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Theory of Molecular Nonlinear OpticsMark G. Kuzyk,1,* Kenneth D. Singer,2 and George I. Stegeman3,4

1Department of Physics and Astronomy, Washington State University, Pullman,Washington 99164-2814, USA

2Department of Physics, Case Western Reserve University, Cleveland,Ohio 44106-7079, USA

3College of Engineering, King Fahd University of Petroleum and Minerals,P.O. Box 5005, Dhahran 31261, Saudi Arabia

4College of Optics and Photonics and CREOL, University of Central Florida,4000 Central Florida Blvd., Florida 32751, USA

*Corresponding author: [email protected]

Received July 16, 2012; revised October 31, 2012; accepted November 1, 2012;published March 26, 2013

The theory of molecular nonlinear optics based on the sum-over-states (SOS)model is reviewed. The interaction of radiation with a single wtpisolated mole-cule is treated by first-order perturbation theory, and expressions are derived forthe linear (αij) polarizability and nonlinear (βijk , γijkl) molecular hyperpolariz-abilities in terms of the properties of the molecular states and the electric dipoletransition moments for light-induced transitions between them. Scale invarianceis used to estimate fundamental limits for these polarizabilities. The crucial roleof the spatial symmetry of both the single molecules and their ordering indense media, and the transition from the single molecule to the dense mediumcase (susceptibilities χ�1�ij , χ�2�ijk , χ

�3�ijkl), is discussed. For example, for βijk , symme-

try determines whether amolecule can support second-order nonlinear processesor not. For asymmetric molecules, examples of the frequency dispersion basedon a two-level model (ground state and one excited state) are the simplest pos-sible for βijk and examples of the resulting frequency dispersion are given. Thethird-order susceptibility is too complicated to yield simple results in terms ofsymmetry properties. It will be shown that whereas a two-level model sufficesfor asymmetric molecules, symmetric molecules require a minimum of threelevels in order to describe effects such as two-photon absorption. The frequencydispersion of the third-order susceptibility will be shown and the importance ofone and two-photon transitions will be discussed. © 2013 Optical Society ofAmerica

OCIS codes: 190.4710, 020.4180

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1. Outline of Review Paper. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1. Definition of the Microscopic Nonlinear Susceptibilities . . . . . . 112.2. Sum-over-States Theory for the Nonlinear Optical Response . . . 12

Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 41943-8206/13/010004-79$15/0$15.00 © OSA

2.2a. Traditional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2b. Dipole-Free SOS Expressions . . . . . . . . . . . . . . . . . . . . 19

2.3. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203. Molecular Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1. Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2. Irreducible Tensor Approach toβ Molecular Nonlinear Optics . . 28

3.3. Two-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3a. Two-Level Model: χ�2� . . . . . . . . . . . . . . . . . . . . . . . . . 373.3b. Two-Level Model: χ�3� . . . . . . . . . . . . . . . . . . . . . . . . . 393.3c. First-Order Effect on χ�3� of Population Changes in Two-Level

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444. Symmetric Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1. General Sum-over-States Model . . . . . . . . . . . . . . . . . . . . . . . 464.2. Three-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5. Transition to Bulk Nonlinear Molecular Optics . . . . . . . . . . . . . . . . 515.1. Local Field Corrections, Linear Susceptibility . . . . . . . . . . . . . 52

5.1a. Continuum Approximation. . . . . . . . . . . . . . . . . . . . . . . 525.1b. Nondipolar Homogeneous Liquids and Solids. . . . . . . . . . 535.1c. Nondipolar Two-Component System . . . . . . . . . . . . . . . . 54

5.2. Oriented Gas Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3. Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4. Electric Field Poled Media . . . . . . . . . . . . . . . . . . . . . . . . . . 595.5. Additional Contributions to Third-Order Nonlinearities . . . . . . . 63

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Appendix A: Cartesian Tensor Decomposition . . . . . . . . . . . . . . . . . . 66Appendix B: More Sophisticated Local Field Effects: Screening and Dressed

Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68B.1. Local Field Model of a Two-Component Dipolar Composite . . . 69

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 5

Theory of Molecular Nonlinear OpticsMark G. Kuzyk, Kenneth D. Singer, and George I. Stegeman

1. Introduction

Kerr, a 19th century experimentalist, was the first to observe nonlinear opticaleffects when he determined the refractive index change of collimated and spec-trally filtered sunlight in response to a voltage applied to organic liquids [1–3].However, the development of the Q-switched ruby laser first suggested byMaiman [4] and realized by McClung and Hellwarth [5] not only marked thebirth of the laser, but also opened the door for the following explosion of interestin nonlinear optics made possible by the sublimely intense light of that laser.Indeed, shortly after that development of the laser, optical harmonic generationin quartz crystal was reported by Franken and collaborators [6]. In this case, lightat twice the incident frequency was observed. Since the second-harmonic signalwas weak, the tiny spot on the photographic film appeared as an imperfection inthe film. Legend has it that the small speck was removed in the production officewhen it was mistakenly attributed to a piece of dirt.

Almost immediately after the first demonstration of second-harmonic generation,Bloembergen, Maker, and their associates [7,8], and later corrections by Hermanand Hayden [9], showed how the interference between propagating light (i.e., thehomogeneous solution to the wave equation) and bound polarization waves (so-lutions to the inhomogeneouswave equation driven by the nonlinear polarization)due to refractive index dispersion leads to interference fringes that can be used todetermine the nonlinear optical response of a slab of material, and that limits thegeneration of a second harmonic. It was soon discovered by Giordmaine [10] thatbirefringence can be used to cancel the dispersion leading to copious phase-matched second-harmonic generation, opening the door to the applications ofnonlinear optics, so that now that speck on the film could be transformed intoan intense laser beam at the second-harmonic frequency. To this day, the a majorapplication of nonlinear optical devices involves tuning of solid-state pulsedlasers using various parametric nonlinear optical devices including harmonicgeneration, as well as parametric oscillation and amplification [11,12].

Bloembergen’s 1965 monograph [7] delineated much of the physics of nonlinearoptics and laid the foundation for a great deal of work in the coming decades. Hiswork earned him the Nobel Prize for nonlinear optics in 1981. A plethora of non-linear optical phenomena described in that monograph, as well as others, has beena rich source of research comprising over 15,000 publications since. Phenomenainclude higher harmonics, intensity-dependent refractive index, multiphotonabsorption, photorefraction, various forms of Raman spectroscopy, and others.These phenomena arise from the nonlinear optical response functions of materi-als, whose study has paralleled those of the nonlinear optical phenomenology.

The initial materials focus, continuing to the present for parametric devices,centers on crystalline materials [12]. Piezoelectric crystals have received themost attention, given that both the lowest-order nonlinear optical effects and


piezoelectricity require materials without an inversion center. Though the sym-metry properties are similar, the physical origin of the nonlinear optical responseis distinct from piezoelectricity. It was obvious from the varying response indifferent materials that methods to understand the physical origin of the responseneeded to be developed. Beyond the classical anharmonic oscillator approachdescribed as a simple model by Bloembergen, attention soon turned to morerealistic models based on the interaction of light with matter using quantummechanical descriptions of materials. This was a daunting task in the case ofcovalent or ionic crystals, as the methods of solid-state physics to determinethe complete band structure for application in perturbation theory had not beendeveloped. As a consequence, phenomenological models grounded in quantumprinciples were developed, such as the polarization potential tensor [13,14] andthe bond-charge model [15].

Miller observed early on that a parameter later known as “Miller’s delta” couldbe used to define the nonlinear optical response of piezoelectric crystals in termsof the linear optical response, so that

χ�2�ijk �2ω� � χ�1�ii �2ω�χ�1�jj �2ω�χ�1�kk �2ω�δ2ωijk ; (1)

where χ�2�ijk �2ω� is the second-harmonic nonlinear optical susceptibility to be de-fined below, χ�1�uu �2ω� the linear susceptibility, and δ2ωijk the Miller’s delta param-eter. Remarkably, for oxide crystals it varied little from crystal to crystal eventhough the nonlinear susceptibility varied over orders of magnitude [16]. Thus,the intrinsic nonlinearity varied little, and the observed differences in frequencyconversion, for example, arose from phonon and crystal structure contributionsto the linear optical susceptibility.

At the same time, Kurtz and Perry developed a simple powder technique forquickly assessing new crystals. First a crystal is ground into fine powders thatare sifted by size. The fine powders of various sizes are pumped with laser light,and the dependence of second-harmonic intensity with crystal size are used toquickly estimate the second-harmonic coefficients and phase-matching potential[17]. This technique was applied to study a series of organic crystals, whereefficient second-harmonic generation was observed [18]. Notably, studies ofthe nonlinear susceptibility of organic single crystals were found to have a dis-tinctly large Miller’s delta, in contrast to inorganic crystals [19,20]. This sparkedthe study of organic materials for nonlinear optics and the elucidation of theunderlying physics of molecular materials.

The early studies of nonlinear optics and materials focused on the inorganicsolids described above, whose structure consists of periodic atoms bound bycovalent or ionic forces. This parallels the development of electronic materialsand the emergence of solid-state physics in the mid-20th century. However,shortly after World War II, conductivity in phthalocyanine was reported by Eley[21], which, along with the earlier discovery of photoconductivity in anthracene[22], helped to spark interest in organic photoconductors for the development ofsafe, low-cost photocopiers. The ensuing studies established the principles ofunderstanding the electronic and optical properties of these materials [23].The distinctive features of these organic crystals are that they are ordered arraysof complex organic molecules with conjugated electronic systems weakly boundin crystals by van der Waals forces (and sometimes hydrogen bonds). These aredefining characteristics of molecular solids. The van der Waals binding between


molecules implies that many of the optical and electronic properties of the solidmaterials can be understood by studying these properties in the constituentmolecules and where the macroscopic properties require the methods ofstatistical physics to relate the molecular properties to the corresponding proper-ties of the solids. Thus, the molecular properties are amenable to descriptionusing quantum mechanics, with statistical physics applied to collections ofmolecules (crystals, glasses, polymers, membranes, etc.), yielding themacroscopic electronic and optical properties.

Conjugated electron systems consist of networks (rings, chains) of alternatingsingle and multiple bonds, with the examples drawn from this work shown inFig. 1. The s‐p orbital hybridization (mixing) results in significant delocalizationof the π-electrons along the conjugated systems. These arrangements result inhighly colored and electronically responsive molecules and solids. Early modelsof the optical and nonlinear optical properties of delocalized π-electron systemsinclude the “particle in a box” by Kuhn [24,25]. As interest focused on lowest-order nonlinear optics and second-harmonic generation, the requirement for a lackof inversion symmetry at the molecular level required molecules with a dipolemoment, so that electron donor and electron acceptor terminated π-electron struc-tures became the model. Clearer connections to the quantummechanical descrip-tions of these structures emerged through connection to charge transfer within theπ-conjugated system and the excited state dipole moment. Phenomenological

Figure 1

Molecules referenced in this work: (a) phthalocyanine, (b) anthracene,(c) disperse red 1 (DR1), and (d) 2-methyl-4-nitroaniline (MNA). Note theπ-conjugated systems in each.


models based on these concepts emerged [26,27]. Finally, the development ofquantum chemical techniques applied to perturbation theory to calculate the mo-lecular nonlinear response by Lalama and Garito opened the door to a completeunderstanding of and a powerful tool for designing newmaterials [28]. This workcombined evolving approaches to quantum chemistry with previously developedsum-over-states approaches to quantum calculations of optical responses[7,29,30]. The field was now poised to take advantage of the profound chemicalsynthesis flexibility to develop new nonlinear optical materials.

To go along with this understanding, methods to measure the molecular non-linear optical response were required. The first such measurement on moleculesin solution were reported by Levine and Bethea using the electric-field-inducedsecond-harmonic (EFISH) generation technique, which had originally been devel-oped to measure the second hyperpolarizability of gasses [31]. In this technique, astatic (or quasi-static) electric field is applied to a liquid solution during measure-ment. This field aligns the molecular dipoles and breaks the inversion symmetry,allowing second-harmonic generation. The molecular response is obtained byproperly taking into account the number density, alignment in the field, and localfield factors, as we discuss below. Later, an examination of the role of local fieldsin such measurements suggested improvements in this technique and confirmedthat Onsager local field models (discussed below) can apply [32]. This work wasfollowed by more extensive measurements of organic molecules [27,33–35].

A second method for measuring the molecular nonlinear optical response wasdeveloped by Persoons and colleagues later, namely hyper-Rayleigh scattering(HRS) [36]. This technique uses incoherent second-harmonic scattering off ofsolutions of dipolar molecules. The orientational fluctuations of the noncentro-symmetric molecules in a centrosymmetric solution generate a small amount ofscattered second-harmonic light, whose scattering distribution and polarizationyields a significant amount of information on the second-order response func-tion. This technique is especially useful in measuring charged molecules thatcannot sustain an applied low-frequency field for EFISH, but especially for mul-tipolar chromophores that may not possess dipole moments, such as octupolesand other lower symmetry molecules [37–40].

Once the molecular response is obtained, one requires a statistical mechanicaltheory that relates the nonlinear optical response of a molecule to the bulksecond-harmonic response. Such theories have been developed for a numberof cases and are collectively known as oriented gas models since, as we showbelow, the nonlinear response is closely tied to rotational symmetries. Thetheory that relates the molecular hyperpolarizability and second-order responseof a crystal was first reported by Oudar and Zyss [41,42]. Later, Singer andassociates showed that a second-order nonlinear susceptibility could be impartedto an isotropic glassy polymer that is doped with aligned molecules that have alarge hyperpolarizability [43]. An electric field is applied to the molecules abovethe polymer’s glass transition temperature to align the dipoles, and the orienta-tional order is locked in place when the polymer is cooled below the glass transi-tion temperature. These results led to a flurry of activity in custom-designingmaterials for nonlinear optical devices [44].

The first model of the thermodynamic poling necessary for the molecular align-ment was reported shortly after the demonstration of dye-doped polymers [45].The data points lay somewhat below the theory, which is attributed to the fact


that the orientational order relaxed somewhat between the time the polymer waspoled and when it was measured. Use of a cross-linked polymer increased thelifetime of the orientational order and corona poling enabled large electric fieldstrengths to be applied [46]. Poled polymers offered the unique advantage ofbeing processable into thin films and fibers for wave-guiding devices[47,48]. Thackara and associates showed that an electro-optic waveguide phasemodulator could be made by using a poled thin film [49]. The fact that polymersprovide a good host material for molecules makes it possible to break down theproblem of designing a material to first identifying molecules with the rightproperties, including them in a polymer to make it optically nonlinear, thenforming the polymer into a device and poling it where required.

Much of the subsequent work in molecular nonlinear optics was aimed athigh-performance materials for applications in information technology andsignal processing. In particular, materials for the linear electro-optic effect[46,49,50] have received a great deal of attention and a remarkable level ofdevelopment, with the latest results having recently been reviewed [51].Similarly, third-order nonlinear optical properties in organic molecular materialshave been investigated over the same period [52,53]. Molecular materials forterahertz components have been shown to lead to an enhanced spectral response,opening up new vistas for terahertz spectroscopy [54,55].

While a great deal of work has focused on organic molecular nonlinear opticalmaterials, studies of nonlinear optics of molecular materials at the nanoscale,mesoscale, and microscale have blossomed into the principal trend of molecularnonlinear optics, and their spatial and symmetry sensitivity are literally illumi-nating the science of interfaces, nanostructures, and biological materials. One ofthe original studies of second-harmonic in monolayers by Heinz et al. [56]illustrates the molecular nature of the nonlinear response and its potential forprobing interfaces [57,58] and nanoparticles [59]. Nonlinear optical microscropyis a rapidly evolving field with important applications in biology and single-molecule detection [60–65]. Another important trend involves multiphotonabsorption, which has generated interest due to the ability to localize intenselight in a small volume with applications in microscopy and even three-dimensional photopatterning [64,66,67].

1.1. Outline of Review Paper

This review article follows a similar bottom-up approach, i.e., starting with mi-croscopic structure leading to bulk materials. We start from the perspective ofthe interaction of a single molecule with electromagnetic fields. First-orderperturbation theory is used to derive the sum-over-states (SOS) model for themolecular linear polarizability αij, the first hyperpolarizability βijk , and thesecond hyperpolarizability γijkl in terms of the electronic excited states (energylevels) of a molecule labeled m, their energy Em − Eg � ℏωmg above the groundstate (Eg), and the electric dipole transition moments between states m and n,μmn. A scale invariance approach is then used to estimate fundamental limits forthese polarizabilities. The effect of the inherent reflection and rotational spatialsymmetry on a molecule’s nonlinear optical properties is then discussed. Thesusceptibility for different second-order processes, such as second-harmonicgeneration, is deduced from the molecular symmetry properties in terms ofthe irreducible tensors that reflect the symmetry properties.


The ordering of the molecules in a dense medium determines the symmetryproperties of a “bulk” medium, and hence the macroscopic susceptibilitiesχ�1�ij ; χ�2�ijk ; χ

�3�ijkl…. This transition from single molecule to bulk medium properties

will be discussed for χ�2�ijk , specifically for crystals and partially ordered media,such as poled polymers. Because the electromagnetically induced dipole fieldsin neighboring molecules augment the local field at a molecule in dense media,approximate local field corrections for the different susceptibilities will be de-rived. However, all the locations of the energy levels and the transition electricdipole moments between them in a typical molecule are not available in general.A simplified two-level model, the ground state plus one excited state for asym-metric molecules, is used to obtain approximate analytical expressions from theSOS for the second-order nonlinear susceptibilities. The frequency dispersionin the application’s frequency regions will be discussed and compared to thepopular anharmonic oscillator models.

The third-order susceptibility is too complicated to yield simple results in termsof symmetry properties (which are tabulated in the literature). The SOS suscept-ibilities for a single molecule will be corrected for local field effects. The role ofeigenstates of the parity operator are shown to strongly affect the nature of thenonlinear optical response. In particular, terms due to single photon (i.e., paritychanging) transitions and multiphoton transitions are identified. The focus willbe on molecular media treated in the simple two- and three-level model approx-imations. It will be shown that symmetric molecules require a minimum of threelevels in order to describe effects such as two-photon absorption. The frequencydispersion of the third-order susceptibility will be shown for simple cases inthree frequency regimes: (1) near and on resonance, (2) off resonance, and(3) in the zero frequency (non-resonant) limit for both the two- and three-levelmodels. The importance of one- and two-photon transitions will be discussed.These approximate theories simplify in the non-resonant limit and it is shownthat there occurs destructive interference between one- and two-photon transi-tions so that the sign of the non-resonant nonlinearity depends on which termsare dominant. Finally, the relative contribution to the nonlinearity caused by asmall population in the excited state due to linear absorption in the two-levelmodel will be addressed.

In the last section we comment on the role that other third-order nonlinearities,principally due to vibrations, play in our understanding of measured subpico-second nonlinearities originally believed to be due to electronic transitions.

2. Theory

2.1. Definition of the Microscopic NonlinearSusceptibilities

The nonlinear interaction of light with matter occurs at the site of individualmolecules. The induced dipole moment of a molecule, p�F�, is a function ofthe applied local electric field, F. In the dipole approximation, p�F� is expressedas a power series of the local field with expansion coefficients, the materialresponse functions, that are called the polarizability (linear term), first hyperpo-larizability (quadratic term), second hyperpolarizability (cubic term), etc. Thenth-order contribution to the induced dipole moment, p�n�i �ω�, in the conventionwe adopt is of the form


p�n�i �ω� � 1

2n−1ε0ξ

�n�ijk…ℓ�−ω;ω1;ω2…ωn�Fj�ω1�…F ℓ�ωn�; (2)

where F�ω�, the electric field at frequency ω at the molecular site, is written inthe form (adopted here for all fields)

F�t� � 1

2F�ω�e−iωt � c:c:; F�−ω� � F�ω�: (3)

ξ�n�ijk…ℓ is also sometimes called the molecular nth-order nonlinear optical suscept-ibility tensor, and ω (ω � �ω1 � ω2;…� ωn) is the frequency of the dipoleoscillation excited by the mixing of n electric fields at frequencies ω1;ω2;…ωn.Here we use the Einstein summation convention (double indices are summedover the three Cartesian coordinates). The local field F�t� will be describedin more detail in Section 5 and Appendix B.

2.2. Sum-over-States Theory for the Nonlinear OpticalResponse

Here we present a brief derivation of the sum-over-states (SOS) quantum theoryof the nonlinear optical response of a single molecule.

2.2a. Traditional Approach

The nonlinear susceptibility of a quantum system starts with the calculation ofthe induced dipole moment as a function of the electric field, which is expandedas a Taylor series in the electric fields. The coefficients of the various powers ofthe field yield the nonlinear susceptibilities [7,11,68,69]. The dipole moment issimply given by the expectation value of the dipole moment using the ground-state wave function of the molecule that includes coupling to the applied electricfields.

The mth energy eigenstate of an atom or molecule in the presence of the localelectric field is given by jψm�F�i, where

F�t� �Xno: incident fields

p�1

Fp�ωp; t�: (4)

At zero temperature, the polarization is given by

P�F� � hψg�F�t��jPjψg�F�t��i; (5)

where jψg�F�t��i is the perturbed ground state. The generalized molecularsusceptibility is then given by

ξ�n�ijk…ℓ�−ω;ω1;ω2;ωn��1

ε0D0

∂n

∂Fj�ω1�∂Fk�ω2�…∂F ℓ�ωn�hψg�F�jPijψg�F�ijF�0

;

(6)

where D0 is the frequency-dependent degeneracy denominator that depends onthe number of distinct frequencies and the number of fields at zero frequency.Since this factor can depend on the convention used [68], it will not be discussedfurther.


