ hyyz

0BB@

1CCA; ð24Þ

where Ns is the number of molecules per unit areaand where a ¼ cos3u; b ¼ 1

4 ðcos u� cos3uÞ; c ¼ 14 ðcos uþ

cos3 uÞ. In addition, the following identities hold:

mð2ÞZXX ¼ mð2Þ

ZYY ;mð2ÞXZX ¼ mð2Þ

YZY ;mð2ÞXXZ ¼ mð2Þ

YYZ :

fNo coherent SHG, whether electric-dipolar or magnetic-dipolar/electric-quadrupolar, can be generated in the bulk of a liquid if the incident fieldsare plane waves. Only near a surface can multipolar ‘‘bulk’’ SHG give riseto a traveling wave.

dThe polarization oscillating at the sum frequency is the source of a newwave, so that one has to consider the interaction of several (incident andgenerated) coupled waves in the medium. Efficient frequency conversionoccurs when the vector sum of the incoming photon momenta matchesthe momentum of the generated wave (phase-matching).eParity, or space inversion, is the symmetry operation that interconvertsthe enantiomers of a chiral molecule. Under space inversion allcoordinates ( x, y, z) are replaced everywhere by (�x,�y,�z).8 It followsthat under parity all the hyperpolarizability tensor components in eq. (23)change sign, e.g., hxyz��!Parity hð�xÞð�yÞð�zÞ ¼ �hxyz. Hence, the antisymmetricsum (h ) in eq. (23) also changes sign under parity.


For chiral molecules, there are additional components

mð2ÞXYZ

mð2ÞYZX

mð2ÞZXY

0B@

1CA ¼ Nj

"0

d e ee d ee e d

24

35 hxyz � hyxz

hyzx � hxzy

hzxy � hzyx

0@

1A; ð25Þ

where d ¼ 12 cos2u; e ¼ 1

4 ð1 � cos2uÞ, and where

mð2ÞXYZ ¼ �mð2Þ

YXZ ;mð2ÞYZX ¼ �mð2Þ

XZY ;mð2ÞZXY ¼ �mð2Þ

ZYX :

The matrix equations above hold for SFG and SHG. Inaddition, the following identities apply in the case of SHG:

mð2ÞXZX ¼ mð2Þ

XXZ ¼ mð2ÞYZY ¼ mð2Þ

YYZ ;

mð2ÞXYZ ¼ �mð2Þ

YXZ ¼ �mð2ÞYZX ¼ mð2Þ

XZY : ð26Þ

Only chiral molecules, i.e., molecules with the point-group symmetry Cn, Dn, O, T, or I, can give rise to thesurface susceptibility elements in eq. (25). Hence anexperimental geometry that probes one of the ‘‘XYZ’’surface-susceptibility elements is a probe of molecularchirality. For example, any Y-polarized SHG that isobserved in reflection from a Cl surface, when the inputbeam is polarized parallel to the plane of incidence (ZX),requires the surface to be chiral. The components of thepolarization from the surface are in this case given by

PY ð2NÞ ¼ 2"0mð2ÞYXZ cos ’ sin ’ E

!0ðNÞ

�� 2; ð27Þ

where u is the angle of incidence. In addition, there will beachiral contributions

PZð2NÞ ¼ "0 mð2ÞZZZ sin2 ’þ mð2Þ

ZXX cos2 ’� ��E!0ðNÞ

��2;PX ð2NÞ ¼ 2"0m

ð2ÞXZX cos ’ sin ’

��E!0ðNÞ��2; ð28Þ

where we have used the identities in eq. (26). The second-harmonic’s plane of polarization is thus seen to ‘‘rotate’’as a function of m(2)

YXZ and as a function of the enantio-meric excess at the interface. This effect has been termed‘‘SHG-ORD.’’19 Similarly, the circular SHG intensity differ-ential has been termed ‘‘SHG-CD,’’ and ‘‘SHG-LD’’describes linear SHG intensity differentials.20 These de-scriptions are widely used, and we follow this practice inthis review. However, we would like to stress that theseSHG processes do not arise from differences in the re-fractive and absorption indices that respectively underlieoptical rotatory dispersion and circular dichroism in linearoptical activity.

This completes our theoretical overview, and we nowturn to the application of these concepts.

SUM-FREQUENCY GENERATIONIN CHIRAL LIQUIDS

In sum-frequency-generation (SFG) spectroscopy, thepulses from a laser are overlapped in a medium and thelight generated at the sum of the two incident frequen-cies is detected. The light at the sum-frequency is a co-

herent and highly directional beam, and SFG is routinelyused in crystals to generate radiation at wavelengthsdifferent from those of the available laser. As discussedin the preceding section, electric-dipolar SFG cannot occurin a medium that has a center of symmetry. SFG istherefore excluded in liquids, unless they are optically ac-tive: The intrinsic symmetry-breaking in chiral moleculescauses a nonracemic liquid to be non-centrosymmetricand, as predicted by Giordmaine, allows for electric-dipolarSFG.21 Sum-frequency generation from chiral liquids hasrecently been reexamined22 and has been observedexperimentally.23 – 28 We now discuss some of its sa-lient features.

The induced electric-dipole polarization radiating at thesum-frequency is in a liquid given by eq. (22),

P!ðN1 þ N2Þ ¼ "0mð2ÞðN1 þ N2Þ E

!ðN1Þ � E

!ðN2Þ

� �;

where E!ðN1Þ and E

!ðN2Þ are monochromatic fields that

oscillate with frequency components N1 and N2, respec-tively. It follows from the vector cross product that theelectric fields at N1, N2, and (N1 + N2) need to span the X,Y, and Z directions of a Cartesian frame. This requires anon-collinear beam geometry with one s-polarized and twop-polarized beams.29 Whereas linear optical activity probeschirality with left- and right-circularly polarized light, thechiral probe in SFG corresponds to the three fielddirections which either form a left-handed or a right-handed coordinate frame. Chiral SFG spectroscopy there-fore needs no circularly polarized light and no polarizationmodulation. Rather, it is the detection of photons at thesum frequency that constitutes the chiral measurement.

