Download Theory of Molecular Nonlinear Optics

Theory of Molecular Nonlinear Optics
Mark G. Kuzyk,1,* Kenneth D. Singer,2 and George I. Stegeman3,4
1
Department of Physics and Astronomy, Washington State University, Pullman,
Washington 99164-2814, USA
2
Department of Physics, Case Western Reserve University, Cleveland,
Ohio 44106-7079, USA
3
College of Engineering, King Fahd University of Petroleum and Minerals,
P.O. Box 5005, Dhahran 31261, Saudi Arabia
4
College 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 molecule is treated by first-order perturbation theory, and expressions are derived for
the linear (αij ) polarizability and nonlinear (βijk , γ ijkl ) molecular hyperpolarizabilities in terms of the properties of the molecular states and the electric dipole
transition moments for light-induced transitions between them. Scale invariance
is used to estimate fundamental limits for these polarizabilities. The crucial role
of the spatial symmetry of both the single molecules and their ordering in
dense media, and the transition from the single molecule to the dense medium
2 3
case (susceptibilities χ 1
ij , χ ijk , χ ijkl ), is discussed. For example, for β ijk , symmetry determines whether a molecule can support second-order nonlinear processes
or not. For asymmetric molecules, examples of the frequency dispersion based
on a two-level model (ground state and one excited state) are the simplest possible for βijk and examples of the resulting frequency dispersion are given. The
third-order susceptibility is too complicated to yield simple results in terms of
symmetry properties. It will be shown that whereas a two-level model suffices
for asymmetric molecules, symmetric molecules require a minimum of three
levels in order to describe effects such as two-photon absorption. The frequency
dispersion of the third-order susceptibility will be shown and the importance of
one and two-photon transitions will be discussed. © 2013 Optical Society of
America
OCIS codes: 190.4710, 020.4180
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Outline of Review Paper. . . . . . . . . . . . . . . . . . . . . . . . .
2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. Definition of the Microscopic Nonlinear Susceptibilities . . .
2.2. Sum-over-States Theory for the Nonlinear Optical Response
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1943-8206/13/010004-79$15/0$15.00 © OSA
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2.2a. Traditional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2b. Dipole-Free SOS Expressions . . . . . . . . . . . . . . . . . . . . 19
2.3. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Molecular Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1. Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⃗⃗
3.2. Irreducible Tensor Approach to β⃗ Molecular Nonlinear Optics . . 28
3.3. Two-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3a. Two-Level Model: χ 2 . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3b. Two-Level Model: χ 3 . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3c. First-Order Effect on χ 3 of Population Changes in Two-Level
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4. Symmetric Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1. General Sum-over-States Model . . . . . . . . . . . . . . . . . . . . . . . 46
4.2. Three-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5. Transition to Bulk Nonlinear Molecular Optics . . . . . . . . . . . . . . . . 51
5.1. Local Field Corrections, Linear Susceptibility . . . . . . . . . . . . . 52
5.1a. Continuum Approximation. . . . . . . . . . . . . . . . . . . . . . . 52
5.1b. Nondipolar Homogeneous Liquids and Solids. . . . . . . . . . 53
5.1c. Nondipolar Two-Component System . . . . . . . . . . . . . . . . 54
5.2. Oriented Gas Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3. Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4. Electric Field Poled Media . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5. Additional Contributions to Third-Order Nonlinearities . . . . . . . 63
6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Appendix A: Cartesian Tensor Decomposition . . . . . . . . . . . . . . . . . . 66
Appendix B: More Sophisticated Local Field Effects: Screening and Dressed
Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B.1. Local Field Model of a Two-Component Dipolar Composite . . . 69
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
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Theory of Molecular Nonlinear Optics
Mark G. Kuzyk, Kenneth D. Singer, and George I. Stegeman
1. Introduction
Kerr, a 19th century experimentalist, was the first to observe nonlinear optical
effects when he determined the refractive index change of collimated and spectrally filtered sunlight in response to a voltage applied to organic liquids [1–3].
However, the development of the Q-switched ruby laser first suggested by
Maiman [4] and realized by McClung and Hellwarth [5] not only marked the
birth of the laser, but also opened the door for the following explosion of interest
in nonlinear optics made possible by the sublimely intense light of that laser.
Indeed, shortly after that development of the laser, optical harmonic generation
in quartz crystal was reported by Franken and collaborators [6]. In this case, light
at twice the incident frequency was observed. Since the second-harmonic signal
was weak, the tiny spot on the photographic film appeared as an imperfection in
the film. Legend has it that the small speck was removed in the production office
when 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 Herman
and Hayden [9], showed how the interference between propagating light (i.e., the
homogeneous solution to the wave equation) and bound polarization waves (solutions to the inhomogeneous wave equation driven by the nonlinear polarization)
due to refractive index dispersion leads to interference fringes that can be used to
determine the nonlinear optical response of a slab of material, and that limits the
generation of a second harmonic. It was soon discovered by Giordmaine [10] that
birefringence can be used to cancel the dispersion leading to copious phasematched second-harmonic generation, opening the door to the applications of
nonlinear optics, so that now that speck on the film could be transformed into
an intense laser beam at the second-harmonic frequency. To this day, the a major
application of nonlinear optical devices involves tuning of solid-state pulsed
lasers using various parametric nonlinear optical devices including harmonic
generation, as well as parametric oscillation and amplification [11,12].
Bloembergen’s 1965 monograph [7] delineated much of the physics of nonlinear
optics and laid the foundation for a great deal of work in the coming decades. His
work earned him the Nobel Prize for nonlinear optics in 1981. A plethora of nonlinear optical phenomena described in that monograph, as well as others, has been
a rich source of research comprising over 15,000 publications since. Phenomena
include higher harmonics, intensity-dependent refractive index, multiphoton
absorption, photorefraction, various forms of Raman spectroscopy, and others.
These phenomena arise from the nonlinear optical response functions of materials, 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 the
most attention, given that both the lowest-order nonlinear optical effects and
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piezoelectricity require materials without an inversion center. Though the symmetry properties are similar, the physical origin of the nonlinear optical response
is distinct from piezoelectricity. It was obvious from the varying response in
different materials that methods to understand the physical origin of the response
needed to be developed. Beyond the classical anharmonic oscillator approach
described as a simple model by Bloembergen, attention soon turned to more
realistic models based on the interaction of light with matter using quantum
mechanical descriptions of materials. This was a daunting task in the case of
covalent or ionic crystals, as the methods of solid-state physics to determine
the complete band structure for application in perturbation theory had not been
developed. As a consequence, phenomenological models grounded in quantum
principles were developed, such as the polarization potential tensor [13,14] and
the bond-charge model [15].
Miller observed early on that a parameter later known as “Miller’s delta” could
be used to define the nonlinear optical response of piezoelectric crystals in terms
of the linear optical response, so that
1
1
1
2ω
χ 2
ijk 2ω χ ii 2ωχ jj 2ωχ kk 2ωδijk ;
(1)
where χ 2
ijk 2ω is the second-harmonic nonlinear optical susceptibility to be de2ω
fined below, χ 1
uu 2ω the linear susceptibility, and δijk the Miller’s delta parameter. Remarkably, for oxide crystals it varied little from crystal to crystal even
though the nonlinear susceptibility varied over orders of magnitude [16]. Thus,
the intrinsic nonlinearity varied little, and the observed differences in frequency
conversion, for example, arose from phonon and crystal structure contributions
to the linear optical susceptibility.
At the same time, Kurtz and Perry developed a simple powder technique for
quickly assessing new crystals. First a crystal is ground into fine powders that
are 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 to
quickly estimate the second-harmonic coefficients and phase-matching potential
[17]. This technique was applied to study a series of organic crystals, where
efficient second-harmonic generation was observed [18]. Notably, studies of
the nonlinear susceptibility of organic single crystals were found to have a distinctly large Miller’s delta, in contrast to inorganic crystals [19,20]. This sparked
the study of organic materials for nonlinear optics and the elucidation of the
underlying physics of molecular materials.
The early studies of nonlinear optics and materials focused on the inorganic
solids described above, whose structure consists of periodic atoms bound by
covalent or ionic forces. This parallels the development of electronic materials
and 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 of
safe, low-cost photocopiers. The ensuing studies established the principles of
understanding the electronic and optical properties of these materials [23].
The distinctive features of these organic crystals are that they are ordered arrays
of complex organic molecules with conjugated electronic systems weakly bound
in crystals by van der Waals forces (and sometimes hydrogen bonds). These are
defining characteristics of molecular solids. The van der Waals binding between
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molecules implies that many of the optical and electronic properties of the solid
materials can be understood by studying these properties in the constituent
molecules and where the macroscopic properties require the methods of
statistical physics to relate the molecular properties to the corresponding properties of the solids. Thus, the molecular properties are amenable to description
using quantum mechanics, with statistical physics applied to collections of
molecules (crystals, glasses, polymers, membranes, etc.), yielding the
macroscopic electronic and optical properties.
Conjugated electron systems consist of networks (rings, chains) of alternating
single and multiple bonds, with the examples drawn from this work shown in
Fig. 1. The s‐p orbital hybridization (mixing) results in significant delocalization
of the π-electrons along the conjugated systems. These arrangements result in
highly colored and electronically responsive molecules and solids. Early models
of the optical and nonlinear optical properties of delocalized π-electron systems
include the “particle in a box” by Kuhn [24,25]. As interest focused on lowestorder nonlinear optics and second-harmonic generation, the requirement for a lack
of inversion symmetry at the molecular level required molecules with a dipole
moment, so that electron donor and electron acceptor terminated π-electron structures became the model. Clearer connections to the quantum mechanical descriptions of these structures emerged through connection to charge transfer within the
π-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 the
π-conjugated systems in each.
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models based on these concepts emerged [26,27]. Finally, the development of
quantum chemical techniques applied to perturbation theory to calculate the molecular nonlinear response by Lalama and Garito opened the door to a complete
understanding of and a powerful tool for designing new materials [28]. This work
combined evolving approaches to quantum chemistry with previously developed
sum-over-states approaches to quantum calculations of optical responses
[7,29,30]. The field was now poised to take advantage of the profound chemical
synthesis flexibility to develop new nonlinear optical materials.
To go along with this understanding, methods to measure the molecular nonlinear optical response were required. The first such measurement on molecules
in solution were reported by Levine and Bethea using the electric-field-induced
second-harmonic (EFISH) generation technique, which had originally been developed to measure the second hyperpolarizability of gasses [31]. In this technique, a
static (or quasi-static) electric field is applied to a liquid solution during measurement. This field aligns the molecular dipoles and breaks the inversion symmetry,
allowing second-harmonic generation. The molecular response is obtained by
properly taking into account the number density, alignment in the field, and local
field factors, as we discuss below. Later, an examination of the role of local fields
in such measurements suggested improvements in this technique and confirmed
that Onsager local field models (discussed below) can apply [32]. This work was
followed by more extensive measurements of organic molecules [27,33–35].
A second method for measuring the molecular nonlinear optical response was
developed by Persoons and colleagues later, namely hyper-Rayleigh scattering
(HRS) [36]. This technique uses incoherent second-harmonic scattering off of
solutions of dipolar molecules. The orientational fluctuations of the noncentrosymmetric molecules in a centrosymmetric solution generate a small amount of
scattered second-harmonic light, whose scattering distribution and polarization
yields a significant amount of information on the second-order response function. This technique is especially useful in measuring charged molecules that
cannot sustain an applied low-frequency field for EFISH, but especially for multipolar chromophores that may not possess dipole moments, such as octupoles
and other lower symmetry molecules [37–40].
Once the molecular response is obtained, one requires a statistical mechanical
theory that relates the nonlinear optical response of a molecule to the bulk
second-harmonic response. Such theories have been developed for a number
of cases and are collectively known as oriented gas models since, as we show
below, the nonlinear response is closely tied to rotational symmetries. The
theory that relates the molecular hyperpolarizability and second-order response
of a crystal was first reported by Oudar and Zyss [41,42]. Later, Singer and
associates showed that a second-order nonlinear susceptibility could be imparted
to an isotropic glassy polymer that is doped with aligned molecules that have a
large hyperpolarizability [43]. An electric field is applied to the molecules above
the polymer’s glass transition temperature to align the dipoles, and the orientational order is locked in place when the polymer is cooled below the glass transition temperature. These results led to a flurry of activity in custom-designing
materials for nonlinear optical devices [44].
The first model of the thermodynamic poling necessary for the molecular alignment 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
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that the orientational order relaxed somewhat between the time the polymer was
poled and when it was measured. Use of a cross-linked polymer increased the
lifetime of the orientational order and corona poling enabled large electric field
strengths to be applied [46]. Poled polymers offered the unique advantage of
being processable into thin films and fibers for wave-guiding devices
[47,48]. Thackara and associates showed that an electro-optic waveguide phase
modulator could be made by using a poled thin film [49]. The fact that polymers
provide a good host material for molecules makes it possible to break down the
problem of designing a material to first identifying molecules with the right
properties, including them in a polymer to make it optically nonlinear, then
forming the polymer into a device and poling it where required.
Much of the subsequent work in molecular nonlinear optics was aimed at
high-performance materials for applications in information technology and
signal processing. In particular, materials for the linear electro-optic effect
[46,49,50] have received a great deal of attention and a remarkable level of
development, with the latest results having recently been reviewed [51].
Similarly, third-order nonlinear optical properties in organic molecular materials
have been investigated over the same period [52,53]. Molecular materials for
terahertz 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 optical
materials, studies of nonlinear optics of molecular materials at the nanoscale,
mesoscale, and microscale have blossomed into the principal trend of molecular
nonlinear optics, and their spatial and symmetry sensitivity are literally illuminating the science of interfaces, nanostructures, and biological materials. One of
the original studies of second-harmonic in monolayers by Heinz et al. [56]
illustrates the molecular nature of the nonlinear response and its potential for
probing interfaces [57,58] and nanoparticles [59]. Nonlinear optical microscropy
is a rapidly evolving field with important applications in biology and singlemolecule detection [60–65]. Another important trend involves multiphoton
absorption, which has generated interest due to the ability to localize intense
light in a small volume with applications in microscopy and even threedimensional photopatterning [64,66,67].
1.1. Outline of Review Paper
This review article follows a similar bottom-up approach, i.e., starting with microscopic structure leading to bulk materials. We start from the perspective of
the interaction of a single molecule with electromagnetic fields. First-order
perturbation theory is used to derive the sum-over-states (SOS) model for the
molecular linear polarizability αij , the first hyperpolarizability βijk , and the
second hyperpolarizability γ ijkl in terms of the electronic excited states (energy
levels) of a molecule labeled m, their energy E m − E g ℏωmg above the ground
state (Eg ), and the electric dipole transition moments between states m and n,
⃗μmn . A scale invariance approach is then used to estimate fundamental limits for
these polarizabilities. The effect of the inherent reflection and rotational spatial
symmetry on a molecule’s nonlinear optical properties is then discussed. The
susceptibility for different second-order processes, such as second-harmonic
generation, is deduced from the molecular symmetry properties in terms of
the irreducible tensors that reflect the symmetry properties.
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The ordering of the molecules in a dense medium determines the symmetry
properties of a “bulk” medium, and hence the macroscopic susceptibilities
2 3
χ 1
ij ; χ ijk ; χ 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 fields
in neighboring molecules augment the local field at a molecule in dense media,
approximate local field corrections for the different susceptibilities will be derived. However, all the locations of the energy levels and the transition electric
dipole 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 asymmetric molecules, is used to obtain approximate analytical expressions from the
SOS for the second-order nonlinear susceptibilities. The frequency dispersion
in the application’s frequency regions will be discussed and compared to the
popular anharmonic oscillator models.