When the electric field of the light is much weaker than the electric fields thathold a molecule together, the wave function of the molecule under the influenceof an optical field can be determined using perturbation theory with the zero-field wave functions as a basis set. The perturbation potential is simply the time-dependent electric dipole coupling energy between the field and the molecule.Defining H0 as the unperturbed Hamiltonian (i.e., with the light turned off), thetime evolution of a state jψi is given by

iℏ∂∂tjψi � H0jψi: (7)

With electric dipole coupling, the time-dependent perturbation potential is givenby

V �t� � −

Xp

μ · Fp�t�; (8)

where μ is the dipole moment (either induced or permanent) of the molecule andp spans all distinct photon fields. The total Hamiltonian,H , is then the sum of themolecular Hamiltonian and the perturbation potential multiplied by a smallperturbation parameter λ,

H � H0 � λV : (9)

With unperturbed eigenstates of the form

jψ �0�m �t�i � jψ �0�

m ie−iωmt; (10)

where ωm � ωm − iτ−1m with ωm � Em∕ℏ, τm is the lifetime of the mth excitedstate, and jψ �0�

m i is the spatial eigenstates of the unperturbed eigenfunctions ψ �0�m ,

the perturbed states can be expressed as a sum of increasing orders ofcorrection, indexed by s, that are labeled λs,

jψm�t�i �X∞s�0

λsjψ �s�m �t�i: (11)

Here the “hat” above the frequency ωm identifies it as a complex quantity.Substituting Eq. (11) into the Schrödinger equation, and keeping only termsof order s,

iℏ∂∂tjψ �s�

m �t�i � H0jψ �s�m �t�i � V �t�jψ �s−1�

m �t�i: (12)

Since the eigenvectors jψ �s�m �t�i can be expressed in terms of the unperturbed

states jψ �0�ℓ �t�i with coefficients a�s�mℓ �t�,

jψ �s�m �t�i �

Xℓ

a�s�mℓ �t�jψ �0�ℓ �t�i; (13)

the ground-state wave function is given by

jψ �s�g �t�i �

Xℓ

a�s�ℓ �t�jψ �0�ℓ �t�i; (14)

where a�s�l �t� � a�s�gl �t�. Substituting Eq. (14) into Eq. (12) yields


iℏ

�Xℓ

_a�s�ℓ �t�jψ �0�ℓ �t�i �

Xℓ

a�s�ℓ �t��−iωℓ�jψ �0�ℓ �t�i

�

�Xℓ

Eℓa�s�ℓ �t�jψ �0�

ℓ �t�i � V �t�Xℓ

a�s−1�ℓ �t�jψ �0�ℓ �t�i: (15)

Operating on Eq. (15) from the left with hψ �0�m j, we get

iℏ_a�s�m �t�e−iωmt � ℏωma�s�m �t�e−iωmt � Ema

�s�m �t�e−iωmt

�Xℓ

hψ �0�m jV �t�jψ �0�

ℓ ia�s−1�ℓ �t�e−iωℓt: (16)

Defining ωmℓ � ωm − ωℓ and Vmℓ�t� � hψ �0�m jV �t�jψ �0�

ℓ i, _a�s�m �t� is given by

_a�s�m �t� � 1

iℏ

Xℓ

Vmℓ�t�a�s−1�ℓ �t�eiωmℓt: (17)

Integration of Eq. (12) gives

a�s�m �t� � 1

iℏ

Xℓ

Zt

−∞Vmℓ�t�a�s−1�ℓ �t�eiωmℓtdt: (18)

With the system initially in its ground state, a�0�ℓ � δℓ;g. Equation (18) with thehelp of Eqs. (8) and (3) gives

a�1�m �t� � 1

iℏ

Xℓ

Zt

−∞μmg ·

1

2

Xp

F�ωp�e−iωpteiωmgtdt: (19)

The integral at negative infinity vanishes since ωmg � ωmg − iτ−1mg, where τ−1mg is

the decay time for electrons in the mth state to decay to the ground state, soEq. (19) yields in the mth state,

a�1�m �t� � 1

2ℏ

Xp

μmg · F�ωp�ωmg − ωp

ei�ωmg−ωp�t: (20)

The coefficient a�2�v �t� is derived by substituting Eq. (20) into Eq. (18),

a�2�v �t� � 1

4ℏ2

Xp;q

Xm

�μvm · F�ωq��μmg · F�ωp��ωvg − ωp − ωq��ωmg − ωp�

ei�ωvg−ωp−ωq�t; (21)

and a�3�d is calculated by substituting Eq. (21) into Eq. (18):

a�3�d �t� � 1

8ℏ3

Xp;q;r

Xd;v;m

�μdv · F�ωr��μvm · F�ωq��μmg · F�ωp��ωdg − ωp − ωq − ωr��ωvg − ωp − ωq��ωmg − ωp�

× ei�ωdg−ωp−ωq−ωr�t: (22)

In the adiabatic approximation, which holds when the optical photon energy(frequency) is much lower than the eigenenergies (Bohr frequencies), the mo-lecule will remain in the perturbed ground state in the presence of the field; but,


because the field is time dependent, the ground-state wave function will evolveaccording to

jψg�t�i � jψ �0�g i � λ

Xm

a�1�m �t�jψ �0�m ieiωmgt � λ2

Xv

a�2�v �t�jψ �0�v ieiωvgt � � � � ;

(23)

where

jψ �0�m �t�i � jψ �0�

m ie−iωmgt: (24)

Using Eq. (24), the expectation value of the dipole moment will be of the form

hμi�t� ��hψ �0�

g j � λXm

a�1�m �t�hψ �0�m jeiω

mgt � � � ��μ

×�jψ �0�

g i � λXm

a�1�m �t�jψ �0�m ie−iωmgt � λ2

Xm

a�2�m �t�jψ �0�m ie−iωmgt

�: (25)

The induced dipole moment to first order in λ from Eq. (25) is

hμi�1��t� �Xm

a�1�m �t�hψ �0�g jμeiω

mgtjψ �0�m i � hψ �0�

m jμXm

a�1�m �t�jψ �0�g ie−iωmgt: (26)

Using Eqs. (26) and (20), the fact that μmg � μgm and ω−q � −ωq, and some

manipulation yields

hμi�1��t� � 1

2ℏ

Xm

Xp

�μgm · F�ωp�μmg

ωmg � ωp

� μmg · F�ωp�μgmωmg − ωp

�e−iωpt � c:c:; (27)

where we have used the shorthand notation μℓ0ℓ � hψ �0�ℓ0 jμjψ �0�

ℓ i for arbitrarystates ℓ0 and ℓ.

Since the linear molecular polarization is usually written in terms of the elec-tromagnetically induced molecular dipole moment given by p�t� � hμi�1��t�,

p�t� � 1

2

Xp

�p�ωp�e−iωpt � p�−ωp�eiωpt�; (28)

p�ωp� �1

ℏ

Xm

�μgm · F�ωp�μmg

ωmg � ωp

� μmg · F�ωp�μgmωmg − ωp

�; (29)

where the summations over p and m are over all of the frequencies in the inputand over all of the discrete states of the molecule, respectively. The contributionto the first-order linear susceptibility [usually called the linear molecular polar-izability α�1�ij �−ωp;ωp�] of the Maxwell field Ei�ωp� � Fi�ωp�∕L�ωp� is calcu-lated by using Eq. (6) and yields

α�1�ij �−ωp;ωp� �1

ε0ℏL�ωp�

�μgm;iμmg;jωmg � ωp

� μgm;jμmg;iωmg − ωp

�; (30)

where μgm;j is the jth Cartesian component of μgm and where by definitionα�1�ij �−ωp;ωp� assigns i to be in the direction of the polarization p�ωp�, and j


to be in the direction of the applied field E�ωp�. Note that the orientation of theaxes for the polarizability is arbitrary. We now fix the axes as those in whichα�1�ij �−ωp;ωp� is a diagonal tensor.

The second- and third-order (and higher order) susceptibilities are calculatedusing a similar approach by calculating hμi�n��t� from hμi�t� to order λn andusing Eq. (25) and projecting out the ω Fourier component. The second-ordermolecular nonlinear polarization p�2��ω� is then given by

pNL�ω � ωp � ωq� �1

ε0ℏ2

Xqp

Xv;m

�μgv�μvm · F��ωq��μmg · F�ωp��ωvg − ωp∓ωq��ωmg − ωp�

� �μgv · F��ωq��μvm�μmg · F�ωp��ω

mg � ωp��ωvg∓ωq�

� �μgv · F��ωq��μvm · F�ωp��μmg�ω

mg � ωp��ωvg � ωp � ωq�

�: (31)

Note that the summations over p and q are both over all of the incident fields andv and m over all of the states.

As first pointed out by Bloembergen and associates, the Maxwell nonlinear po-larization is not given by simply multiplying each of the incident local fields by alocal field correction factor because there are fields present at all the frequencies,

including at the frequency generated by the nonlinear interaction, E�ω� [7].Consider the local field problem for which a second-order, nonlinear Maxwell

polarization P�ω � ωp � ωq� exists throughout the medium at the nonlinearlygenerated frequency ω � ωp�ωq due to the nonlinear interaction of theMaxwell field with the medium. The local field factor L�ω�, as described morefully in Section 5 and Appendix B, is given by

F�ω� � E�ω� � 1

3ε0P�ω� � L�ω�E�ω�: (32)

Including now the nonlinear polarization field pNL�ω� induced at the moleculeby the mixing of fields at the molecule and the “cavity” field at the molecule atfrequency ω due to contribution from all the other molecules outside the cavity,the total Maxwell polarization at the molecule is

→ p�ω� � N α ·

�E�ω� � 1

3ε0P�ω�

�� NpNL�ω�; (33)

where N is the molecular density. Hence the total molecular polarization atfrequency ω, including the Lorentz–Lorenz local field factor, is

→ p�ω� ��εr�ω� � 2

3

�hα · E�ω� � ~pNL�ω�

i: (34)

Now defining the nonlinear polarization component p�2��r; t� in the usual way asp�2��r; t� � 1

2

Xq;p

p�2��ωp � ωq�e−i�ωp�ωq�t � c:c:

� 1

4ε0Xq:p

β�−�ωp � ωq�;ωp;�ωq�∶E�ωp�E��ωq�e−i�ωp�ωq�t � c:c:;

(35)


whereβ�−�ωp � ωq�;ωp;�ωq� is defined as the second-order susceptibility with

p�2��ωp � ωq� �1

2ℏ2

�εr�ωp � ωq� � 2

3

�Xn;m

�~μgn�~μnm · F��ωq��~μmg · F��ωp��ωng∓ωq − ωp��ωmg − ωp�

� �~μgn · F��ωq��~μnm�~μmg · F��ωp��ωng � ωq��ω

ng − ωp��

� �~μgn · F��ωq��~μnm · F��ωp�~μmg�ω

mg � ωq � ωp��ωng � ωq�

�; (36)

yields

βijk�−�ωp � ωq�;ωp;�ωq� �1

ℏ2ε0L�ωp � ωq�L�ωp�L��ωq�

×Xnm

�μgn;iμnm;k μmg;j

� ˆωng∓ωq − ωp�� ˆωmg − ωp�

� μgn;kμnm;iμmg;j� ˆω

ng � ωq�� ˆωmg − ωp�

� μnm;jμgn;k μmg;j� ˆω

mg � ωq � ωp�� ˆωng � ωq�

�: (37)

Note that the nonlinear local field correction is

L�ωp � ωq�L�ωp�L��ωq� �εr�ωp � ωq� � 2

3

εr�ωp� � 2

3

εr�ωq� � 2

3; (38)

that is, it contains an extra factor relative to the linear case at the generatedfrequency ωp � ωq. After some manipulations, the ground state is found tobe excluded from the sum, leading to [29,70]

βijk�−�ωp � ωq�;ωp;�ωq� �1

ε0ℏ2L�ωp � ωq�L�ωp�L��ωq�

×X0

nm

�μgn;i�μnm;k − μgg;k�μmg;j

�ωng − ωq∓ωp��ωmg − ωp�

� μgn;k�μnm;j − μgg;j�μmg;i�ω

ng � ωq��ωmg � ωp � ωq�

� μgn;k�μnm;i − μgg;i�μmg;j�ω

ng � ωq��ωmg � ωq�

�; (39)

where the prime over the summations indicates that the sum excludes the groundstate, and terms like μvm;j have been replaced by μvm;j − μgg;j.

An alternative definition for the second-order susceptibility is given by

p�2��r; t� � 1

2

Xq;p

p�2��ωp � ωq�e−i�ωp�ωq�t � c:c:

� 1

4ε0PI

β�−�ωp � ωq�;ωp;�ωq�∶E�ωp�E��ωq�e−i�ωp�ωq�t � c:c:;

(40)


where PI is the “intrinsic permutation operator” that directs us to take theaverage over all permutations of ωq and ωp with simultaneous permutationsof the Cartesian components. For example, the first term in brackets inEq. (39) under permutation yields

PI

μgv;iμvm;jμmg;k�ωvg − ωp − ωq��ωmg − ωp�

� 1

2

�μgv;iμvm;jμmg;k

�ωvg − ωp − ωq��ωmg − ωp�

� μgv;iμvm;kμmg;j�ωvg − ωq − ωp��ωmg − ωq�

�: (41)

Using the same approach, the third-order susceptibility with the minor change innotation that the excited states are m, n, and v is given by

γijkl�−ω;ωp;ωq;ωr� �1

ε0ℏ3L�ω�L�ωp�L�ωq�L�ωr�PI

×

"X0

v;n;m

x

�μgv;i�μνn;l − μgg;l��μnm;k − μgg;k�μmg;j

�ωνg −ωp −ωq −ωr��ωng −ωp −ωq��ωmg −ωp�

� μgv;j�μvn;k − μgg;k��μnm;i − μgg;i�μmg;l�ω

νg �ωp��ωng �ωp�ωq��ωmg −ωr�

� μgv;l�μvn;i − μgg;i��μnm;k − μgg;k�μmg;j�ω

νg �ωr��ωng −ωp −ωq��ωmg −ωp�

� μgv;j�μνn;k − μgg;k��μnm;l − μgg;l�μmg;i�ω

νg �ωp��ωng �ωp�ωq��ω

mg �ωp�ωq�ωr�

�

−

X0

n;m

�μgn;iμng;lμgm;kμmg;j

�ωng −ωp −ωq −ωr��ωng −ωr��ωmg −ωp�

� μgn;iμng;lμgm;kμmg;j�ω

mg �ωq��ωng −ωr��ωmg −ωp�� μgn;lμng;iμgm;jμmg;k�ω

ng �ωr��ωmg �ωp��ωmg −ωq�

� μgn;lμng;iμgm;jμmg;k�ω

ng �ωr��ωmg �ωp��ω

ng �ωp�ωq�ωr�

�#; (42)

with the permutation parameter PI again signifying a summation over all thefields three times, i.e.,

Pp

Pq

Pr.

An important limit for both the nonlinear susceptibilities is the zero frequencylimit, called the Kleinman limit in the literature, in which all of the inputfrequencies are set to zero. With ωngτng ≫ 1, which is usually the case, Eqs. (37)and (42) are greatly simplified, namely,

βijk�−�ωp � ωq�;ωp;ωq�!ωng≫ωp 1

ε0ℏ2L�ωp � ωq�L�ωp�L��ωq�

×X0

n;m

1

ωngωmgfμgn;i�μnm;k − μgg;k�μmg;j � μgn;k�μnm;j − μgg;j�μmg;i

� μgn;k�μnm;i − μgg;i�μmg;jg; (43)


γijkl�−ω;ωp;ωq;ωr� �1

ε0ℏ3L�ω�L�ωp�L�ωq�L�ωr�PI

X0

v;n;m

1

ωvgωngωmg

× fμgn;i�μvn;l − μgg;l��μnm;k − μgg;k�μmg;j� μgv;j�μvn;k − μgg;k��μnm;i − μgg;i�μmg;l� μgv;l�μvn;i − μgg;i��μnm;k − μgg;k�μmg;j� μgv;j�μvn;k − μgg;k��μnm;l − μgg;l�μmg;ig

− 2X0

n;m

1

ωngωmgfμgn;iμng;lμgm;kμmg;j

� μgn;lμng;iμgm;jμmg;kg; (44)

respectively.

It is now useful to write the total polarization of a molecule up to third order inpowers of the Maxwell electric field as

pi � p0i � αijEj � βijkEjEk � γijklEjEkEl � � � � ; (45a)

where p0i is the ground-state dipole moment, αij is the polarizability tensor,and βijk and γijkl are the first and second hyperpolarizability tensors,respectively.

2.2b. Dipole-Free SOS Expressions

As we have seen above, the nonlinear susceptibilities are derived from perturba-tion theory and depend on the matrix elements of the dipole operator and theenergy eigenvalues of the Hamiltonian. The complexity of this expression makesit difficult to ascertain how the nonlinear-optical susceptibility depends on theunderlying system’s properties. One method of simplification is to truncate theSOS expressions to include only a finite number of states. Indeed, the two andthree-level models have been quite successful at modeling the first and secondhyperpolarizability of molecules.

A more rigorous simplification comes about from the fact that there are funda-mental relationships among the transition moments and energy eigenvalues. Assuch, the dipole matrix elements and energies may not be arbitrarily variedwithout violating quantum mechanical principles. The sum rules can be usedto simplify the SOS expression by re-expressing all dipole moments (the diag-onal elements of the dipole matrix) in terms of transition dipole moments (thenondiagonal elements). The result is commonly referred to as the dipole-freeexpression and is derived as follows.

All solutions of the Schrödinger equation must obey the Thomas–Kuhn sumrules [70–72]. The generalized sum rules relate the matrix elements and energiesto each other according to [73]

X∞n�0

�En −

1

2�Em � Ev�

�μmnμnv �

ℏ2Ne

2meδm;v; (45b)

where me is the mass of the electron and Ne is the number of electrons in themolecule. The sum, indexed by n, is over all states of the system. Equation (45b)represents an infinite number of equations, one for each value of m and v. Thus,


the parameters in Eqs. (30), (41), and (42) (i.e., dipole matrix elements andenergies) cannot be independently varied because they are related to each other.Indeed, it was shown that an N -level model for the nonlinear response, whichdepends on a dipole matrix with elements and N energy levels, has enough sumrule equations to reduce the total number of parameters to N − 1 [74]. For thesecond-order nonlinearity,

βxxx�−ω;ω1;ω2� � −

N

2ε0ℏL�ω1�L�ω2�L�ω�PI

X0∞

n�0

X0∞

m≠n

μgn;xμnm;xμmg;xˆωng�ωi� ˆωmg�ωj�

×

�1 −

ωmg�ωj�f2ωmg − ωnggˆωng�ωj�ωmg

�� ; (46)

where ˆωng�ωi� � ωng − ωi for any general state n and frequency ωi and where allpermutations of ωi and ωj are permuted over all frequencies ω1, ω2, and ω. Thepermutation operator PI directs us to average over the exchange of the two inputfrequencies ωi and ωj. The second term in brackets is the dispersion term thatresults when the sum rules are used to re-express the dipolar terms (i.e., the oneswith dipole moment differences) in terms of the transition moments.

In the standard SOS expression, the simplest approximation is the two-levelmodel, with parameters μ10;x, Δμ10;x � μ11;x − μ00;x, E10, and E20. The simplestapproximation for Eq. (46) is the three-level model, with parameters μ10;x, μ21;x,E10, and E20. In contrast, the standard SOS expression in the three-level approx-imation has two additional terms, which include a dipole moment differencebetween the ground state and the first excited state and a dipole moment dif-ference between the ground state and the second excited state. Thus, thedipole-free expression has fewer parameters when truncated to the same numberof states. This reduction in parameters results from the sum rules, makingthe dipole-free form more parsimonious when fitting data. However, theSOS and dipole-free expressions usually differ when truncated. Which onebetter describes the data depends on the quantum system involved.

The third-order nonlinear optical susceptibility can also be rewritten in dipole-free form. Given the complexity of the result, we refer the reader to the literature[75]. As a result of these simplifications, the dipole-free expression is a usefultool since it requires the determination of fewer parameters when modeling thenonlinear susceptibility.

Now the researcher has two choices for modeling the nonlinear optical responseof a quantum system. Interestingly, when the SOS expression is truncated to afinite number of terms, the traditional expression and the dipole-free expressiondiffer. Only in the infinite number of state limit do the two converge. One ex-pression may be more useful or accurate than the other, depending on the systemand how it is being applied. For example, for a system whose nonlinear responseis at the fundamental limit, both expressions should again converge in the three-level model. In the analyses that follow, we will use the traditional SOSexpression.

2.3. Scaling

Historically, the motivation for understanding of the nonlinear-optical propertiesof molecules was fueled by their potential usefulness in making materials for


nonlinear-optical devices such as electro-optic modulators and harmonicgenerators, and more recently, in applications of 3D photolithography andbioimaging. As such, the focus has been on the magnitude of the nonlinearsusceptibilities.

The nonlinear susceptibility increases with the size of the molecule and the num-ber of electrons within. A more interesting fundamental question pertains towhat properties of a molecule affect its intrinsic nonlinearity. As a case in point,perhaps the shape of the confining potential is important, but with size effectsproperly taken into account. Size effects are best accounted for by using scalinglaws. These laws determine how the nonlinear susceptibilities grow with the sizeof the system.

The size of a system is not a well-defined quantum-mechanical property.However, the wavelength of the electron in the atom in its ground state is a goodestimate of size. Similarly, adding extra electrons increases the strength of thenonlinearity. Once the effects of “size” can be removed from the equation,molecules of all shapes and sizes can be compared with each other to searchfor the fundamental properties that most affect the nonlinear-optical response.As an example, the susceptibility of a harmonic oscillator depends on the springconstant, but, once scaling is taken into account, all harmonic oscillators areequivalent.

At issue is the fact that most molecules are too complex to break the problemdown into such a simple argument. The intrinsic nonlinear-optical susceptibilitystrips away the unimportant stuff and leaves behind the core of the nonlinearresponse. Once the core properties are known, the molecule can be “scaledup” to yield a large absolute nonlinear susceptibility. The following derivationrigorously determines the scaling laws, and from these defines the intrinsichyperpolarizabilities. As we will see, the energy difference between the groundand first excited state and the number of electrons define the scale of a quantumsystem.

The Hamiltonian of any N -electron system, such as a molecule or charges in amultiple quantum well, depends on the potential V , which can contain coulombrepulsion terms, such as −e2∕4πε0jr1 − r2j, spin interactions, spin–orbit cou-pling, and external electric fields, and the vector potential can describe interac-tions with external electric and magnetic fields.

The N -electron Schrödinger equation is of the form

Hψ�r1; r2…rN � � Eψ�r1; r2…rN �: (47)

Scaling can be understood by transforming Eq. (47) with rk → εrk , where ε isthe scaling parameter, into

1

2m

XNk

�pk −

e

cεA�εr1;…�

�2

ψ�εr1;…�

� ε2V �εr1;…; s1;…; L1;…�ψ�εr1;…�� Eψ�εr1;…�; (48)

where A is the vector potential and si, Li, and ri are the spin, orbital angularmomentum, and position of the ith electron. Thus, ψ�εr1…�, which has the same


shape as ψ�εr1…�, aside from being spatially compressed by a factor 1∕ε, is asolution of the Schrödinger equation with pk → pk∕ε2, E → Eε2,V �r1…� → V �εr1…�ε2, and A�r1…� → εA�εr1;…�. In effect, this causes com-pression of the potentials spatially by a factor of 1∕ε, rescaling the energy by ε2,and rescaling the vector potential by ε but leaving the shape of the wave func-tions unchanged. Rescaling is then, by definition, the transformation of Eq. (47)into Eq. (48) with the associated rescaling of the energies and vector potential.