One can immediately see that there can be no chiralprobe if two of the three waves have the same frequency,as is the case for SHG. The electric field vectors of the twofrequency-degenerate fields add, and the three waves nolonger make a coordinate frame. This is also borne out bythe quantum mechanical expression for h which is pro-portional to N1 � N2 and therefore goes to zero for SHG,where N1 = N2.30

As discussed in the introductory material, m(2) is therotationally averaged component of the second-order sus-ceptibility and is a pseudoscalar. For a solution that con-tains only two optically active molecular species, namely,the R- and S-enantiomers of a chiral molecule, we can writethe isotropic part of the electric-dipolar second-ordersusceptibility as

mð2Þ ¼ 1000NA

"0ð½R� � ½S�ÞhR ; ð29Þ

where the square brackets denote a concentration inmol/l, and where NA is Avogadro’s number. hR is the hof the R-enantiomer and is given by

hR ¼ 1

6ðhR; xyz � hR; xzy þ hR; yzx � hR; yxz þ hR; zxy

� hR; zyxÞ; ð30Þ


as is required for a pseudoscalar, hR ¼ �hS, so that mð2Þ iszero for a racemic solution (where [R] = [S]). The intensityat the sum frequency is proportional to jmð2Þj2 , and thesum-frequency signal thus depends quadratically on thedifference in concentration of the two enantiomers. A sum-frequency experiment therefore measures the chirality ofa solution and not its handedness, and SFG does not dis-tinguish between optical isomers.31 This is shown inFigure 1, where a quadratic dependence of a SFG signal onthe (fractional) concentration difference of the R-(+) andS-(�) enantiomers of 1,1V-bi-2-naphthol (BN) is ob-served.25 Within the noise of the experiment, no signal isrecorded for the racemic mixture. SFG is thus effectivelybackground free: any achiral signals (solvent, higher-ordermultipolar contributions) are either too weak to be ob-served or can be eliminated by choosing appropriate beampolarizations. As seen in Table 1, this is in contrast to linearoptical activity phenomena, which always contain anachiral and a chiral response. Several other differencesbetween optical activity and chiral SFG are also listed inTable 1. For instance, linear optical activity phenomenarequire both electric-dipolar as well as magnetic-dipolartransitions, whereas SFG from an optically active liquid isentirely electric-dipolar. This is significant, as magnetic-dipole (and electric-quadrupole) transitions are typicallymuch weaker than electric-dipole transitions. Neverthe-less, the absolute strength of the SFG signals is low. Inprinciple, this is not a problem, as it is possible to detect lowlight levels with single-photon counting methods, espe-cially as there is little or no background in SFG. However,in practice at least one of the three frequencies needs to benear or on (electronic or vibrational) resonance for there tobe a measurable SFG signal. The concomitant linearabsorption further limits the conversion efficiency, whichis already low as the SFG process cannot be phase-matchedin liquids.

Ab initio calculations and h of binaphthol. Quantumchemical calculations on several chiral molecules confirmthat the SFG pseudoscalar h is, even near resonance,much weaker than a regular nonzero tensor component ofthe first hyperpolarizability.22,26,32 – 34 Since h has no staticlimit, its dispersion is, however, much more dramatic.Figure 2 shows the enhancement by many orders ofmagnitude that the SFG signal (which is proportional tothe square of h) from BN experiences over a relativelysmall wavelength range.26 The mechanism of the non-linear response in BN has been computed ab initio,26,35,36

Fig. 1. Structures of 1,1V-bi-2-naphthol (BN) and continuous titration of two 0.5 M solutions of R-(+)-BN and S-(�)-BN in tetrahydrofuran. The sum-frequency intensity is observed at 266 nm with the incident beams at 800 and 400 nm. (Adapted from ref. 25).

TABLE 1. Comparison between linear optical activity andnonlinear optical sum-frequency generation as probes of

molecular chirality in liquids

Optical activity Nonlinear optics (SFG)

pseudoscalar G V / NIm½A!gm �m!mg �electric- andmagnetic-dipolar

h / ðN1 � N2Þ A!gm�½A!mn � A

!ng �

electric-dipolar

signal chiral and achiralresponse

only chiral response,no background

experiment:chiral probe

cp light linearly polarized beams

observable different responseto cp light

fGV

intensity at sumfrequency

fjhj2


and its (vibronic) exciton chirality26,35 has been discussedin terms of coupled-oscillator models.35,37 – 39,109 The non-linearity of BN is predominantly electric-dipolar35 (see alsothe section ‘‘Molecular origin of chiral SHG’’ ). Recentexperiments show that nonlocal higher-order multipolar(magnetic and electric quadrupolar) contributions to SFGfrom BN are measurable and that they are indeed muchweaker than the electric-dipolar nonlinearity.26 We stressthat selecting the polarization of all three fields in bulkSFG allows the chiral electric-dipolar signals to bediscerned from any achiral (magnetic-dipolar and elec-tric-quadrupolar) SFG signals. In two recent ab initiostudies on helical p-conjugated molecules, such ashelicenes and heliphenes, Champagne and co-workerspredict that the nonresonant sum frequency jhj can beincreased by functionalizing these molecules with suitable‘‘push–pull’’ chromophores.34,40

SFG in the presence of a dc-electric field. As thesum-frequency signal is proportional to the square of the

enantiomeric concentration difference, SFG can in gen-eral not distinguish between optical isomers. However,Buckingham and Fischer have predicted that the appli-cation of a static electric field to a SFG process allowsthe enantiomers of a chiral solute to be distinguishedand their absolute configuration to be determined.41 Thestatic field gives rise to an electric-field induced contribu-tion to coherent sum-frequency generation. This is an achi-ral electric-dipolar third-order contribution, such that thecombined sum-frequency polarization along x is given by