The third-order susceptibility is too complicated to yield simple results in terms
of symmetry properties (which are tabulated in the literature). The SOS susceptibilities for a single molecule will be corrected for local field effects. The role of
eigenstates of the parity operator are shown to strongly affect the nature of the
nonlinear optical response. In particular, terms due to single photon (i.e., parity
changing) transitions and multiphoton transitions are identified. The focus will
be on molecular media treated in the simple two- and three-level model approximations. It will be shown that symmetric molecules require a minimum of three
levels in order to describe effects such as two-photon absorption. The frequency
dispersion of the third-order susceptibility will be shown for simple cases in
three 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-level
models. The importance of one- and two-photon transitions will be discussed.
These approximate theories simplify in the non-resonant limit and it is shown
that there occurs destructive interference between one- and two-photon transitions so that the sign of the non-resonant nonlinearity depends on which terms
are dominant. Finally, the relative contribution to the nonlinearity caused by a
small population in the excited state due to linear absorption in the two-level
model 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 subpicosecond nonlinearities originally believed to be due to electronic transitions.
2. Theory
2.1. Definition of the Microscopic Nonlinear
Susceptibilities
The nonlinear interaction of light with matter occurs at the site of individual
⃗ is a function of
molecules. The induced dipole moment of a molecule, ⃗pF,
⃗ In the dipole approximation, ⃗pF
⃗ is expressed
the applied local electric field, F.
as a power series of the local field with expansion coefficients, the material
response functions, that are called the polarizability (linear term), first hyperpolarizability (quadratic term), second hyperpolarizability (cubic term), etc. The
nth-order contribution to the induced dipole moment, pn
i ω, in the convention
we adopt is of the form
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pn
i ω 1
ε ξn −ω; ω1 ; ω2 …ωn F j ω1 …F ℓ ωn ;
n−1 0 ijk…ℓ
2
(2)
⃗
where Fω,
the electric field at frequency ω at the molecular site, is written in
the form (adopted here for all fields)
1⃗
−iωt
⃗
Ft
Fωe
c:c:;
2
⃗
⃗
F−ω
Fω:
(3)
ξn
ijk…ℓ is also sometimes called the molecular nth-order nonlinear optical susceptibility tensor, and ω (ω ω1 ω2 ; … ωn ) is the frequency of the dipole
oscillation excited by the mixing of n electric fields at frequencies ω1 ; ω2 ; …ωn .
Here we use the Einstein summation convention (double indices are summed
⃗
over the three Cartesian coordinates). The local field Ft
will be described
in more detail in Section 5 and Appendix B.
2.2. Sum-over-States Theory for the Nonlinear Optical
Response
Here we present a brief derivation of the sum-over-states (SOS) quantum theory
of the nonlinear optical response of a single molecule.
2.2a. Traditional Approach
The nonlinear susceptibility of a quantum system starts with the calculation of
the induced dipole moment as a function of the electric field, which is expanded
as a Taylor series in the electric fields. The coefficients of the various powers of
the field yield the nonlinear susceptibilities [7,11,68,69]. The dipole moment is
simply given by the expectation value of the dipole moment using the groundstate wave function of the molecule that includes coupling to the applied electric
fields.
The mth energy eigenstate of an atom or molecule in the presence of the local
⃗
electric field is given by jψ m Fi,
where
⃗
Ft
no: incident
X fields
F⃗ p ωp ; t:
(4)
p1
At zero temperature, the polarization is given by
⃗ F
⃗ hψ g Ftj
⃗
⃗ g Fti;
⃗
P
Pjψ
(5)
⃗
is the perturbed ground state. The generalized molecular
where jψ g Fti
susceptibility is then given by
ξn
ijk…ℓ −ω;ω1 ;ω2 ;ωn 1
∂n
⃗
⃗
hψ FjP
;
⃗
i jψ g FijF0
ε0 D0 ∂F j ω1 ∂F k ω2 …∂F ℓ ωn g
(6)
where D0 is the frequency-dependent degeneracy denominator that depends on
the 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 discussed
further.
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When the electric field of the light is much weaker than the electric fields that
hold a molecule together, the wave function of the molecule under the influence
of an optical field can be determined using perturbation theory with the zerofield wave functions as a basis set. The perturbation potential is simply the timedependent electric dipole coupling energy between the field and the molecule.
Defining H 0 as the unperturbed Hamiltonian (i.e., with the light turned off), the
time evolution of a state jψi is given by
iℏ
∂
jψi H 0 jψi:
∂t
(7)
With electric dipole coupling, the time-dependent perturbation potential is given
by
X
V t −
⃗μ · F⃗ p t;
(8)
p
where ⃗μ is the dipole moment (either induced or permanent) of the molecule and
p spans all distinct photon fields. The total Hamiltonian, H, is then the sum of the
molecular Hamiltonian and the perturbation potential multiplied by a small
perturbation parameter λ,
H H 0 λV :
(9)
With unperturbed eigenstates of the form
0 −iω̂m t
jψ 0
;
m ti jψ m ie
(10)
where ω̂m ωm − iτ−1
m with ωm E m ∕ℏ, τ m is the lifetime of the mth excited
0
state, and jψ 0
m i is the spatial eigenstates of the unperturbed eigenfunctions ψ m ,
the perturbed states can be expressed as a sum of increasing orders of
correction, indexed by s, that are labeled λs ,
jψ m ti ∞
X
λs jψ s
m ti:
(11)
s0
Here the “hat” above the frequency ωm identifies it as a complex quantity.
Substituting Eq. (11) into the Schrödinger equation, and keeping only terms
of order s,
iℏ
∂ s
s−1
jψ m ti H 0 jψ s
ti:
m ti V tjψ m
∂t
(12)
Since the eigenvectors jψ s
m ti can be expressed in terms of the unperturbed
0
states jψ ℓ ti with coefficients as
mℓ t,
X s
jψ s
amℓ tjψ 0
(13)
m ti ℓ ti;
ℓ
the ground-state wave function is given by
X s
aℓ tjψ 0
jψ s
g ti ℓ ti;
(14)
ℓ
s
where as
l t agl t. Substituting Eq. (14) into Eq. (12) yields
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iℏ
X
ℓ
X
0
a_ s
ℓ tjψ ℓ ti ℓ
0
as
ℓ t−iωℓ jψ ℓ ti
X s
X s−1
E ℓ aℓ tjψ 0
aℓ tjψ 0
ℓ ti V t
ℓ ti:
ℓ
(15)
ℓ
Operating on Eq. (15) from the left with hψ 0
m j, we get
−iω̂m t
−iω̂m t
−iω̂m t
iℏ_as
ℏωm as
Em as
m te
m te
m te
X 0
s−1
hψ m jV tjψ 0
te−iω̂ℓ t : (16)
ℓ iaℓ
ℓ
0
s
_m
t is given by
Defining ω̂mℓ ω̂m − ω̂ℓ and V mℓ t hψ 0
m jV tjψ ℓ i, a
a_ s
m t 1X
V tas−1
teiω̂mℓ t :
ℓ
iℏ ℓ mℓ
(17)
Integration of Eq. (12) gives
as
m t
1X
iℏ ℓ
Z
t
−∞
V mℓ tas−1
teiω̂mℓ t dt:
ℓ
(18)
With the system initially in its ground state, a0
ℓ δℓ;g . Equation (18) with the
help of Eqs. (8) and (3) gives
Z
1X t
1X⃗
1
am t ⃗μmg ·
(19)
Fωp e−iωp t eiω̂mg t dt:
iℏ ℓ −∞
2 p
−1
The integral at negative infinity vanishes since ω̂mg ωmg − iτ−1
mg , where τmg is
the decay time for electrons in the mth state to decay to the ground state, so
Eq. (19) yields in the mth state,
a1
m t ⃗
1 X ⃗μmg · Fω
p iω̂mg −ωp t
e
:
2ℏ p ω̂mg − ωp
(20)
The coefficient a2
v t is derived by substituting Eq. (20) into Eq. (18),
a2
v t ⃗
⃗
1 X X ⃗μvm · Fω
q ⃗μmg · Fωp iω̂vg −ωp −ωq t
e
;
4ℏ2 p;q m ω̂vg − ωp − ωq ω̂mg − ωp (21)
and a3
d is calculated by substituting Eq. (21) into Eq. (18):
a3
d t ⃗
⃗
⃗
⃗μdv · Fω
1 XX
r ⃗μvm · Fωq ⃗μmg · Fωp 3
8ℏ p;q;r d;v;m ω̂dg − ωp − ωq − ωr ω̂vg − ωp − ωq ω̂mg − ωp × eiω̂dg −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 molecule will remain in the perturbed ground state in the presence of the field; but,
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14
because the field is time dependent, the ground-state wave function will evolve
according to
X 2
X 1
iω̂ t
2
iω̂ t
jψ g ti jψ 0
am tjψ 0
av tjψ 0
g iλ
m ie mg λ
v ie vg ;
m
v
(23)
where
0 −iω̂mg t
jψ 0
:
m ti jψ m ie
(24)
Using Eq. (24), the expectation value of the dipole moment will be of the form
X 1
0 iω̂mg t
h ⃗μit hψ 0
j
λ
a
thψ
je
⃗μ
g
m
m
m
X
X 1
0 −iω̂mg t
−iω̂mg t
am tjψ 0
λ2 a2
: (25)
× jψ 0
g iλ
m ie
m tjψ m ie
m
m
The induced dipole moment to first order in λ from Eq. (25) is
X 1
X 1
0
iω̂ t 0
−iω̂mg t
h ⃗μi1 t am thψ 0
am tjψ 0
: (26)
g j ⃗μe mg jψ m i hψ m j ⃗μ
g ie
m
m
Using Eqs. (26) and (20), the fact that ⃗μmg ⃗μgm and ω−q −ωq , and some
manipulation yields
1 XX
h ⃗μi t 2ℏ m p
1
⃗
⃗
⃗μgm · Fω
⃗μmg · Fω
p ⃗μmg
p ⃗μgm
e−iωp t c:c:; (27)
ω̂mg ωp
ω̂mg − ωp
0
where we have used the shorthand notation ⃗μℓ0 ℓ hψ 0
ℓ0 j ⃗μjψ ℓ i for arbitrary
0
states ℓ and ℓ.
Since the linear molecular polarization is usually written in terms of the electromagnetically induced molecular dipole moment given by ⃗pt h ⃗μi1 t,
⃗pt 1X
⃗pωp e−iωp t ⃗p−ωp eiωp t ;
2 p
⃗
⃗
⃗μmg · Fω
1 X ⃗μgm · Fω
p ⃗μmg
p ⃗μgm
⃗pωp ;
ω̂mg ωp
ω̂mg − ωp
ℏ m
(28)
(29)
where the summations over p and m are over all of the frequencies in the input
and over all of the discrete states of the molecule, respectively. The contribution
to the first-order linear susceptibility [usually called the linear molecular polarizability α1
ij −ωp ; ωp ] of the Maxwell field E i ωp F i ωp ∕Lωp is calculated by using Eq. (6) and yields
μgm;i μmg;j μgm;j μmg;i
1
1
Lωp ;
(30)
αij −ωp ; ωp ε0 ℏ
ω̂mg ωp ω̂mg − ωp
where μgm;j is the jth Cartesian component of ⃗μgm and where by definition
α1
ij −ωp ; ωp assigns i to be in the direction of the polarization ⃗pωp , and j
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15
⃗
to be in the direction of the applied field Eω
p . Note that the orientation of the
axes for the polarizability is arbitrary. We now fix the axes as those in which
α1
ij −ωp ; ωp is a diagonal tensor.
The second- and third-order (and higher order) susceptibilities are calculated
using a similar approach by calculating h ⃗μin t from h ⃗μit to order λn and
using Eq. (25) and projecting out the ω Fourier component. The second-order
molecular nonlinear polarization ⃗p2 ω is then given by
⃗
⃗
1 X X ⃗μgv ⃗μvm · Fω
q ⃗μmg · Fωp ⃗pN L ω ωp ωq 2
ω̂vg − ωp ∓ωq ω̂mg − ωp ε0 ℏ qp v;m
⃗
⃗
⃗μgv · Fω
q ⃗μvm ⃗μmg · Fωp ω̂mg ωp ω̂vg ∓ωq ⃗
⃗
⃗μgv · Fω
q ⃗μvm · Fωp ⃗μmg
:
ω̂mg ωp ω̂vg ωp ωq (31)
Note that the summations over p and q are both over all of the incident fields and
v and m over all of the states.
As first pointed out by Bloembergen and associates, the Maxwell nonlinear polarization is not given by simply multiplying each of the incident local fields by a
local field correction factor because there are fields present at all the frequencies,
⃗
including at the frequency generated by the nonlinear interaction, Eω
[7].
Consider the local field problem for which a second-order, nonlinear Maxwell
⃗
polarization Pω
ωp ωq exists throughout the medium at the nonlinearly
generated frequency ω ωp ωq due to the nonlinear interaction of the
Maxwell field with the medium. The local field factor Lω, as described more
fully in Section 5 and Appendix B, is given by
1 ⃗
⃗
⃗
⃗
Pω LωEω:
(32)
Fω
Eω
3ε0
Including now the nonlinear polarization field ⃗pNL ω induced at the molecule
by the mixing of fields at the molecule and the “cavity” field at the molecule at
frequency ω due to contribution from all the other molecules outside the cavity,
the total Maxwell polarization at the molecule is
1
⃗
⃗
⃗
(33)
→ ⃗pω N ⃗α · Eω
Pω
N ⃗pN L ω;
3ε0
where N is the molecular density. Hence the total molecular polarization at
frequency ω, including the Lorentz–Lorenz local field factor, is
i
εr ω 2 h ⃗ ⃗
⃗α · Eω p̄~NL ω :
(34)
→ ⃗pω 3
Now defining the nonlinear polarization component ⃗p2 ⃗r; t in the usual way as
1 X 2
⃗p ωp ωq e−iωp ωq t c:c:
⃗p2 ⃗r; t 2 q;p
1 X ⃗⃗⃗
−iωp ωq t
⃗
⃗
ε0 β−ω
c:c:;
p ωq ; ωp ; ωq ∶Eωp Eωq e
4 q:p
(35)
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16
⃗⃗
⃗
where β−ω
p ωq ; ωp ; ωq is defined as the second-order susceptibility with
⃗
⃗
~
1 εr ωp ωq 2 X μ̄~ gn μ̄~ nm · Fω
q μ̄mg · Fωp ⃗p2 ωp ωq 2
3
ω̂ng ∓ωq − ωp ω̂mg − ωp 2ℏ
n;m
⃗
⃗
~ ~
μ̄~ gn · Fω
q μ̄nm μ̄mg · Fωp ω̂ng ωq ω̂ng − ωp ⃗
⃗
~
~
μ̄~ gn · Fω
q μ̄nm · Fωp μ̄mg
;
ω̂mg ωq ωp ω̂ng ωq (36)
yields
βijk −ωp ωq ; ωp ; ωq 1
Lωp ωq Lωp Lωq ℏ ε0
X
μgn;i μnm;k μ̄mg;j
×
ˆ
ˆ
nm ω̄ng ∓ ωq − ωp ω̄mg − ωp 2
μgn;k μnm;i μ̄mg;j
ˆ
ˆ mg − ωp ω̄ng ωq ω̄
μnm;j μgn;k μ̄mg;j
:
ˆ mg ωq ωp ω̄ˆ ng ωq ω̄
(37)
Note that the nonlinear local field correction is
Lωp ωq Lωp Lωq εr ωp ωq 2 εr ωp 2 εr ωq 2
;
3
3
3
(38)
that is, it contains an extra factor relative to the linear case at the generated
frequency ωp ωq . After some manipulations, the ground state is found to
be excluded from the sum, leading to [29,70]
βijk −ωp ωq ; ωp ; ωq 1
Lωp ωq Lωp Lωq ε 0 ℏ2
X0 μgn;i μnm;k − μgg;k μmg;j
×
ω̂ng − ωq ∓ ωp ω̂mg − ωp nm
μgn;k μnm;j − μgg;j μmg;i
ω̂ng ωq ω̂mg ωp ωq μgn;k μnm;i − μgg;i μ̄mg;j
;
ω̂ng ωq ω̂mg ωq (39)
where the prime over the summations indicates that the sum excludes the ground
state, and terms like μvm;j have been replaced by μvm;j − μgg;j .