Upon rescaling, the position and energy product r · rE is invariant. The dipolemoment, μ, is defined as

μ � −eXNi�1

ri; (49)

where ri is the position of the ith electron, and −e is the electron charge. Thus, theposition operator is given by

r � −

μ

e: (50)

When electrons are added to the system in a way that does not change theeigenenergies, then the invariance relation can be generalized to the form

r · rE � kN ; (51)

where k is a constant. The components of the position operator obey the sumrules given by Eq. (45b). By convention, the x direction is along the largestdiagonal component of the hyperpolarizability tensor. Given Eq. (51), μmn;xis called the transition moment along the x axis between states n and m.

Using the sum rules and the three-level ansatz, it can be shown that the polariz-ability, hyperpolarizability, and higher-order hyperpolarizabilities are bounded[76–79]. The off-resonance polarizability (or zeroth-order hyperpolarizability) is

α≤ αmax0 �

�eℏffiffiffiffim

p�

2 N

ε0E210

; (52)

the fundamental limit of the hyperpolarizability (also called the first hyperpo-larizability) is

jβj≤ βmax0 �

ffiffiffi3

4p �

eℏffiffiffiffim

p�

3 N 3∕2e

ε0E7∕210

; (53)

and the fundamental limit of the second hyperpolarizability is

jγj≤ γmax0 � 4

�eℏffiffiffiffim

p�

4 N 3∕2e

ε0E7∕210

: (54a)

We note that there are no approximations used in calculating the fundamentallimits, so they are exact and therefore cannot be exceeded. While the three-levelansatz has not been proven rigorously, it appears to hold always; when a quan-tum system has a nonlinear response at the limit, only three states (ground andtwo excited states) contribute. Note that this does not imply that, if the three-level ansatz holds, the system must have a nonlinear response at the limit.However, when the nonlinear response is small, usually many states contribute.


Similarly, the transition electric dipole moment to any excited state is boundedand for the first excited state it is given by

jμ10j≤ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2Neℏ2

2ε20meE10

s≡ μ max

10 : (54b)

The fundamental limit of the nth hyperpolarizability is of the form

η�n��max� ∝N �n�2�∕2

e

ε0E�3n�4�∕210

; (55)

where η�−1�max is the fundamental limit of jμ10j, η�0�max is the fundamental limit of α,η�1�max is the fundamental limit of jβj, etc.The nth hyperpolarizability is of the form

η�n� ∝�μvm�n�2

ε0�Em�n�1; (56)

where �μvm� represents products of transition dipole moments of the form μvm and�Em� represents products of energy differences of the form Emg � Em − Eg. Thus,rescaling the nth hyperpolarizability according to Eq. (48) and using Eq. (51) yields

η�n� � k�n�2�∕2

ε0�E��3n�4�∕2 N�n�2�∕2: (57)

The intrinsic nth hyperpolarizability is then given by

η�n�int �η�n�

η�n�max

∝ k�n�2�∕2�E10

�E�

��3n�4�∕2; (58)

wherewehaveusedEqs. (69) and (71).Theparameter η�n�int is unchangedunder simplescaling and independent of the number of electrons, making it a scale-invariant quan-tity. The intrinsic nth-order hyperpolarizability is invariant upon rescaling under thesame transformation that leaves the Schrödinger equation invariant, as de-scribed above.

The intrinsic hyperpolarizabilities are quantities that remove the effects of scal-ing, and are a measure of a molecule’s nonlinear optical efficiency, independentof the number of electrons or energy gap. While larger molecules with moreelectrons will generally interact more strongly with light than smaller, electron-poor systems, the intrinsic hyperpolarizabilities remove such effects, allowingone to focus on the structural properties that affect the response. This allows oneto determine whether all of the electrons are contributing to the nonlinearresponse with maximal efficiency. Only then can truly new paradigms be devel-oped for making large molecules with exceptionally enhanced response.

The approach of using the intrinsic hyperpolarizability as a scale-invariant measureof the nonlinearity of a molecule has been used by some groups to compare mo-lecules, but is not as generally appreciated given its fundamental importance. Theintrinsic hyperpolarizability is important because it is a fundamental intensivequantity that helps us to better understand the strength of light–matter interactions,and can more intelligently guide molecular engineering strategies. The approach


for optimizing a molecule is to first determine the intrinsic properties that make amolecule good, then to scale it up while preserving the critical parameters.

Using this approach, making a molecule with a large hyperpolarizability (or sec-ond hyperpolarizability) starts by first identifying a molecule that has a largeintrinsic nonlinearity and then making the molecule larger to take advantageof scaling. A molecule that has a large intrinsic nonlinearity that is well abovethe intrinsic nonlinear response of most others usually represents a new mole-cular paradigm.

As an example, the series of molecules studied by Liao and associates (labeled“[2]” in Fig. 2) have been found to have larger hyperpolarizabilities than formost other molecules [80]. As such, their technological impact is clear. How-ever, given their already large size, making such molecules even larger clearlywill suffer from diminishing returns in terms of volume fraction.

Potential energy optimization studies [74,81] suggested that to improve the hy-perpolarizability, the conjugation path between the donor and acceptor shouldcontain a mixture of atoms, called modulation of conjugation by the authors.An example of a modulated conjugation path is shown in molecule P-7 inFig. 2, where molecule P-4 is the homologue without the heterocyclic bridge.The molecule with modulation of conjugation (P-7) was found to break theapparent limit, which is shown as the horizontal blue line. The apparent limitis a factor of 30 below the fundamental limit and is defined by the largestintrinsic hyperpolarizabilities observed in all molecules prior to the more recentdiscovery of better molecules.

Figure 2

Intrinsic hyperpolarizability of some representative molecules that have largenonlinear optical responses, plotted as a function of the energy differenceE10. Reproduced with permission of www.nlosource.com. The numbers inbrackets label a series of similar molecules.


Molecules with a twist in the bridge, as represented by the series labeled “[1],”have the highest intrinsic nonlinear optical response ever measured [82]. Thisclass of molecules is clearly a new paradigm that stands out from all others.Making them bigger and incorporating them in a bulk material would yieldan unparalleled nonlinear optical susceptibility.

The intrinsic nonlinear response also sheds insights into scaling of the first andsecond hyperpolarizability with length of the molecule. Slepkov and associatesmeasured the length dependence of the second hyperpolarizabilities of a series ofpolyynes, as shown in the inset of Fig. 3 [83]. Figure 3 also shows a plot of thescaling law (curve) as predicted by Eq. (57) given the number of electrons andthe measured energy difference between the ground and excited states. Notethat the theory assumes that the measured values are the off-resonance ones.Resonance enhancement may affect the results.

The fact that the scaling laws, predicted using the fundamental limits, are similarto the observed scaling law makes polyynes a promising motif for makingmolecules with ever larger second hyperpolarizability. Indeed, Biaggio and as-sociates have investigated a series of planar molecules with high densities oftriple bond conjugation as in the polyynes and also find critical scaling alongwith a large second hyperpolarizability [84,85].

Structure-property studies seek to determine how the nonlinear optical responseof a molecule depends on a parameter that may be easy to control experimen-tally. Marder and associates found that the first and second hyperpolarizabilitywere peaked functions of bond order alternation (BOA) [86,87], the differenceof bond lengths between adjacent single and double bonds in a conjugated struc-ture. As a result, this suggested that molecules synthesized to have a BOA at thepeak would have a large nonlinear optical response.

BOA has been used extensively by many researchers as an aid in designingmolecules, though it was never clear at the time if this paradigm could be usedto make molecules that reach the fundamental limit. Figure 4 shows a plot of thecalculated hyperpolarizability as a function of BOA (solid curve) for the

Figure 3

Plot of the measured second hyperpolarizability as a function of N (points) andthe scaling predicted by the fundamental limits (curve) [158,161]. Reproducedwith permission of J. Mat. Chem.


molecule shown in the inset. Figure 4 also shows a plot of the fundamental limitof the hyperpolarizability calculated from Eq. (53). The shape of this curveoriginates in a shift in the energy E10 as a function of BOA.

The fundamental limit of the hyperpolarizability peaks at a BOA of about −0.08where the actual hyperpolarizability predicted by the BOA model is small, il-lustrating that the peak hyperpolarizability at a BOA of 0.2 and −0.25 is not theglobal maximum. Furthermore, the BOA metric is not the ideal paradigm be-cause it does not follow the fundamental limit curve, and thus will not allow amaterial to be optimized to its full potential through the power of scaling. Wenote that, in principle, there is no limit to the achievable nonlinear responsethrough scaling, but other effects may limit the maximum effect length.

There are two competing effects in designing bulk materials from molecules.Molecules with larger nonlinear optical susceptibilities occupy more space,so fewer molecules will fit in a fixed volume of material. The question is whicheffect wins. In quasi-one-dimensional materials, such as the polyenes, Rustagiand Ducuing showed that the polarizability and hyperpolarizabilities scale as afunction of length according to [88]

αRD ∝ L3; γRD ∝ L5; (59)

whereas, in comparison, the number of one-dimensional molecules that can beadded to a material grows as 1∕L. Thus, increasing the size is a winning strategyto improve the nonlinearity of the molecule provided that the scaling law can bemade to hold over longer distances. Once the molecular properties are opti-mized, a bulk material made of these molecules can take advantage of the mo-lecular hyperpolarizabilities if interactions between molecules are small enoughto be taken into account using local field models.

Greene et al. showed that the nonlinear response of polydiacetylene can bemodeled on and near the one-photon resonance as an exciton that is confinedto one dimension [89]. In the crystalline material poly-[2,4-hexadiyn-1,6 diol bis(p-toluene sulfonate)] (PTS) that they characterized, the polymer length was in

Figure 4

Plot of the calculated second hyperpolarizability of the molecule in the inset as afunction of bond-order alternation.


principle infinite but the exciton length was determined to be about 3.3 nm. Thusin PTS, the effective length of the polymer chain was less than the full chainlength. The implication is that the power laws may saturate in most oligomerswhen the chain lengths become long enough.

Interestingly, the maximum scaling of a one-dimensional system [90], assumingthat the conjugated path can be modeled using particle-in-a-box states with Pauliexclusion, i.e., the technique of Kuhn [24,25], yields [90]

αscaling ∝ L3; βscaling ∝ L5; γscaling ∝ L7: (60)

Thus, the maximum power law given by simple scaling that originates from thetheory of the fundamental limits is even greater than for the polyenes. Theimplications are that it may be possible to make even better materials, but newbreakthroughs in molecular engineering may be required. In both cases, thepower laws far exceed dilution effects associated with placing large moleculesin a fixed volume.

We have provided in this section the quantum mechanical basis for calculatinghyperpolarizabilities and limits to them without much regard to the tensorialnature of these quantities and the underlying symmetries affecting them. Wenow turn our attention to understanding how spatial symmetry affects thenonlinear optical response.

3. Molecular Symmetry

As mentioned in Section 1, the presence or absence of a center of inversion canplay a crucial role in the nonlinear optical response. In particular, the even-ordernonlinear optical responses are identically zero in the electric dipole approxima-tion. This section focuses on molecules lacking an inversion center and, thus,mostly on the second-order nonlinear optical response. The susceptibilityβijk�−ω;ωp;ωq� is analyzed in terms of the irreducible third-order tensors ofthe symmetry classes, yielding insight into the various structures that can exhibitnonzero elements in βijk�−ω;ωp;ωq�. Calculations of the dispersion with fre-quency of β�−ω;ωp;ωq� based on a single excited state, i.e., a two-level model,are presented. Such a two-level model is the simplest possible for asymmetricmolecules. The frequency dispersion of γijkl�−ω;ωp;ωq;ωr� based on this modelis also discussed. The simplest model for symmetric molecules involves threestates and will be discussed in Section 4.

3.1. Selection Rules

Molecules that are centrosymmetric have spatial energy eigenfunctions uℓ�r �that are either symmetric or antisymmetric under the parity operation, oruℓ�−r � � �uℓ�r �. Spatial wave functions that exhibit no spatial symmetrycan be expressed as a linear superposition of a symmetric and antisymmetricpart. The electric dipole transition moment between states ℓ and ℓ0, definedby μℓℓ0 �

R∞−∞ uℓ �r �μ�r �uℓ0 �r �dr, will not vanish if the spatial wave functions

uℓ�r � and uℓ0 �r � of states ℓ and ℓ0 are of opposite parity by virtue of the fact thatμ�r � � er has odd parity. In general, μℓℓ0 ≠ 0 when the electronic states are ex-pressible as a superposition of even and odd parity. The case ℓ � ℓ0 gives thepermanent dipole moment in the ℓ0th state. Therefore the terms in the numeratorof Eq. (39), namely μgv, μvn, μnm, and μmg, are all nonzero.


For centrosymmetric molecules, the molecular spatial wave functions exhibiteither even (gerade, subscript g) or odd (ungerade, subscript u) symmetry withthe ground state having even symmetry. Since the electric dipole transitionsmoments are non-zero only if the states are of opposite symmetry, substitutingfor the symmetry of the wave functions as either even (g) or odd (u) into thenumerators of Eq. (39) gives μgu�μuu − μgg�μug. A similar argument can be madefor the three-level model, which shows that a nonzero octupole moment isrequired as discussed previously [91]. For centrosymmetric molecules the per-manent dipole moments and permanent octupole moments are zero and henceall χ�2� � 0.

In the preceding section, the general sum-over-states (SOS) model for thenonlinear susceptibilities of molecules was discussed. Although it is inprinciple the most accurate model available for calculating the second- andthird-order nonlinear response of molecules, frequently only a fraction ofthe information needed, such as the electronic states and the transition electricdipole moments between them, is known. One- and two-photon absorptionspectroscopy can in principle yield this information. Even if not all of thediscrete states and their transition moments can be measured with reasonableaccuracy, the spectra do identify the transitions with the largest probabilitiesthat can be used in models involving just a few states with the largest transitionmoments.

3.2. Irreducible Tensor Approach to

β MolecularNonlinear Optics

The nonlinear optical tensors for both molecules and the macroscopic mediathey comprise involve, in principle, 3n components, where n − 1 is the orderof the nonlinear optical process. Nonlinear optical interactions depend on var-ious symmetries intrinsic to the materials response tensor, but also involve theparticular experimental or thermodynamic conditions. These symmetries in-clude spatial transformations related to the molecular or material symmetry,permutation of the tensor indices, and the permutation of the frequencies in-volved in the process [69]. The difficulties with dealing with so many coef-ficients, symmetries, and possible coordinate systems can be effectivelyaddressed by applying the methods of group theory to the nonlinear opticaltensors [37,92–97]. This powerful framework provides a mechanism for(1) most efficiently representing the physical properties for a given set ofsymmetry operations, (2) providing insight into the contributions to the re-sponse to aid in the design of improved materials, (3) providing figuresof merit to compare materials and molecules of differing symmetry, and(4) conveniently connecting the molecular to the macroscopic response. Inthis section, we will provide the groundwork for this analysis and apply itto the second-order nonlinear optical response. On this basis, we can suggestmaterials approaches at the molecular and supramolecular level for efficientnonlinear optics, given the various ways to produce noncentrosymmetricmolecules and materials.

The starting point of this analysis is noting that the macroscopic and molecularnonlinear responses, as fundamental material properties, should not depend onthe coordinate frame describing them. That is, they are translationally invariantand physical properties should be invariant with respect to rotations of the


coordinate system; for example, the inner product of vectors should not dependon the choice of coordinate system.

For illustration, consider a rank 2 tensor, Tij. Any rank 2 tensor can be expressedas a direct product of two vectors, A and B:

Tij � AiBj: (61)

Equation (61) can be equivalently expressed as

Tij �A · B

3δij �

�AiBj − AjBi�2

��AiBj � AjBi�

2−

A · B

3δij

�: (62)

Equation (62), while appearing more complex, is useful because each termrepresents a particular type of symmetry.

The first term, being the trace of Tij, is invariant upon rotation, and is therefore ascalar. The second term corresponds to the three components of the cross pro-duct A × B, so represents an axial vector. As such, it is invariant under the parityoperation, i.e., it is unchanged when the coordinate frame is transformed accord-ing to x → −x, y → −y, and z → −z. Finally, the last term represents a tracelesssymmetric tensor, which is described by five independent parameters.

Recall that Tij is a 3 × 3 tensor, so if it is real, it has nine independent parameters.According to Eq. (62), it can be represented by the sum of three terms: a scalar(one parameter), a vector (three parameters), and a traceless symmetric tensor(five parameters). The angular momentum operators are the generators of rota-tions, each with a multiplicity of 2j� 1. Thus, the three terms of Eq. (62) can beassociated with the angular-momentum-like quantum numbers j � 0, j � 1,and j � 2.

In this illustration, we have represented the tensor in Cartesian form. Other formsare possible, such as the spherical tensors, which have a one-to-one correspon-dence to the spherical harmonic functions. We can express Eq. (62) in a tensor-independent form to represent the decomposition into the three terms as follows:

T �2�ij ∼ 1 ⊗ 1 ∼ 0 ⊕ 1 ⊕ 2; (63)

where 1 ⊗ 1 represents the direct product of two vectors (i.e., rank 1 tensors; thedirect tensor product results in a tensor whose rank is the sum of the ranks of thefactors, e.g., A ⊗ B � C; AiBj � Cij in Cartesian coordinates for the direct pro-duct of two vectors resulting in a second-rank tensor.) The meaning of 0 ⊕ 1 ⊕2 is that any rank 2 tensor can be represented by the sum of the three rank 2tensors of scalar (0), axial vector (1), and symmetric (2) character. In analogy tothe addition of the angular momentum of two spin 1 particles, i.e., j1 � 1 andj2 � 2, the possible results are j � jj1 − j2j to j � jj1 � j2j, or 0, 1, 2. Equa-tion (62) is the statement that any rank 2 tensor can be written in terms of threeirreducible tensors with a scalar, pseudovector, and symmetric tensor.

The susceptibility χ�1� is a second-rank tensor, so it too can be expressed in termsof the three irreducible representations. Given that χ�1�ij � χ�1�ji , the vector partvanishes, so


χ�1� ∼ 0 ⊕ 2: (64)

As a consequence, the susceptibility is independent of the polar order. The χ�1� isthus nonzero for a centrosymmetric material, originating in the scalar part “0,”and anisotropy is described by uniaxial order, which originates in the symmetricpart “2.”

The same principles apply for higher order tensors, where intuition may fail us.In these cases, each tensor is expressed in terms of irreducible representationsand symmetries that can be used to simplify the expression for the nonlinearoptical response. The formal derivations follow.

In the case of the nonlinear responses associated with molecules, unit cells, ormacroscopic materials, the higher-order tensors can be simplified using geome-trical symmetries that are summarized in the tensor forms associated with the 32point groups, well known in crystallography, and derived from group theory.These point groups are subgroups of the full orthogonal rotation group, whichis itself the direct product of the three-dimensional rotation group with the in-version group related to the symmetry axes of the system (molecule, unit cell,macroscopic material). This is well known in that the 32 point groups are com-pletely characterized only by various rotations and inversions [98]. We note thatinversion can be represented as a rotation followed by a reflection through aplane perpendicular to that axis, known as an improper rotation. A proper rota-tion is an ordinary coordinate rotation (without inversion or reflection). The Car-tesian form for the response tensor as usually tabulated is obtained by applyingthe various rotation and reflection symmetries to homogeneous polynomials ofdegree n as described, for example, by Nye [98].

Our analysis proceeds by considering the reduction of the response tensor to itsirreducible forms; these forms comprise the decomposition into a series of ten-sors of rank n and lower that do not mix under any three-dimensional rotationand thus reflect the necessary rotational invariance [92]. This is a generalizationof Eq. (61) and the discussion following. Here, we consider the tensors in theirgeneral form. An irreducible rank n tensor is labeled by its “weight” J having(2J � 1) independent components consistent with the three-dimensional rota-tion group. This formalism is reminiscent of the quantum mechanical additionof angular momenta. We also note that there may be an irreducible tensor ofweight J labeled by another parameter t. According to the scheme of angularmomenta, the product of two irreducible tensors of weight J 1 and J 2 generates aseries of tensors from jJ 1 − J 2j to jJ 1 � J 2j. Starting with the fact that the mostgeneral third-rank second-order nonlinear optical tensor defined in Eq. (45), butin spherical form, for a molecule (the first hyperpolarizability) is related to thehomogeneous polynomial presented as a direct tensor product of three vectorsdenoted each with their rank:

β ∼ 1 ⊗ 1 ⊗ 1 ∼ 1 ⊗ �0 ⊕ 1 ⊕ 2� ∼ 0 ⊕ �1 ⊕ 1 ⊕ 1� ⊕ �2 ⊕ 2� ⊕ 3: (65)

Note that, in Eq. (65), the initial transformation involving the product of twoweight 1 tensors results in the sum of irreducible tensors of weights 0, 1, and 2,in keeping with the remarks just above. Equation (65) assumes no intrinsic per-mutation symmetry and thus describes parametric light scattering or three-wavemixing in contrast with second-harmonic generation and the linear electro-opticeffect, which possess permutation symmetries brought about by degenerate


frequencies. Note that, with 2J � 1 components for each of seven tensors (1scalar, 3 vectors, 2 second-rank tensors, and 1 third-rank tensor), there are up toa total of 27 components as required for a third-order tensor, depending on thepoint group. For the cases of second-harmonic generation and the linear electro-optic effect, the hyperpolarizabilities have two indistinguishable frequencies inthe three-wave processes. In these cases, βSHG

ijk �−2ω;ω;ω� � βSHGikj �−2ω;ω;ω�

and βLEOijk �� ω; 0;ω� � βLEOjik �−ω;ω; 0�, and, for example,

βSHG ∼ 1 ⊗ �1 ⊗ 1� ∼ 1 ⊗ �0 ⊕ 2� ∼ �1 ⊕ 1� ⊕ 2 ⊕ 3: (66)

The 2J � 1 components for these four tensors yield up to 18 components,depending on the point group.WhenKleinman (full Cartesian index permutation)symmetry applies, the decomposition yields

βKS ∼ 1 ⊕ 3; (67)

yielding eight independent components comprising one vector and one third-ranktensor. The disappearance of the weight 1 irreducible tensor in the product of thetwo rank 1 tensors comes about from the permutation symmetry of second-harmonic generation. The details of the reduction in the number of irreduciblecomponents in Eqs. (66) and (67) relative to Eq. (65) will become clear whenthe Cartesian forms are discussed below.