PxðN1 þ N2Þ ¼ "0

�mð2ÞEyðN1ÞEzðN2Þ

chiral

gþmð3ÞEyðN1ÞEyðN2ÞExð0Þ

achiral

�g ; ð31Þ

where we assume that the x1 beam travels along the zdirection and has its electric field vector oscillating along y,and the N2 beam to be plane polarized in the yz plane (seeFig. 3). The SFG signal is fjP!ðN1 þ N2Þj2 , and the cross-term between the chiral m(2) and the achiral electric-fieldinduced x(3) gives a contribution to the sum-frequency in-tensity that is linear in the static field, which is linear in m(2):

SFGðEÞ / mð2Þ mð3ÞEð0Þ; ð32Þ

where we denote the contribution to the intensity at thesum frequency that is linear in the static field by SFG(E).Thus the absolute sign of the pseudoscalar m(2) (and henceh) can be obtained. It, in turn, is needed to determine theabsolute configuration of the chiral solutes in the opticallyactive liquid.

Fischer et al. have recently observed the effect insolutions of 1,1V-bi-2-naphthol and have demonstrated thatit allows nonlinear optical sum-frequency generation to beused to determine the absolute configuration of a chiralsolute via an electric-dipolar optical process.28 Figure 3shows that the SFG(E) signals depend linearly on the

Fig. 2. The dispersion of the chirality specific pseudoscalar h is shownrelative to that of the (achiral) vector component of the first hyper-polarizability bO for the sum-frequency generation process 3N = 2N + N inoptically active 1,1V-bi-2-naphthol (configuration action singles sum-over-states computation; Hartree – Fock theory with a cc-pVDZ basis set).(Adapted from ref. 26).

Fig. 3. (a) SFG(E) signals measured as a function of the fractional concentration difference in R-(+)- and S-(�)-1,1V-bi-2-naphthol in tetrahydrofuran.(b) SFG beam geometry. (Adapted from ref. 28).


strength of the dc field and that they change sign withthe enantiomer.

Vibrational SFG. Sum-frequency generation canprobe electronic as well as vibrational transitions. If oneof the lasers is tunable in the infrared, SFG may be used torecord the vibrational spectrum of a chiral molecule. SFGvibrational spectra from neat limonene in the region ofthe CH-stretch have been reported by Shen and co-workers.23,42 The strength of the chiral CH-stretch modesin the limonene solutions jmð2Þj was reported to be about3 orders of magnitude smaller than a typical achiralstretch.23 Chiral spectra are measured with p-polarizedinput beams and a s-polarized sum-frequency beam,and with one s- and one p-polarized incident beam anda p-polarized sum-frequency. The liquids from the twoenantiomers can be distinguished if beam polarizationsare used that permit the observation of mixed chiral/achi-ral SFG.23

The scattering amplitude for vibrational SFG in a liquidis proportional to the (generally weak) antisymmetricRaman tensor associated with the vibrational mode whichis probed by the infrared resonance. Belkin et al. haveshown that the antisymmetric part of the Raman tensorcan become comparable in strength to the symmetric partunder conditions of double-resonance (vibrational andelectronic),43 and strong enhancement in doubly resonantSFG was recently observed in the vibrational spectrum ofBN in solution and on a surface.27

Time-resolved vibrational SFG optical activity mea-surements with circularly polarized infrared light havebeen proposed.44

SURFACE NONLINEAR OPTICS

Second harmonic generation (SHG) is a surface spe-cific spectroscopy that probes molecular order at sur-faces and interfaces, e.g., the liquid/air interface.45,46 InSHG, a lightwave from a laser is incident on a surfaceor an interface at a fixed angle with respect to the sur-face normal. The incident field at the frequency N in-duces dipole moments that oscillate at 2N. Coherentaddition of these moments from the surface leads to amacroscopic polarization

P!ð2NÞ ¼ "0m

$ð2ÞðNþ NÞE!ðNÞE

!ðNÞ; ð33Þ

which radiates a lightwave at the second harmonic (2N) inthe specular direction. The intrinsic surface specificity ofSHG originates from the symmetry breaking that occurs atan interface. Whereas m$ð2ÞðNþ NÞ vanishes in the bulk ofisotropic media, it is nonzero at surfaces and interfaceswhere the molecules may adopt a preferred orientation.Compared to linear optical spectroscopies, SHG can probea single layer of molecules on a surface with littlebackground signal from the bulk. Furthermore, as SHGis a nonlinear optical process, it relies on several pa-rameters, such as the directions or the polarizations of theincoming light beams, and is therefore experimentallymore versatile than a linear optical technique.

In 1993, Hicks and co-workers discovered that circularintensity differences in SHG are a probe of surface chi-rality.20 The application of SHG to the study of chiral mol-ecules has been pioneered by Hicks and co-workers20,47

on 1,1V-bi-2-naphthol (BN) and by Persoons and co-workers48 on a functionalized poly(isocyanide) polymer.Numerous studies on chiral-surface SHG have since ap-peared in the literature, and we review some of them here.

In the case of SHG from a surface layer of chiralmolecules, symmetry breaking is 2-fold: First, the inter-face breaks centrosymmetry such that surface SHG mayoccur, and second, the symmetry breaking in chi-ral molecules gives rise to additional contributions to theSHG signal. Both of these features contribute to surfaceSHG from chiral molecules. As described in the theoreticalbackground material, SHG can be observed from anachiral surface as well as a chiral surface. Even for achiral surface, the dominant signals can be electric-dipolar(or ‘‘local’’)

P!ð2Þð2NÞ ¼ "0m

$eee

E!ðNÞE

!ðNÞ; ð34Þ

where we label the susceptibility by the operators thatenter its quantum mechanical expression (here a productof three electric-dipole (e) transition moments). In contrastto linear optical activity phenomena, no magnetic dipolaror electric quadrupolar (‘‘nonlocal’’) contributions arerequired in SHG in order to observe a chiral opticalresponse. In this case, surface SHG probes chirality,similar to sum-frequency generation from the bulk of aliquid, via susceptibility tensor elements that depend onthe three orthogonal spatial directions ‘‘XYZ.’’ We discusshow SHG can be used to determine the origin of amolecule’s optical activity, and we shall see that the chiralSHG response for certain molecules is determined byelectric-dipolar (local) susceptibilities and for others bymagnetic-dipolar (nonlocal) susceptibilities.