An alternative definition for the second-order susceptibility is given by
⃗p2 ⃗r; t 1 X 2
⃗p ωp ωq e−iωp ωq t c:c:
2 q;p
⃗⃗
1
−iωp ωq t
⃗
⃗
⃗
ε0 PI β−ω
c:c:;
p ωq ; ωp ; ωq ∶Eωp Eωq e
4
(40)
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17
where PI is the “intrinsic permutation operator” that directs us to take the
average over all permutations of ωq and ωp with simultaneous permutations
of the Cartesian components. For example, the first term in brackets in
Eq. (39) under permutation yields
μgv;i μvm;j μmg;k
μgv;i μvm;j μmg;k
1
PI
ω̂vg − ωp − ωq ω̂mg − ωp 2 ω̂vg − ωp − ωq ω̂mg − ωp μgv;i μvm;k μmg;j
:
ω̂vg − ωq − ωp ω̂mg − ωq (41)
Using the same approach, the third-order susceptibility with the minor change in
notation that the excited states are m, n, and v is given by
1
LωLωp Lωq Lωr PI
ε 0 ℏ3
"
X0 μgv;i μνn;l − μgg;l μnm;k − μgg;k μmg;j
×
x
ω̂νg − ωp − ωq − ωr ω̂ng − ωp − ωq ω̂mg − ωp v;n;m
γ ijkl −ω; ωp ; ωq ; ωr μgv;j μvn;k − μgg;k μnm;i − μgg;i μmg;l
ω̂νg ωp ω̂ng ωp ωq ω̂mg − ωr μgv;l μvn;i − μgg;i μnm;k − μgg;k μmg;j
ω̂νg ωr ω̂ng − ωp − ωq ω̂mg − ωp μgv;j μνn;k − μgg;k μnm;l − μgg;l μmg;i
ω̂νg ωp ω̂ng ωp ωq ω̂mg ωp ωq ωr X0 μgn;i μng;l μgm;k μmg;j
−
ω̂ng − ωp − ωq − ωr ω̂ng − ωr ω̂mg − ωp μgn;i μng;l μgm;k μmg;j
ω̂mg ωq ω̂ng − ωr ω̂mg − ωp μgn;l μng;i μgm;j μmg;k
ω̂ng ωr ω̂mg ωp ω̂mg − ωq #
μgn;l μng;i μgm;j μmg;k
; (42)
ω̂ng ωr ω̂mg ωp ω̂ng ωp ωq ωr n;m
with the permutation parameter
P P PPI again signifying a summation over all the
fields three times, i.e., p q r .
An important limit for both the nonlinear susceptibilities is the zero frequency
limit, called the Kleinman limit in the literature, in which all of the input
frequencies are set to zero. With ωng τng ≫ 1, which is usually the case, Eqs. (37)
and (42) are greatly simplified, namely,
ωng ≫ωp
βijk −ωp ωq ; ωp ; ωq !
×
X0
n;m
1
Lωp ωq Lωp Lωq ε0 ℏ 2
1
fμ μ
− μgg;k μmg;j μgn;k μnm;j − μgg;j μmg;i
ωng ωmg gn;i nm;k
μgn;k μnm;i − μgg;i μmg;j g;
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(43)
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γ ijkl −ω; ωp ; ωq ; ωr X0
1
1
LωLωp Lωq Lωr PI
3
ω
ω
ω
ε0 ℏ
v;n;m vg ng mg
× 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;i g
X0 1
fμ μ μ μ
−2
ωng ωmg gn;i ng;l gm;k mg;j
n;m
μgn;l μng;i μgm;j μmg;k g;
(44)
respectively.
It is now useful to write the total polarization of a molecule up to third order in
powers of the Maxwell electric field as
pi p0i αij Ej βijk E j Ek γ ijkl Ej E k El ;
(45a)
where p0i is the ground-state dipole moment, αij is the polarizability tensor,
and βijk and γ ijkl 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 perturbation theory and depend on the matrix elements of the dipole operator and the
energy eigenvalues of the Hamiltonian. The complexity of this expression makes
it difficult to ascertain how the nonlinear-optical susceptibility depends on the
underlying system’s properties. One method of simplification is to truncate the
SOS expressions to include only a finite number of states. Indeed, the two and
three-level models have been quite successful at modeling the first and second
hyperpolarizability of molecules.
A more rigorous simplification comes about from the fact that there are fundamental relationships among the transition moments and energy eigenvalues. As
such, the dipole matrix elements and energies may not be arbitrarily varied
without violating quantum mechanical principles. The sum rules can be used
to simplify the SOS expression by re-expressing all dipole moments (the diagonal elements of the dipole matrix) in terms of transition dipole moments (the
nondiagonal elements). The result is commonly referred to as the dipole-free
expression and is derived as follows.
All solutions of the Schrödinger equation must obey the Thomas–Kuhn sum
rules [70–72]. The generalized sum rules relate the matrix elements and energies
to each other according to [73]
∞ X
1
ℏ2 N e
δ ;
(45b)
En − Em Ev μmn μnv 2me m;v
2
n0
where me is the mass of the electron and N e is the number of electrons in the
molecule. 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,
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19
the parameters in Eqs. (30), (41), and (42) (i.e., dipole matrix elements and
energies) cannot be independently varied because they are related to each other.
Indeed, it was shown that an N -level model for the nonlinear response, which
depends on a dipole matrix with elements and N energy levels, has enough sum
rule equations to reduce the total number of parameters to N − 1 [74]. For the
second-order nonlinearity,
∞ X
∞
X
0
0 μgn;x μnm;x μmg;x
N
Lω1 Lω2 LωPI
βxxx −ω; ω1 ; ω2 −
ˆ mg ωj 2ε0 ℏ
ω̄ˆ ωi ω̄
n0 m≠n ng
ω̂mg ωj f2ω̂mg − ω̂ng g
;
(46)
× 1−
ˆ ng ωj ω̂mg
ω̄
where ω̄ˆ ng ωi ω̂ng − ωi for any general state n and frequency ωi and where all
permutations of ωi and ωj are permuted over all frequencies ω1 , ω2 , and ω. The
permutation operator PI directs us to average over the exchange of the two input
frequencies ωi and ωj . The second term in brackets is the dispersion term that
results when the sum rules are used to re-express the dipolar terms (i.e., the ones
with dipole moment differences) in terms of the transition moments.
In the standard SOS expression, the simplest approximation is the two-level
model, with parameters μ10;x , Δμ10;x μ11;x − μ00;x , E10 , and E20 . The simplest
approximation 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 approximation has two additional terms, which include a dipole moment difference
between the ground state and the first excited state and a dipole moment difference between the ground state and the second excited state. Thus, the
dipole-free expression has fewer parameters when truncated to the same number
of states. This reduction in parameters results from the sum rules, making
the dipole-free form more parsimonious when fitting data. However, the
SOS and dipole-free expressions usually differ when truncated. Which one
better describes the data depends on the quantum system involved.
The third-order nonlinear optical susceptibility can also be rewritten in dipolefree 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 useful
tool since it requires the determination of fewer parameters when modeling the
nonlinear susceptibility.
Now the researcher has two choices for modeling the nonlinear optical response
of a quantum system. Interestingly, when the SOS expression is truncated to a
finite number of terms, the traditional expression and the dipole-free expression
differ. Only in the infinite number of state limit do the two converge. One expression may be more useful or accurate than the other, depending on the system
and how it is being applied. For example, for a system whose nonlinear response
is at the fundamental limit, both expressions should again converge in the threelevel model. In the analyses that follow, we will use the traditional SOS
expression.
2.3. Scaling
Historically, the motivation for understanding of the nonlinear-optical properties
of molecules was fueled by their potential usefulness in making materials for
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20
nonlinear-optical devices such as electro-optic modulators and harmonic
generators, and more recently, in applications of 3D photolithography and
bioimaging. As such, the focus has been on the magnitude of the nonlinear
susceptibilities.
The nonlinear susceptibility increases with the size of the molecule and the number of electrons within. A more interesting fundamental question pertains to
what 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 effects
properly taken into account. Size effects are best accounted for by using scaling
laws. These laws determine how the nonlinear susceptibilities grow with the size
of 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 good
estimate of size. Similarly, adding extra electrons increases the strength of the
nonlinearity. Once the effects of “size” can be removed from the equation,
molecules of all shapes and sizes can be compared with each other to search
for the fundamental properties that most affect the nonlinear-optical response.
As an example, the susceptibility of a harmonic oscillator depends on the spring
constant, but, once scaling is taken into account, all harmonic oscillators are
equivalent.
At issue is the fact that most molecules are too complex to break the problem
down into such a simple argument. The intrinsic nonlinear-optical susceptibility
strips away the unimportant stuff and leaves behind the core of the nonlinear
response. Once the core properties are known, the molecule can be “scaled
up” to yield a large absolute nonlinear susceptibility. The following derivation
rigorously determines the scaling laws, and from these defines the intrinsic
hyperpolarizabilities. As we will see, the energy difference between the ground
and first excited state and the number of electrons define the scale of a quantum
system.
The Hamiltonian of any N -electron system, such as a molecule or charges in a
multiple quantum well, depends on the potential V , which can contain coulomb
repulsion terms, such as −e2 ∕4πε0 j⃗r1 − r⃗ 2 j, spin interactions, spin–orbit coupling, and external electric fields, and the vector potential can describe interactions with external electric and magnetic fields.
The N -electron Schrödinger equation is of the form
Hψ⃗r1 ; r⃗ 2 …⃗rN Eψ⃗r1 ; r⃗ 2 …⃗rN :
(47)
Scaling can be understood by transforming Eq. (47) with r⃗ k → ε⃗rk , where ε is
the scaling parameter, into
N
1 X
2m k
e ⃗
⃗pk − εAε⃗
r1 ; …
c
2
ψε⃗r1 ; …
ε2 V ε⃗r1 ; …; s⃗ 1 ; …; L⃗ 1 ; …ψε⃗r1 ; …
Eψε⃗r1 ; …;
(48)
where A⃗ is the vector potential and s⃗ i , L⃗ i , and r⃗ i are the spin, orbital angular
momentum, and position of the ith electron. Thus, ψε⃗r1 …, which has the same
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21
shape as ψε⃗r1 …, aside from being spatially compressed by a factor 1∕ε, is a
solution of the Schrödinger equation with ⃗pk → ⃗pk ∕ε2 , E → Eε2 ,
⃗ r1 … → εAε⃗
⃗ r1 ; …. In effect, this causes comV ⃗r1 … → V ε⃗r1 …ε2 , and A⃗
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 functions 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 r⃗ · r⃗ E is invariant. The dipole
moment, ⃗μ, is defined as
⃗μ −e
N
X
r⃗ i ;
(49)
i1
where r⃗ i is the position of the ith electron, and −e is the electron charge. Thus, the
position operator is given by
⃗μ
r⃗ − :
e
(50)
When electrons are added to the system in a way that does not change the
eigenenergies, then the invariance relation can be generalized to the form
r⃗ · r⃗ E kN ;
(51)
where k is a constant. The components of the position operator obey the sum
rules given by Eq. (45b). By convention, the x direction is along the largest
diagonal component of the hyperpolarizability tensor. Given Eq. (51), ⃗μmn;x
is 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 polarizability, hyperpolarizability, and higher-order hyperpolarizabilities are bounded
[76–79]. The off-resonance polarizability (or zeroth-order hyperpolarizability) is
eℏ 2 N
max
;
(52)
α ≤ α0 pffiffiffiffi
m ε0 E 210
the fundamental limit of the hyperpolarizability (also called the first hyperpolarizability) is
ffiffiffi eℏ 3 N 3∕2
p
e
4
max
;
(53)
jβj ≤ β0 3 pffiffiffiffi
m ε0 E7∕2
10
and the fundamental limit of the second hyperpolarizability is
eℏ 4 N 3∕2
e
max
:
jγj ≤ γ 0 4 pffiffiffiffi
m ε0 E7∕2
10
(54a)
We note that there are no approximations used in calculating the fundamental
limits, so they are exact and therefore cannot be exceeded. While the three-level
ansatz has not been proven rigorously, it appears to hold always; when a quantum system has a nonlinear response at the limit, only three states (ground and
two excited states) contribute. Note that this does not imply that, if the threelevel ansatz holds, the system must have a nonlinear response at the limit.
However, when the nonlinear response is small, usually many states contribute.
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Similarly, the transition electric dipole moment to any excited state is bounded
and for the first excited state it is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e 2 N e ℏ2
≡ ⃗μ10max :
(54b)
j ⃗μ10 j ≤
2ε20 me E10
The fundamental limit of the nth hyperpolarizability is of the form
ηn
max ∝
N n2∕2
e
ε0 E3n4∕2
10
;
(55)
0
where η−1
max is the fundamental limit of jμ10 j, ηmax is the fundamental limit of α,
1
ηmax is the fundamental limit of jβj, etc.
The nth hyperpolarizability is of the form
ηn ∝
μ̄vm n2
;
ε0 Em n1
(56)
where μ̄vm represents products of transition dipole moments of the form μ̄vm and
E m represents products of energy differences of the form E mg Em − Eg . Thus,
rescaling the nth hyperpolarizability according to Eq. (48) and using Eq. (51) yields
ηn k n2∕2
N n2∕2 :
ε0 E3n4∕2
The intrinsic nth hyperpolarizability is then given by
3n4∕2
ηn
n
n2∕2 E 10
ηint n ∝ k
;
E
ηmax
(57)
(58)
where we have used Eqs. (69) and (71). The parameter ηn
int is unchanged under simple
scaling and independent of the number of electrons, making it a scale-invariant quantity. The intrinsic nth-order hyperpolarizability is invariant upon rescaling under the
same transformation that leaves the Schrödinger equation invariant, as described above.
The intrinsic hyperpolarizabilities are quantities that remove the effects of scaling, and are a measure of a molecule’s nonlinear optical efficiency, independent
of the number of electrons or energy gap. While larger molecules with more
electrons will generally interact more strongly with light than smaller, electronpoor systems, the intrinsic hyperpolarizabilities remove such effects, allowing
one to focus on the structural properties that affect the response. This allows one
to determine whether all of the electrons are contributing to the nonlinear
response with maximal efficiency. Only then can truly new paradigms be developed for making large molecules with exceptionally enhanced response.
The approach of using the intrinsic hyperpolarizability as a scale-invariant measure
of the nonlinearity of a molecule has been used by some groups to compare molecules, but is not as generally appreciated given its fundamental importance. The
intrinsic hyperpolarizability is important because it is a fundamental intensive
quantity that helps us to better understand the strength of light–matter interactions,
and can more intelligently guide molecular engineering strategies. The approach
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23
for optimizing a molecule is to first determine the intrinsic properties that make a
molecule good, then to scale it up while preserving the critical parameters.
Using this approach, making a molecule with a large hyperpolarizability (or second hyperpolarizability) starts by first identifying a molecule that has a large
intrinsic nonlinearity and then making the molecule larger to take advantage
of scaling. A molecule that has a large intrinsic nonlinearity that is well above
the intrinsic nonlinear response of most others usually represents a new molecular 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 for
most other molecules [80]. As such, their technological impact is clear. However, given their already large size, making such molecules even larger clearly
will suffer from diminishing returns in terms of volume fraction.
Potential energy optimization studies [74,81] suggested that to improve the hyperpolarizability, the conjugation path between the donor and acceptor should
contain a mixture of atoms, called modulation of conjugation by the authors.
An example of a modulated conjugation path is shown in molecule P-7 in
Fig. 2, where molecule P-4 is the homologue without the heterocyclic bridge.
The molecule with modulation of conjugation (P-7) was found to break the
apparent limit, which is shown as the horizontal blue line. The apparent limit
is a factor of 30 below the fundamental limit and is defined by the largest
intrinsic hyperpolarizabilities observed in all molecules prior to the more recent
discovery of better molecules.