Given that the various irreducible forms arise from proper and improper rota-tions as defined above, they can be characterized by the parity under those rota-tions. For vectors, these define polar and axial vectors, for example describingelectric and magnetic fields, respectively. In our example of a second-rank tensorof Eq. (61), the vector component is an axial vector as it is the cross product oftwo-vectors. Correspondingly, one can define true tensors and pseudotensorsrelative to their parity under improper rotations. For the former, parity is givenby π � �−1�n and, for the latter, π � �−1�n�1. For the irreducible tensors ofweight J derived from a rank n tensor, π � �−1�J describes an irreducible truetensor, and π � �−1�J�1 an irreducible pseudotensor. Thus, when reducing atrue (or pseudo) Cartesian tensor, the irreducible parts with �n� J � even fortensors (or odd for pseudo-tensors) are true tensors, and those with �n� J �odd for tensors (or even for pseudo-tensors) are pseudo-tensors. Thus, in thereduction spectrum defined in Eq. (66) for second-harmonic generation above,the 0 weight component is a pseudo-scalar and the 2 weight component is apseudo-tensor. In physical terms, for example, this implies that in Eqs. (65)and (66), for a general three-wave mixing, the presence (Eq. (65)) and absence(Eq. (66)) of the 0 weight pseudoscalar implies that a liquid lacking an inversioncenter (e.g., chiral) will exhibit such wave mixing, while second harmonic willnot be observed.

Insight into the structure of these tensors can be obtained by consideringthe decomposition of the Cartesian forms of these tensors as carried out inAppendix A with results given below [94,95]. We will seek the reduction spec-trum implied by Eqs. (65)–(67), where the irreducible forms are labeled by therank n � 3, and the weights J . Note that there are may be more than one tensorof a given weight, which can be labeled with another parameter, but we will findit convenient to label them by their properties under permutations of their indicesas will also be described in Appendix A.


Before we describe the application of the formalism of Appendix A in the im-portant case of second-harmonic generation, we can now point out how thispicture leads to useful figures of merit to analyze the nonlinear optical responsecorresponding to cases involving various symmetries, both intrinsic materialsymmetries and symmetries corresponding to a particular nonlinear optical pro-cess. This is done by considering the norm of the nonlinear optical tensor asdescribed by Eqs. (A2) and (A3), that is, as the sum of the irreducible compo-nents embedded in a third-rank tensor space. One can show that the tensors ofweight J span an n-dimensional orthogonal space, so that the norm of the sum ofthe tensors of weight J is equal to the sum of the norms yielding a generalizedPythagorean Theorem [92]. Thus,

‖βijk‖2 �XJ ;m

‖β�J ;m�ijk ‖2: (68)

This sum will contain contributions from each value for J and m, as well as“interference” terms given by the generalized inner products of pairs of irredu-

cible tensors of the same weight but differing m. The quantities ‖β�J ;m�ijk ‖ and the

inner products β�J ;m�ijk · β�J ;m0�

ijk are invariants and are figures of merit that can be

measured by using hyper-Rayleigh scattering. Consequently, these figures ofmerit are useful for characterizing molecular hyperpolarizability componentsthat directly relate to both the molecular response and the supramolecular(macroscopic) response as the irreducible components do not mix when per-forming the orientational average in relating the molecular to supramolecularresponse using an oriented gas model. Descriptions of the techniques for mea-suring these figures of merit for second-harmonic generation have beendescribed in the literature in both the Kleinman symmetric [37,38,99] andKleinman disallowed cases [39,40,100]. We can now describe how this form-alism provides insight into contributions to the nonlinear optical response at boththe molecular and supramolecular levels, which we examine in the case ofsecond-harmonic generation. As both the linear electro-optic effect and second-harmonic generation share a common index pair permutation symmetry due to apair of degenerate electromagnetic field frequencies, our analysis pertainsto both.

We begin by applying the permutation projection of Eq. (A4) in Appendix Ato the second-order nonlinear optical tensor expressed as a sum of the irre-ducible components embedded in the rank 3 tensor space given in Eqs. (A6)and (A7), yielding forms convenient for analysis in the case of a pair of de-generate frequencies. Equation (A5) applies to the general second-order pro-cess [94,97]. We now consider the sum of Eq. (66) relevant for secondharmonic and the linear electro-optic effect with the permutation projection.This yields

βijk � β�3s�ijk � β�2m�ijk � β�1s�ijk � β�1m�ijk : (69)

The indices s and m describe the index permutation symmetry for the fullysymmetric and mixed symmetry cases, respectively, as described in Appen-dix A. The absence of a and m0 components reflects the index pair permuta-tion symmetry. This is the most convenient way to differentiate the two J � 1

irreducible tensors. The embedded forms are given by


β�1s�ijk � 1

5�β�1s�i δjk � β�1s�j δik � β�1s�k δij�;

β�1m�ijk � 1

5�2β�1m�i δjk − β�1m�j δik − β�1m�k δij�;

β�2m�ijk � 1

5�2εijlβ�2m�ik � β�2m�il εljk �. (70)

The octupolar component (3s) is not shown but is a fully symmetric tracelessthird-rank tensor obtained by symmetrizing the hyperpolarizability and subtract-ing the Kleinman symmetric part containing traces. The embedded irreducibleforms in Eq. (70) are given by

β�1s�i � 1

3�βijkδjk � βjikδjk � βjkiδjk �;

β�1m�i � 1

3�2βijkδjk − βjikδjk − βjkiδjk �;

β�2m�ij � 1

2�εiklβklj � εjklβkli�;

β�3s�ijk � 1

3�βijk � βjki � βkij� − β�1s�ijk . (71)

The SOS expressions of the irreducible tensors in Eqs. (71) are the essentialquantities to be analyzed and whose norms of the embedded forms of Eqs. (70)are the figures of merit measured in hyper-Rayleigh scattering. We can use theseequations along with the SOS expressions for βijk introduced in Section 2 togarner insight into the origin of the response and to guide the design of molecularmaterials.

The irreducible representations of the second-harmonic hyperpolarizability(ignoring losses) are given by

β�1s�i � 1

ℏ2

�Xn≠g

ω2ng

�ω2ng−ω

2��ω2ng−4ω

2��Δμingjμgnj2�2μign�Δμng ·μgn��

−

Xn≠g

m≠n≠g

�2ω4�ω2

mg−4ωmgωng�ω2ng��ω2ωmgωng�3ω2

mg−ωmgωng�3ω2ng�−ω3

mgω3ng

�ω2ng−ω

2��ω2ng−4ω

2��ω2mg−ω

2��ω2mg−4ω

2�

×�μinm��μgn ·μgm��2μign�μnm ·μgm��

; (72)

β�1m�i � 2ω2

ℏ2

�Xn≠g

2

�ω2ng −ω2��ω2

ng − 4ω2� �μgn × �μgn ×Δμng��i

�Xn≠g

m≠n≠g

�8ω4

− 2ω2�3ω2mg � 2ωmgωng� −ωmgωng�ω2

mg −ωmgωng − 2ω2ng�

�ω2ng −ω2��ω2

ng − 4ω2��ω2mg −ω2��ω2

mg − 4ω2�

× �μgn × �μgn ×Δμng��i��

; (73a)


β�2m�ij � 3ω2

4ℏ2

(Xn≠g

4

�ω2ng − ω2��ω2

ng − 4ω2� �μgn × Δμng�iμjgn

�Xn≠g

m≠n≠g

"2�2ω2

− ωmgωng − ω2ng�

�ω2ng − ω2��ω2

ng − 4ω2��ω2mg − ω2� �μnm × μgn�iμjgm

−

�2ω2 � ωmgωng��ω2mg − ω2

ng�2�ω2

ng − ω2��ω2ng − 4ω2��ω2

mg − ω2��ω2mg − 4ω2�

× �μgm × μgn�iμjnm#� i↔j

); (73b)

β�3s�ijk � 1

ℏ2

�Xn≠g

ω2ng

�ω2ng−ω

2��ω2ng−4ω

2�

×

�1

2Ps�Δμingμjgnμkgn�−

1

5�jμ2gnj�Δμingδjk�Δμjngδik�Δμkngδij�

−2�Δμng·μgn��Δμignδjk�Δμjgnδik�Δμkgnδij��

×Xn≠g

m≠n≠g

2ω4�ω2mg−4ωmgωng�ω2

ng��ω2ωmgωng�3ω2mg−ωmgωng�3ω2

ng�−ω3mgω

3ng

�ω2ng−ω

2��ω2ng−4ω

2��ω2mg−ω

2��ω2mg−4ω

2�

×

�1

2Ps�μignμjnmμkgm�−

1

5��μnm·μgm��μinmδjk�μjnmδik�μknmδij�

�2�μnm·μgm��μignδjk�μjgnδik�μkgnδij��

; (74)

where Δμng � μnn − μgg denotes the change in dipole moment between the twostates n and g, and Ps is the fully symmetric permutation operator interchangingi, j, and k. Equations (72) through (74) are displayed so that the first term corre-sponds to the two-level expression involving the ground state and single excitedstates.

We now describe the physical implications of these hyperpolarizabilities. First,we recall that all components require an absence of inversion symmetry. The 1scomponent is a fully symmetric tensor of rank 1, or, in other words, a polar vector,and it transforms in that manner. Spatially and mathematically it is described as avector, often associated with a one-dimensional molecule. This is by far the mostwidely appreciated contribution to the hyperpolarizability as it is the componentdirectlymeasured in electric-field-induced second-harmonic generation (EFISH),and the one most often described in relation to organic second-order nonlinearoptical materials. It is the component that contributes to the nonlinear optical re-sponse in poled polymer materials and so called one-dimensional materials. Thispart is described thoroughly in the next section. The other symmetric componentis the 3s component, which is also widely known as the octupolar component ofthe hyperpolarizability [37]. This component is also noncentrosymmetric, but,instead requires a two-dimensional or three-dimensional response, associatedwith the prototypical structures shown in Fig. 5.

Kleinman or full-permutation symmetry is reflected in the irreducible represen-tations, as shown in Eq. (67), where only the contributions given in Eqs. (72)


and (74) will be nonzero. This is consistent with our expression of the irreducibletensor components in terms of those of the permutation operations. That is, sinceKleinman symmetric components must be fully symmetric, those representa-tions in Eqs. (73a) and (73b) of mixed symmetry are identically zero. Theconditions for Kleinman symmetry center on thermodynamic arguments imply-ing the simultaneous full permutation symmetry of both Cartesian indices andfrequencies [69] for the special case of all frequencies far from resonance. In thiscase, dispersion of the response is negligible and permutation of the Cartesianindices is decoupled from permutation of the frequencies. It has recently beenpointed out that in the realm of molecular nonlinear optics, this conditiondoes not often apply [101]. Given the typical absorption frequencies and non-linear interaction frequencies, especially in organic chromophores, the far-from-resonance condition generally is not satisfied, and, thus, Kleinmansymmetry is often broken.

However, a symmetry equivalent to Kleinman symmetry can arise from spatialsymmetries in the nonlinear optical medium. In particular, in the case of the one-dimensional molecule, only one component of the hyperpolarizability tensor isnonzero, and Kleinman symmetry trivially and necessarily holds regardless offrequency. Thus, for molecular nonlinear optics, the extent that Kleinman sym-metry does not hold is usually a measure of the departure of the molecule fromone-dimensionality. In the case, of polar nonlinear optics described above andthe next section, it is often the case that nearly one-dimensional molecules areemployed due to their large vector component of the hyperpolarizability. Ofcourse, for octupolar nonlinear optics, a multidimensional molecule is requiredto reflect that symmetry, so Kleinman symmetry would not be expected and themixed symmetry components will contribute to an extent consistent with theparticular molecular structure and thermodynamic considerations. In the caseof other molecular symmetries of higher dimension, the expressions for theirreducible hyperpolarizability components provide insight.

These insights can be gleaned from the forms of the transition dipole vectors inEqs. (72)–(74). In the 1s and 3s components, only dot products of the dipolevectors appear, while in the 1m and 2m components, cross products appear.Thus, these forms imply a requirement for multidimensional molecules in orderfor the 1m and 2m components to be nonzero, the same condition for Kleinmansymmetry breaking. It also explains why one-dimensional molecules are favoredin the polar case (1s) since the dot product is most easily maximized in that case.

Figure 5

Prototypical structures having octupolar symmetry in (a) two and (b) threedimensions. The colors/shapes represent distinct chemical moieties.


The octupolar case is more complex, but is maximized under threefold rotationsymmetry in two dimensions [102].

We now consider the example of multidimensional molecules of symmetry C2v,which, as noted above are generally Kleinman nonsymmetric for organic chro-mophores [94,103–105]. We start by considering the extent of summationneeded in the SOS expressions. As we discuss below, restriction of the sumto the ground and one excited states has been shown successful for understand-ing one-dimensional push–pull chromophores. This is inadequate when intramo-lecular charge transfer in multiple dimensions contributes to the nonlinearoptical response. This has been established both heuristically and experimen-tally, through measurements of the dispersion of the nonlinear optical response[91,94]. In addition, when complex multidimensional molecules are considered,the spatial symmetry can dictate state degeneracies as is the case withoctupolar molecules [37].

The C2v symmetry case will require at least three levels, but considerable insightcan be obtained by considering the response in light of our analysis above. Thisanalysis indicates that, for the case of C2v symmetry, transition dipoles are par-allel and perpendicular to the symmetry axis, and consequently Δμ will be mostimportant. This can be analyzed best in terms of group theory as above. If themolecule is invariant under certain symmetry operations, the Hamiltonian willcommute with the group elements so that the quantum states are eigenfunctionsbelonging to that group. The ground state belongs to the fully symmetric (trivial)representation (A1). The irreducible representations corresponding to the excitedstates will define the nature of the transitions involved. Consider the charactertable for C2v symmetry shown as Table 1.

Table 1 is interpreted as follows. The columns are the symmetry operations:identity (360° rotation), twofold rotation (180° rotation), mirror x–z plane, andmirror y–z plane. The rows are symbols for the irreducible representations.The entries in the table are known as characters, and are defined as the traceof the transformation matrix representing that symmetry operation. The char-acters in this case are either symmetric (�1) or antisymmetric (−1) under thesymmetry operations as noted for each representation. These characters canbe understood by the rotation in Hilbert space reflecting the twofoldsymmetry of the molecules. The eigenvalues of an n-fold rotation are givenby Rmjψi � exp�2πin∕m�jψi where R is the operator for a rotation by 2π∕m.So, the wave function accumulates a phase (for n � 1) of exp�2πi∕2� � −1.Similarly, a molecule with a mirror plane will also acquire a phase ofexp�2πi∕2� � −1 upon reflection since the reflection operator σ has the prop-erty that σ2 � 1.

Thus the A representations are symmetric under onefold and twofold rotations,and the B representations antisymmetric under those rotations. This can be

Table 1. Character Table for C2v Symmetry

E C2 σx σy

A1 1 1 1 1A2 1 1 −1 −1

B1 1 −1 1 −1

B2 1 −1 −1 1


understood by considering the rotation of the transition dipole moment, where acomponent of a dipole moment parallel to the symmetry axis z is invariant undertwofold rotation (A-type), whereas components perpendicular to z change sign(B-type) under 180° rotation. The two mutually exclusive cases for the transitionmatrix dipole have an excited state with A character with transition moment par-allel to z, and an excited state of B character with transition moment perpendi-cular to z. This implies that a two-level expression involving only the ground andfirst excited molecular electronic states can either describe a diagonal Cartesiancomponent (βzzz) with an A state, or an off-diagonal �βxxz; βyyz� component with aB state, but not both. This explains why a two-level model can apply chromo-phores whose nonlinear response involves only a single direction, describinglinear, one-dimensional chromophores, but not to a twofold symmetry (two-dimensional) molecule. We note that a two-state model might describe certainlow-symmetry, two-dimensional chromophores.

This analysis can now be combined with Eqs. (72)–(74) by considering thecross products of the transition moments with the dipole moment changes(Δμ). By symmetry the Δμ must be aligned with the symmetry axis. This im-plies that the Kleinman nonsymmetric components 1m and 2m must have tran-sition moments perpendicular to the symmetry axis due to the cross products,while the Kleinman symmetric 1s and 3s components must have transitionmoments along the symmetry axis. Thus, A states contribute to the 1s and3s components, while B states will contribute to the 1m and 2m components.As we will show in Section 5.2, the molecular alignment scheme to produce anon-centrosymmetric bulk medium will determine whether A or B states areimportant, thus confirming that this analysis has provided a method for design-ing molecules to best optimize the molecular and macroscopic nonlinear opticalresponse.

3.3. Two-Level Model

Based on Eq. (39), only one excited state and a permanent dipole moment aresufficient to produce a nonzero χ�2� so that a two-level model is the simplestmodel possible; see Fig. 6. Despite its simplicity, this model has proven veryvaluable for understanding general trends of susceptibilities with frequency fornoncentrosymmetric molecules. It also provides a good representation of chargetransfer molecules [11].

3.3a. Two-Level Model: χ�2�

For two states, the ground state and one excited state, the SOS expressionbecomes

Figure 6

(a) (b)

Two-level model for calculating (a) χ�2� and (b) χ�3� for noncentrosymmetricmolecules.


χ�2�ijk �−ω;ωP;ωq� �N

ε0ℏ2L�ωp�L�ωq�L�ω�

�μ01;i�μ11;k −μ00;k�μ10;j

�ω10 −ωp −ωq��ω10 −ωp�

� μ01;k�μ11;j −μ00;j�μ10;i�ω

10�ωq��ω10�ωp�ωq�

� μ01;k�μ11;i −μ00;i�μ10;j�ω

10�ωq��ω10 −ωp�

�: (75)

The permanent dipole moments in the ground and excited states are written asμ00 and μ11, respectively, and the transition dipole moment is μ10 � μ01:

The two-level model provides some insight into the frequency dispersion of thesecond-order susceptibility. In order to avoid efficiency limiting losses at thefundamental and harmonic frequencies, in the specific examples discussed next,the input frequency is chosen to be in the off-resonance or non-resonant regimes.In the off-resonance case, the τ−110 part of ω10 in the denominators can be ne-glected, i.e., ω10 is real. For the non-resonance regime, all input frequenciesare set to zero.

Assume that periodically poled lithium niobate (PPLN), for example, can beusefully described by a two-level model. The dominant second-order nonlinear-ity lies along the z axis [106]. For this case, the off-resonance result in the two-level model is

χ�2�zzz �−2ω;ω;ω� � N

ε0ℏ2L2�ω�L�2ω�jμ10j2�μ11;z − μ00;z�

3ω210

�ω210 − ω2��ω2

10 − 4ω2� :

(76)

It is useful to compare this result with that obtained using the anharmonic os-cillator model, which can be found in any textbook [7,11]. That result away fromresonance is

χ�2�zzz�−2ω;ω;ω� � Ne3

ε0m3e

k�2�zzz

�ω210 − 4ω2��ω2

10 − ω2�2 : (77)

Not only can Eq. (77) not yield specific values because there is no method tocalculate the nonlinear force constant k�2�zzz , but it also predicts a stronger reso-nance at the fundamental frequency �ω2

10 − ω2�−2 than at the second harmonic�ω2

10 − 4ω2�−1. Unfortunately, there are no measurements of the dispersion of thenonlinearity over a sufficiently wide spectral range to make a useful comparisonbetween experiment and theory.

Another example for comparison with the anharmonic oscillator model is sumfrequency generation for which there are two input frequencies, namely ω1 andω2. The SOS result is

χ�2�zzz �−�ω2 � ω1�;ω1;ω2� � χ�2�zzz �−�ω2 � ω1�;ω2;ω1�

� ω210N

ℏ2L�ω1�L�ω2�L�ω2 � ω1�

× jμ01;zj2�μ11;z − μ00;z�ω210�3ω2

10 − �ω1 � ω2�2 � ω1ω2��ω2

10 − �ω1 � ω2�2��ω210 � ω2

1��ω210 � ω2

2�: (78)

Again, the frequency dispersion is different from the anharmonic oscillator resultwhich has no frequency dependence in the numerator as in Eq. (78), i.e.,


χ�2�zzz�−�ω2 � ω1�;ω1;ω2� � χ�2�zzz�−�ω2 � ω1�;ω2;ω1�

� 2Ne3

ε0m3e

k�2�zzz

�ω210 − �ω1 � ω2�2��ω2

10 − ω21��ω2

10 − ω22�: (79)

One can conclude that the anharmonic oscillator model, although widely used, isnot strictly correct. Nor should one have expected it to be, since it is not based ona physical model for a molecule.

The electro-optical response of a material is another manifestation of a χ�2�

process, namely Realfχ�2�ijk �−ω;ω; 0� � χ�2�ijk �−ω; 0;ω�g. For example, for allfields polarized along the x axis and far enough away from the resonancesso that the relaxation time can be neglected, in the two-level approximation,

Realfχ�2�xxx�−ω; 0;ω� � χ�2�xxx�−ω;ω;0�g � N

ε0ℏ2L2�ω�L�0�fμ10;x�μ11;x − μ00;x�μ01;xg

×2�3ω2

10 −ω2�

�ω210 −ω

2�2 : (80)

3.3b. Two-Level Model: χ�3�

The frequency dispersion of χ�3� and the sign of the non-resonant nonlinearityhave been a source of speculation since the early days of nonlinear optics. Thetwo-level model can be used to evaluate the third-order nonlinearity in a firstapproximation for molecules that have permanent dipole moments. FromEq. (42), the third-order susceptibility is

χ�3�ijkl�−�ωp � ωq � ωr�;ωp;ωq;ωr�

� N

ε0ℏ3L�ωp�L�ωq�L�ωr�L�ωp � ωq � ωr�

×

��μ01;i�μ11;l − μ00;l��μ11;k − μ00;k�μ10;j

�ω10 − ωp − ωq − ωr��ω10 − ωq − ωp��ω10 − ωp�

� μ01;j�μ11;k − μ00;k��μ11;i − μ00;i�μ10;l�ω

10 � ωp��ω10 � ωq � ωp��ω10 − ωr�

� μ01;l�μ11;i − μ00;i��μ11;k − μ00;k�μ10;j�ω

10 � ωr��ω10 − ωq − ωp��ω10 − ωp�

� μ01;j�μ11;k − μ00;k��μ11;l − μ00;l�μ10;i�ω

10 � ωp��ω10 � ωq � ωp��ω

10 � ωp � ωq � ωr�

�

−

�μ01;iμ01;lμ01;kμ01;j

�ω10 − ωp − ωq − ωr��ω10 − ωr��ω10 − ωp�� μ01;iμ01;lμ01;kμ01;j

�ω10 � ωq��ω10 − ωr��ω10 − ωp�

� μ01;lμ01;iμ01;jμ01;k�ω

10 � ωr��ω10 � ωp��ω10 − ωq�

� μ01;lμ01;iμ01;jμ01;k�ω

10 � ωr��ω10 � ωp��ω

10 � ωp � ωq � ωr�

��: (81)


The terms in the first summation correspond to single transitions between theexcited state and the ground state and a second transition involving the change inthe permanent dipole moments. The second set of terms involve two successiveone-photon transitions to the excited state and back to the ground state.