Chiral SHG: experimental setup. The basic exper-imental geometry is depicted in Figure 4. The beam

Fig. 4. Surface SHG geometry. The surface normal is the Z directionin the laboratory frame. The angle of incidence is constant at 45j.


polarizations are controlled either by polarizers, wave-plates, or Babinet–Soleil compensators. The lightwave atthe second harmonic is typically detected with a photo-multiplier tube. Its intensity is proportional to

��P!ð2NÞ��2.

Three types of experiments have been devised to studythe response of a chiral surface that is rotationally in-variant with respect to the surface normal (Cl symmetry).In SHG optical rotatory dispersion (SHG-ORD),g theincident beam is p-polarizedh and the detection of any‘‘rotated’’ s-polarizedh polarization component at thesecond-harmonic indicates the presence of chiral mole-cules on the surface. Similarly, a difference in the SHGintensity as the circularity of a circularly polarized incidentbeam is reversed is a signature of chiral molecules. Thisdifference in SHG conversion efficiency is termed ‘‘second-harmonic generation circular difference’’ (SHG-CD).

gA

second-harmonic generation linear difference (SHG-LD)measurement consists of measuring the SHG intensity forthe incident fundamental beam polarized at F45j withrespect to the plane of incidence.49 Here again, theobservation of an intensity difference is a sign of chiralmolecules. In SHG-CD and SHG-LD experiments, thepolarization states of the incident light can be continuouslymodulated with a rotating quarter-wave or half-wave plate,respectively, which dramatically improves the signal-to-noise ratio in these measurements.

The intensity differences in SHG experiments on chiralsurfaces can be relatively much larger than commonlyobserved in typical linear optical ORD or CD experiments.For example, Hicks et al. measured SHG circular inten-sity differences of order unity from a monolayer of theR-enantiomer of BN,50 while in linear optics the differencebetween the left and right circularly polarized absorptivitydivided by their sum is typically 10�3.9 Hicks and co-workers also showed that the origin of the SHG signalsfrom BN can be explained within the electric-dipoleapproximation, as their experimental data agreed wellwith a theoretical calculation of the electric-dipolar non-linear surface susceptibility.35 However, it is reasonable toexpect that SHG signals from (certain) chiral moleculeswill, in general, also contain magnetic dipolar (and electricquadrupolar) contributions.51

Local and nonlocal contributions to SHG. We showthat magnetic-dipolar and electric-quadrupolar contribu-tions are indeed required to understand SHG in somechiral molecules. Nonlocal susceptibilities should thus beincluded in the SHG response. In general, magnetic-dipolar as well as electric-quadrupolar terms ought to beconsidered separately.8,52 Instead, one usually writes allthe nonlocal terms collectively as a single, general-ized ‘‘magnetic contribution.’’48 This is justified, as it isnot possible to experimentally distinguish between the

magnetic-dipolar and the electric-quadrupolar terms in atypical surface second-harmonic experiment.37 The totalpolarization is then expressed as

P!ð2Þð2NÞ ¼ "0m

$eee E!ðNÞE

!ðNÞ þ "0m

$eem E!ðNÞB

!ðNÞ; ð35Þ

and it is necessary to introduce the nonlinear magne-tization

M!ð2Þð2NÞ ¼ "0m

$mee E!ðNÞE

!ðNÞ: ð36Þ

The nonlocal m$mee and m$eem susceptibilities are alsothird-rank tensors, but their symmetry differs from thelocal, electric-dipolar susceptibility, due to the presenceof a magnetic-dipole transition moment (m) in place of anelectric-dipole transition moment (e). The nonzero com-ponents of these susceptibilities for achiral and for chiralsurfaces can be found in reference 52. In order to establishthe extent to which the nonlocal susceptibilities con-tribute to the chiral SHG signals, one needs to devise anexperiment that can, on the one hand, distinguish betweenlocal and nonlocal signals and, on the other hand, betweenchiral and achiral signals. Should the local (electric-dipolar) contributions dominate, then a counter-propagat-ing beam geometry can be used to separate the chiral fromthe achiral contributions.53 However, to some extent thevarious contributions remain mixed.

It is possible to separate the achiral and chiralcontributions by examining the field at the secondharmonic if it is written in the following form:

E!

s;pð2NÞ ¼ fs;pE!2

pðNÞ þ gs;pE!2

s ðNÞ þ hs;pE!

pðNÞE!

sðNÞ: ð37Þ

The advantage of this formulation is that the parametersfp, gp, hs are functions of only the achiral componentswhereas fs, gs, hp depend exclusively on chiral susceptibil-ities. Exact expressions for these coefficients as a functionof the susceptibility tensors can be found in reference 48.Note that these coefficients are complex-valued and thateach coefficient contains electric as well as magneticcontributions—with the exception of gs which involvesonly magnetic terms. Plotting the SHG intensity as afunction of the waveplate angle allows one to extract thef, g, and h coefficients.54 An example of such a mea-surement is given in Figure 5. Fitting of these curves(for explicit fitting formulas, see Schanne-Klein et al.55)yields the real and imaginary parts of the coefficients.These measurements only reveal the relative phase of thevarious coefficients and are therefore not quite sufficientto unambiguously separate the electric and the magneticcontributions in the surface SHG signal. However, achiralcontributions are generally electric-dipolar, such that thephase of the achiral coefficients, e.g., fp, gp, hs, can serve asa reference. It is then possible to separate the electric fromthe magnetic contributions in the chiral coefficients: thosein phase with the achiral coefficients are electric dipolar,and those in quadrature correspond to the magnetic

gSHG-ORD and SHG-CD are unrelated to ‘‘optical rotatory dispersion’’ and‘‘circular dichroism’’ in linear optics (see ‘‘Theoretical Background’’).hThe electric field vector is orthogonal to the plane of incidence fors-polarized light and parallel to the plane of incidence in p-polarized light.


terms. The phase relationship is expected to hold even onresonance, provided the electric- and magnetic-dipolartransitions involve the same energy levels.