Figure 2
Intrinsic hyperpolarizability of some representative molecules that have large
nonlinear optical responses, plotted as a function of the energy difference
E10 . Reproduced with permission of www.nlosource.com. The numbers in
brackets label a series of similar molecules.
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24
Molecules with a twist in the bridge, as represented by the series labeled “[1],”
have the highest intrinsic nonlinear optical response ever measured [82]. This
class of molecules is clearly a new paradigm that stands out from all others.
Making them bigger and incorporating them in a bulk material would yield
an unparalleled nonlinear optical susceptibility.
The intrinsic nonlinear response also sheds insights into scaling of the first and
second hyperpolarizability with length of the molecule. Slepkov and associates
measured the length dependence of the second hyperpolarizabilities of a series of
polyynes, as shown in the inset of Fig. 3 [83]. Figure 3 also shows a plot of the
scaling law (curve) as predicted by Eq. (57) given the number of electrons and
the measured energy difference between the ground and excited states. Note
that 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 similar
to the observed scaling law makes polyynes a promising motif for making
molecules with ever larger second hyperpolarizability. Indeed, Biaggio and associates have investigated a series of planar molecules with high densities of
triple bond conjugation as in the polyynes and also find critical scaling along
with a large second hyperpolarizability [84,85].
Structure-property studies seek to determine how the nonlinear optical response
of a molecule depends on a parameter that may be easy to control experimentally. Marder and associates found that the first and second hyperpolarizability
were peaked functions of bond order alternation (BOA) [86,87], the difference
of bond lengths between adjacent single and double bonds in a conjugated structure. As a result, this suggested that molecules synthesized to have a BOA at the
peak would have a large nonlinear optical response.
BOA has been used extensively by many researchers as an aid in designing
molecules, though it was never clear at the time if this paradigm could be used
to make molecules that reach the fundamental limit. Figure 4 shows a plot of the
calculated hyperpolarizability as a function of BOA (solid curve) for the
Figure 3
Plot of the measured second hyperpolarizability as a function of N (points) and
the scaling predicted by the fundamental limits (curve) [158,161]. Reproduced
with permission of J. Mat. Chem.
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25
molecule shown in the inset. Figure 4 also shows a plot of the fundamental limit
of the hyperpolarizability calculated from Eq. (53). The shape of this curve
originates in a shift in the energy E10 as a function of BOA.
The fundamental limit of the hyperpolarizability peaks at a BOA of about −0.08
where the actual hyperpolarizability predicted by the BOA model is small, illustrating that the peak hyperpolarizability at a BOA of 0.2 and −0.25 is not the
global maximum. Furthermore, the BOA metric is not the ideal paradigm because it does not follow the fundamental limit curve, and thus will not allow a
material to be optimized to its full potential through the power of scaling. We
note that, in principle, there is no limit to the achievable nonlinear response
through 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 which
effect wins. In quasi-one-dimensional materials, such as the polyenes, Rustagi
and Ducuing showed that the polarizability and hyperpolarizabilities scale as a
function of length according to [88]
αRD ∝ L3 ;
γ RD ∝ L5 ;
(59)
whereas, in comparison, the number of one-dimensional molecules that can be
added to a material grows as 1∕L. Thus, increasing the size is a winning strategy
to improve the nonlinearity of the molecule provided that the scaling law can be
made to hold over longer distances. Once the molecular properties are optimized, a bulk material made of these molecules can take advantage of the molecular hyperpolarizabilities if interactions between molecules are small enough
to be taken into account using local field models.
Greene et al. showed that the nonlinear response of polydiacetylene can be
modeled on and near the one-photon resonance as an exciton that is confined
to 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 a
function of bond-order alternation.
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26
principle infinite but the exciton length was determined to be about 3.3 nm. Thus
in PTS, the effective length of the polymer chain was less than the full chain
length. The implication is that the power laws may saturate in most oligomers
when the chain lengths become long enough.
Interestingly, the maximum scaling of a one-dimensional system [90], assuming
that the conjugated path can be modeled using particle-in-a-box states with Pauli
exclusion, i.e., the technique of Kuhn [24,25], yields [90]
αscaling ∝ L3 ;
βscaling ∝ L5 ;
γ scaling ∝ L7 :
(60)
Thus, the maximum power law given by simple scaling that originates from the
theory of the fundamental limits is even greater than for the polyenes. The
implications are that it may be possible to make even better materials, but new
breakthroughs in molecular engineering may be required. In both cases, the
power laws far exceed dilution effects associated with placing large molecules
in a fixed volume.
We have provided in this section the quantum mechanical basis for calculating
hyperpolarizabilities and limits to them without much regard to the tensorial
nature of these quantities and the underlying symmetries affecting them. We
now turn our attention to understanding how spatial symmetry affects the
nonlinear optical response.
3. Molecular Symmetry
As mentioned in Section 1, the presence or absence of a center of inversion can
play a crucial role in the nonlinear optical response. In particular, the even-order
nonlinear optical responses are identically zero in the electric dipole approximation. This section focuses on molecules lacking an inversion center and, thus,
mostly on the second-order nonlinear optical response. The susceptibility
βijk −ω; ωp ; ωq is analyzed in terms of the irreducible third-order tensors of
the symmetry classes, yielding insight into the various structures that can exhibit
nonzero elements in βijk −ω; ωp ; ωq . Calculations of the dispersion with frequency 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 asymmetric
molecules. The frequency dispersion of γ ijkl −ω; ωp ; ωq ; ωr based on this model
is also discussed. The simplest model for symmetric molecules involves three
states 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, or
uℓ −⃗r uℓ ⃗r . Spatial wave functions that exhibit no spatial symmetry
can be expressed as a linear superposition of a symmetric and antisymmetric
part. The Relectric dipole transition moment between states ℓ and ℓ0 , defined
∞ uℓ ⃗r ⃗μ⃗r uℓ0 ⃗r d⃗r, will not vanish if the spatial wave functions
by ⃗μℓℓ0 −∞
uℓ ⃗r and uℓ0 ⃗r of states ℓ and ℓ0 are of opposite parity by virtue of the fact that
⃗μ⃗r e⃗r has odd parity. In general, ⃗μℓℓ0 ≠ 0 when the electronic states are expressible as a superposition of even and odd parity. The case ℓ ℓ0 gives the
permanent dipole moment in the ℓ0 th state. Therefore the terms in the numerator
of Eq. (39), namely ⃗μgv , ⃗μvn , ⃗μnm , and ⃗μmg , are all nonzero.
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For centrosymmetric molecules, the molecular spatial wave functions exhibit
either even (gerade, subscript g) or odd (ungerade, subscript u) symmetry with
the ground state having even symmetry. Since the electric dipole transitions
moments are non-zero only if the states are of opposite symmetry, substituting
for the symmetry of the wave functions as either even (g) or odd (u) into the
numerators of Eq. (39) gives ⃗μgu ⃗μuu − ⃗μgg ⃗μug . A similar argument can be made
for the three-level model, which shows that a nonzero octupole moment is
required as discussed previously [91]. For centrosymmetric molecules the permanent dipole moments and permanent octupole moments are zero and hence
all χ 2 0.
In the preceding section, the general sum-over-states (SOS) model for the
nonlinear susceptibilities of molecules was discussed. Although it is in
principle the most accurate model available for calculating the second- and
third-order nonlinear response of molecules, frequently only a fraction of
the information needed, such as the electronic states and the transition electric
dipole moments between them, is known. One- and two-photon absorption
spectroscopy can in principle yield this information. Even if not all of the
discrete states and their transition moments can be measured with reasonable
accuracy, the spectra do identify the transitions with the largest probabilities
that can be used in models involving just a few states with the largest transition
moments.
⃗⃗
3.2. Irreducible Tensor Approach to β⃗ Molecular
Nonlinear Optics
The nonlinear optical tensors for both molecules and the macroscopic media
they comprise involve, in principle, 3n components, where n − 1 is the order
of the nonlinear optical process. Nonlinear optical interactions depend on various symmetries intrinsic to the materials response tensor, but also involve the
particular experimental or thermodynamic conditions. These symmetries include spatial transformations related to the molecular or material symmetry,
permutation of the tensor indices, and the permutation of the frequencies involved in the process [69]. The difficulties with dealing with so many coefficients, symmetries, and possible coordinate systems can be effectively
addressed by applying the methods of group theory to the nonlinear optical
tensors [37,92–97]. This powerful framework provides a mechanism for
(1) most efficiently representing the physical properties for a given set of
symmetry operations, (2) providing insight into the contributions to the response to aid in the design of improved materials, (3) providing figures
of merit to compare materials and molecules of differing symmetry, and
(4) conveniently connecting the molecular to the macroscopic response. In
this section, we will provide the groundwork for this analysis and apply it
to the second-order nonlinear optical response. On this basis, we can suggest
materials approaches at the molecular and supramolecular level for efficient
nonlinear optics, given the various ways to produce noncentrosymmetric
molecules and materials.
The starting point of this analysis is noting that the macroscopic and molecular
nonlinear responses, as fundamental material properties, should not depend on
the coordinate frame describing them. That is, they are translationally invariant
and physical properties should be invariant with respect to rotations of the
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coordinate system; for example, the inner product of vectors should not depend
on the choice of coordinate system.
For illustration, consider a rank 2 tensor, T ij . Any rank 2 tensor can be expressed
⃗
as a direct product of two vectors, A⃗ and B:
T ij Ai Bj :
(61)
Equation (61) can be equivalently expressed as
Ai Bj − Aj Bi Ai Bj Aj Bi A⃗ · B⃗
A⃗ · B⃗
−
T ij δ δ :
2
2
3 ij
3 ij
(62)
Equation (62), while appearing more complex, is useful because each term
represents a particular type of symmetry.
The first term, being the trace of T ij , is invariant upon rotation, and is therefore a
scalar. The second term corresponds to the three components of the cross pro⃗ so represents an axial vector. As such, it is invariant under the parity
duct A⃗ × B,
operation, i.e., it is unchanged when the coordinate frame is transformed according to x̂ → −x̂, ŷ → −ŷ, and ẑ → −ẑ. Finally, the last term represents a traceless
symmetric tensor, which is described by five independent parameters.
Recall that T ij 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 rotations, each with a multiplicity of 2j 1. Thus, the three terms of Eq. (62) can be
associated 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 forms
are possible, such as the spherical tensors, which have a one-to-one correspondence to the spherical harmonic functions. We can express Eq. (62) in a tensorindependent 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; the
direct tensor product results in a tensor whose rank is the sum of the ranks of the
factors, e.g., A ⊗ B C; Ai Bj C ij in Cartesian coordinates for the direct product 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 2
tensors of scalar (0), axial vector (1), and symmetric (2) character. In analogy to
the addition of the angular momentum of two spin 1 particles, i.e., j1 1 and
j2 2, the possible results are j jj1 − j2 j to j jj1 j2 j, or 0, 1, 2. Equation (62) is the statement that any rank 2 tensor can be written in terms of three
irreducible tensors with a scalar, pseudovector, and symmetric tensor.
The susceptibility χ 1 is a second-rank tensor, so it too can be expressed in terms
1
of the three irreducible representations. Given that χ 1
ij χ ji , the vector part
vanishes, so
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29
χ 1 ∼ 0 ⊕ 2:
(64)
As a consequence, the susceptibility is independent of the polar order. The χ 1 is
thus nonzero for a centrosymmetric material, originating in the scalar part “0,”
and anisotropy is described by uniaxial order, which originates in the symmetric
part “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 representations
and symmetries that can be used to simplify the expression for the nonlinear
optical response. The formal derivations follow.
In the case of the nonlinear responses associated with molecules, unit cells, or
macroscopic materials, the higher-order tensors can be simplified using geometrical symmetries that are summarized in the tensor forms associated with the 32
point groups, well known in crystallography, and derived from group theory.
These point groups are subgroups of the full orthogonal rotation group, which
is itself the direct product of the three-dimensional rotation group with the inversion 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 completely characterized only by various rotations and inversions [98]. We note that
inversion can be represented as a rotation followed by a reflection through a
plane perpendicular to that axis, known as an improper rotation. A proper rotation is an ordinary coordinate rotation (without inversion or reflection). The Cartesian form for the response tensor as usually tabulated is obtained by applying
the various rotation and reflection symmetries to homogeneous polynomials of
degree n as described, for example, by Nye [98].
Our analysis proceeds by considering the reduction of the response tensor to its
irreducible forms; these forms comprise the decomposition into a series of tensors of rank n and lower that do not mix under any three-dimensional rotation
and thus reflect the necessary rotational invariance [92]. This is a generalization
of Eq. (61) and the discussion following. Here, we consider the tensors in their
general form. An irreducible rank n tensor is labeled by its “weight” J having
(2J 1) independent components consistent with the three-dimensional rotation group. This formalism is reminiscent of the quantum mechanical addition
of angular momenta. We also note that there may be an irreducible tensor of
weight J labeled by another parameter t. According to the scheme of angular
momenta, the product of two irreducible tensors of weight J 1 and J 2 generates a
series of tensors from jJ 1 − J 2 j to jJ 1 J 2 j. Starting with the fact that the most
general third-rank second-order nonlinear optical tensor defined in Eq. (45), but
in spherical form, for a molecule (the first hyperpolarizability) is related to the
homogeneous polynomial presented as a direct tensor product of three vectors
denoted 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 two
weight 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 permutation symmetry and thus describes parametric light scattering or three-wave
mixing in contrast with second-harmonic generation and the linear electro-optic
effect, which possess permutation symmetries brought about by degenerate
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frequencies. Note that, with 2J 1 components for each of seven tensors (1
scalar, 3 vectors, 2 second-rank tensors, and 1 third-rank tensor), there are up to
a total of 27 components as required for a third-order tensor, depending on the
point group. For the cases of second-harmonic generation and the linear electrooptic effect, the hyperpolarizabilities have two indistinguishable frequencies in
SHG
the three-wave processes. In these cases, βSHG
ijk −2ω; ω; ω βikj −2ω; ω; ω
LEO
and βLEO
ijk ω; 0; ω βjik −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. When Kleinman (full Cartesian index permutation)
symmetry applies, the decomposition yields
βKS ∼ 1 ⊕ 3;
(67)
yielding eight independent components comprising one vector and one third-rank
tensor. The disappearance of the weight 1 irreducible tensor in the product of the
two rank 1 tensors comes about from the permutation symmetry of secondharmonic generation. The details of the reduction in the number of irreducible
components in Eqs. (66) and (67) relative to Eq. (65) will become clear when
the Cartesian forms are discussed below.
Given that the various irreducible forms arise from proper and improper rotations as defined above, they can be characterized by the parity under those rotations. For vectors, these define polar and axial vectors, for example describing
electric and magnetic fields, respectively. In our example of a second-rank tensor
of Eq. (61), the vector component is an axial vector as it is the cross product of
two-vectors. Correspondingly, one can define true tensors and pseudotensors
relative to their parity under improper rotations. For the former, parity is given
by π −1n and, for the latter, π −1n1 . For the irreducible tensors of
weight J derived from a rank n tensor, π −1J describes an irreducible true
tensor, and π −1J 1 an irreducible pseudotensor. Thus, when reducing a
true (or pseudo) Cartesian tensor, the irreducible parts with n J even for
tensors (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 the
reduction spectrum defined in Eq. (66) for second-harmonic generation above,
the 0 weight component is a pseudo-scalar and the 2 weight component is a
pseudo-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 inversion
center (e.g., chiral) will exhibit such wave mixing, while second harmonic will
not be observed.
Insight into the structure of these tensors can be obtained by considering
the decomposition of the Cartesian forms of these tensors as carried out in
Appendix A with results given below [94,95]. We will seek the reduction spectrum implied by Eqs. (65)–(67), where the irreducible forms are labeled by the
rank n 3, and the weights J . Note that there are may be more than one tensor
of a given weight, which can be labeled with another parameter, but we will find
it convenient to label them by their properties under permutations of their indices
as will also be described in Appendix A.