The simplest example of a third-order effect is third-harmonic generation with asingle z-polarized input and output beam:

χ�3�zzzz�−3ω;ω;ω;ω� � N

ε0ℏ3L3�ω�L�3ω�

�� μ11;z − μ00;z�2jμ10;zj2�ω10 − 3ω��ω10 − 2ω��ω10 − ω�

� �μ11;z − μ00;z�2jμ10;zj2�ω

10 � ω��ω10 � 2ω��ω10 − ω�

� �μ11;z − μ00;z�2jμ10;zj2�ω

10 � ω��ω10 − 2ω��ω10 − ω�

� �μ11;z − μ00;z�2jμ10;zj2�ω

10 � ω��ω10 � 2ω��ω

10 � 3ω�

�

−

� jμ10;zj4�ω10 − 3ω��ω10 − ω��ω10 − ω�

� jμ10;zj4�ω

10 � ω��ω10 − ω��ω10 − ω�

� jμ10;zj4�ω

10 � ω��ω10 � ω��ω10 − ω�

� jμ10;zj4�ω

10 � ω��ω10 � ω��ω

10 � 3ωr�

��: (82)

For materials in which a two-level system would be valid, it is evident fromEq. (82) that third-harmonic resonance peaks occur for 3ω � ω10, 2ω � ω10,and ω � ω10.

There are no symmetry restrictions on nonlinear refraction and absorption sincethey are χ�3� processes. These phenomena occur in all materials. The startingpoint for the two-level analysis is Eq. (42). As shown in Fig. 7, there are threeχ�3�, each corresponding to a different ordering of the frequencies �ω;ω;−ω� thatcontribute. The sum of the three is the physically relevant quantity with the pos-sibility of strong interferences between the contributing terms. Cases I and II gothrough the ground state with a DC response in an intermediate step, whereasCase III has a two-photon resonance.

Figure 7

|g> -

),,;()3( −−xxxx

Case II:

|g>

-

),,;()3( −−xxxxCase III:

|g>

),,;()3( −−xxxx

Case I:

ω

ω

ω ω

ω ω ω ω

ω ω

ω ω ωχ

χ

χ

ωω

ω ω

ω

The three χ�3� contributions to nonlinear absorption and refraction. The upwardarrows correspond to absorption and the downward ones to emission.


Substituting for the ground state and the excited state with x-polarized light [11],Case I: χ�3�xxxx�−ω;ω;−ω;ω�

χ�3�xxxx�−ω;ω;−ω;ω� � N

ε0ℏ3L3�ω�L�−ω�

�jμ10j2�μ11 − μ00�2

�1

�ω10 − ω�2ω10

� 1

�ω10 � ω�ω10�ω10 − ω� �

1

�ω10 � ω�ω

10�ω10 − ω�

� 1

�ω10 � ω�2ω

10

�

− jμ01j4�

1

�ω10 − ω�3 �1

�ω10 − ω��ω10 − ω�2

� 1

�ω10 � ω�2�ω10 � ω� �

1

�ω10 � ω�3

��: (83)

Case II: χ�3�xxxx�−ω;−ω;ω;ω�

χ�3�xxxx�−ω;−ω;ω;ω�� N


�jμ10j2�μ11 −μ00�2

�1

�ω210 −ω

2�ω10

� 1

�ω10�ω�ω10�ω10�ω��

1

�ω10 −ω�ω

10�ω10 −ω�

� 1

�ω210 −ω

2�ω10

�

− jμ01j4�

1

�ω210 −ω

2�

�1

�ω10 −ω�� 1

�ω10�ω�

�

� 1

�ω210 −ω

2�

�1

�ω10 −ω�� 1

�ω10�ω�

��. (84)

Case III: χ�3�xxxx�−ω;ω;ω;−ω�

χ�3�xxxx�−ω;ω;ω;−ω� � N


�jμ10j2�μ11 − μ00�2

×

�1

�ω10 − 2ω��ω10 − ω�

�1

ω10 − ω� 1

�ω10 − ω�

�

� 1

�ω10 � 2ω��ω

10 � ω�

�1

�ω10 � ω� �1

�ω10 � ω�

��

− jμ01j4�

1

�ω210 − ω2�

�1

�ω10 − ω� �1

�ω10 � ω�

�

� 1

�ω210 − ω2�

�1

�ω10 − ω� �1

�ω10 � ω�

��: (85)

This last case (Case III) is the only one that gives rise to a two-photon resonancepeak. As a result, all the terms proportional to �μ11 − μ00�2 are labeled as two-photon contributions. In the two-level model, two-photon absorption requires amolecule with a permanent dipole moment. We will see later that the three-levelmodel has a two-photon transition even when the dipole moment vanishes.


Although the detailed frequency dispersion depends on the relative magnitude of�μ11 − μ00�2 and jμ10j2, it is useful to examine the typical frequency dependenceof the different contributions to the total third-order susceptibility in the specialcase where jμ10j2 � �μ11 − μ00�2, as shown in Fig. 8 [11]. The one-photon con-tributions (∝ jμ10j4), both real and imaginary, are negative at all frequencies.However the two-photon contributions (∝ jμ10j2�μ11 − μ00�2) can be eitherpositive or negative, depending on the frequency. Between ω � ω10 and thetwo-photon dispersion resonance at ω � ω10∕2, Real�χ�3�� χ�3�R is negativeand after the resonance it is positive all the way to the non-resonant limitω � 0. The two-photon imaginary component �χ�3�I � starts out negative atω � ω10, changes sign before it reaches the two-photon peak at ω � ω10∕2,and remains positive out to the non-resonant limit, where it falls to zero. Whetherthe non-resonant real value is positive or negative for the sum of the two con-tributions depends on which process, i.e., one- or two-photon transitions, dom-inates. Note that, in Fig. 8, the real part of the total susceptibility goes to zero inthe non-resonant limit since �μ11 − μ00�2 � jμ10j2 is assumed. As will be shownanalytically later, the sign of the real component of the non-resonant nonlinearityis determined by the sign of �μ11 − μ00�2 − jμ10j2.The detailed formulas are complicated [11]. It is instructive here to examineapproximate formulas that are valid in each of the four frequency regimes de-fined below, namely, near the one- and two-photon resonances, off resonanceand non-resonant [11,107]:

1. On and near resonance

a. one-photon resonance (jω − ω10jτ10 ≤ 5) and

b. two-photon resonance (j2ω − ω10jτ10 ≤ 5)

Figure 8

Generic dependence on normalized frequency of the real and imaginary com-ponents in arbitrary units of the one- and two-photon terms of the third-ordersusceptibility in the two-level model. The blue curves are for the total of the one-photon terms (∝ jμ10j4) and the red curves are for the total two-photon terms�∝ jμ10j2�μ11 − μ00�2�. The regions of positive and negative suscepti-bility are identified. The upper curves show the dispersion of the two-photonresonance terms on a linear scale.


For near and on the one-photon resonance, i.e., ω ≈ ω10,

χ�3�xxxx�−ω;ω;ω;−ω� � χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;−ω;ω;ω�

� −

2N

ε0ℏ3L3�ω�L�−ω� ×

� �ω10 − ω��ω10 − ω�2 � τ−210 �2

×

�jμ10j2�μ11 − μ00�2

τ−210ω210

� jμ10j4�ω10 − ω�2

��ω10 − ω�2 � τ−210 �

�

− iτ−110�ω10 − ω�2

��ω10 − ω�2 � τ−210 �2�29

6jμ10j2�μ11 − μ00�2

1

ω210

− jμ10j22

��ω10 − ω�2 � τ−210 �

��. (86)

Note that cancellation effects between the different contributing terms occur inthe two-photon contributions [∝ �μ11 − μ00�2], causing the real leading term tobe proportional to τ−210 at resonance. As a result, the total response is dominatedby the triply resonant terms in χ�3�xxxx�−ω;ω;−ω;ω� ∝ μ410 associated with Case Iunless the permanent dipole moment differences are unphysically large. There-fore, the predictions of this model are that searching for materials with largepermanent dipole moments is not expected to produce large on-resonancethird-order nonlinearities.

However, near the two-photon resonance, i.e., 2ω ≈ ω10, only thejμ10j2�μ11 − μ00�2 terms in χ�3�xxxx�−ω;ω;ω;−ω� are enhanced and they dominatethe nonlinear response, i.e.,

χ�3�xxxx�−ω;ω;ω;−ω� ≅ 8N

εℏ3L4�ω�jμ10j2�μ11 − μ00�2

�ω10 − 2ω� � iτ−110ω210��ω10 − 2ω�2 � τ−210 �

. (87)

2. Off-resonance (jω − ω10jτ10 > 5 and j2ω − ω10jτ10 > 5)

In this region, the damping term in the resonance denominators can be ignored,greatly simplifying the analysis. The result is

χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;ω;ω − ω� � χ�3�xxxx�−ω;−ω;ω;ω�

� 4N

εℏ3L3�ω�L�−ω�

��jμ10j2�μ11 − μ00�23ω10

�1

�ω210 − 4ω2��ω2

10 − ω2�

� iωτ−110�5ω2

10 − 8ω2��ω2

10 − 4ω2�2�ω210 − ω2�2

�

− jμ10j4ω10�3ω210 � ω2� �ω

210 − ω2� � 4iτ−110ω

�ω210 − ω2�4

��. (88)

These formulas are essentially valid for the “tails” of the response on both thelow- and high-frequency sides of the one- and two-photon resonances, andthe region between them. It is important to note that the imaginary componentof the third-order susceptibility is proportional to the product of the frequency


and the inverse of the excited state lifetime and goes to zero in the Kleinmanlimit. Note also that the real part of the nonlinearity for frequencies below thetwo-photon resonance can be positive if �μ11 − μ00�2 can be tuned independentlyof jμ10j2, as appears to be possible for chromophores based on charge transferstates [84,85].

3. Non-resonant (ω → 0)

Mathematically, the non-resonant limit corresponds to ω210 ≫ ω2. The relevant

physics, as shown in Fig. 9, is that all the transitions are essentially very closeto the ground state and all that remains in the denominators is the transitionfrequency. As a result, the third-order susceptibility reaches a constant value.That is, all terms in the summation proportional to jμ10j2�μ11 − μ00�2 contributeequally. Similarly, all the terms proportional to jμ10j4 also contribute equally.As a result, the relative contribution due to the permanent dipole momentsis orders of magnitude larger here than in the near- and on-resonance case.Here,

χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;ω;ω;−ω�

� χ�3�xxxx�−ω;−ω;ω;ω�!ω→0 12N

ε0ℏ3L4�0� jμ10j

2

ω310

f�μ11 − μ00�2 − jμ10j2g.

(89)

Note that, if there are any low-frequency dielectric processes present, the limitω → 0 refers to frequencies far below the electronic resonances and far abovethe inverse of the dielectric relaxation times. Equation (89) indicates that the twocontributions interfere, which can result in a reduced nonlinearity for molecules.Typically molecules optimized for a large χ�2� will exhibit a positive non-resonant n2. Note that, as ω → 0, Imag�χ�3�xxxx ∝ ω → 0�.

3.3c. First-Order Effect on χ�3� of Population Changes in Two-LevelSystems

It was assumed at the outset of the SOS derivation that initially all the electronswere in the ground state and excited state populations were neglected. In fact, itis commonly believed that, for low input intensities, the nonlinear contributionto any population produced in the excited state is negligible compared to

Figure 9

n=m ν= =|1>

g

ω

ω

ω

=|0>

1

10t≈ ∆t≈ ∆

t≈ ∆

1

10

1

10

Non-resonant case for the interaction of low-energy photons with the two-levelsystem. Reproduced with permission of John Wiley and Sons publishers [10].


the Kerr effect just discussed. We now show that this is not necessarily thecase.

The probability for an absorption process to occur is proportional to the popula-tion difference between the excited and ground states. As shown in Fig. 10,when a photon of frequency ω ≈ ω10 is incident on a two-level system with allof its electrons initially in the ground state, it can be absorbed with a probabilityproportional to the intensity, thus raising one electron (per absorbed photon) tothe excited state via stimulated absorption. Furthermore there is stimulated emis-sion, proportional to the intensity, by which excited electrons are returned to theground state. As discussed previously, there is also spontaneous emissionquantified by the natural lifetime which governs the decay of the excited stateelectrons back to the ground state.

Absorption and emission lead to changes in the population (number density) ofthe states. Defining the initial (total) electron density as N , N0 as in the groundstate, and N1 as in the excited state density, ΔN � N0 − N 1 and N � N0 � N1.For intensities well below the saturation intensity I sat (ω), the usual steady-staterate equations give, for the first-order susceptibility [11],

χ�1�ii �−ω;ω� � NL�ω�1� I�ω�∕I sat�ω�

jμ10;ij2ℏε0

2ω10

×

�ω210 − ω2 � τ−210 � 2iτ−110ω

��ω10 − ω�2 � τ−210 ��ω10 � ω�2 � τ−210 �

�: (90)

For small intensities, i.e., I�ω� ≪ I sat�ω�, �1�I�ω�∕I sat�ω��−1≅1−I�ω�∕I sat�ω�.The contribution due to the intensity can be written as [11]

χ�3�eff �−ω;ω;−ω;ω� � −16NL4�ω�jμ10j4ω2

10ω

n�ω�ε0cℏ3

×

�ω210 − ω2 � τ−210 � 2iτ−110ω

��ω10 − ω�2 � τ−210 �2��ω10 � ω�2 � τ−210 �2�: (91)

The real and imaginary contributions to the effective susceptibility off resonancedecrease with decreasing ω and ω2, respectively, because the linear absorptionresponsible for this contribution goes to zero in this limit and hence they are bothzero in the non-resonant limit.

Figure 10

1N

0N 0Nτ

1NBIBI

I

II

III

10

Nωh

(a) Two-level system with all electrons initially in the ground state. (b) A singleincoming photon is absorbed and an electron is raised to the excited state. (c) Si-tuation after many photons have been absorbed. Process I refers to stimulatedemission, II refers to stimulated absorption, and III to spontaneous emission.Reproduced with permission of John Wiley and Sons [10].


The relative contribution to the total third-order susceptibility χ�3�xxxx�−ω;ω;−ω;ω�due to this population effect needs to be included in certain limits, for example, forcw inputs or for pulses with pulse width Δt ≫ τ10 in which there is a steady statepopulation in the excited state. For pulses τ10 ≫ Δt the excited state populationwill typically be very small and can be neglected. For intermediate pulse widths,the evolution in time of the excited state population must be taken into account.For the cw and long pulse cases, the frequency responses of the real and imaginarycomponents of the total χ�3�xxxx�−ω;ω;−ω;ω� are shown in Fig. 11 as the dashedcurves. For comparable one and two-photon contributions, the population effectscomplicate further the frequency spectrum. For example, the cancellation of thetotal nonlinearity at �ω10 − ω�∕ω10 � 0.64 due to interference between the Kerrelectronic nonlinearity and the saturation contribution can occur. The lesson hereis that the frequency dispersion ofReal�χ�3�� ∝ n2 can exhibit multiple changes inthe sign, as well as depend on the pulse width of the laser.

4. Symmetric Molecules

Symmetric molecules by definition have no permanent dipole moments in anyelectronic states. In principle their third-order and higher susceptibility proper-ties can be analyzed in terms of the molecular symmetry properties and irredu-cible tensors as discussed above for χ�2�. This procedure is a very complex

problem in the case of χ�3�ijk . It was important for the second-order susceptibility

because symmetry properties can lead to χ�2�ijk � 0. This is not the case for the χ�3�ijk

tensor, which has nonzero elements, even for isotropic media.

4.1. General Sum-over-States Model

The general formula given by Eq. (44) specific to symmetric molecules (nopermanent dipole moments in the ground and excited states) is [108]

Figure 11

Relative contributions to the total nonlinearity (solid curves) of the Kerr elec-tronic nonlinearity (dotted curve) of the saturation contribution (dashed curve),all for the case �μ11 − μ00�2 � jμ10j2. (a) The real part, which also shows the totalnonlinearity for �μ11 − μ00�2 � 1.2jμ10j2 as a dashed–dotted curve. (b) The ima-ginary part of the third-order nonlinearity in arbitrary units. The� signs identifywhether the nonlinearity is positive or negative. The vertical lines indicate wherethe nonlinearity changes sign. Reproduced with permission of John Wiley andSons [11].


χ�3�ijkl�−�ωp � ωq � ωr�;ωp;ωq;ωr�

� N

ε0ℏ3L�ωp�L�ωq�L�ωr�L�ωp � ωq � ωr�

×

"X0

v;n;m

x

(μgv;iμvn;lμnm;kμmg;j

�ωvg − ωp − ωq − ωr��ωng − ωq − ωp��ωmg − ωp�

� μgv;jμvn;kμnm;iμmg;l�ω

vg � ωp��ωng � ωq � ωp��ωmg − ωr�

� μgv;lμvn;iμnm;kμmg;j�ω

vg � ωr��ωng − ωq − ωp��ωmg − ωp�

� μgv;jμvn;kμnm;lμmg;i�ω

vg � ωp��ωng � ωq � ωp��ω

mg � ωp � ωq � ωr�

)

−

X0

n;m

(μgn;iμng;lμgm;kμmg;j

�ωng − ωp − ωq − ωr��ωng − ωr��ωmg − ωp�

� μgn;iμng;lμgm;kμmg;j�ω

mg � ωq��ωng − ωr��ωmg − ωp�� μgn;lμng;iμgm;jμmg;k

�ωng � ωr��ω

mg � ωp��ωmg − ωq�

� μgn;lμng;iμgm;jμmg;k�ω

ng � ωr��ωmg � ωp��ω

ng � ωp � ωq � ωr�

)#: (92)

Although it is useful to obtain analytical formulas for Eq. (92) in order to studytrends due to molecular engineering, etc., these equations are just too complex tobe solved analytically in the general case, or even near and on resonance. As aresult, numerical methods need to be used if sufficient information is availableabout the states and the transition dipole moments. However, for copolarizedinputs and outputs, it has proven possible to derive general analytical formulasfor the real part of the nonlinear susceptibility for nonlinear refraction in the limitthat the inverse state lifetimes can be neglected relative to the difference betweenthe photon frequencies and the resonance frequency. This limits the validity tothe off-resonance and non-resonant regimes [108].

For the off-resonance case,

Realfχ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;ω;ω;−ω� � χ�3�xxxx�−ω;−ω;ω;ω�g

� N

ε0ℏ3L4�ω�

�4X0

m

X0

v

X0

n

μgvμvnμnmμmgωng�ω2

ng − 4ω2��ω2vg − ω2��ω2

mg − ω2�× f3ω2

ngωvgωmg � ω2ng�ωvg − ωmg�ω

� �ω2ng � 2ωng�ωvg � ωmg� − 8ωvgωmg�ω2

− 4�ωvg − ωmg�ω3g

− 2X0

n

X0

m

jμngj2jμmgj2�ω2

ng − ω2�2�ω2mg − ω2�2 f�ωmg � ωng�f3ω2

mgω2ng

� �2ω2ng � 2ω2

mg − 7ωngωmg�ω2− ω4g � ω2�ω3

mg � ω3ng�g

�: (93)

Noting the complexity of these equations, it is clear that the net nonlinearity canchange sign multiple times with frequency.


As discussed previously, the energy levels for symmetric molecules have eitherodd (“ungerade,” Bu) or even (“gerade,” Ag) symmetry. Therefore, one-photonelectric dipole transitions between states with the same symmetry are not al-lowed. Thus, in the first summation, the symmetric excited states ℓ can bereached only by intermediate coupling to an odd-symmetry state ℓ0 via two elec-tric dipole transition moments, namely, μℓ0g and μℓℓ0 . The “pathways” correspond-ing to the terms in the first and second summations for one-photon processes inEqs. (92) and (93) are shown in Fig. 12 as solid and dashed lines, respectively[108]. Therefore, even-symmetry excited states (called “two-photon” states) canbe accessed only by the simultaneous absorption of two photons. However,the second summation involves only one-photon transitions from the even-symmetry ground state to odd-symmetry excited states, sometimes called“one-photon” states.

For the non-resonant case, Eq. (93) simplifies further to [108]

χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;ω;ω;−ω� � χ�3�xxxx�−ω;−ω;ω;ω�

� NL4�ω → 0�ε0ℏ3

×

�12X0

v

X0

n

X0

m

μgvμvnμnmμmgωngωvgωmg

− 6X0

n

X0

m

jμngj2jμmgj2ω2ngω

2mg

�ωmg � ωng��:

(94)

If the contributions of the one-photon transitions, which are always negative, arelarger than the positive contribution from the two-photon terms, then the non-resonant nonlinearity will be negative, and vice versa [108]. The interferencedetermines the sign of the nonlinearity in the limit ω → 0. This conclusionis critical since even-symmetry states and their transition moments do notcontribute to the linear susceptibility and must be evaluated by nonlinearspectroscopy.

Figure 12

(a) (b)

Ag

Ag

Ag

Ag

Bu

Bu

Bu

Bu

Examples of the different pathways possible for (a) the second summation(dashed lines) and (b) the first summation (solid lines) in Eq. (93) by whichthe even-symmetry excited states can be reached only via intermediate odd-symmetry excited states. Reproduced by permission of the Optical Society ofAmerica [108].


4.2. Three-Level Model

A minimum of three states are required to describe the nonlinear optics ofsymmetric molecules for which general analytical results have been obtained[11,108–112]. The three levels are shown in Fig. 13. In such a model, thefirst state is the even-symmetry ground state 1Ag, the second is the odd-symmetry excited state, labeled 1Bu, which is most strongly coupled tothe ground state via μ1Bu←1Ag

, and the third is the lowest lying even-symmetryexcited state mAg with strong coupling to 1Bu via a large transition dipolemoment μmAg ←1Bu

.