Molecular origin of chiral SHG. The questionwhether magnetic (nonlocal) contributions to surfaceSHG are measurable, arose early on. Byers et al.35 showedthat their results on 1,1V-bi-2-naphthol could be fully ex-plained within the electric-dipole approximation, whereasKauranen et al. showed that magnetic terms were es-sential to account for their measurements on a polymer(poly-isocyanide) film.56 The presence of magnetic con-

tributions was also clearly observed in a SHG study ofchiral polythiophene, where a nonzero gs was measured.56

This finding was confirmed by Schanne-Klein et al., whoalso measured a nonzero gs in a pentamethinium salt.57

These findings do not contradict each other if oneconsiders the molecular basis for optical activity in thesemolecules and its implications for second-order nonlinearoptics. Two principal mechanisms are responsible for achiral molecule’s optical activity57: (i) a one-electronmechanism in which the electric and the magnetictransitions are mixed on the same chromophore due toits asymmetric environment; and (ii) a coupled-oscillatormechanism, in which two electric-dipole-allowed transi-tions belonging to two separate chromophores in themolecule are asymmetrically coupled. Hache et al.37 haveused a classical model to calculate surface SHG signalsand have shown that the dominant chiral surface sus-ceptibilities have a magnetic origin for molecules thatexhibit ‘‘one-electron optical activity,’’ whereas for mole-cules with ‘‘coupled-oscillator optical activity’’ the electriccontributions dominate. These calculations also explainthe experimental findings: 1,1V-bi-2-naphthol, studied byHicks et al., shows coupled-oscillator chirality, and it istherefore not necessary to consider nonlocal (magnetic)contributions to SHG. This has been confirmed by Fischeret al., who used SFG to discriminate between the nonlocaland the local response and have shown that the latterclearly dominates in 1,1V-bi-2-naphthol.26 Nonlocal suscep-tibilities are, however, essential in order to explain theresults obtained with the molecules studied by Kauranenet al. and Schanne-Klein et al. which pertain to the one-electron case. Interestingly, further conclusions can bedrawn from the model calculations (see Table 2). Whenexamining the signal obtained in surface SHG experi-ments, one can see that SHG-ORD and SHG-LD will give aspecific chiral signal only if the second-order susceptibilityis electric-dipolar, and these two experimental configura-tions are thus not sufficient to unambiguously detect thepresence of chiral molecules on a surface. SHG-CD gives achiral signal irrespective of the mechanism. The mecha-

Fig. 5. Second-harmonic generation from a surface of chiral stilbene(small squares – black dashed line) and of a Troger’s base (large squares –red solid line) as a function of the analyzer angle. No polarization rotationis observable for the stilbene, whereas a 64j rotation is measured for theTroger’s base.

TABLE 2. Second-order nonlinear optical response modeled by a one-electron mechanism (represented by a helix) and acoupled-oscillator mechanism (represented by two anharmonic oscillators)37

Model Example Relevant parameters Expected effects

fp, gp, and hs (achiral) = electric dipolar SHG-CDfs, gs, and hp (chiral) = magnetic dipolar

fp, gp, and hs (achiral) = electric dipolar +electric quadrupolar + magnetic dipolar

SHG-CD +SHG-LD, SHG-ORD

fs, gs, and hp (chiral) = electric dipolar

Respective examples are from experiments on a chiral stilbene59 and a Troger’s base.58


nism of a chiral molecule’s optical activity may thereforebe determined by examining both SHG-ORD and SHG-LDexperiments. This is not possible in linear optics, whichdemonstrates the potential of nonlinear optical spectros-copies to distinguish between ‘‘one-electron’’ and ‘‘coupled-oscillator’’ chirality. The mechanism of optical activity inan acridine-substituted Troger’s base58 and a chiralstilbene59 have been determined with SHG measurements(see Fig. 6). A large rotation of the SHG polarization hasbeen recorded for the Troger’s base, whereas no rotationcould be observed in the chiral stilbene. Hache et al.report a SHG rotation of 64j in the Troger’s base, which isthe largest rotation observed to date in a surface SHGexperiment. It is instructive to compare SHG-ORD withconventional, linear optical rotatory dispersion (ORD).The chiral surface SHG signals originate from a thin filmof chiral molecules. The same number of molecules wouldgive rise to an ORD angle that is far below the limit ofdetection. We estimate that the rotation measured in thefilm is 6 orders of magnitude larger in the surface SHGexperiment than it would be in conventional ORD. Thisdemonstrates the excellent sensitivity of surface SHG todetect chiral molecules.