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Before we describe the application of the formalism of Appendix A in the important case of second-harmonic generation, we can now point out how this
picture leads to useful figures of merit to analyze the nonlinear optical response
corresponding to cases involving various symmetries, both intrinsic material
symmetries and symmetries corresponding to a particular nonlinear optical process. This is done by considering the norm of the nonlinear optical tensor as
described by Eqs. (A2) and (A3), that is, as the sum of the irreducible components embedded in a third-rank tensor space. One can show that the tensors of
weight J span an n-dimensional orthogonal space, so that the norm of the sum of
the tensors of weight J is equal to the sum of the norms yielding a generalized
Pythagorean Theorem [92]. Thus,
‖βijk ‖2 X
J ;m
J ;m 2
‖βijk
‖:
(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 irreduJ ;m
‖ and the
cible tensors of the same weight but differing m. The quantities ‖βijk
0
;m
;m · βJ
are invariants and are figures of merit that can be
inner products βJ
ijk
ijk
measured by using hyper-Rayleigh scattering. Consequently, these figures of
merit are useful for characterizing molecular hyperpolarizability components
that directly relate to both the molecular response and the supramolecular
(macroscopic) response as the irreducible components do not mix when performing the orientational average in relating the molecular to supramolecular
response using an oriented gas model. Descriptions of the techniques for measuring these figures of merit for second-harmonic generation have been
described in the literature in both the Kleinman symmetric [37,38,99] and
Kleinman disallowed cases [39,40,100]. We can now describe how this formalism provides insight into contributions to the nonlinear optical response at both
the molecular and supramolecular levels, which we examine in the case of
second-harmonic generation. As both the linear electro-optic effect and secondharmonic generation share a common index pair permutation symmetry due to a
pair of degenerate electromagnetic field frequencies, our analysis pertains
to both.
We begin by applying the permutation projection of Eq. (A4) in Appendix A
to the second-order nonlinear optical tensor expressed as a sum of the irreducible 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 degenerate frequencies. Equation (A5) applies to the general second-order process [94,97]. We now consider the sum of Eq. (66) relevant for second
harmonic and the linear electro-optic effect with the permutation projection.
This yields
2m
1s
1m
βijk β3s
ijk βijk βijk βijk :
(69)
The indices s and m describe the index permutation symmetry for the fully
symmetric and mixed symmetry cases, respectively, as described in Appendix A. The absence of a and m0 components reflects the index pair permutation symmetry. This is the most convenient way to differentiate the two J 1
irreducible tensors. The embedded forms are given by
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1 1s
1s
1s
β1s
ijk βi δjk βj δik βk δij ;
5
1 1m
δjk − β1m
δik − β1m
δij ;
β1m
j
ijk 2βi
k
5
1
2m
β2m
β2m
ijk 2εijl βik
il εljk .
5
(70)
The octupolar component (3s) is not shown but is a fully symmetric traceless
third-rank tensor obtained by symmetrizing the hyperpolarizability and subtracting the Kleinman symmetric part containing traces. The embedded irreducible
forms in Eq. (70) are given by
1
βijk δjk βjik δjk βjki δjk ;
β1s
i
3
1
1m
βi
2βijk δjk − βjik δjk − βjki δjk ;
3
1
2m
βij εikl βklj εjkl βkli ;
2
1
3s
βijk βijk βjki βkij − β1s
ijk .
3
(71)
The SOS expressions of the irreducible tensors in Eqs. (71) are the essential
quantities 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 these
equations along with the SOS expressions for βijk introduced in Section 2 to
garner insight into the origin of the response and to guide the design of molecular
materials.
The irreducible representations of the second-harmonic hyperpolarizability
(ignoring losses) are given by
ω2ng
1 X
Δμing jμ⃗ gn j2 2μign Δμ⃗ ng · μ⃗ gn ℏ2 n≠g ω2ng −ω2 ω2ng −4ω2 X 2ω4 ω2mg −4ωmg ωng ω2ng ω2 ωmg ωng 3ω2mg −ωmg ωng 3ω2ng −ω3mg ω3ng
−
ω2ng −ω2 ω2ng −4ω2 ω2mg −ω2 ω2mg −4ω2 n≠g
m≠n≠g
i
i
×μnm μ⃗ gn · μ⃗ gm 2μgn μ⃗ nm · μ⃗ gm ;
(72)
β1s
i
β1m
i
2ω2 X
2
⃗μgn × ⃗μgn × Δ ⃗μng i
2
2
2
2
2
ℏ
−
ω
ω
−
4ω
ω
ng
ng
n≠g
X 8ω4 − 2ω2 3ω2mg 2ωmg ωng − ωmg ωng ω2mg − ωmg ωng − 2ω2ng ω2ng − ω2 ω2ng − 4ω2 ω2mg − ω2 ω2mg − 4ω2 n≠g
m≠n≠g
i
× ⃗μgn × ⃗μgn × Δ ⃗μng ;
(73a)
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β2m
ij
3ω2
2
4ℏ
(
X
X
n≠g
m≠n≠g
−
ω2ng
n≠g
"
4
⃗μgn × Δ ⃗μng i μjgn
2
2
− ω ωng − 4ω 2
22ω2 − ωmg ωng − ω2ng ⃗μnm × ⃗μgn i μjgm
ω2ng − ω2 ω2ng − 4ω2 ω2mg − ω2 2ω2 ωmg ωng ω2mg − ω2ng 2ω2ng − ω2 ω2ng − 4ω2 ω2mg − ω2 ω2mg − 4ω2 #
)
× ⃗μgm × ⃗μgn i μjnm i↔j ;
1
β3s
ijk 2
(73b)
X
ω2ng
ω2ng −ω2 ω2ng −4ω2 n≠g
ℏ
1
1
× Ps Δμing μjgn μkgn − jμ2gn jΔμing δjk Δμjng δik Δμkng δij 2
5
−2Δμ⃗ ng ·μ⃗ gn Δμign δjk Δμjgn δik Δμkgn δij ×
X 2ω4 ω2mg −4ωmg ωng ω2ng ω2 ωmg ωng 3ω2mg −ωmg ωng 3ω2ng −ω3mg ω3ng
n≠g
m≠n≠g
ω2ng −ω2 ω2ng −4ω2 ω2mg −ω2 ω2mg −4ω2 1
1
× Ps μign μjnm μkgm − μ⃗ nm ·μ⃗ gm μinm δjk μjnm δik μknm δij 2
5
2μ⃗ nm ·μ⃗ gm μign δjk μjgn δik μkgn δij ;
(74)
where Δ ⃗μng ⃗μnn − ⃗μgg denotes the change in dipole moment between the two
states n and g, and Ps is the fully symmetric permutation operator interchanging
i, j, and k. Equations (72) through (74) are displayed so that the first term corresponds to the two-level expression involving the ground state and single excited
states.
We now describe the physical implications of these hyperpolarizabilities. First,
we recall that all components require an absence of inversion symmetry. The 1s
component 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 a
vector, often associated with a one-dimensional molecule. This is by far the most
widely appreciated contribution to the hyperpolarizability as it is the component
directly measured in electric-field-induced second-harmonic generation (EFISH),
and the one most often described in relation to organic second-order nonlinear
optical materials. It is the component that contributes to the nonlinear optical response in poled polymer materials and so called one-dimensional materials. This
part is described thoroughly in the next section. The other symmetric component
is the 3s component, which is also widely known as the octupolar component of
the hyperpolarizability [37]. This component is also noncentrosymmetric, but,
instead requires a two-dimensional or three-dimensional response, associated
with the prototypical structures shown in Fig. 5.
Kleinman or full-permutation symmetry is reflected in the irreducible representations, as shown in Eq. (67), where only the contributions given in Eqs. (72)
Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004
34
Figure 5
Prototypical structures having octupolar symmetry in (a) two and (b) three
dimensions. The colors/shapes represent distinct chemical moieties.
and (74) will be nonzero. This is consistent with our expression of the irreducible
tensor components in terms of those of the permutation operations. That is, since
Kleinman symmetric components must be fully symmetric, those representations in Eqs. (73a) and (73b) of mixed symmetry are identically zero. The
conditions for Kleinman symmetry center on thermodynamic arguments implying the simultaneous full permutation symmetry of both Cartesian indices and
frequencies [69] for the special case of all frequencies far from resonance. In this
case, dispersion of the response is negligible and permutation of the Cartesian
indices is decoupled from permutation of the frequencies. It has recently been
pointed out that in the realm of molecular nonlinear optics, this condition
does not often apply [101]. Given the typical absorption frequencies and nonlinear interaction frequencies, especially in organic chromophores, the farfrom-resonance condition generally is not satisfied, and, thus, Kleinman
symmetry is often broken.
However, a symmetry equivalent to Kleinman symmetry can arise from spatial
symmetries in the nonlinear optical medium. In particular, in the case of the onedimensional molecule, only one component of the hyperpolarizability tensor is
nonzero, and Kleinman symmetry trivially and necessarily holds regardless of
frequency. Thus, for molecular nonlinear optics, the extent that Kleinman symmetry does not hold is usually a measure of the departure of the molecule from
one-dimensionality. In the case, of polar nonlinear optics described above and
the next section, it is often the case that nearly one-dimensional molecules are
employed due to their large vector component of the hyperpolarizability. Of
course, for octupolar nonlinear optics, a multidimensional molecule is required
to reflect that symmetry, so Kleinman symmetry would not be expected and the
mixed symmetry components will contribute to an extent consistent with the
particular molecular structure and thermodynamic considerations. In the case
of other molecular symmetries of higher dimension, the expressions for the
irreducible hyperpolarizability components provide insight.
These insights can be gleaned from the forms of the transition dipole vectors in
Eqs. (72)–(74). In the 1s and 3s components, only dot products of the dipole
vectors appear, while in the 1m and 2m components, cross products appear.
Thus, these forms imply a requirement for multidimensional molecules in order
for the 1m and 2m components to be nonzero, the same condition for Kleinman
symmetry breaking. It also explains why one-dimensional molecules are favored
in the polar case (1s) since the dot product is most easily maximized in that case.
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35
The octupolar case is more complex, but is maximized under threefold rotation
symmetry in two dimensions [102].
We now consider the example of multidimensional molecules of symmetry C 2v ,
which, as noted above are generally Kleinman nonsymmetric for organic chromophores [94,103–105]. We start by considering the extent of summation
needed in the SOS expressions. As we discuss below, restriction of the sum
to the ground and one excited states has been shown successful for understanding one-dimensional push–pull chromophores. This is inadequate when intramolecular charge transfer in multiple dimensions contributes to the nonlinear
optical response. This has been established both heuristically and experimentally, 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 with
octupolar molecules [37].
The C 2v symmetry case will require at least three levels, but considerable insight
can be obtained by considering the response in light of our analysis above. This
analysis indicates that, for the case of C 2v symmetry, transition dipoles are parallel and perpendicular to the symmetry axis, and consequently Δ ⃗μ will be most
important. This can be analyzed best in terms of group theory as above. If the
molecule is invariant under certain symmetry operations, the Hamiltonian will
commute with the group elements so that the quantum states are eigenfunctions
belonging to that group. The ground state belongs to the fully symmetric (trivial)
representation (A1 ). The irreducible representations corresponding to the excited
states will define the nature of the transitions involved. Consider the character
table for C 2v 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, and
mirror 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 trace
of the transformation matrix representing that symmetry operation. The characters in this case are either symmetric (1) or antisymmetric (−1) under the
symmetry operations as noted for each representation. These characters can
be understood by the rotation in Hilbert space reflecting the twofold
symmetry of the molecules. The eigenvalues of an n-fold rotation are given
by Rm jψi exp2πin∕mjψi where R is the operator for a rotation by 2π∕m.
So, the wave function accumulates a phase (for n 1) of exp2πi∕2 −1.
Similarly, a molecule with a mirror plane will also acquire a phase of
exp2πi∕2 −1 upon reflection since the reflection operator σ has the property that σ 2 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 C 2v Symmetry
A1
A2
B1
B2
E
C2
σx
σy
1
1
1
1
1
1
−1
−1
1
−1
1
−1
1
−1
−1
1
Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004
36
understood by considering the rotation of the transition dipole moment, where a
component of a dipole moment parallel to the symmetry axis z is invariant under
twofold rotation (A-type), whereas components perpendicular to z change sign
(B-type) under 180° rotation. The two mutually exclusive cases for the transition
matrix dipole have an excited state with A character with transition moment parallel to z, and an excited state of B character with transition moment perpendicular to z. This implies that a two-level expression involving only the ground and
first excited molecular electronic states can either describe a diagonal Cartesian
component (βzzz ) with an A state, or an off-diagonal βxxz ; βyyz component with a
B state, but not both. This explains why a two-level model can apply chromophores whose nonlinear response involves only a single direction, describing
linear, one-dimensional chromophores, but not to a twofold symmetry (twodimensional) molecule. We note that a two-state model might describe certain
low-symmetry, two-dimensional chromophores.
This analysis can now be combined with Eqs. (72)–(74) by considering the
cross products of the transition moments with the dipole moment changes
(Δ ⃗μ). By symmetry the Δ ⃗μ must be aligned with the symmetry axis. This implies that the Kleinman nonsymmetric components 1m and 2m must have transition moments perpendicular to the symmetry axis due to the cross products,
while the Kleinman symmetric 1s and 3s components must have transition
moments along the symmetry axis. Thus, A states contribute to the 1s and
3s 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 a
non-centrosymmetric bulk medium will determine whether A or B states are
important, thus confirming that this analysis has provided a method for designing molecules to best optimize the molecular and macroscopic nonlinear optical
response.
3.3. Two-Level Model
Based on Eq. (39), only one excited state and a permanent dipole moment are
sufficient to produce a nonzero χ 2 so that a two-level model is the simplest
model possible; see Fig. 6. Despite its simplicity, this model has proven very
valuable for understanding general trends of susceptibilities with frequency for
noncentrosymmetric molecules. It also provides a good representation of charge
transfer molecules [11].
3.3a. Two-Level Model: χ 2
For two states, the ground state and one excited state, the SOS expression
becomes
Figure 6
(a)
(b)
Two-level model for calculating (a) χ 2 and (b) χ 3 for noncentrosymmetric
molecules.
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37
χ 2
ijk −ω; ωP ; ωq μ01;i μ11;k − μ00;k μ10;j
N
Lω
Lω
Lω
p
q
2
ω̂10 − ωp − ωq ω̂10 − ωp ε0 ℏ
μ01;k μ11;j − μ00;j μ10;i
μ01;k μ11;i − μ00;i μ10;j
: (75)
ω̂10 ωq ω̂10 ωp ωq ω̂10 ωq ω̂10 − ωp The permanent dipole moments in the ground and excited states are written as
⃗μ00 and ⃗μ11 , respectively, and the transition dipole moment is ⃗μ10 ⃗μ01 :
The two-level model provides some insight into the frequency dispersion of the
second-order susceptibility. In order to avoid efficiency limiting losses at the
fundamental and harmonic frequencies, in the specific examples discussed next,
the input frequency is chosen to be in the off-resonance or non-resonant regimes.
^ 10 in the denominators can be neIn the off-resonance case, the τ−1
10 part of ω
glected, i.e., ω10 is real. For the non-resonance regime, all input frequencies
are set to zero.
Assume that periodically poled lithium niobate (PPLN), for example, can be
usefully described by a two-level model. The dominant second-order nonlinearity lies along the z axis [106]. For this case, the off-resonance result in the twolevel model is
χ 2
zzz −2ω; ω; ω N 2
3ω210
2 μ
L
ωL2ωjμ
j
−
μ
:
10
11;z
00;z
ω210 − ω2 ω210 − 4ω2 ε 0 ℏ2
(76)
It is useful to compare this result with that obtained using the anharmonic oscillator model, which can be found in any textbook [7,11]. That result away from
resonance is
χ 2
zzz −2ω; ω; ω N e3
k 2
zzz
:
ε0 m̄3e ω210 − 4ω2 ω210 − ω2 2
(77)
Not only can Eq. (77) not yield specific values because there is no method to
calculate the nonlinear force constant k 2
zzz , but it also predicts a stronger resonance at the fundamental frequency ω210 − ω2 −2 than at the second harmonic
ω210 − 4ω2 −1 . Unfortunately, there are no measurements of the dispersion of the
nonlinearity over a sufficiently wide spectral range to make a useful comparison
between experiment and theory.