The key question is which excited states to use. This can be decided by bruteforce using numerical ab initio calculations of the states and transition mo-ments. These are necessarily limited to simple molecules even when using themost powerful computers [113]. Alternatively, the dominant linear absorptionpeak can be used to evaluate the transition moments to the 1Bu state. Two-photon absorption and third-harmonic generation spectroscopy can likewisebe used via measurement of the dominant two-photon peak to evaluatethe location of mAg and μmAg ←1Bu

[114,115]. If there is more than one domi-nant peak, then it is necessary to resort to numerical evaluation of Eq. (92)[116]. In some cases mAg may represent a clustered grouping of even-sym-metry excited states if mAg falls in a quasi-continuum of even-symmetrystates [110,117]. Finally, spontaneous decay to the ground state is not allowedfrom even-symmetry states, and the state mAg can only decay to 1Bu via τ21with subsequent decay to ground state via τ10. The effective decay time τeffvia this coupling is given by τ−1eff � τ−121 � τ−110 .

Assuming x polarized incident light polarized parallel to the symmetry axis, ana-lytical formulas for the leading terms in the four limits defined in Section 3.3aare [11]

χ�3�xxxx�−ω; −ω;ω;ω� � χ�3�xxxx�−ω;ω; −ω;ω� � χ�3�xxxx�−ω;ω;ω; −ω�

� N

ε0ℏ3L4�ω�jμ10;xj2

�jμ21;xj2

×

�2�ω2

10 − ω2��ω20 − ω10��ω20 − 2ω10��ω210 − ω2� � 2iω10τ

−110 � � i�τ−121 � τ−110 �ω20�ω2

10 − ω2�22ω2

10ω20�ω20 − 2ω10�2��ω10 − ω�2 � τ−210 �2�

− jμ10;xj22�ω10 − ω�3 � 4iτ−110�ω10 − ω�2

��ω10 − ω�2 � τ−210 �3�

(95)

and

Figure 13

gmA

uB1

gA1

1

0

2

10µ

21µ τ

τ

21

10

The three energy levels, the electric dipole matrix elements, and the excited statelifetimes for the three-level model of a centrosymmetric system.


χ�3�xxxx�−ω;ω;ω;−ω�

� N

ε0ℏ3L4�ω�jμ01;xj2jμ12;xj2

�4�ω2

20 − 4ω2�

ω20�2ω10 −ω20�2��ω20 − 2ω�2 � �τ−121 � τ−110�2�

� 8i�τ−121 � τ−110 �ω20�4ω2

10 −ω220��ω20 � 2ω10�2 � τ−110�ω2

20 − 4ω2��ω20 � 2ω10�3

ω20�2ω10 −ω20�3��ω20 − 2ω�2 � �τ−121 � τ−110�2��2ω10 �ω20�3�

(96)

for the one- and two-photon resonances, respectively. For the off-resonancecase,

χ�3�xxxx�−ω;ω;ω;−ω� � χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;−ω;ω;ω�

� 4N

ε0ℏ3L4�ω�

�jμ10;xj2jμ21;xj2

�3ω2

10ω220 � ω2�ω2

20 � 4ω20ω10 − 8ω210�

ω20�ω210 − ω2�2�ω2

20 − 4ω2�

� iω

�τ−110

ω20ω210�7ω20 � 2ω10� � ω2�ω20 � 8ω10��ω20 − 2ω10�

ω20�ω210 − ω2�3�ω2

20 − 4ω2�

� 2�τ−121 � τ−110 ��ω20 � 2ω10��ω20ω10 � 2ω2�

�ω210 − ω2�2�ω2

20 − 4ω2�2��

− jμ10j4ω10�3ω210 � ω2� ��ω

210 − ω2� � 4iωτ−110 ��ω2

10 − ω2�4�: (97)

Finally, for the non-resonant case,

χ�3�xxxx�−ω;ω;−ω;ω� � χ�3�xxxx�−ω;ω;ω;−ω� � χ�3�xxxx�−ω;−ω;ω;ω�

� 12NL4�ω�jμ10j2ε0ℏ3ω2

10

�jμ21j2ω20

−

jμ10j2ω10

�: (98)

The sign of the nonlinearity is determined by the ratio jμ21j2ω10∕jμ10j2ω20.When it is greater than unity, the net nonlinearity is positive, and vice versa.This conclusion has been verified experimentally in a number of cases, includingpolydiacetylenes and squaraines in which the transition dipole moments werecalculated and the signs of the non-resonant nonlinearities were found to bein good agreement with the predictions of Eq. (98) [110–112].

The typical frequency dependence of the different contributions to the totalthird-order susceptibility is shown in Fig. 14 [11]. On and near resonance,the one-photon resonance, χ�3�xxxx has both negative real and imaginary compo-nents, subscripts R and I, respectively. The one-photon contributions are al-ways negative. There is a dispersion type of resonance at ω � ω20∕2 that canlead to a positive χ�3�R for the two-photon contribution for frequencies below thetwo-photon resonance. The net result can be either a positive or negative χ�3�R asthe non-resonant limit is approached. [The parameter range that leads to positivevalues is given by Eq. (98).] The two-photon contribution to χ�3�I is positivethroughout the whole frequency range, increasing with decreasing frequencyup to ω � ω20∕2, where it peaks and then decreases as the zero frequency limitis approached. Whether the non-resonant value of χ�3�R is positive or negativedepends on which process, i.e., one- or two-photon transitions, dominates;see Fig. 15, for example.


5. Transition to Bulk Nonlinear Molecular Optics

In this section, we go beyond the single molecule response and focus on con-densed media. As we previously discussed, molecular materials are marked bythe dominance of the molecular response, even in condensed media, due to thefact that they are bound loosely in the condensed state, an example being van derWaals crystals. Local field corrections as “perturbations” on the molecular re-sponse are often sufficient to describe the molecular response in the condensed

Figure 14

Relative contributions to the total nonlinearity (solid curve) of the Kerr two-photon transitions (dotted curve) and the Kerr one-photon transitions(dashed curve) for ω20 � 1.33ω10, ω10τ10 � 0.001, ω10τ21 � 0.01, andjμ21j2ω10∕jμ10j2ω20 � 1.25. (a) The real part and (b) the imaginary part ofthe third-order nonlinearity in arbitrary units. The � signs identify whetherthe nonlinearity is positive or negative. The vertical lines indicate where thenonlinearity changes sign. Reproduced with permission of John Wiley andSons [11].

Figure 15

x100→

→

(3) (

arbi

trar

y un

its)

−

10

10

1.00.80.60

0.2 0.4ω

ωχ

ω

Calculation of n2 ∝ ℜealfχ�3�� in arbitrary units versus the normalizedfrequency �ω10 − ω�∕ω10 for the three-level model with ω20 � 1.33ω10,ω10τ21 � 0.01, ω10τ10 � 0.001, and jμ21j2ω10∕jμ10j2ω20 � 0.75 (dashedcurve), jμ21j2ω10∕jμ10j2ω20 � 1.25 (solid curve) [108].


state and are discussed first. These molecular media can be defined by the abilityto be described by a local field-corrected oriented gas model. In this description,as described in Section 3, rotations are sufficient for transformation between themolecular and condensed states. The oriented gas model and applications tocrystals and poled polymers are then described. Finally, collective phenomenacontributing to the third-order response are described.

5.1. Local Field Corrections, Linear Susceptibility

The incident or emitted electric “Maxwell” fields, E�ωi�, are field averages overvolumes that contain many molecules but are small over all of the wavelengthsinvolved in the interaction. (They are called the Maxwell fields since they appearin Maxwell’s equations.) For a molecule in a host material or in a collection ofsimilar molecules, the electric field F�ω� at the molecular site is a superpositionof the applied electric fields E�ωi� and the local electric fields due to the dipolesinduced by the Maxwell fields in the nearby material. In dilute matter, such asgases, where the molecular density is sufficiently small that the Maxwell fieldE�ω� at a molecule is much larger than the total of the fields at the site of themolecule due to the dipoles induced in neighboring molecules, F�ω� � E�ω�.The local electric field is one of four types:

1. the applied electric field polarizes the surrounding material which creates afield that acts on the molecule;

2. the surrounding material is the source of an electric field even in the absenceof an applied field due to local fluctuations in charge density of an otherwiseelectrically neutral material;

3. if the molecule has a ground-state moment of any order (such as a dipolemoment, quadrupole moment, etc.) the electric field associated with the mo-lecule will polarize nearby material, which results in a reaction field; that is, afield that acts back on the molecule; or

4. an applied electric field induces a moment in a molecule which polarizes thematerial surrounding the molecule and results in a reaction field that changesthe dipole moment of the molecule.

Clearly the local electric field at a molecular site is a complex phenomenon andinvolves the use of self-consistent methods to be evaluated properly. Here, wereview the two simplest models; the Lorentz–Lorenz local field model and theOnsager model. Recall that the Onsager model is central to nonlinear optics, asshown in the early work of Levine and Bethea [33] and Oudar [34]. Screening ofembedded dipoles, reaction fields, the effect of an external electric field on ascreened dipole, and radiating dipoles embedded in a dielectric are also brieflydescribed in Appendix B. The sections that follow are based on the commonlyavailable literature and textbooks on electrostatics [118] and local electric fields[68]. Much of our presentation closely follows a development geared towardpractitioners of nonlinear optics.

5.1a. Continuum Approximation

Consider the local fields in a dipolar liquid or solid solution. In the simple modeldepicted in Fig. 16(a), the motion of each dipole is random and on averagesweeps out a spherical volume. Averaged over long enough time scales, thesystem will appear to be continuous, as represented in Fig. 16(b). A uniform


and static electric field, when applied to the isotropic and homogeneous solution,yields a time-averaged polarization,

P�1� � ε0χ �1�

· E � ε0�εr − 1�E; (99)

where εr is the relative dielectric function of the solution and is the macroscopiclinear susceptibility. Since measurement time scales are long compared with col-lisional times, the time-averaged polarization is typically the measured quantity.Thus, the material can often be viewed as a continuous dielectric in which thesubstituent molecules are considered spherical.

5.1b. Nondipolar Homogeneous Liquids and Solids

Consider a molecule in a liquid that is approximated by a dielectric sphere.Removing it, and assuming that the charges remain frozen in place, the electricfield that would be required to produce the observed polarization in the moleculeis the local electric field. Any textbook on electrostatics describes the electricfield of a polarized dielectric sphere [118].

The dashed circle in Fig. 16(b) represents the molecule, and Fig. 16(c) shows itafter being removed from the dielectric under the assumption that the uniformpolarization is “frozen in.” Both the surfaces of the dielectric sphere and thecavity are necessarily charged. The induced dipole moment of a dielectricsphere, p, in a uniform field F is

p � ε0αF � ε0εr − 1

εr � 2a3F; (100)

where α is the polarizability of the sphere, assuming for simplicity a scalarmedium. The polarization, P, is the dipole moment per unit volume given by

Figure 16

E

P a

(a)

(b)

(c)

(a) A dipolar liquid in which the molecules (arrows) sweep out a spherical vo-lume. (b) The shaded region represents the dielectric in an electric field (arrows)which is modeled as a continuum. (c) A spherical piece of the dielectric isremoved with the charges frozen in place.


P � p43πa3

� ε03

4π

εr − 1

εr � 2F: (101)

Setting Eq. (101) equal to Eq. (96) and solving for F, we get

F � εr � 2

3E: (102)

This is known as the Lorentz–Lorenz local field model. The local field factor isdefined as

L � εr � 2

3; (103)

where F � LE. Note that the local field factor is often labeled f rather than L.For an anisotropic material, the local field factor is a tensor. As a quick approx-imation, many researchers use the Lorentz–Lorenz form of the local field factors,which in principle holds only for a one-component system with no dispersion. Inmany typical cases, this will yield a reasonable approximation to the true localelectric field, but only in isotropic materials.

5.1c. Nondipolar Two-Component System

The local electric field at the site of an individual molecule (or chunk of dielec-tric) in a mixture, for example, in a liquid solution or dye-doped polymer at lowconcentration, can be viewed as a solute molecule that is embedded in a smoothdielectric. Figure 17(a) shows the electric field lines of a dielectric sphere em-bedded in another dielectric under the influence of a uniform electric field. Thereε1 and ε2 are the dielectric constants of the surrounding medium (solvent or host)and sphere (solute or guest), respectively.

The polarization of the sphere is distinct from the polarization of the surroundingdielectric. Figure 17(b) shows the dielectric with the sphere removed. The po-larizations P1 and P2 are those of the surrounding medium far from the sphereand inside the sphere, respectively. In analogy to Eq. (3), the polarization insidethe sphere is

P2 � ε0�ε2r − 1�Ein; (104)

Figure 17

E

P1

P2

(a) (b)

(a) Electric field of a dielectric sphere of dielectric constant ε2 embedded in adielectric of constant ε1 and (b) with the charges fixed in place and the dielectricsphere removed.


where Ein is the electric field inside the sphere. Ein is related to the uniformelectric field far away from and outside the sphere, E [68],

Ein ��

3ε1r2ε1r � ε2r

�E. (105)

Using Eqs. (104) and (105), the polarization inside the sphere in terms of theexternal electric field is

P2 � ε0�ε2r − 1��

3ε1r2ε1r � ε2r

�E. (106)

Again, the local electric field is the field that is required to induce a polari-zation P2 in the sphere. Using an argument similar to that leading to Eq. (102)with εr → ε2r, the relationship between the local field F and the polarization ofthe sphere is

P2 � 3ε0

�ε2r − 1

ε2r � 2

�F: (107)

Setting Eqs. (106) and (107) equal, the local electric field is given by

F ��

3ε1r2ε1r � ε2r

��ε2r � 2

3

�E: (108)

This expression is similar to the Onsager local field formula described inAppendix B. Note that we get the Lorentz local field when ε1r � ε2r.

It is clear that models that account for nonspherical cavities and the tensor natureof the dielectric function of both the molecule and the surrounding material arefar more complex. In such cases, the local field factor is a second-rank tensor andrelates F to E according to

Fi � LijEj: (109)

When the ensemble average principal axes coincide with the principal axes ofthe bulk system, the local field tensor is diagonal, which greatly simplifies theproblem. It is straightforward to generalize the local field tensor to the opticalregime when the guest molecule or solute particle is small compared to thewavelength of the illuminating source. Then, the optical field is approximatelyspatially uniform in the vicinity of the sphere so that the same formalism applies.For the two-component system, then, the scalar local field factor at frequency ω,L�ω� is

L�ω� ��

3ε1r�ω�2ε1r�ω� � ε2r�ω�

��ε2r�ω� � 2

3

�: (110)

5.2. Oriented Gas Model

We now turn attention to the usefulness of the group theoretical approach ofSection 3.2 for conveniently connecting the microscopic to macroscopic


response while maintaining the efficiency derived from its convenient represen-tation of molecular properties. In materials where the interaction between themolecular units is small compared to intramolecular forces, such as it is indye-doped polymers and molecular crystals, the bulk nonlinear optical suscept-ibility is given by a tensor sum over the local-field corrected hyperpolarizabil-ities of the molecules. The hyperpolizability is related to the macroscopicnonlinear optical susceptibility in an oriented gas model through [119,120]

χ�2�ijk �−�ωp � ωq�;ωp;�ωq� � NL�ωp � ωq�L�ωp�L��ωq�× hβIJK�−�ωp � ωq�;ωp;�ωq�iijk ; (111)

where N is the molecular number density and hβIJKiijk (dropping the frequencynotation for convenience) is the orientationally averaged hyperpolarizabilitytensor connecting the molecular coordinate system IJK to the macroscopic(laboratory) system ijk. The quantity hβIJKiijk can be written as

hβIJKiijk � hRijk;IJKiβIJK . (112)

The tensor R↔is the rotation transformation matrix, which can be written in terms

of the Euler angles defined in Fig. 18 as

R↔�

0@ cos ϕ cos ψ − cos θ sin ϕ sin θ − cos θ cos ψ sin ϕ − cos ϕ sin ψ sin θ sin ϕ

cos ψ sin ϕ� cos θ cos ϕ sin ψ cos θ cos ϕ cos ψ − sin ϕ sin ψ − cos ϕ sin θsin θ sin ψ cos ψ sin θ cos θ

1A.

�113�

An important simplification is that, because of the rotational invariance, uponaveraging the irreducible representation of the hyperpolarizability, the tensors ofvarious weight do not mix. This means

Figure 18

(a) Euler angles relating molecular to macroscopic frames. (b) Geometry of C2v

electron donor–acceptor–donor (D-A-D) chromophores.


χ�2�ijk ∝ hβIJKiijk � hβ�1s�IJKiijk � hβ�1m�IJK iijk � hβ�2m�IJK iijk � hβ�3s�IJKiijk∝ χ�1s�ijk � χ�1m�ijk � χ�2m�ijk � χ�3s�ijk . (114)

This fact is used routinely for averaging one-dimensional molecules to expressthe resulting macroscopic 1s component; for example, in poled polymers therelevant macroscopic component is the 1s, and results from the orientationallyaverage vector hyperpolarizability. We now provide a more sophisticated exam-ple building on our treatment of molecules of C2v symmetry as depicted inFig. 18(b) [119]. Consider an axially aligned chiral (handed) macroscopic med-ium, such as the one based on nematic-like (nematic refers to specific liquidcrystal class) alignment of helices decorated with C2v chromophores (Fig. 19).The macroscopic symmetric in this case is D∞. In Cartesian coordinates, theonly nonzero components of the hyperpolarizability of this point group areχD∞

� χxyz � χxzy � −χyzx � −χyxz. In this case, the only irreducible tensorcontributing to this response is the 2m one, so that we write

χ�2m� � χxyz

0@−1 0 0

0 −1 0

0 0 2

1A: (115)

Thus, the susceptibility will depend on the magnitude of the 2m tensor and theorientation average. Noting that the hyperpolarizability can be written as

β�2m�C2v�

0@ 0 Δβ∕2 0

Δβ∕2 0 0

0 0 0

1A; (116)

where Δβ � βzxx − βxxz, we can then specialize Eq. (111) to

χ�2m� � 1

6NΔβh2RzxRzy − RxxRxy − RyxRyyi �

1

4NΔβhsin2 θ sin 2ψi. (117)

This simple expression indicates that the macroscopic response is maximized bythe structure shown in Fig. 19 when ψ � �π∕4 and θ � π∕2 [119]. The sign ofψ should be uniform to maintain a chiral medium.

Figure 19

Optimum alignment of C2v chromophores on a helix.


5.3. Crystals

Whether a medium can be χ�2�-active, i.e., it exhibits inversion symmetry on thescale of an optical wavelength, depends on two factors: (1) the symmetry proper-ties of the molecules as just discussed and (2) the molecular alignment. There arebasically three possibilities in crystals, each depending on the symmetry proper-ties of the crystal’s unit cell formed on crystallization. The different cases aresummarized in Fig. 20 for the example of molecules with 1D second-order non-linearities, i.e., βiii ≠ 0, along some axis “i.” The simplest case of a single non-centrosymmetric molecule per unit cell that leads to χ�2�iii ≠ 0 is shown inFig. 20(c). In crystals, if the unit cells have no inversion symmetry, i.e., no per-manent dipole moment or no permanent octupole moment, the medium is notχ�2�-active. This can occur if the molecules themselves have inversion symme-try, as indicated in Fig. 20(a), for which the unit cell has no permanent dipolemoment. Alternatively, if the individual molecules do have permanent dipolemoments (and hence βijk ≠ 0 for some combination of ijk for an individual mo-lecule) but are aligned within a unit cell so that the unit cell has a zero net dipolemoment (i.e.,

Pmolecules in unit cell βijk � 0); hence, the medium will not be χ�2�-

active. An example is shown in Fig. 20(b) for counteraligned molecules. Finally,if the molecules are noncentrosymmetric and if the unit cell contains multiplemolecules whose alignment of the molecules in the unit cell results in a non-centrosymmetric unit cell, then some elements of χ�2�ijk ≠ 0. One such arrange-ment is three dipoles in an equilateral triangle, which have no net dipolemoment, but the centrosymmetry is broken and a second-order response due tothe octupole term occurs [91]. Which optical field components will result innonzero second-order parametric processes—such as second-harmonic genera-tion or sum frequency generation—depends on the crystal symmetry.

When there are strong interactions between molecules, for example, in micro-scopic [90,121–127] and macroscopic cascading [128–130], Eq. (111) needs tobe generalized. For example, in organic crystal lattices, when the van der Waalsinteraction or hydrogen bond energies responsible for intermolecular cohesionare several orders of magnitude smaller than intramolecular chemical bondenergies, the bulk nonlinear optical response will be a sum of the local-field-corrected molecular units. Zyss and Oudar calculated the second-harmonic coef-ficient (proportional to the second-order nonlinear optical susceptibility) in

Figure 20

Examples of unit cells containing molecules with different properties. The redarrows portray one-dimensional second-order nonlinear coefficients. (a) Unitcells containing a centrosymmetric molecule (dipole moment represented bythe dot). (b) Unit cells containing two counteraligned noncentrosymmetricmolecules. (c) Unit cells with a single noncentrosymmetric molecule.


terms of the hyperpolarizability using Eq. (111) with fixed molecular orienta-tions [41,52], as is found in molecular crystals.

Zyss and Oudar applied this model to 2-methyl,4-nitro analine (MNA) for whichthe crystal structure, electro-optic coefficient [52], and second-harmonic coeffi-cients [17,20,31,42,131] are known. They found good agreement between themodel and measurements of the bulk nonlinearities.

5.4. Electric Field Poled Media

The organic nonlinear optics community has invented a different way to align non-centrosymmetricmolecules to produce artificial “crystals”with uniaxial symmetry.Molecules with large permanent dipole moments in the ground state have beenengineered. “Charge transfer”molecules are synthesized by attaching groups withdifferent electron affinity at opposite ends of a “bridge” whose function is to facil-itate transfer of electrons between the end groups as shown in Fig. 21. Because thegroups have different electron affinity, charge is transferred from the electron donorgroup (D) to the electron acceptor group (A), producing a noncentrosymmetric lin-ear molecule. Typically, linear chains of carbon atoms whose pz orbitals overlap toform new delocalized π orbitals allow the electrons to move more easily betweenthe end groups. These molecules are either bonded somewhere in a polymer chainor “dissolved” inside a polymer as “guest” molecules. This results in randomlyoriented charge transfer molecules inside the bulk of a polymer.

A common technique for aligning charge transfer molecules is electric field pol-ing. When strong electric fields are applied, some net orientation of the mole-cules can be induced via the permanent dipole moment at elevated temperaturesas shown in Fig. 22. This requires first “softening” of the polymer above theglass transition temperature Tgl for the host polymer by heating, followed byapplying a DC field to produce partial orientation of the molecules. When thestructure is cooled to below the glass transition temperature, a partial net orienta-tion is effectively “frozen in” and the resulting medium has uniaxial symmetryaround the poling direction. This is performed on thin films that can be used aswaveguides for various applications. The most common of these is for electro-optics. Although various “tricks” have been used to achieve phase matching, theefficiency for second-harmonic generation achieved with poled films was neverlarge enough to be practical [132].