All of the above-mentioned capabilities of surface SHGmeasurements on chiral molecules require that the

surface be symmetric with respect to the surface normal.Should the distribution of molecules be anisotropic withinthe surface plane (i.e., not Cl), then circular and linearintensity differences can also be observed from achiralmolecules, as has been shown by Verbiest and col-leagues.60,61 Electric-dipole-allowed SHG (and SFG) fromachiral molecules—that are arranged such that thesurface is chiral—have been studied.62,63 A carefultensor analysis of the response from anisotropic chiralfilms has shown that it is nevertheless possible to sepa-rate the effects due to anisotropy from those due to chi-ral molecules.64,65

The molecular aspects of chiral second-order nonlinearoptical spectroscopies (SHG and SFG) have also beendiscussed in a recent mini-review by Simpson.66

Sum-frequency generation from chiral surfaces. Sur-face second-order optical measurements are not restrictedto SHG. Sum-frequency generation (SFG) is equallyversatile.62 It has the advantage that the incidentfrequencies can be tuned independently. Shen and co-workers measured the SFG spectrum of a monolayer ofBN on water between 310 and 360 nm with a laser beamat 1.06 mm and another beam tunable from 450 to550 nm.67 The sum frequency was resonant with thelowest-lying absorption of BN with rich spectral content,and the polarization of the SFG provided additionalinformation on the molecules’ orientation.67 Should oneof the lasers be tunable in the infrared, then vibrationalsum-frequency spectra can be recorded. Large enhance-ments of the chiral response have been reported in recentdoubly-resonant experiments.27 These have allowed thevibrational sum-frequency spectrum of a BN monolayer tobe recorded.27

Applications of chiral surface spectroscopies: non-linear optics and proteins. Apart from the funda-mental questions addressing the molecular mechanismthat underlie surface SHG in a number of chiral mole-cules, several applications of surface SHG have recentlybeen reported.

Taking advantage of the fact that the second-ordersusceptibility changes sign with the opposite enantiomerof a chiral molecule, Busson et al. have demonstratedquasi-phase matching in a structure composed of alternat-ing stacks of helicenebisquinone.68 Interesting applica-tions may arise when chiral molecules are used inmaterials for nonlinear optical frequency conversion.Research in this direction is pursued in a number oflaboratories.69,70

Chiral-surface SHG has also been observed frommolecules of biological interest. A measurement on adipeptide (tert-butyloxycarbonyl-tryptophan-tryptophan) atthe air/water interface has been reported.71 SHG-CD andSHG-ORD measurements revealed the handedness of thedipeptide conformation. The handedness was assigned toa twisted conformation of the two tryptophans. SHG-CDhas also been used, together with a surface-enhancedresonance Raman study, to monitor the oxidation state of

Fig. 6. SFG-CD: p-polarized component of the SHG signal from asurface of Troger’s base as a function of the rotation angle of a quarter-waveplate. (Top) (+)-Enantiomer; (bottom) (�)-enantiomer. Arrows at+45j and �135j indicate that the polarization of the fundamental laserbeam is right circularly polarized, and correspondingly arrows at �45jand +135j that it is left circularly polarized. The lines are theoretical fits ofthe fp, gp, and hp functions to the data.


cytochrome c adsorbed on model membranes surfaces.72

SHG-CD was shown to be a particularly sensitive probe ofthe redox state of the heme group, which is known to beessential for the electron transfer process that occurs incytochrome c. Simpson and co-workers used chiral SHGexperiments to observe the dynamics of protein binding(bovine serum albumin) to a surface.73 Their experimentalresults are shown in Figure 7. Recently, chiral SHG signalsfrom the peptide melittin allowed Conboy et al. to detect itsbinding to a lipid bilayer.53 The same group has alsodeveloped a chirality-specific form of SHG microscopywhich used a counterpropagating geometry53 to image apatterned surface.74

THIRD-ORDER NONLINEAR CHIROPTICALEFFECTS IN A LIQUID

Thus far we have discussed nonlinear optical effectswhich are quadratic in the optical fields and which giverise to the generation of light at a new frequency thatis different from that of the incident laser. We now turn

to those nonlinear optical effects that can modify theoptical properties of an optically active liquid. As shownin the theoretical background material, this may occurat third order, where the optical response depends onthe cube of the optical fields and has a componentthat oscillates at the frequency of the laser. The so-called ‘‘optical Kerr effect’’ occurs in any medium andgives rise to a nonlinear contribution to the refractiveindex. It depends on the (achiral) hyperpolarizability g$

(cf. eq. (15)), and we can write the corresponding macro-scopic polarization as

P!ðNÞ ¼ "0m

$eeeeðN� Nþ NÞ��E!ðNÞ��2 E!ðNÞ; ð38Þ

where we have indicated that the third-order susceptibilityis electric-dipolar, and where we have given its frequencyargument N � N + N (which is = N). Apart from theintensity-dependent refractive index (eq. (17)), manyimportant nonlinear optical phenomena, including self-focusing, self-phase modulation, and two-photon absorp-tion, are directly related to the susceptibility in eq. (38).3

Fig. 7. Chiral-specific approach for label-free measurements of bovine serum albumin (BSA) binding using a rhodamine dye as a local probe forinterfacial chirality. In the absence of the protein, the SHG generated at the interface was suppressed by passing the beam through an appropriatelyoriented quarter-wave plate (QWP)/half-wave plate (HWP)/polarizer (Pol) combination (a). Upon exposure to an aqueous protein solution, the changein the polarization state of the SHG (presumably, through changes in the orientation distribution of the rhodamine dye) resulted in an SHG signal scalingwith the square of the surface protein density (b). Solid lines in (b) are fits of the data to exponential rises to maxima. In the upper curve (correspondingto s-polarized detection with a p-polarized incident beam), the polarization combination used exclusively probes the chiral-specific SHG response of theinterface. (Figure provided by Prof. G.J. Simpson, see also reference 73).


Laser beams with two different frequency componentscan also interact in a medium via a third-order suscepti-bility and give rise to frequency conversion processes, ormodify each other’s refractive index, e.g., N1 � N2 + N2.The latter also describes pump–probe experiments, wherethe fields at N1 and N2 are short pulses that are incidentwith variable time delays in order to observe any effectresulting from the induced polarization as a function of thetime between the pulses.3

We now discuss how the optical Kerr effect and all therelated achiral nonlinear optical phenomena may becomespecific to chiral molecules.