Another example for comparison with the anharmonic oscillator model is sum
frequency generation for which there are two input frequencies, namely ω1 and
ω2 . The SOS result is
2
χ 2
zzz −ω2 ω1 ; ω1 ; ω2 χ zzz −ω2 ω1 ; ω2 ; ω1 ω210 N
Lω1 Lω2 Lω2 ω1 ℏ2
ω2 3ω2 − ω1 ω2 2 ω1 ω2 :
× jμ01;z j2 μ11;z − μ00;z 2 10 10
ω10 − ω1 ω2 2 ω210 ω21 ω210 ω22 (78)
Again, the frequency dispersion is different from the anharmonic oscillator result
which has no frequency dependence in the numerator as in Eq. (78), i.e.,
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38
2
χ 2
zzz −ω2 ω1 ; ω1 ; ω2 χ zzz −ω2 ω1 ; ω2 ; ω1 2
N e3
k 2
zzz
:
ε0 m3e ω210 − ω1 ω2 2 ω210 − ω21 ω210 − ω22 (79)
One can conclude that the anharmonic oscillator model, although widely used, is
not strictly correct. Nor should one have expected it to be, since it is not based on
a physical model for a molecule.
The electro-optical response of a material is another manifestation of a χ 2
2
process, namely Realfχ 2
ijk −ω; ω; 0 χ ijk −ω; 0; ωg. For example, for all
fields polarized along the x axis and far enough away from the resonances
so that the relaxation time can be neglected, in the two-level approximation,
2
Realfχ 2
xxx −ω; 0; ω χ xxx −ω; ω; 0g N 2
L ωL0fμ10;x μ11;x − μ00;x μ01;x g
ε 0 ℏ2
×
23ω210 − ω2 :
ω210 − ω2 2
(80)
3.3b. Two-Level Model: χ 3
The frequency dispersion of χ 3 and the sign of the non-resonant nonlinearity
have been a source of speculation since the early days of nonlinear optics. The
two-level model can be used to evaluate the third-order nonlinearity in a first
approximation for molecules that have permanent dipole moments. From
Eq. (42), the third-order susceptibility is
χ 3
ijkl −ωp ωq ωr ; ωp ; ωq ; ωr N
Lωp Lωq Lωr Lωp ωq ωr ε 0 ℏ3
μ01;i μ11;l − μ00;l μ11;k − μ00;k μ10;j
×
ω̂10 − ωp − ωq − ωr ω̂10 − ωq − ωp ω̂10 − ωp μ01;j μ11;k − μ00;k μ11;i − μ00;i μ10;l
ω̂10 ωp ω̂10 ωq ωp ω̂10 − ωr μ01;l μ11;i − μ00;i μ11;k − μ00;k μ10;j
ω̂10 ωr ω̂10 − ωq − ωp ω̂10 − ωp μ01;j μ11;k − μ00;k μ11;l − μ00;l μ10;i
ω̂10 ωp ω̂10 ωq ωp ω̂10 ωp ωq ωr μ01;i μ01;l μ01;k μ01;j
−
ω̂10 − ωp − ωq − ωr ω̂10 − ωr ω̂10 − ωp μ01;i μ01;l μ01;k μ01;j
ω̂10 ωq ω̂10 − ωr ω̂10 − ωp μ01;l μ01;i μ01;j μ01;k
ω̂10 ωr ω̂10 ωp ω̂10 − ωq μ01;l μ01;i μ01;j μ01;k
:
ω̂10 ωr ω̂10 ωp ω̂10 ωp ωq ωr Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004
(81)
39
The terms in the first summation correspond to single transitions between the
excited state and the ground state and a second transition involving the change in
the permanent dipole moments. The second set of terms involve two successive
one-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 a
single z-polarized input and output beam:
χ 3
zzzz −3ω; ω; ω; ω
N 3
L ωL3ω
ε 0 ℏ3
μ11;z − μ00;z 2 jμ10;z j2
ω̂10 − 3ωω̂10 − 2ωω̂10 − ω
μ11;z − μ00;z 2 jμ10;z j2
ω̂10 ωω̂10 2ωω̂10 − ω
μ11;z − μ00;z 2 jμ10;z j2
ω̂10 ωω̂10 − 2ωω̂10 − ω
μ11;z − μ00;z 2 jμ10;z j2
ω̂10 ωω̂10 2ωω̂10 3ω
jμ10;z j4
−
ω̂10 − 3ωω̂10 − ωω̂10 − ω
jμ10;z j4
ω̂10 ωω̂10 − ωω̂10 − ω
jμ10;z j4
ω̂10 ωω̂10 ωω̂10 − ω
jμ10;z j4
ω̂10 ωω̂10 ωω̂10 3ωr :
(82)
For materials in which a two-level system would be valid, it is evident from
Eq. (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 since
they are χ 3 processes. These phenomena occur in all materials. The starting
point for the two-level analysis is Eq. (42). As shown in Fig. 7, there are three
χ 3 , each corresponding to a different ordering of the frequencies ω; ω; −ω that
contribute. The sum of the three is the physically relevant quantity with the possibility of strong interferences between the contributing terms. Cases I and II go
through the ground state with a DC response in an intermediate step, whereas
Case III has a two-photon resonance.
Figure 7
(3)
( − ω ;−ω , ω , ω )
Case II: χ xxxx
ω
ω
-ω
|g>
(3)
χ
( − ω ; ω ,− ω , ω )
Case I: xxxx
-ω
ω
|g>
ω
|g>
Case III: χ xxxx (−ω ; ω , ω ,− ω )
(3)
The three χ 3 contributions to nonlinear absorption and refraction. The upward
arrows correspond to absorption and the downward ones to emission.
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40
Substituting for the ground state and the excited state with x-polarized light [11],
Case I: χ 3
xxxx −ω; ω; −ω; ω
N 3
1
3
2
2
L ωL−ω jμ10 j μ11 − μ00 χ xxxx −ω; ω; −ω; ω 3
ε0 ℏ
ω̂10 − ω2 ω̂10
1
1
ω̂10 ωω̂10 ω̂10 − ω ω̂10 ωω̂10 ω̂10 − ω
1
ω̂10 ω2 ω̂10
1
1
4
− jμ01 j
3
ω̂10 − ω
ω̂10 − ωω̂10 − ω2
1
1
:
(83)
ω̂10 ω2 ω̂10 ω ω̂10 ω3
Case II: χ 3
xxxx −ω; −ω; ω; ω
χ 3
xxxx −ω; −ω; ω; ω N 3
1
2
2
L ωL−ω jμ10 j μ11 − μ00 3
2
ε0 ℏ
ω̂10 − ω2 ω̂10
1
1
ω̂10 ωω̂10 ω̂10 ω ω̂10 − ωω̂10 ω̂10 − ω
1
2
ω̂10 − ω2 ω̂10
1
1
1
4
− jμ01 j
ω̂210 − ω2 ω̂10 − ω ω̂10 ω
1
1
1
.
(84)
2
ω̂10 − ω2 ω̂10 − ω ω̂10 ω
Case III: χ 3
xxxx −ω; ω; ω; −ω
χ 3
xxxx −ω; ω; ω; −ω
N 3
L ωL−ω jμ10 j2 μ11 − μ00 2
ε0 ℏ 3
1
1
1
×
ω̂10 − 2ωω̂10 − ω ω̂10 − ω ω̂10 − ω
1
1
1
ω̂10 2ωω̂10 ω ω̂10 ω ω̂10 ω
1
1
1
4
− jμ01 j
ω̂210 − ω2 ω̂10 − ω ω̂10 ω
1
1
1
2
:
(85)
ω̂10 − ω2 ω̂10 − ω ω̂10 ω
This last case (Case III) is the only one that gives rise to a two-photon resonance
peak. As a result, all the terms proportional to μ11 − μ00 2 are labeled as twophoton contributions. In the two-level model, two-photon absorption requires a
molecule with a permanent dipole moment. We will see later that the three-level
model has a two-photon transition even when the dipole moment vanishes.
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41
Although the detailed frequency dispersion depends on the relative magnitude of
μ11 − μ00 2 and jμ10 j2 , it is useful to examine the typical frequency dependence
of the different contributions to the total third-order susceptibility in the special
case where jμ10 j2 μ11 − μ00 2 , as shown in Fig. 8 [11]. The one-photon contributions (∝ jμ10 j4 ), both real and imaginary, are negative at all frequencies.
However the two-photon contributions (∝ jμ10 j2 μ11 − μ00 2 ) can be either
positive or negative, depending on the frequency. Between ω ω10 and the
two-photon dispersion resonance at ω ω10 ∕2, Realχ 3 χ 3
R is negative
and after the resonance it is positive all the way to the non-resonant limit
ω 0. The two-photon imaginary component χ 3
I starts out negative at
ω 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. Whether
the non-resonant real value is positive or negative for the sum of the two contributions depends on which process, i.e., one- or two-photon transitions, dominates. Note that, in Fig. 8, the real part of the total susceptibility goes to zero in
the non-resonant limit since μ11 − μ00 2 jμ10 j2 is assumed. As will be shown
analytically later, the sign of the real component of the non-resonant nonlinearity
is determined by the sign of μ11 − μ00 2 − jμ10 j2 .
The detailed formulas are complicated [11]. It is instructive here to examine
approximate formulas that are valid in each of the four frequency regimes defined below, namely, near the one- and two-photon resonances, off resonance
and non-resonant [11,107]:
1. On and near resonance
a. one-photon resonance (jω − ω10 jτ10 ≤ 5) and
b. two-photon resonance (j2ω − ω10 jτ10 ≤ 5)
Figure 8
Generic dependence on normalized frequency of the real and imaginary components in arbitrary units of the one- and two-photon terms of the third-order
susceptibility in the two-level model. The blue curves are for the total of the onephoton terms (∝ jμ̄10 j4 ) and the red curves are for the total two-photon terms
∝ jμ̄10 j2 μ̄11 − μ̄00 2 . The regions of positive and negative susceptibility are identified. The upper curves show the dispersion of the two-photon
resonance terms on a linear scale.
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42
For near and on the one-photon resonance, i.e., ω ≈ ω10 ,
3
3
χ 3
xxxx −ω; ω; ω; −ω χ xxxx −ω; ω; −ω; ω χ xxxx −ω; −ω; ω; ω
2N 3
ω10 − ω
−
L ωL−ω ×
2
ε 0 ℏ3
ω10 − ω2 τ−2
10 −2
ω10 − ω2
2
2 τ 10
4
× jμ10 j μ11 − μ00 2 jμ10 j
ω10
ω10 − ω2 τ−2
10 2
ω10 − ω
29
1
jμ10 j2 μ11 − μ00 2 2
− iτ−1
10
2
−2 2
6
ω10 − ω τ10 ω10
2
.
− jμ10 j2
ω10 − ω2 τ−2
10 (86)
Note that cancellation effects between the different contributing terms occur in
the two-photon contributions [∝ μ11 − μ00 2 ], causing the real leading term to
be proportional to τ−2
10 at resonance. As a result, the total response is dominated
4
by the triply resonant terms in χ 3
xxxx −ω; ω; −ω; ω ∝ μ10 associated with Case I
unless the permanent dipole moment differences are unphysically large. Therefore, the predictions of this model are that searching for materials with large
permanent dipole moments is not expected to produce large on-resonance
third-order nonlinearities.
However, near the two-photon resonance, i.e., 2ω ≈ ω10 , only the
jμ10 j2 μ11 − μ00 2 terms in χ 3
xxxx −ω; ω; ω; −ω are enhanced and they dominate
the nonlinear response, i.e.,
χ 3
xxxx −ω; ω; ω; −ω ≅
8N 4
ω10 − 2ω iτ−1
10
2
2
. (87)
L
ωjμ
j
μ
−
μ
10
11
00
ω210 ω10 − 2ω2 τ−2
εℏ3
10 2. Off-resonance (jω − ω10 jτ10 > 5 and j2ω − ω10 jτ10 > 5)
In this region, the damping term in the resonance denominators can be ignored,
greatly simplifying the analysis. The result is
3
3
χ 3
xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω − ω χ xxxx −ω; −ω; ω; ω
4N 3
1
2
2
3 L ωL−ω jμ10 j μ11 − μ00 3ω10
2
2
εℏ
ω10 − 4ω ω210 − ω2 5ω210 − 8ω2 iωτ−1
10
ω210 − 4ω2 2 ω210 − ω2 2
ω210 − ω2 4iτ−1
10 ω
4
2
2
− jμ10 j ω10 3ω10 ω .
ω210 − ω2 4
(88)
These formulas are essentially valid for the “tails” of the response on both the
low- and high-frequency sides of the one- and two-photon resonances, and
the region between them. It is important to note that the imaginary component
of the third-order susceptibility is proportional to the product of the frequency
Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004
43
and the inverse of the excited state lifetime and goes to zero in the Kleinman
limit. Note also that the real part of the nonlinearity for frequencies below the
two-photon resonance can be positive if μ11 − μ00 2 can be tuned independently
of jμ10 j2 , as appears to be possible for chromophores based on charge transfer
states [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 close
to the ground state and all that remains in the denominators is the transition
frequency. As a result, the third-order susceptibility reaches a constant value.
That is, all terms in the summation proportional to jμ10 j2 μ11 − μ00 2 contribute
equally. Similarly, all the terms proportional to jμ10 j4 also contribute equally.
As a result, the relative contribution due to the permanent dipole moments
is orders of magnitude larger here than in the near- and on-resonance case.
Here,
3
χ 3
xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω; −ω
ω→0
χ 3
xxxx −ω; −ω; ω; ω!12
N 4 jμ10 j2
L 0 3 fμ11 − μ00 2 − jμ10 j2 g.
ω10
ε 0 ℏ3
(89)
Note that, if there are any low-frequency dielectric processes present, the limit
ω → 0 refers to frequencies far below the electronic resonances and far above
the inverse of the dielectric relaxation times. Equation (89) indicates that the two
contributions interfere, which can result in a reduced nonlinearity for molecules.
Typically molecules optimized for a large χ 2 will exhibit a positive nonresonant n2 . Note that, as ω → 0, Imagχ 3
xxxx ∝ ω → 0.
3.3c. First-Order Effect on χ 3 of Population Changes in Two-Level
Systems
It was assumed at the outset of the SOS derivation that initially all the electrons
were in the ground state and excited state populations were neglected. In fact, it
is commonly believed that, for low input intensities, the nonlinear contribution
to any population produced in the excited state is negligible compared to
Figure 9
n=m= ν=|1>
1
1
ω 10
≈ ∆t
ω 10
≈ ∆t
g=|0>
1
ω 10
≈ ∆t
Non-resonant case for the interaction of low-energy photons with the two-level
system. Reproduced with permission of John Wiley and Sons publishers [10].
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44
the Kerr effect just discussed. We now show that this is not necessarily the
case.
The probability for an absorption process to occur is proportional to the population 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 all
of its electrons initially in the ground state, it can be absorbed with a probability
proportional to the intensity, thus raising one electron (per absorbed photon) to
the excited state via stimulated absorption. Furthermore there is stimulated emission, proportional to the intensity, by which excited electrons are returned to the
ground state. As discussed previously, there is also spontaneous emission
quantified by the natural lifetime which governs the decay of the excited state
electrons back to the ground state.
Absorption and emission lead to changes in the population (number density) of
the states. Defining the initial (total) electron density as N , N 0 as in the ground
state, and N 1 as in the excited state density, ΔN N 0 − N 1 and N N 0 N 1 .