In crystalline materials, the orientations of the molecules in the unit cell andthe structure of the crystal are well defined. In a doped polymer, the molecularorientations are continuously distributed and there is one molecule per unit cell,so that X

S

β�s�IJK�−2ω;ω;ω� → βIJK�−2ω;ω;ω�: (118)

Figure 21

Prototype charge transfer molecule with acceptor and donor groups separated bya π-electron bridge.


The distribution of molecules is usually represented by an orientational distribu-tion function, which can be found when the orienting forces acting on themolecules are known. The nonlinear optical properties of the doped polymerare then calculated using Eq. (111).

The second-order nonlinear optical susceptibility of a dye-doped isotropic poly-mer is usually imparted with an electric field that is applied above Tgl, andcooled below Tgl to lock in the orientational order [43]. The thermodynamicmodel assumes that the dye molecules freely rotate in response to the appliedelectric field above Tgl. The orientational distribution function is derivable froma Gibbs distribution function with T � Tgl—the point at which the molecularreorientations are slow. The Gibbs distribution yields the orientational distribu-tion function G�Ω; Epol�:

G�Ω; Epol� �exp

h1

kBT�−μ · Epol�

iR�1−1 dΩ exp

h1

kBT�−μ⋅Epol�

i ; (119)

where μ is the dipole moment of each molecule in the ensemble, Ω represents thethree Euler angles, Epol the applied poling field, and kB is Boltzmann’s constant.Using Eqs. (111) and (119) and taking the result to first order in μ⋅Epol∕kBTyields

χ�2�ijk �−2ω;ω;ω� �Nβ�s�IJK�−2ω;ω;ω�

8π2

ZdΩ

μ · Epol

kBTaiI�Ω�ajJ �Ω�akK�Ω�;

(120)

where aiI�Ω�… is the Euler rotation matrix discussed previously. Assuming thatthe molecule is one-dimensional, that is, the only nonvanishing component ofμ and β�s�IJK�−2ω;ω;ω� are μZ and β�s�ZZZ�−2ω;ω;ω�, respectively, then the twoindependent tensor components of the bulk response are

Figure 22

(a) In-plane poling and (b) parallel plate poling of the charge transfer layer. Therole of the buffer materials and glass substrate is to inhibit current flow betweenelectrodes, which diminishes the poling field and causes dielectric breakdown.(c) Random orientation of molecules prior to poling. (d) Partial alignment ofmolecules by field. (e) “Frozen-in” structure.


x�2�zzz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω�μEpol

5kBT; (121)

χ�2�xxz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω�μEpol

15kBT: (122)

This thermodynamic model for poling was first applied to Disperse Red azo 1dye (DR1) in poly(methyl methacrylate) (PMMA), and the theory—using themeasured value of β�s�ZZZ�−2ω;ω;ω� for this dye in solution, correctly predictedthe bulk second-harmonic coefficient of DR1/PMMA within experimentaluncertainties for a wide range of poling field and number density [43].

While the above model is useful for a poled isotropic polymer, many materialsare anisotropic. Examples of anisotropic materials include liquid crystals andstretched polymers. For stretched polymers and nonferroelectric materials, theorientational order is uniaxial, in contrast to a poled polymer, which has polarorder. The additional force can be added to the distribution function according to

G�Ω; Epol� �exp

h1

kBT�−μ · Epol � Ufcos�Θ�g�

iR�1−1 dΩ exp

h1

kBT�−μ · Epol � Ufcos�Θ�g�

i ; (123)

where Ufcos�Θ�g is the axial ordering potential. With no poling field applied,the order parameter of the dyes is given by

hPii �R�1−1 d cos�θ�Pi�cos�θ�� exp

h−

U�cos�θ��kBT

iR�1−1 dΩ exp

h−

U�cos�θ��kBT

i ; (124)

where Pi�cos θ� is the ith Legendre polynomial. Because the axial forces repre-sented by U�cos Θ� are centrosymmetric, only even-order order parameters willbe nonzero. Equations (123) and (124), with the help of Eq. (111) to first order inμEpol∕kBT , then lead to

x�2�zzz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω�μEpol

kBT

�1

5� 4

7hP2i �

8

35hP4i

�; (125)

x�2�xxz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω�μEpol

kBT

�1

15� 1

21hP2i −

4

35hP4i

�: (126)

We stress that the above equations relate the orientational order of the materialbefore poling as quantified by hP2i and hP4i to the second-order susceptibilityafter the material is poled. Without a poling field

Epol � 0

�, all tensor

components of βijk vanish. Errors in this general theory were later correctedby Ghebremichael and associates [133].

This formalism was applied to the study of dye-doped polymers with applied uni-axial stress during the poling process. It was shown that, by adjusting the polingfield and the applied stress, the tensor ratio χ�2�xxz�−2ω;ω;ω�∕χ�2�zzz�−2ω;ω;ω� couldbe controlled to make the ratio near unity, making these materials useful in


electro-optic devices whose operation is independent of the polarization of thelight beam [134].

A large applied electric field can affect the order parameters, making the cal-culations more complex because the small field approximation is no longer va-lid. Then, higher powers in Epol must be included when evaluating the integrals.Furthermore, large fields can induce the material to undergo a phase transition,which results in a drastic change in the order parameters. Under these conditions,a formalism that includes the axial interaction between molecules under a meanfield approximation must be applied.

Van der Vorst and Picken extended the thermodynamic model of poling to thehigh-field regime [135]. They included in their potential function,U , the effect ofpoling to second order in the electric field and an effective single particle potential:

U�Θ� � −μ · Epol −1

2αE2

pol −1

3ΔαE2P2�cos Θ� − εhP2iP2�cos Θ�; (127)

where the first term corresponds to dipolar poling and the second term to theenergy shift of an isotropic material in response to the field (here α representsthe isotropic average of the polarizability). The third term represents poling of theinduced dipolemoment (Δα is the difference in the polarizability between the longand short axes of the cylindrical molecule), and P2 is the second-order Legendrepolynomial. The fourth term is due to internal liquid crystalline forces as origin-ally modeled byMaier and Saupe [136–138]. Here ε represents the strength of themean field single particle potential and hP2i the second-order order parameter,which depends on the strength of the poling field and the single particle potential.But, because hP2i is a parameter in the single particle potential, the order para-meters must be calculated self-consistently. The consequence of such a self-consistent calculation is that the poling field affects the order parameter hP2iwhich can induce a phase transition between the isotropic and nematic phaseof a liquid crystal.

Poled ferroelectric materials have odd-order order parameters that are predeter-mined: for example, a ferroelectric liquid crystal or a Langmuir Blodgett film[139–142]. Because an even-order response of order n depends only on thematerial’s odd-order order parameters less than n� 2, the second-order responsewill depend only on hP1i and hP3i. For the one-dimensional molecule, the twoindependent tensor components are

x�2�zzz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω��3

5hP1i �

2

5hP3i

�; (128)

x�2�xxz�−2ω;ω;ω� � Nβ�s�ZZZ�−2ω;ω;ω��1

5hP1i −

1

5hP3i

�: (129)

All the models above relate the molecular hyperpolarizabilities to the bulk non-linearity. By choosing appropriate molecules and polymers and poling themunder well-defined conditions, stretching them, and using naturally presentinternal forces, a wide range of nonlinear optical properties result. Alignedχ�2�-active films have also been produced by electric field poling of moleculeswith octupolar symmetry or by using Langmuir–Blodgett techniques to depositmultiple monolayers [91,139–142]


5.5. Additional Contributions to Third-OrderNonlinearities

In the preceding sections only the ultrafast electronic hyperpolarizability leadingto χ�3�ijkl�−�ωp � ωq � ωr�;ωp;ωq;ωr� was considered. However, there are manyother slower effects that lead to nonlinear refractive index and absorptionchanges in molecular media [11,110]. They mostly arise from collectivephenomena and include

1. nuclear (vibrational) contributions to n2,

2. single reorientation of molecules with anisotropic polarizabilities,

3. collective reorientation of molecules with anisotropic polarizabilities,

4. photorefractive effects,

5. electrostriction,

6. thermal nonlinearities, and

7. cascading of second-order nonlinearities.

Of these, the most important ones for time scales below 1 ns are the first two.They both involve the usual molecular degrees of freedom, namely, vibrationand rotation. With the exception of the first one, the others have been discussedin detail in a recent review paper [110]. Single molecule rotational nonlinearitiesare well understood, and little progress has been made in this field over the lastdecade.

Vibrational contributions were discussed first in the 1970s for glasses, and theywere found to contribute up to 20% of the Kerr nonlinearity [110]. As a resultthere was only limited interest in these contributions. In the 1990s large vibroniccontributions were observed and discussed theoretically in the linear absorptionspectrum of linear molecules and conjugated polymers [143,144]. They havealso been observed in the nonlinear optics of such materials [115,144–146].Although no quantitative estimate of the vibronic contribution to the two-photonabsorption spectrum of the polydiacetylene PTS was reported in 1996, it is clearthat the integrated intensity of the vibronic subbands with picosecond pulses waslarger than for the contribution of the nonvibronic peak [115].

In 2000 Chernyak and associates showed that in second and third harmonic gen-eration experiments, when the wavelengths are tuned below the lowest-energy,excited electronic state, the purely electronic hyperpolarizabilities account for90%–95% of the total [146]. A theoretical analysis of the Chernyak experimen-tal results came to the same conclusion [147]. From the early 1990s, when thefirst theoretical papers began to appear, most of the theoretical calculations havebeen in the static limit (non-resonant limit) where the one- and two-photon con-tributions to the nonlinear refractive index interfere destructively, as discussed inthe preceding two sections [148,149]. Bishop pointed out that, in this case, thevibronic component could dominate the Kerr component.

The most recent experiments on carbon disulphide with pulses ranging in widthfrom 30 fs to ∼10 ps at frequencies close to the non-resonant regime have shownthat the vibrational and rotational contributions can indeed be dominant, asshown in Fig. 23 [150]. Additional experiments showed a lack of dispersion withwavelength in the nonlinear data and confirm that indeed the conditions approx-imate the non-resonant limit. The vibrational contribution appears at pulse


widths comparable to the vibrational frequencies of the molecule. Note that thevibrational contribution is three times the Kerr, and the rotational contribution isthree times the vibrational one.

Clearly there is much still to be learned from further experiments in thenon-resonant regime.

6. Conclusions

In molecular materials, the nonlinear optical properties of the bulk medium aredetermined from a sum over the nonlinear optical properties of the molecules byvirtue of the weak interactions between them. This is in contrast to some inor-ganic crystals (for example, semiconductors) where strong interactions leadto delocalization of the wave functions, yielding band structures. Molecularmaterials made of organic molecules offer a vast choice of molecular structuresthat can be custom designed through organic synthesis. This combination ofcustom tailoring molecular structure and molecular assembly into a bulk struc-ture allows for the ultimate in “bottom-up” organic materials engineering.

The first organic crystal engineered specifically for nonlinear optics using this“bottom-up” approach was MNA.While organic crystals could in principle havea large nonlinear optical response because the molecules can be engineered toindividually have large nonlinearities, the molecules cannot always be arrangedin a unit cell in a way to optimize the bulk response. This has led to a differentway to use molecules, namely doped and functionalized into host polymers inwhich molecular alignment is achieved by electric field poling to optimize thesecond-order susceptibility [51]. Furthermore, such polymers can be made intothin films, fibers, and moldable components, making them amenable to high-volume manufacturing.

Since the strength of the nonlinearity in dipolar molecules derives from the samephysics as the dipole moment, i.e., the charge transfer mechanism, the tendency ofstrong dipoles is to be counteraligned in the crystal phase. Only 20%–30% oforganic crystals containing highly nonlinear molecules exhibit any bulk non-linearity at all. Furthermore, the stronger the charge transfer mechanism, thenarrower the transmission window in the visible and near-infrared, another char-acteristic of the charge transfer. Inorganic crystals have also been shown to be up tothe task for frequency conversion, while this is not the case for integrated

Figure 23

Z-scan measurement of η2;eff from 30 fs to 9 ps for liquid CS2. Courtesy of Prof.E. VanStryland and Dr. H. Hu, University of Central Florida [150].


electro-optic devices. As a result, the major application of molecular nonlinearoptics is in poled polymer devices used for electro-optics for operation in thenear-infrared.

There has been steady progress to understand the nonlinear mechanism throughquantum modeling, for example the SOS approach with feedback from experi-ments, as first applied by Garito and associates [151]. Given the complexityof quantum modeling, attempts were made to connect structures with nonlinearoptical properties in an effort to identify motifs for synthetic chemists to follow.These included work on length scaling [152–156], bond-length alternation inconjugated molecules [86,87], and symmetry [157], which found that the bestthird-order materials are made from centrosymmetric molecules [157]. Describ-ing the molecules in terms of the irreducible tensors associated with a molecule’sspatial symmetry has been proven to be a very powerful tool for understandingtheir microscopic and macroscopic optical properties. More recently, a broaderunderstanding is being developed using scaling arguments [158–160], which arebased on fundamental limits of the nonlinear optical response [161].

Once good molecules are identified and built into materials, models of the localelectric fields need to be used to predict the bulk response. Conversely, deter-mination of molecular properties relies on local field models.

In this article, we have reviewed this bottom-up approach, and shown how lim-ited state models can be used to describe the generic dispersion of the nonlinearoptical response and how the dispersion can be used as a probe of the symmetryof the system [162]. Such symmetry arguments can be used to determine whichclass of states contribute. As a result, for example, given just the sign of the off-resonant response of a centrosymmetric system, one can show that a nonlinearMiller’s delta approach does not apply.

The magnitude and sign of the nonlinearities change with frequency. Armedwith knowledge of the locations of the excited states relative to the ground stateand the electric transition dipole moments, it is possible to predict the magnitudeand sign of the nonlinearity. However, this detailed knowledge is usually re-stricted to just a few states and transition moments, those which dominatethe linear and nonlinear absorption spectrum. For asymmetric molecules,primarily useful for electro-optics, a two-level model (the ground state andone excited state, both measurable by linear absorption spectroscopy) providesuseful information about the nonlinear dispersion, sign, and magnitude of thenonlinearity. A key result is that, for χ�3�, there is a cancellation betweenone- and two-photon transitions in the region of lower loss.

Symmetric molecules require a three-level model. The symmetry separates outthe one from the two-photon transitions. The three levels consist of an even-symmetry ground state and two excited states, one with even-symmetry wavefunctions and one with odd-symmetry wave functions. Also required are two-dipole transition moments, one from the ground state to the odd-symmetry ex-cited state and one from that excited state to the even-symmetry excited state.Predictions of such a three-level model have been surprisingly accurate in ex-plaining the frequency dispersion and sign of the nonlinearity, and most impor-tant the sign of the non-resonant nonlinearity. If the molecule exhibits more thanone strong peak in the linear and nonlinear absorption spectrum, multiple excitedstates contribute significantly to the nonlinearity requiring “brute force” numer-ical calculations to understand the nonlinearity.


Molecular nonlinear optics has come a long way over the last three decades inthe design of better materials and in better understanding the physics behindlight–matter interactions important for nonlinear optical imaging, interfacial stu-dies, and other applications of nonlinear optics. Given the richness of materialsavailable through organic synthesis and materials processing, and the complex-ity of nonlinear interactions, new phenomena and applications surely awaitdiscovery.

Appendix A: Cartesian Tensor Decomposition

In this appendix, we will describe the decomposition of the hyperpolarizabilityinto its Cartesian form. We will also express the components of various weightsin terms of the elements of the permutation operations. In decomposing a tensorinto its irreducible components in Cartesian form, we will need to have a methodfor reducing the rank of the tensor since the irreducible components have a rankeither the same as or lower than the tensor being decomposed. Following reduc-tion, we can recover the rank n � 3 tensor by embedding the irreducible tensorsinto this higher rank, arriving at a description where the nonlinear optical tensoris a sum of its embedded irreducible components. Both of these operations de-rive from the two rotationally invariant Cartesian tensor forms: (1) the sym-metric second-rank form δij, the Kronecker delta whose components areunity when i � j and zero otherwise, and (2) the antisymmetric Levi–Civitastensor εijk whose components are �1 (−1) if ijk is an even (odd) permutationof 123 and 0 otherwise. Note that the Levi–Civitas tensor is a pseudotensor.Contraction of a tensor with δij (a double contraction, i.e., with both i and j)extracts the trace and lowers its rank, keeping the weight and parity unchanged.The double contraction with εijk extracts the antisymmetric part of a tensor andalso lowers its rank. Thus, since irreducible forms are extracted with these δ andε tensors, the irreducible form of n � J , known as its natural form, must betraceless and symmetric, and appear as appropriate linear combinations of var-ious permutations and combinations of the reduction products with the δ and εtensors. Returning to our example in Eqs. (61) and (62), we see that the first termin Eq. (62) follows from contracting T with δ as Tijδij, the second term as Tijεijk ,and so forth.

To this end, the three-wave mixing tensor yields one scalar,

β�0� � βijkεijk�� βijkεjklδil�; (A1)

three vectors (J � 1) as traces

β�1;1�i � βijkδjk ; β�1;2�i � βjikδjk β�1;3�i � βjkiδjk ; (A2)

two traceless symmetric second-rank tensors (J � 2),

β�2;1�ij � 1

2�εiklβklj � εjklβkli� −

1

3β�0�δij;

β�2;2�ij � 1

2�βiklεklj � βjklεkli� −

1

3β�0�δij; (A3)

and a single third-rank tensor (J � 3):


β�3�ijk � 1

6�βijk � βikj � remaining permutations� − traces: (A4)

Note where symmetrizing sums and subtracted traces appear in these equations.Equations (A2)–(A4) illustrate the reduction procedure yielding the irreduciblerepresentations in Cartesian coordinates using the δij and εijk tensors. We canthen write the hyperpolarizability tensor in terms of these irreducible representa-tions by embedding them in a third-rank tensor using the same two tensors yield-ing a sum with terms related to the irreducible tensors:

βijk � β�3�ijk � β�2;1�ijk � β�2;2�ijk � β�1;1�ijk � β�1;2�ijk � β�1;3�ijk � β�0�ijk ; (A5)

where

β�0�ijk � 1

6β�0�εijk ;

β�1;1�ijk � 1

10�4β�1;1�i δjk − β�1;1�j δik − β�1;1�k δij�;

β�1;2�ijk � 1

10�−β�1;2�i δjk � 4β�1;2�j δik − β�1;2�k δij�;

β�1;3�ijk � 1

10�−β�1;3�i δjk − β�1;3�j δik � 4β�1;3�k δij�;

β�2;2�ijk � 1

3�2εijlβ�2;1�lk � β�2;1�il εljk�;

β�2;1�ijk � 1

3�εijlβ�2;2�lk � 2β�2;2�il εljk�: (A6)

In deriving these irreducible representations, we merely labeled the tensors ofcommon rank by m, the second index in the superscript of β�J ;m�ijk indicated inEqs. (A2)–(A6). However, when proceeding to consider the second-harmonicand Kleinman symmetric cases, it is more useful to label these m in terms oftheir behavior under Cartesian index permutation symmetry. This can be doneby considering the irreducible representation of the permutation group of threeobjects. This group has three irreducible representations (two 1D and one 2D)yielding four possible permutation projection operators:

Ps �1

6�1� �1↔2� � �1↔3� � �2↔3� � �1 → 2 → 3� � �1 → 3 → 2��;

Pa �1

6�1� �1 → 2 → 3� � �1 → 3 → 2� − �1↔2� − �1↔3� − �2↔3��;

Pm � 1

6�2� 2�2↔3� − �1↔2� − �1↔3� − �1 → 2 → 3� − �1 → 3 → 2��;

Pm0 � 1

6�2� 2�2↔3� � �1↔2� � �1↔3� − �1 → 2 → 3� − �1 → 3 → 2��: (A7)

The two-digit sequences denote the exchange of the two indices, while the three-digit sequences are cyclic index permutations, where, for example, �1↔2�βijk �βjik and �1↔2↔3�βijk � βkij. These are orthogonal operators which extracttensors that are fully symmetric (s), fully antisymmetric (a), both being one-dimensional operators, and two of mixed symmetry (m), (m0) correspondingto the two-dimensional operator. The factors ensure that PiPj � Piδij. Since they


belong to the same representation, they are chosen so that (m) is symmetric withrespect to one pair of indices and (m0) is antisymmetric in one pair of indices.Thus, expressing the irreducible representations of βijk in terms of the permuta-tion group is especially convenient since, for second-harmonic generation andthe linear electro-optic effect, both are symmetric in one pair of indices, thusyielding (m0) and (a) components that are identically zero. Then, the (a) com-ponent is applicable to parametric processes (three waves of different fre-quency), but does not contribute to second-harmonic generation, and the (s)projection contributes to the fully Kleinman symmetric case. This, then, explainsthe forms of Equations (66) and (67).

Appendix B: More Sophisticated Local Field Effects:Screening and Dressed Dipoles

This treatment follows treatments of local fields that can be found in the litera-ture [68]. The dielectric surrounding a molecule will also affect the molecule’sstatic dipole moment. For a molecular vacuum moment μ in a cavity made of adielectric of relative dielectric constant ε1r, a charge is induced on the cavitywall, as shown in Fig. 24. The surface charge will modify the electric field insideand outside the cavity. The electric potentials inside the cavity, φin, and outsidethe cavity, φout, are given by

φin �1

4πε0

�μ

r2cos Θ −

2�ε1r − 1�2ε1r � 1

μ

a3r cos Θ

�; (B1)

φout �1

4πε0

�μ

r2cos Θ −

2�ε1r − 1�2ε1r � 1

μ

r2r cos Θ

�� 1

4πε0

3

2ε1r � 1

μ

r2cos Θ;

(B2)

where Θ is the angle between the field and the z axis, r is the distance to the fieldpoint, and a is the cavity radius. The first term in Eq. (B1) is the dipole due to μand the second term is the uniform reaction field due to the induced surface

Figure 24

(b)

rz

(a)

P0

(c) (d)

(a) A dipole in a cavity within a dielectric and (b) a representation of the charges,including the dipole-induced surface charge on the cavity wall and the dipole inthe cavity. (c) Molecule represented as a point dipole and in a dielectric of per-mittivity, ε1r (d) represented as a dielectric sphere of permittivity ε2r and polar-ization P0. No electric fields or induced polarization are shown.


charge. Similarly, the first term in Eq. (B2) is the dipole potential of the baredipole μ and the second term is the dipole potential from the induced charge onthe cavity surface, with the sum of the two defining the dipole field outside thecavity.