Nonlinear optical activity. Chiroptical effects at third-order have first been discussed by Akhmanov75 andBarron76 in the early years of nonlinear optics, and aspectsof the theory have since been developed further.77,78

Optical activity arises in the nonlinear response if one goesbeyond the electric dipole approximation in eq. (38). In-stead of the electric-dipolar susceptibility m$eeee, one needsto consider nonlocal susceptibilities that contain amagnetic-dipolar or electric-quadrupolar transition in placeof an electric-dipole transition (e), such as meeem (N� N + N)and similar expressions for the electric-quadrupolarsusceptibilities. A rigorous introduction of these suscepti-bilities and a discussion of their properties can be found inthe book by Svirko and Zheludev.51 From the phenom-enological discussion in the introductory material, itfollows that one expects nonlinear optical activity effectsin refraction and absorption according to eq. (18)

n� � ðn0 þ n2IÞ � ðg0 þ g2IÞ;

where the nonlinear optical activity index, g2, gives rise tononlinear optical rotatory dispersion (NLORD) and non-linear circular dichroism (NLCD), respectively.

There are only two experimental studies of NLORD todate: a light-induced optical rotation due to thermaleffects,79 and a nonlinear optical rotation study on uridineand sucrose by Cameron and Tabisz.80 The first exper-imental evidence of NLCD has recently been obtained byMesnil and Hache from solutions of ruthenium(II)tris(bipyridyl) (RuTB), where, in agreement with theoret-ical predictions, an intensity-dependent contribution to thecircular dichroism spectrum has been observed.81,82 Asrequired, the NLCD spectrum changed sign with theenantiomer and vanished for the racemic mixture.Extension of these measurements to a pump–probeconfiguration has recently been reported.83 Two indepen-dently tunable lasers have been used to simultaneouslyacquire both linear and nonlinear CD (and absorption)spectra of RuTB (see Fig. 8), demonstrating the potentialof this technique for spectroscopic applications.

Nonlinear optical activity is expected to be widely usedin the future, and a modified Z-scan technique has recentlybeen proposed that allows nonlinear chiroptical phenom-ena to be studied with a simple experimental setup.84

Many more nonlinear techniques will undoubtedly be

developed in the future, as exemplified by a recenttheoretical study of two-dimensional circularly polarizedpump–probe spectroscopy.85

Nonlinear chiroptical properties could yield applicationsfor signal processing. One study reported experiments onan optical switch based on light-induced isomerization.86

Recently, use of nonlinear chiral photonic bandgapstructures for all-optical switching was proposed.87

Investigation of biomolecules by time-resolved chi-roptical techniques. In one form of pump–probe experi-ments, an intense laser pulse is used to bring moleculesinto an excited (electronic or vibrational) state. Evolutionof the excited molecules is then monitored by a second,weaker pulse which probes the state of the molecules.Most often, it is the absorption of the probe which givesthe relevant information. Measurements are carried out asa function of the time delay between the two pulses, andthe time resolution is a function of the pulse duration. Thispopular technique permits resolution of processes atultrafast time scales. Extension of these techniques to

Fig. 8. (a) Linear (black squares) and nonlinear (blue dots) absorptionand, (b) linear (green squares) and nonlinear (red dots) CD measure-ments in a pump – probe experiment as a function of the probewavelength. The wavelength of the pump is 465 nm. The linear spectraare measured in a 2-mm-thick cuvette; the nonlinear coefficients are givenin absolute units (W�1 cm2). The solid lines are the spectra measuredusing spectrophotometers.


chiral molecules and more specifically to biomolecules isindeed appealing because it allows one to probe dynamicstereochemical properties. Optical activity is often ameasure of global structure in biomolecules and cantherefore provide information that is not accessible toother pump–probe techniques.

The most advanced applications of chiral pump–probespectroscopy concern time-resolved circular dichroismstudies which have been used to follow the structuraldynamics of protein conformation.88 Pump-induced linearCD measurements (i.e., linear optical activity and notNLCD) have been reported of the photolysis of carbon-monoxymyoglobin at nanosecond89 and picosecond90 timescales, and recently sub-100-picosecond CD dynamics hasbeen observed in the same protein.91 The observed con-formational changes upon photolysis of CO from myoglo-bin have been interpreted with the help of CD calculationsbased on a classical polarizability theory.92

Given the numerous applications of electronic CDspectroscopy for secondary structure determination ofproteins, it is expected that time-resolved CD will play animportant role in the elucidation of ultrafast protein foldingand structural dynamics.

Coherent Raman optical activity. Nonlinear exten-sions of natural optical activity phenomena are notrestricted to optical rotation and circular dichroism.Nonlinear optical analogs of Raman optical activity(ROA) have also been considered and their theory hasbeen discussed.93,94 ROA measures a small difference inthe vibrational Raman spectrum for left- and right-circularly polarized incident light.95,96 Although Ramanoptical activity is now routinely used to study chiral(bio)molecules in aqueous solution,95 it is an incoherentspectroscopy and the signals are thus weak. SpontaneousRaman scattering can easily be masked by light emittedfrom competing processes such as fluorescence. Non-linear Raman spectroscopies could be of value in the studyof vibrational optical activity, as they make it possible toprobe Raman resonances through coherent rather thanincoherent scattering.94 The signal strength in nonlinearRaman spectroscopies, such as coherent anti-StokesRaman scattering (CARS) and Raman-induced Kerr effectspectroscopy (RIKES), is therefore typically at least 3orders of magnitude larger than in spontaneous Ramanscattering.97 Nonlinear Raman spectroscopy should makeit possible to eliminate interference from fluorescence, andto obtain previously unavailable temporal informationabout a molecule’s vibrations. Nonlinear Raman opticalactivity is described by nonlocal third-order susceptibilitiesof the form �eeem (N1 � N2 + N1).93,94 In CARS, the laser atN2 is tuned, and when the difference in angular fre-quencies N1 � N2 is proportional to a vibrational energyh- Nvib, a resonantly enhanced signal is observed at N1 +Nvib. To our knowledge, there is only one report of asuccessful nonlinear Raman optical activity measurementfrom a chiral liquid to date. Spiegel and Schneider haveobserved Raman optical activity with CARS in a liquid of(+)-trans-pinane and report chiral signals that are f10�3 of

the conventional electric-dipolar CARS intensity.98 Theyused a band-shape analysis of three ROA bands between750 and 870 cm�1 in the CARS spectrum and reportnonlinear ROA spectra in excellent agreement withtheory.98 The nonlinear ROA effects are predicted to bemuch larger near electronic resonance.