For intensities well below the saturation intensity I sat (ω), the usual steady-state
rate equations give, for the first-order susceptibility [11],
χ 1
ii −ω; ω jμ10;i j2
N Lω
2ω10
1 Iω∕I sat ω ℏε0
−1
ω210 − ω2 τ−2
10 2iτ 10 ω
:
×
2
−2
ω10 − ω2 τ−2
10 ω10 ω τ 10 (90)
For small intensities, i.e., Iω ≪ I sat ω, 1Iω∕I sat ω−1 ≅1−Iω∕I sat ω.
The contribution due to the intensity can be written as [11]
N L4 ωjμ10 j4 ω210 ω
χ 3
eff −ω; ω; −ω; ω −16
nωε0 cℏ3
−1
ω210 − ω2 τ−2
10 2iτ 10 ω
×
: (91)
2
2
−2 2
ω10 − ω2 τ−2
10 ω10 ω τ 10 The real and imaginary contributions to the effective susceptibility off resonance
decrease with decreasing ω and ω2 , respectively, because the linear absorption
responsible for this contribution goes to zero in this limit and hence they are both
zero in the non-resonant limit.
Figure 10
N1
hω
N
N0
II
BI
BI
I
N1
τ10
N0
III
(a) Two-level system with all electrons initially in the ground state. (b) A single
incoming photon is absorbed and an electron is raised to the excited state. (c) Situation after many photons have been absorbed. Process I refers to stimulated
emission, II refers to stimulated absorption, and III to spontaneous emission.
Reproduced with permission of John Wiley and Sons [10].
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45
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, for
cw inputs or for pulses with pulse width Δt ≫ τ10 in which there is a steady state
population in the excited state. For pulses τ10 ≫ Δt the excited state population
will 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 imaginary
components of the total χ 3
xxxx −ω; ω; −ω; ω are shown in Fig. 11 as the dashed
curves. For comparable one and two-photon contributions, the population effects
complicate further the frequency spectrum. For example, the cancellation of the
total nonlinearity at ω10 − ω∕ω10 0.64 due to interference between the Kerr
electronic nonlinearity and the saturation contribution can occur. The lesson here
is that the frequency dispersion of Realχ 3 ∝ n2 can exhibit multiple changes in
the 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 any
electronic states. In principle their third-order and higher susceptibility properties can be analyzed in terms of the molecular symmetry properties and irreducible 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
3
because symmetry properties can lead to χ 2
ijk 0. This is not the case for the χ 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 (no
permanent dipole moments in the ground and excited states) is [108]
Figure 11
Relative contributions to the total nonlinearity (solid curves) of the Kerr electronic nonlinearity (dotted curve) of the saturation contribution (dashed curve),
all for the case μ11 − μ00 2 jμ10 j2 . (a) The real part, which also shows the total
nonlinearity for μ11 − μ00 2 1.2jμ10 j2 as a dashed–dotted curve. (b) The imaginary part of the third-order nonlinearity in arbitrary units. The signs identify
whether the nonlinearity is positive or negative. The vertical lines indicate where
the nonlinearity changes sign. Reproduced with permission of John Wiley and
Sons [11].
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46
χ 3
ijkl −ωp ωq ωr ; ωp ; ωq ; ωr N
Lωp Lωq Lωr Lωp ωq ωr ε0 ℏ3
(
"
X0
μgv;i μvn;l μnm;k μmg;j
x
×
ω̂vg − ωp − ωq − ωr ω̂ng − ωq − ωp ω̂mg − ωp v;n;m
μgv;j μvn;k μnm;i μmg;l
ω̂vg ωp ω̂ng ωq ωp ω̂mg − ωr μgv;l μvn;i μnm;k μmg;j
ω̂vg ωr ω̂ng − ωq − ωp ω̂mg − ωp )
μgv;j μvn;k μnm;l μmg;i
ω̂vg ωp ω̂ng ωq ωp ω̂mg ωp ωq ωr (
X0
μgn;i μng;l μgm;k μmg;j
−
ω̂ng − ωp − ωq − ωr ω̂ng − ωr ω̂mg − ωp n;m
μgn;i μng;l μgm;k μmg;j
μgn;l μng;i μgm;j μmg;k
ω̂mg ωq ω̂ng − ωr ω̂mg − ωp ω̂ng ωr ω̂mg ωp ω̂mg − ωq )#
μgn;l μng;i μgm;j μmg;k
:
(92)
ω̂ng ωr ω̂mg ωp ω̂ng ωp ωq ωr Although it is useful to obtain analytical formulas for Eq. (92) in order to study
trends due to molecular engineering, etc., these equations are just too complex to
be solved analytically in the general case, or even near and on resonance. As a
result, numerical methods need to be used if sufficient information is available
about the states and the transition dipole moments. However, for copolarized
inputs and outputs, it has proven possible to derive general analytical formulas
for the real part of the nonlinear susceptibility for nonlinear refraction in the limit
that the inverse state lifetimes can be neglected relative to the difference between
the photon frequencies and the resonance frequency. This limits the validity to
the off-resonance and non-resonant regimes [108].
For the off-resonance case,
3
3
Realfχ 3
xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω; −ω χ xxxx −ω; −ω; ω; ωg
XXX
μgv μvn μnm μmg
0
0
0
N 4
L ω 4
3
2
2
2
2
2
2
ε0 ℏ
m
v
n ωng ωng − 4ω ωvg − ω ωmg − ω × f3ω2ng ωvg ωmg ω2ng ωvg − ωmg ω
ω2ng 2ωng ωvg ωmg − 8ωvg ωmg ω2 − 4ωvg − ωmg ω3 g
X0 X0
−2
jμng j2 jμmg j2
fωmg ωng f3ω2mg ω2ng
2
2 2
2
2 2
ω
−
ω
ω
−
ω
ng
mg
n
m
2
2
2
4
2
3
3
2ωng 2ωmg − 7ωng ωmg ω − ω g ω ωmg ωng g :
(93)
Noting the complexity of these equations, it is clear that the net nonlinearity can
change sign multiple times with frequency.
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47
As discussed previously, the energy levels for symmetric molecules have either
odd (“ungerade,” Bu ) or even (“gerade,” Ag ) symmetry. Therefore, one-photon
electric dipole transitions between states with the same symmetry are not allowed. Thus, in the first summation, the symmetric excited states ℓ can be
reached only by intermediate coupling to an odd-symmetry state ℓ0 via two electric dipole transition moments, namely, μℓ0 g and μℓℓ0 . The “pathways” corresponding to the terms in the first and second summations for one-photon processes in
Eqs. (92) and (93) are shown in Fig. 12 as solid and dashed lines, respectively
[108]. Therefore, even-symmetry excited states (called “two-photon” states) can
be accessed only by the simultaneous absorption of two photons. However,
the second summation involves only one-photon transitions from the evensymmetry ground state to odd-symmetry excited states, sometimes called
“one-photon” states.
For the non-resonant case, Eq. (93) simplifies further to [108]
3
3
χ 3
xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω; −ω χ xxxx −ω; −ω; ω; ω
N L4 ω → 0
ε 0 ℏ3
XXX
X0 X0 jμng j2 jμmg j2
0
0
0 μgv μvn μnm μmg
× 12
−6
ω
ω
mg
ng :
ωng ωvg ωmg
ω2ng ω2mg
v
n
m
n
m
(94)
If the contributions of the one-photon transitions, which are always negative, are
larger than the positive contribution from the two-photon terms, then the nonresonant nonlinearity will be negative, and vice versa [108]. The interference
determines the sign of the nonlinearity in the limit ω → 0. This conclusion
is critical since even-symmetry states and their transition moments do not
contribute to the linear susceptibility and must be evaluated by nonlinear
spectroscopy.
Figure 12
(a)
(b)
Bu
Ag
Bu
Bu
Ag
Ag
Bu
Ag
Examples of the different pathways possible for (a) the second summation
(dashed lines) and (b) the first summation (solid lines) in Eq. (93) by which
the even-symmetry excited states can be reached only via intermediate oddsymmetry excited states. Reproduced by permission of the Optical Society of
America [108].
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48
4.2. Three-Level Model
A minimum of three states are required to describe the nonlinear optics of
symmetric molecules for which general analytical results have been obtained
[11,108–112]. The three levels are shown in Fig. 13. In such a model, the
first state is the even-symmetry ground state 1Ag , the second is the oddsymmetry excited state, labeled 1Bu , which is most strongly coupled to
the ground state via μ1Bu ←1Ag , and the third is the lowest lying even-symmetry
excited state mAg with strong coupling to 1Bu via a large transition dipole
moment μmAg ←1Bu .
The key question is which excited states to use. This can be decided by brute
force using numerical ab initio calculations of the states and transition moments. These are necessarily limited to simple molecules even when using the
most powerful computers [113]. Alternatively, the dominant linear absorption
peak can be used to evaluate the transition moments to the 1Bu state. Twophoton absorption and third-harmonic generation spectroscopy can likewise
be used via measurement of the dominant two-photon peak to evaluate
the location of mAg and μmAg ←1Bu [114,115]. If there is more than one dominant 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-symmetry excited states if mAg falls in a quasi-continuum of even-symmetry
states [110,117]. Finally, spontaneous decay to the ground state is not allowed
from even-symmetry states, and the state mAg can only decay to 1Bu via τ21
with subsequent decay to ground state via τ10 . The effective decay time τeff
−1
−1
via this coupling is given by τ−1
eff τ21 τ 10 .
Assuming x polarized incident light polarized parallel to the symmetry axis, analytical formulas for the leading terms in the four limits defined in Section 3.3a
are [11]
3
3
χ 3
xxxx −ω; −ω; ω; ω χ xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω; −ω
N 4
2
jμ21;x j2
L
ωjμ
j
10;x
ε 0 ℏ3
−1
−1
2
2 2
2ω210 − ω2 ω20 − ω10 ω20 − 2ω10 ω210 − ω2 2iω10 τ−1
10 iτ 21 τ10 ω20 ω10 − ω ×
2
2ω210 ω20 ω20 − 2ω10 2 ω10 − ω2 τ−2
10 2
2ω10 − ω3 4iτ−1
10 ω10 − ω
− jμ10;x j2
(95)
2
−2
3
ω10 − ω τ10 and
Figure 13
2
mA g
µ 21
τ 21
µ10
τ 10
1
0
1Bu
1A g
The three energy levels, the electric dipole matrix elements, and the excited state
lifetimes for the three-level model of a centrosymmetric system.
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49
χ 3
xxxx −ω; ω; ω; −ω
N 4
4ω220 − 4ω2 2
2
L
ωjμ
j
jμ
j
01;x
12;x
−1 2
ε0 ℏ3
ω20 2ω10 − ω20 2 ω20 − 2ω2 τ−1
21 τ10 8i
−1
2
2
2
−1
2
2
3
τ−1
21 τ10 ω20 4ω10 − ω20 ω20 2ω10 τ10 ω20 − 4ω ω20 2ω10 3
2
−1
−1 2
3
ω20 2ω10 − ω20 ω20 − 2ω τ21 τ10 2ω10 ω20 (96)
for the one- and two-photon resonances, respectively. For the off-resonance
case,
3
3
χ 3
xxxx −ω; ω; ω; −ω χ xxxx −ω; ω; −ω; ω χ xxxx −ω; −ω; ω; ω
2 2
2
2
2
4N 4
2
2 3ω10 ω20 ω ω20 4ω20 ω10 − 8ω10 L
ω
jμ
j
jμ
j
10;x
21;x
ε 0 ℏ3
ω20 ω210 − ω2 2 ω220 − 4ω2 ω20 ω210 7ω20 2ω10 ω2 ω20 8ω10 ω20 − 2ω10 iω τ−1
10
ω20 ω210 − ω2 3 ω220 − 4ω2 2
−1 ω20 2ω10 ω20 ω10 2ω 2τ−1
τ
21
10
ω210 − ω2 2 ω220 − 4ω2 2
2
2
−1
4
2
2 ω10 − ω 4iωτ10 − jμ10 j ω10 3ω10 ω :
(97)
ω210 − ω2 4
Finally, for the non-resonant case,
3
3
χ 3
xxxx −ω; ω; −ω; ω χ xxxx −ω; ω; ω; −ω χ xxxx −ω; −ω; ω; ω
L4 ωjμ10 j2 jμ21 j2 jμ10 j2
−
12N
:
ω20
ω10
ε0 ℏ3 ω210
(98)
The sign of the nonlinearity is determined by the ratio jμ21 j2 ω10 ∕jμ10 j2 ω20 .
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, including
polydiacetylenes and squaraines in which the transition dipole moments were
calculated and the signs of the non-resonant nonlinearities were found to be
in good agreement with the predictions of Eq. (98) [110–112].
The typical frequency dependence of the different contributions to the total
third-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 components, subscripts R and I, respectively. The one-photon contributions are always negative. There is a dispersion type of resonance at ω ω20 ∕2 that can
lead to a positive χ 3
R for the two-photon contribution for frequencies below the
two-photon resonance. The net result can be either a positive or negative χ 3
R as
the non-resonant limit is approached. [The parameter range that leads to positive
values is given by Eq. (98).] The two-photon contribution to χ 3
I is positive
throughout the whole frequency range, increasing with decreasing frequency
up to ω ω20 ∕2, where it peaks and then decreases as the zero frequency limit
is approached. Whether the non-resonant value of χ 3
R is positive or negative
depends on which process, i.e., one- or two-photon transitions, dominates;
see Fig. 15, for example.
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50
Figure 14
Relative contributions to the total nonlinearity (solid curve) of the Kerr twophoton transitions (dotted curve) and the Kerr one-photon transitions
(dashed curve) for ω20 1.33ω10 , ω10 τ10 0.001, ω10 τ21 0.01, and
jμ21 j2 ω10 ∕jμ10 j2 ω20 1.25. (a) The real part and (b) the imaginary part of
the third-order nonlinearity in arbitrary units. The signs identify whether
the nonlinearity is positive or negative. The vertical lines indicate where the
nonlinearity changes sign. Reproduced with permission of John Wiley and
Sons [11].
5. Transition to Bulk Nonlinear Molecular Optics
In this section, we go beyond the single molecule response and focus on condensed media. As we previously discussed, molecular materials are marked by
the dominance of the molecular response, even in condensed media, due to the
fact that they are bound loosely in the condensed state, an example being van der
Waals crystals. Local field corrections as “perturbations” on the molecular response are often sufficient to describe the molecular response in the condensed
χ (3) (arbitrary units)
Figure 15
x100→
0
0.2
0.4
0.6
ω 10 −ω
→
ω 10
0.8
1.0
Calculation of n2 ∝ ℜealfχ 3 in arbitrary units versus the normalized
frequency ω10 − ω∕ω10 for the three-level model with ω20 1.33ω10 ,
ω10 τ21 0.01, ω10 τ10 0.001, and jμ21 j2 ω10 ∕jμ10 j2 ω20 0.75 (dashed
curve), jμ21 j2 ω10 ∕jμ10 j2 ω20 1.25 (solid curve) [108].
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51
state and are discussed first. These molecular media can be defined by the ability
to be described by a local field-corrected oriented gas model. In this description,
as described in Section 3, rotations are sufficient for transformation between the
molecular and condensed states. The oriented gas model and applications to
crystals and poled polymers are then described. Finally, collective phenomena
contributing to the third-order response are described.
5.1. Local Field Corrections, Linear Susceptibility
⃗
The incident or emitted electric “Maxwell” fields, Eω
i , are field averages over
volumes that contain many molecules but are small over all of the wavelengths
involved in the interaction. (They are called the Maxwell fields since they appear
in Maxwell’s equations.) For a molecule in a host material or in a collection of
⃗
similar molecules, the electric field Fω
at the molecular site is a superposition
⃗
of the applied electric fields Eωi and the local electric fields due to the dipoles
induced by the Maxwell fields in the nearby material. In dilute matter, such as
gases, where the molecular density is sufficiently small that the Maxwell field
⃗
Eω
at a molecule is much larger than the total of the fields at the site of the
⃗
⃗
molecule due to the dipoles induced in neighboring molecules, Fω
Eω.