Here we follow the approach of Kuzyk and Dirk [68]. The electric field iscalculated from the potential according to E � ε1r∇φ1, so, from Eq. (B2), theeffective dipole moment is given by

μe �3ε1r

2ε1r � 1μ: (B3)

The electric field inside the cavity is a superposition of a dipole field and a uni-form electric field due to the surface charge that is induced by the dipole and iscalled the reaction field R. If the molecule is polarizable, the reaction field canact on the molecule and change its dipole moment, which in turn can change thereaction field. The total dipole moment, μ0, is then the sum of the permanentdipole moment and the induced dipole moment:

μ0 � μ� ε0α2R � μ

1 −�α23

��2�ε1r−1�2ε1r�1

� . (B4)

Applying the Onsager approximation 4πNa3∕3 � 1 and the Clausius–Mossottiequation for the polarizability α2, the effective internal dipole moment is givenby [68]

μ0 � μ

1 −�

n22�0�−1

�n22�0��2�

��2�ε1r�0�−1�2ε1r�0��1

� � μ�2ε1r�0� � 1��n22�0� � 2�

3�2ε1r�0� � n22�0��; (B5)

where n � ffiffiffiffiεr

p, and n�0� refers to the zero-frequency limit of the fast electronic

response of the medium.

Just as the electric field from the vacuum dipole μ is screened by the dielectric,this effective internal dipole moment is also screened by the dielectric. In anal-ogy to Eq. (B4), the effective dipole moment measured by an observer whoseperspective is external to the cavity is

μ0e � μ03ε1r�0�

2ε1r�0� � 1� μ

�n22�0� � 2�ε1r�0�n22�0� � 2ε1r�0�

: (B6)

B.1. Local Field Model of a Two-Component DipolarComposite

The dipole moment of an orientationally fixed molecule can be written as a ser-ies expansion in the electric field, where the first two terms for a dipole in avacuum are

pi � μi �1

ε0αijFj: (B7)

If the molecule freely rotates, the orientational average of the permanent dipolemoment vanishes at nonzero temperatures. An applied electric field will induce


an electronic polarization due to electron cloud deformation and a reorientationalpolarization due to partial alignment of the permanent dipole moments willoccur, leading to [69]

p � ε0

�jμj2F3kBT

� α · F

�; (B8)

where α is the orientationally averaged polarizability. (For a spherical molecule,α is a scalar.) We can then rewrite Eq. (B8) as

p � ε0αeffF � ε0�αor � α�F; (B9)

where the effective polarizability αeff is the sum of the orientational and elec-tronic parts.

For a dipole inside a dielectric, the local electric field F has two sources,

F � Ec � R; (B10)

the cavity field Ec, and the reaction field R. The cavity field is the sum ofthe applied electric field and the field due to the induced charge on the surfaceof the cavity, while the reaction field is due to the surface charge that is inducedby the dipole inside the cavity. The reaction field is always along the axis of thedipole, so it can never reorient the molecule but it can polarize the electron cloud.The cavity field, on the other hand, will both reorient and polarize the electroncloud. The induced dipole moment of a molecule in a dielectric is thus of theform

p � ε0�αorEc � αF�; (B11)

where α is the polarizability of the embedded molecule. For a spherical cavity,the electric field inside is given by Eq. (106) with ε2r � 1 and

Ec �3ε1r

2ε1r � 1E. (B12)

The induced dipole moment can thus be written in terms of the applied electricfield as

p � ε0αor

�3ε1r

2ε1r � 1

�E � α

�2�ε1r − 1��2ε1r � 1� p

�� ε0

3ε1r2ε1r � 1

E: (B13)

Solving this self-consistent expression for the dipole moment, p, we get

p � ε0

24 3ε1r2ε1r�1

�αor � α2�1 − 2�ε1r−1�α2

�2ε1r�1�a3

35E: (B14)

According to Eq. (100), the effective polarizability of the spherical molecule isrelated to its dielectric function by


αeff �ε2r − 1

ε2r � 2a3: (B15)

The electronic polarizability is related to the “fast” part of the response and therefractive index to the electronic part of the dielectric constant according ton22 � ε2r. With the understanding that n2 refers to the electronic part of εr,

α2 �n22 − 1

n22 � 2a3. (B16)

Substituting Eqs. (B15) and (B16) into Eq. (B14) yields

p � ε0ε1r�ε2r − 1��n22 � 2��ε2r � 2��n22 � 2ε1r�

a3E: (B17)

Using Equation (104) with εr → ε2r and p → p2 and setting this equal toEq. (B17) gives

F � ε1r�n22 � 2�n22 � 2ε1r

E: (B18)

This is the Onsager local field model. Note that this derivation is for the two-component system, such as a dye-doped polymer or liquid solution, so that ε1r isthe dielectric constant of the host and n2 is the refractive index of the guest. Thesingle component Onsager expression is obtained by removing the subscriptsfrom Eq. (B18).

The local field calculations above neglect the fact that the polarizability andnonlinear susceptibility of a molecule change in the presence of an electric field.The linear and nonlinear optical susceptibilities are peaked at the optical fre-quency corresponding to resonant excitations of the molecules. The fluctuationsin the local electric field can result in peak broadening while a static local electricfield can affect both the shape and positions of these peaks. The broadening iscommonly described by a phenomenological width (or excited state decay timeτ) parameter. The effect of the reaction field on the susceptibility through itseffect on the structure of the molecule is important.

The reaction acts on a molecule and changes its energy levels, normally by onlya small amount. The molecular susceptibilities are functions of the transitionfrequencies ωvm � ωv − ωm and the transition moments μvm of the molecule,where m and v label the energy eigenstates. The energy levels of a moleculein a dielectric material such as a polymer shift and the transition momentschange. The most pronounced effect on the linear absorption spectrum is a shiftin the wavelength of maximum absorbance. Such a shift also appears in the non-linear optical spectrum. When the transition moments are not strongly affectedby the reaction field, the shifts can be formally introduced by an energy shiftoperator, O�ωm;ω0

m�, which affects any arbitrary function, f �ω0m�, as follows:

f �ωm� � O�ωm;ω0m�f �ω0

m�; (B19)

where ℏω0m is the vacuum energy of level m. Clearly, the shift operator depends

on the dielectric properties of the host matrix.


Accounting for the local fields using a local field tensor as described byEq. (112), Eq. (2) becomes

p�n�i �ω� � 1

2n−1ε0�ΠmO�ωm;ω0

m��ξ�n�ijk…ℓ�−ω;ω1;ω2…ωn�Ljj0 �ω1�Ej0 �ω1�

× Lkk 0 �ω2�Ek 0 �ω2� × � � � × Lℓℓ0 �ωn�Eℓ0 �ωn�; (B20)

where the index m spans all the energy eigenstates of the molecule and theoperator Πm represents the product over all energy eigenstates. MultiplyingEq. (B20) by Lii0 �ω�, we get

p0�n�i �ω� � 1

2n−1ε0ξ

0�n�ijk…ℓ�−ω;ω1;ω2;…;ωn�Ej�ω1�Ek�ω2� × � � � × Eℓ�ωn�;

(B21)

where the quantities with the primes (0) are called the “dressed” induced dipolemoment and susceptibility:

p0�n�i �ω� � p�n�i �ω�Li0i�ω�; (B22)

ξ0�n�ijk…ℓ�−ω;ω1;ω2;…;ωn� � �ΠmO�Ωm;Ω0m��ξ�n�ijk…ℓ�−ω;ω1;ω2…ωn�

× Lii0 �ω�Ljj0 �ω1� × Lkk 0 �ω2� ×…Lℓℓ0 �ωn�: (B23)

Because p0�n�i �ω� now contains the local field factor, it corresponds to a“Maxwell” polarization when multiplied by N , the number of molecules per unitvolume. For a two-component isotropic system, the local field factors are givenby the Onsager local fields, which, for an isotropic medium associated with theradiation field at frequency ω, are of the form [68]

Lii0 �ω� � δii03ε1r�ω�

2ε1r�ω� � 1: (B24)

The remaining local field factors are of the Onsager form, which, for an inputfrequency ω1, for example, is

Ljj0 �ω1� � δjj0

�3ε1r�ω1�

2ε1r�ω1� � n22�ω1�

��n22�ω1� � 2

3

�: (B25)

In the dressed susceptibility formalism, the dressed dipole moment’s depen-dence on the applied electric field is identical in form to the vacuum relationship.

Acknowledgments

MGK thanks the NSF (ECCS-1128076) and the AFOSR (Grant No: FA9550-10-1-0286), KDS acknowledges support by the NSF Center for Layered Poly-meric Systems (DMR-0423914) and helpful discussions with Prof. RolfePetschek, and GIS thanks his IRA and KFUPM for supporting this work.


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105. L. Sanguinet, R. J. Twieg, G. Wiggers, G. Mao, K. D. Singer, and R. G.Petschek, “Synthesis and spectral characterization of bisnaphthylmethyland trinaphthylmethyl cations,” Tetrahedron Lett. 46(31), 5121–5125(2005).

106. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J.Quantum Electron. 28(11), 2631–2654 (1992).

107. C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-levelmodel for describing the molecular third-order nonlinear optical suscept-ibility,” Int. J. Quantum Chem. 43(1), 27–36 (1992).

108. G. I. Stegeman, M. G. Kuzyk, D. G. Papazoglou, and S. Tzortzakis,“Off-resonance and non-resonant dispersion of Kerr nonlinearity forsymmetric molecules [Invited],” Opt. Express 19(23), 22486–22495(2011).


109. M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, “Mechanisms of quadratic elec-trooptic modulation of dye-doped polymer systems,” J. Opt. Soc. Am. B7(5), 842–858 (1990).

110. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W.Van Stryland, “Nonlinear refraction and absorption: mechanisms andmagnitudess,” Adv. Opt. Photon. 2(1), 60–200 (2010).

111. G. Stegeman and H. Hu, “Refractive nonlinearity of linear symmetricmolecules and polymers revisited,” Photon. Lett. Poland 1, 148–150(2009).

112. G. I. Stegeman, “Nonlinear optics of conjugated polymers and linearmolecules,” Nonlinear Opt. Quantum Opt. 43(1), 143158 (2012).

113. D. Jacquemin, B. Champagne, and B. Kirtman, “Ab initio static polariz-ability and first hyperpolarizability of model polymethineimine chains. II.Effects of conformation and of substitution by donor/acceptor end groups,”J. Chem. Phys. 107(13), 5076–5087 (1997).

114. J. H. Andrews, J. D. V. Khaydarov, K. D. Singer, D. L. Hull, and K. C.Chuang, “Characterization of excited states of centrosymmetric and non-centrosymmetric squaraines by third-harmonic spectral dispersion,” J. Opt.Soc. Am. B 12(12), 2360–2371 (1995).

115. W. E. Torruellas, B. L. Lawrence, G. I. Stegeman, and G. Baker, “Two-photon saturation in the band gap of a molecular quantum wire,” Opt. Lett.21(21), 1777–1779 (1996).

116. D. M. Bishop, B. Kirtman, and B. Champagne, “Differences between theexact sum-over-states and the canonical approximation for the calculationof static and dynamic hyperpolarizabilities,” J. Chem. Phys. 107(15),5780–5784 (1997).

117. P. McWilliams, P. Hayden, and Z. Soos, “Theory of even-parity stateand two-photon spectra of conjugated polymers,” Phys. Rev. B 43(12),9777–9791 (1991).

118. For example, J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley,1996).

119. V. Ostroverkhov, K. D. Singer, and R. G. Petschek, “Second-harmonic gen-eration in nonpolar chiral materials: relationship between molecular andmacroscopic properties,” J. Opt. Soc. Am. B 18(12), 1858–1865 (2001).

120. D. Wanapun, V. J. Hall, N. J. Begue, J. G. Grote, and G. J. Simpson,“DNA-based polymers as chiral templates for second-order nonlinear op-tical materials,” Chem. Phys. Chem. 10(15), 2674–2678 (2009).

121. M. G. Kuzyk, “Third order nonlinear optical processes in organic liquids,”Ph.D. dissertation (University of Pennsylvania, 1985).

122. J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Pair correlations, cas-cading, and local-field effects in nonlinear optical susceptibilities,” Phys.Rev. A 46(7), 4172–4184 (1992).

123. J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Molecular orientation,pair correlations and cascading in nonlinear optical susceptibilties,” Mol.Cryst. Liq. Cryst. 223(1), 143–150 (1992).

124. A. Baev, J. Autschbach, R. W. Boyd, and P. N. Prasad, “Microscopic cas-cading of second-order molecular nonlinearity: new design principles forenhancing third-order nonlinearity,” Opt. Express 18(8), 8713–8721(2010).

125. G. R. Meredith, “Local field cascading in third-order non-linear opticalphenomena of liquids,” Chem. Phys. Lett. 92(2), 165–171 (1982).


126. G. R. Meredith, “Second-order cascading in third-order nonlinear opticalprocesses,” J. Chem. Phys. 77(12), 5863–5871 (1982).

127. N. J. Dawson, B. R. Anderson, J. L. Schei, and M. G. Kuzyk, “Quantummechanical model of the upper bounds of the cascading contribution to thesecond hyperpolarizability,” Phys. Rev. A 84(4), 043407 (2011).

128. G. R. Meredith, “Cascading in optical third-harmonic generation by crys-talline quartz,” Phys. Rev. B 24(10), 5522–5532 (1981).

129. G. I. Stegeman, D. J. Hagan, and L. Torner, “Cascading phenomena andtheir applications to all-optical signal processing, mode-locking, pulse com-pression and solitons,” Opt. Quantum Electron. 28(12), 1691–1740 (1996).

130. M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, “All-optical switching byuse of cascading of phase-matched sum-frequency-generation anddifference-frequency-generation processes in periodically poledLiNbO3,” Opt. Lett. 22(5), 274–276 (1997).

131. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison oftheory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys.41(4), 1667–1681 (1970).

132. M. Canva and G. I. Stegeman, “Parametric interactions in organicwaveguides,” Adv. Polym. Sci. 158, 87–121 (2002).

133. F. Ghebremichael, M. G. Kuzyk, K. D. Singer, and J. H. Andrews,“Relationship between the second-order microscopic and macroscopicnonlinear optical susceptibilities of poled dye-doped polymers,” J. Opt.Soc. Am. B 15(8), 2294–2297 (1998).

134. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second ordernonlinear optical tensor properties of poled films under stress,” J. Opt. Soc.Am. B 6(4), 742–752 (1989).

135. C. P. J. M. van der Vorst and S. J. Picken, “Electric field poling of acceptor–donor molecules,” J. Opt. Soc. Am. B 7(3), 320–325 (1990).

136. W. Maier and A. Saupe, “Eine einfache molekulare theorie des nematischenkristallinflussigen zustandes,” Z. Naturforsch. A 13, 564–566 (1958).

137. W. Maier and A. Saupe, “Eine einfache molekular-statistische theorie dernematischen kristallinflussigen phase 1,” Z. Naturforsch. A 14, 882–889(1959).

138. W. Maier and A. Saupe, “Eine einfache molekular-statistische theorie dernematischen kristallinflussigen phase 2,” Z. Naturforsch. A 15, 287–292(1960).

139. I. R. Girling, N. A. Cade, P. V. Kolinsky, and C. M. Montgomery,“Observation of second-harmonic generation from a Langmuir-Blodgettmonolayer of merocyanine dye,” Electron. Lett. 21(5), 169–170 (1985).

140. I. R. Girling, P. V. Kolinsky, N. A. Cade, J. D. Earls, and I. R. Peterson,“Second harmonic generation from alternating Langmuir-Blodgett films,”Opt. Commun. 55(4), 289–292 (1985).

141. G. J. Ashwell, T. Handa, and R. Ranjan, “Improved second-harmonic gen-eration from homomolecular Langmuir-Blodgett films of a transparentdye,” J. Opt. Soc. Am. B 15(1), 466–470 (1998).

142. I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, R. A. Hann, P. F. Gordon, B. D.Bothwell, S. K. Gupta, S. Allen, P. Robin, E. Chastaing, and J. C. Dubois,“Second harmonic generation by Langmuir-Blodgett multilayers of anorganic azo dye,” Europhys. Lett. 3, 803–809 (1987).

143. A. Painelli, “Vibronic contribution to static NLO properties: exact resultsfor the DA dimer,” Chem. Phys. Lett. 285(5–6), 352–358 (1998).


144. S. Polyakov, F. Yoshino, M. Liu, and G. I. Stegeman, “Nonlinear refractionand multi-photon absorption in polydiacetylenes from 1200 to 2200 nm,”Phys. Rev. B 69(11), 115421 (2004).

145. D. M. Bishop, B. Champagne, and B. Kirtman, “Relationship betweenstatic vibrational and electronic hyperpolarizabilities of π-conjugatedpush-pull molecules within the two-state valence-bond charge-transfermodel,” J. Chem. Phys. 109(22), 9987–9994 (1998).

146. V. Chernyak, S. Tretiak, and S. Mukamel, “Electronic versus vibrationaloptical nonlinearities of push-pull polymers,” Chem. Phys. Lett. 319(3–4),261–264 (2000).

147. D. M. Bishop, B. Champagne, and B. Kirtman, “Comment on ‘Electronicversus vibrational optical nonlinearities of push–pull polymers,’” Chem.Phys. Lett. 329(3–4), 329–330 (2000).

148. G. P. Das, A. T. Yeates, and D. Dudis, “Vibronic contribution to staticmolecular hyperpolarizabilties,”Chem. Phys. Lett. 212(6), 671–676 (1993).

149. B. Kirtman and B. Champagne, “Nonlinear optical properties of quasilinearconjugated oligomers, polymers and organic molecules,” Int. Rev. Phys.Chem. 16(4), 389–420 (1997).

150. H. Hui, S. Webster, D. Hagan, and E. Van Stryland, CREOL, University ofCentral Florida, are working on a manuscript, title and journal to bedetermined.

151. S. J. Lalama, K. D. Singer, A. F. Garito, and K. N. Desai, “Exceptionalsecond-order non-linear optical susceptibilities of quinoid systems,” Appl.Phys. Lett. 39(12), 940–942 (1981).

152. J. W. Wu, J. R. Heflin, R. A. Norwood, K. Y. Wong, O. Zamani-Khamiri,A. F. Garito, P. Kalyanaraman, and J. Sounik, “Nonlinear optical processesin lower-dimensional conjugated structures,” J. Opt. Soc. Am. B 6(4),707–720 (1989).

153. J.R.Heflin,Y.M.Cai,andA.F.Garito,“Dispersionmeasurementsofelectric-field-induced second-harmonic generation and third-harmonic generation inconjugated linear chains,” J. Opt. Soc. Am. B 8(10), 2132–2147 (1991).

154. D. C. Rodenberger, J. R. Heflin, and A. F. Garito, “Excited-state enhance-ment of third-order nonlinear optical responses in conjugated organicchains,” Phys. Rev. A 51(4), 3234–3245 (1995).

155. J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito,“Symmetry-controlled electron correlation mechanism for third order non-linear optical properties of conjugated linear chains,” Mol. Cryst. Liq.Cryst. 160, 37–51 (1988).

156. J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, “Non-linear optical properties of linear chains and electron-correlation effects,”Phys. Rev. B 38(2), 1573–1576 (1988).

157. M. G. Kuzyk and C. W. Dirk, “Effects of centrosymmetry on the nonre-sonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A41(9), 5098–5109 (1990).

158. S. Shafei and M. G. Kuzyk, “Critical role of the energy spectrum in de-termining the nonlinear optical response of a quantum system,” J. Opt. Soc.Am. B 28(4), 882–891 (2011).

159. M. G. Kuzyk, “A bird’s-eye view of nonlinear optical processes: unificationthrough scale invariance,” Nonlinear Opt. Quantum Opt. 40, 1–13 (2010).

160. J. Pérez-Moreno and M. G. Kuzyk, “Comment on ‘Organometallic com-plexes for nonlinear optics. 45. Dispersion of the third-order nonlinear


optical properties of triphenylamine-cored alkynylruthenium dendrimers’—Increasing the nonlinear optical response by two orders of magnitude,”Adv. Mater. 23(12), 1428–1432 (2011).

161. M. G. Kuzyk, “Using fundamental principles to understand and optimizenonlinear optical materials,” J. Mater. Chem. 19(40), 7444–7465 (2009).

162. J. Pérez-Moreno, S.-T. Hung, M. G. Kuzyk, J. Zhou, S. K. Ramini, and K.Clays, “Experimental verification of a self-consistent theory of the first-,second-, and third-order (non)linear optical response,” Phys. Rev. A 84(3),033837 (2011).

Mark G. Kuzyk received the B.A. (1979), M.S. (1981), andPh.D. (1985) degrees in physics from the University ofPennsylvania. He was a Member of Technical Staff atAT&T Bell Laboratories until 1990, then became a facultymember at Washington State University, Pullman, where hewas also the Boeing Distinguished Professor of Physics andMaterials Science and is now Regents Professor. He is a Fel-low of the Optical Society of America, the American Phy-

sical Society, and SPIE; was an Associate Chair of Physics and the Chair ofthe Materials Science Program; and presented the 2005 WSU Distinguished Fa-culty Address. He served as topical editor for JOSA B and is one of the foundersof the ICONO conferences on organic nonlinear optics. In his spare time, heplays ice hockey with The Geezers.

Kenneth Singer is Ambrose Swasey Professor of Physicsand Director of the Engineering Physics Program at CaseWestern Reserve University. He received his B.S. summacum laude in physics from the Ohio State University in1975 and Ph.D. in physics from the University ofPennsylvania in 1981. He was a Member of Technical Staffat Bell Laboratories from 1982 to 1989, and DistinguishedMember of Technical Staff from 1989 to 1990. From 1990 to

1993 he was the Warren E. Rupp Associate Professor of Physics at Case. Singeris a Fellow of both the American Physical Society and the Optical Society ofAmerica and has served as topical editor of JOSA B.

George I. Stegeman received his Ph.D. from the Universityof Toronto and is the first recipient of the Cobb Family Chairin Optical Sciences and Engineering. The principal interestof Dr. Stegeman’s research group is the experimental studyof nonlinear optics in waveguide structures, especially theproperties of spatial solitons in various regions of the elec-tromagnetic spectrum. Of particular interest are solitons inphotonic crystals, in semiconductor optical amplifiers, in

quasi-phase-matched doubling crystals, and in the discrete systems affordedby coupled arrays of channel waveguides. He is a Fellow of the Optical Societyof America and has received the Hertzberg Medal for Achievement in Physics ofthe Canadian Association of Physicists and the R. Woods Prize of the OpticalSociety of America.


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