CHIRAL SPECTROSCOPIES AT FOURTH-ORDER

Similar to linear optical activity, the spectroscopies atthird-order are characterized by signals that are primarilyachiral and, in the case of optically active liquids, may havean additional, small chirality-specific contribution. Thelatter is observed as an intensity difference signal thatrequires accurate polarization modulation or the mea-surement of small angles of rotation. This is in contrastto sum-frequency generation at second order, which iselectric-dipolar and is essentially background-free. Exten-sion of third-order processes in optically active liquids tofourth-order, as first proposed by Koroteev,99 would againallow for the observation of a background-free, electric-dipolar chiral signal. Although m(4) is in general muchweaker than m(2), a fourth-order process would have thedistinct advantage that it can be realized in a phase-matched geometry. The nonlinear polarization of a fourth-order process has a quartic dependence on the electricfields and is—in a liquid—given by

P!ð3N1 � N2Þ ¼ "0mð4Þ ð3N1 � N2Þ

ðE!ðN1Þ � E

!ðN2ÞÞðE

!ðN1Þ � E

!ðN2ÞÞ; ð39Þ

for the incident waves at N1 and N2.99,100 There is anexperimental report of such a process in liquids. Using twononcollinear frequency-degenerate beams (N1 = N2),Shkurinov et al. report the observation of weak signalsfrom concentrated aqueous solutions of d- and l-arabinosethat exhibited polarization characteristics as predicted byeq. (39).101 We note that recent attempts to reproduceearlier reports of SFG from arabinose101,102 have not beensuccessful and have shown that the m(2) of arabinose ismuch weaker than previously thought.22,24

A new coherent chiral Raman spectroscopy, that ariseswhen 3N1 � N2 in eq. (39) is equal to the angular frequencyof a vibration, has been proposed by Koroteev who gave itthe acronym ‘‘BioCARS’’ to indicate that this spectroscopywould constitute an extension of CARS to chirality, andhence biology.103 BioCARS would exclusively probe chiralRaman vibrations that are simultaneously Raman andhyper-Raman active.103,104 The spectroscopy does notrequire the use of circularly polarized light to detectchirality and would allow for a complete rejection offluorescence, but it could not distinguish between opticalisomers. BioCARS has not yet been observed. Apart fromthe inherent weakness of a nonlinear optical susceptibilityresponsible for BioCARS, fourth-order processes (andSFG) in liquids are weak as they require a noncollinearbeam geometry (due to the vector cross product in eqs.


(22) and (39)) which further reduces the interactionlength. The use of a waveguide, as suggested by Zheltikovet al., may constitute a promising alternative as it wouldallow for a collinear beam geometry and therefore a moreefficient implementation of BioCARS.105

CONCLUSION AND OUTLOOK

The principles of nonlinear optical probes of chiral mol-ecules in solution and on surfaces have been reviewed,and several selected applications of the various spectros-copies have been presented. Whereas a few of thenonlinear optical processes discussed in this review havenot yet, or only recently, been observed, the majority ofnonlinear spectroscopies that can be specific to chiralmolecules, including second-harmonic generation, sum-frequency generation, and, to some extent, nonlinear op-tical activity, are now well understood. The nonlineartechniques are not as general and efficient as conventionallinear optical activity measurements—they require pulsedlasers and often some form of resonance enhancement—but nonlinear processes can reveal aspects of molecularchirality not accessible to linear optics. We have shownthat nonlinear optics can be more sensitive than conven-tional optical activity probes. A monolayer of chiralmolecules may for instance be studied by second-harmonic and sum-frequency generation at surfaces withlittle contribution from the bulk of the liquid. Thecorresponding chiral nonlinear optical observables canbe many orders of magnitude larger than they would be intypical linear optical ORD or CD experiments. Second-order processes can also be used to experimentallydetermine whether a molecule’s optical activity is primarilydue to a one-electron or due to a coupled-oscillatormechanism. Nonlinear optical activity measures intensity-dependent contributions to optical rotation and circulardichroism and makes it possible to study the conforma-tion of chiral molecules at ultrafast time scales. Somenonlinear optical phenomena, such as sum-frequencygeneration (and BioCARS) in optically active liquids, haveno direct analogue in linear optics. They arise in the ab-sence of magnetic-dipole transitions and without any back-ground signals, so that the frequency conversion in itselfis a measure of a solution’s enantiomeric excess.

Novel applications of nonlinear chiral spectroscopies tothe study of biological interfaces, the determination ofabsolute configuration, and protein dynamics have re-cently been reported, and we expect that nonlinear opticalspectroscopies will become important tools in the inves-tigation of molecular chirality.

Note added in proof: The following papers haveappeared after completion of this review: A paper byWang et al. that reports the observation of chiral vibra-tional SFG spectra of proteins at the solid/liquid inter-face away from electronic resonance.106 A SHG studyof polypeptide alpha-helices at the air water interface.107

A theoretical paper that examines chiral third-orderspectroscopies.108

ACKNOWLEDGMENTS

The authors thank Prof. G.J. Simpson for providingFigure 7. P.F. is grateful for a Junior Research Fellowshipfrom the Rowland Institute at Harvard.

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Review Article Nonlinear Optical Spectroscopy of …...Review Article Nonlinear Optical Spectroscopy of Chiral Molecules PEER FISCHER1* and FRANC¸OIS HACHE2 1The Rowland Institute