The local electric field is one of four types:
1. the applied electric field polarizes the surrounding material which creates a
field that acts on the molecule;
2. the surrounding material is the source of an electric field even in the absence
of an applied field due to local fluctuations in charge density of an otherwise
electrically neutral material;
3. if the molecule has a ground-state moment of any order (such as a dipole
moment, quadrupole moment, etc.) the electric field associated with the molecule will polarize nearby material, which results in a reaction field; that is, a
field that acts back on the molecule; or
4. an applied electric field induces a moment in a molecule which polarizes the
material surrounding the molecule and results in a reaction field that changes
the dipole moment of the molecule.
Clearly the local electric field at a molecular site is a complex phenomenon and
involves the use of self-consistent methods to be evaluated properly. Here, we
review the two simplest models; the Lorentz–Lorenz local field model and the
Onsager model. Recall that the Onsager model is central to nonlinear optics, as
shown in the early work of Levine and Bethea [33] and Oudar [34]. Screening of
embedded dipoles, reaction fields, the effect of an external electric field on a
screened dipole, and radiating dipoles embedded in a dielectric are also briefly
described in Appendix B. The sections that follow are based on the commonly
available literature and textbooks on electrostatics [118] and local electric fields
[68]. Much of our presentation closely follows a development geared toward
practitioners of nonlinear optics.
5.1a. Continuum Approximation
Consider the local fields in a dipolar liquid or solid solution. In the simple model
depicted in Fig. 16(a), the motion of each dipole is random and on average
sweeps out a spherical volume. Averaged over long enough time scales, the
system will appear to be continuous, as represented in Fig. 16(b). A uniform
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52
and static electric field, when applied to the isotropic and homogeneous solution,
yields a time-averaged polarization,
1
⃗
P⃗ 1 ε0 ⃗⃗χ · E⃗ ε0 εr − 1E;
(99)
where εr is the relative dielectric function of the solution and is the macroscopic
linear susceptibility. Since measurement time scales are long compared with collisional times, the time-averaged polarization is typically the measured quantity.
Thus, the material can often be viewed as a continuous dielectric in which the
substituent 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 electric
field that would be required to produce the observed polarization in the molecule
is the local electric field. Any textbook on electrostatics describes the electric
field of a polarized dielectric sphere [118].
The dashed circle in Fig. 16(b) represents the molecule, and Fig. 16(c) shows it
after being removed from the dielectric under the assumption that the uniform
polarization is “frozen in.” Both the surfaces of the dielectric sphere and the
cavity are necessarily charged. The induced dipole moment of a dielectric
sphere, ⃗p, in a uniform field F⃗ is
ε −1 3⃗
⃗p ε0 αF⃗ ε0 r
a F;
εr 2
(100)
where α is the polarizability of the sphere, assuming for simplicity a scalar
⃗ is the dipole moment per unit volume given by
medium. The polarization, P,
Figure 16
(a)
E
(b)
P
(c)
a
(a) A dipolar liquid in which the molecules (arrows) sweep out a spherical volume. (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 is
removed with the charges frozen in place.
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⃗p
3 εr − 1 ⃗
P⃗ 4 3 ε0
F:
4π
εr 2
πa
3
(101)
⃗ we get
Setting Eq. (101) equal to Eq. (96) and solving for F,
ε 2 ⃗
F⃗ r
E:
3
(102)
This is known as the Lorentz–Lorenz local field model. The local field factor is
defined as
L
εr 2
;
3
(103)
⃗ Note that the local field factor is often labeled f rather than L.
where F⃗ LE.
For an anisotropic material, the local field factor is a tensor. As a quick approximation, 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. In
many typical cases, this will yield a reasonable approximation to the true local
electric 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 dielectric) in a mixture, for example, in a liquid solution or dye-doped polymer at low
concentration, can be viewed as a solute molecule that is embedded in a smooth
dielectric. Figure 17(a) shows the electric field lines of a dielectric sphere embedded in another dielectric under the influence of a uniform electric field. There
ε1 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 surrounding
dielectric. Figure 17(b) shows the dielectric with the sphere removed. The polarizations P⃗ 1 and P⃗ 2 are those of the surrounding medium far from the sphere
and inside the sphere, respectively. In analogy to Eq. (3), the polarization inside
the sphere is
P⃗ 2 ε0 ε2r − 1E⃗ in ;
(104)
Figure 17
E
P2
P1
(a)
(b)
(a) Electric field of a dielectric sphere of dielectric constant ε2 embedded in a
dielectric of constant ε1 and (b) with the charges fixed in place and the dielectric
sphere removed.
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54
where E⃗ in is the electric field inside the sphere. E⃗ in is related to the uniform
electric field far away from and outside the sphere, E⃗ [68],
3ε1r
⃗Ein ⃗
E.
(105)
2ε1r ε2r
Using Eqs. (104) and (105), the polarization inside the sphere in terms of the
external electric field is
3ε
1r
⃗
E.
(106)
P⃗ 2 ε0 ε2r − 1
2ε1r ε2r
Again, the local electric field is the field that is required to induce a polarization P⃗ 2 in the sphere. Using an argument similar to that leading to Eq. (102)
with εr → ε2r , the relationship between the local field F⃗ and the polarization of
the sphere is
ε −1 ⃗
F:
(107)
P⃗ 2 3ε0 2r
ε2r 2
Setting Eqs. (106) and (107) equal, the local electric field is given by
3ε1r
ε2r 2 ⃗
⃗F E:
(108)
2ε1r ε2r
3
This expression is similar to the Onsager local field formula described in
Appendix 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 nature
of the dielectric function of both the molecule and the surrounding material are
far more complex. In such cases, the local field factor is a second-rank tensor and
relates F⃗ to E⃗ according to
F i Lij E j :
(109)
When the ensemble average principal axes coincide with the principal axes of
the bulk system, the local field tensor is diagonal, which greatly simplifies the
problem. It is straightforward to generalize the local field tensor to the optical
regime when the guest molecule or solute particle is small compared to the
wavelength of the illuminating source. Then, the optical field is approximately
spatially 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,
Lω is
3ε1r ω
ε2r ω 2
Lω :
(110)
2ε1r ω ε2r ω
3
5.2. Oriented Gas Model
We now turn attention to the usefulness of the group theoretical approach of
Section 3.2 for conveniently connecting the microscopic to macroscopic
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55
response while maintaining the efficiency derived from its convenient representation of molecular properties. In materials where the interaction between the
molecular units is small compared to intramolecular forces, such as it is in
dye-doped polymers and molecular crystals, the bulk nonlinear optical susceptibility is given by a tensor sum over the local-field corrected hyperpolarizabilities of the molecules. The hyperpolizability is related to the macroscopic
nonlinear optical susceptibility in an oriented gas model through [119,120]
χ 2
ijk −ωp ωq ; ωp ; ωq N Lωp ωq Lωp Lωq × hβIJ K −ωp ωq ; ωp ; ωq iijk ;
(111)
where N is the molecular number density and hβIJ K iijk (dropping the frequency
notation for convenience) is the orientationally averaged hyperpolarizability
tensor connecting the molecular coordinate system IJ K to the macroscopic
(laboratory) system ijk. The quantity hβIJ K iijk can be written as
hβIJ K iijk hRijk;IJ K iβIJ K .
(112)
↔
The tensor R is the rotation transformation matrix, which can be written in terms
of the Euler angles defined in Fig. 18 as
0
cos ϕ cos ψ − cos θ sin ϕ sin θ
R @ cos ψ sin ϕ cos θ cos ϕ sin ψ
sin θ sin ψ
↔
− cos θ cos ψ sin ϕ − cos ϕ sin ψ
cos θ cos ϕ cos ψ − sin ϕ sin ψ
cos ψ sin θ
1
sin θ sin ϕ
− cos ϕ sin θ A.
cos θ
113
An important simplification is that, because of the rotational invariance, upon
averaging the irreducible representation of the hyperpolarizability, the tensors of
various weight do not mix. This means
Figure 18
(a) Euler angles relating molecular to macroscopic frames. (b) Geometry of C 2v
electron donor–acceptor–donor (D-A-D) chromophores.
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1s
1m
2m
3s
χ 2
ijk ∝ hβIJ K iijk hβIJ K iijk hβIJ K iijk hβIJ K iijk hβIJ K iijk
1m
2m
3s
∝ χ 1s
ijk χ ijk χ ijk χ ijk .
(114)
This fact is used routinely for averaging one-dimensional molecules to express
the resulting macroscopic 1s component; for example, in poled polymers the
relevant macroscopic component is the 1s, and results from the orientationally
average vector hyperpolarizability. We now provide a more sophisticated example building on our treatment of molecules of C 2v symmetry as depicted in
Fig. 18(b) [119]. Consider an axially aligned chiral (handed) macroscopic medium, such as the one based on nematic-like (nematic refers to specific liquid
crystal class) alignment of helices decorated with C 2v chromophores (Fig. 19).
The macroscopic symmetric in this case is D∞ . In Cartesian coordinates, the
only nonzero components of the hyperpolarizability of this point group are
χ D∞ χ xyz χ xzy −χ yzx −χ yxz . In this case, the only irreducible tensor
contributing to this response is the 2m one, so that we write
1
0
−1 0 0
(115)
χ 2m χ xyz @ 0 −1 0 A:
0
0 2
Thus, the susceptibility will depend on the magnitude of the 2m tensor and the
orientation average. Noting that the hyperpolarizability can be written as
1
0
0
Δβ∕2 0
@ Δβ∕2
0
0 A;
(116)
β2m
C 2v 0
0
0
where Δβ βzxx − βxxz , we can then specialize Eq. (111) to
1
1
χ 2m N Δβh2Rzx Rzy − Rxx Rxy − Ryx Ryy i N Δβhsin2 θ sin 2ψi.
6
4
(117)
This simple expression indicates that the macroscopic response is maximized by
the structure shown in Fig. 19 when ψ π∕4 and θ π∕2 [119]. The sign of
ψ should be uniform to maintain a chiral medium.
Figure 19
Optimum alignment of C 2v chromophores on a helix.
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5.3. Crystals
Whether a medium can be χ 2 -active, i.e., it exhibits inversion symmetry on the
scale of an optical wavelength, depends on two factors: (1) the symmetry properties of the molecules as just discussed and (2) the molecular alignment. There are
basically three possibilities in crystals, each depending on the symmetry properties of the crystal’s unit cell formed on crystallization. The different cases are
summarized in Fig. 20 for the example of molecules with 1D second-order nonlinearities, i.e., βiii ≠ 0, along some axis “i.” The simplest case of a single noncentrosymmetric molecule per unit cell that leads to χ 2
iii ≠ 0 is shown in
Fig. 20(c). In crystals, if the unit cells have no inversion symmetry, i.e., no permanent dipole moment or no permanent octupole moment, the medium is not
χ 2 -active. This can occur if the molecules themselves have inversion symmetry, as indicated in Fig. 20(a), for which the unit cell has no permanent dipole
moment. Alternatively, if the individual molecules do have permanent dipole
moments (and hence βijk ≠ 0 for some combination of ijk for an individual molecule) but are P
aligned within a unit cell so that the unit cell has a zero net dipole
moment (i.e., molecules 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 multiple
molecules whose alignment of the molecules in the unit cell results in a noncentrosymmetric unit cell, then some elements of χ 2
ijk ≠ 0. One such arrangement is three dipoles in an equilateral triangle, which have no net dipole
moment, but the centrosymmetry is broken and a second-order response due to
the octupole term occurs [91]. Which optical field components will result in
nonzero second-order parametric processes—such as second-harmonic generation or sum frequency generation—depends on the crystal symmetry.
When there are strong interactions between molecules, for example, in microscopic [90,121–127] and macroscopic cascading [128–130], Eq. (111) needs to
be generalized. For example, in organic crystal lattices, when the van der Waals
interaction or hydrogen bond energies responsible for intermolecular cohesion
are several orders of magnitude smaller than intramolecular chemical bond
energies, the bulk nonlinear optical response will be a sum of the local-fieldcorrected molecular units. Zyss and Oudar calculated the second-harmonic coefficient (proportional to the second-order nonlinear optical susceptibility) in
Figure 20
Examples of unit cells containing molecules with different properties. The red
arrows portray one-dimensional second-order nonlinear coefficients. (a) Unit
cells containing a centrosymmetric molecule (dipole moment represented by
the dot). (b) Unit cells containing two counteraligned noncentrosymmetric
molecules. (c) Unit cells with a single noncentrosymmetric molecule.
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terms of the hyperpolarizability using Eq. (111) with fixed molecular orientations [41,52], as is found in molecular crystals.
Zyss and Oudar applied this model to 2-methyl,4-nitro analine (MNA) for which
the crystal structure, electro-optic coefficient [52], and second-harmonic coefficients [17,20,31,42,131] are known. They found good agreement between the
model and measurements of the bulk nonlinearities.
5.4. Electric Field Poled Media
The organic nonlinear optics community has invented a different way to align noncentrosymmetric molecules to produce artificial “crystals” with uniaxial symmetry.
Molecules with large permanent dipole moments in the ground state have been
engineered. “Charge transfer” molecules are synthesized by attaching groups with
different electron affinity at opposite ends of a “bridge” whose function is to facilitate transfer of electrons between the end groups as shown in Fig. 21. Because the
groups have different electron affinity, charge is transferred from the electron donor
group (D) to the electron acceptor group (A), producing a noncentrosymmetric linear molecule. Typically, linear chains of carbon atoms whose pz orbitals overlap to
form new delocalized π orbitals allow the electrons to move more easily between
the end groups. These molecules are either bonded somewhere in a polymer chain
or “dissolved” inside a polymer as “guest” molecules. This results in randomly
oriented charge transfer molecules inside the bulk of a polymer.
A common technique for aligning charge transfer molecules is electric field poling. When strong electric fields are applied, some net orientation of the molecules can be induced via the permanent dipole moment at elevated temperatures
as shown in Fig. 22. This requires first “softening” of the polymer above the
glass transition temperature T gl for the host polymer by heating, followed by
applying a DC field to produce partial orientation of the molecules. When the
structure is cooled to below the glass transition temperature, a partial net orientation is effectively “frozen in” and the resulting medium has uniaxial symmetry
around the poling direction. This is performed on thin films that can be used as
waveguides for various applications. The most common of these is for electrooptics. Although various “tricks” have been used to achieve phase matching, the
efficiency for second-harmonic generation achieved with poled films was never
large enough to be practical [132].
In crystalline materials, the orientations of the molecules in the unit cell and
the structure of the crystal are well defined. In a doped polymer, the molecular
orientations are continuously distributed and there is one molecule per unit cell,
so that
X s
βIJ K −2ω; ω; ω → βIJ K −2ω; ω; ω:
(118)
S
Figure 21
Prototype charge transfer molecule with acceptor and donor groups separated by
a π-electron bridge.
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Figure 22
(a) In-plane poling and (b) parallel plate poling of the charge transfer layer. The
role of the buffer materials and glass substrate is to inhibit current flow between
electrodes, which diminishes the poling field and causes dielectric breakdown.
(c) Random orientation of molecules prior to poling. (d) Partial alignment of
molecules by field. (e) “Frozen-in” structure.
The distribution of molecules is usually represented by an orientational distribution function, which can be found when the orienting forces acting on the
molecules are known. The nonlinear optical properties of the doped polymer
are then calculated using Eq. (111).
The second-order nonlinear optical susceptibility of a dye-doped isotropic polymer is usually imparted with an electric field that is applied above T gl , and
cooled below T gl to lock in the orientational order [43]. The thermodynamic
model assumes that the dye molecules freely rotate in response to the applied
electric field above T gl . The orientational distribution function is derivable from
a Gibbs distribution function with T T gl —the point at which the molecular
reorientations are slow. The Gibbs distribution yields the orientational distribu⃗ E⃗ pol :
tion function GΩ;
h
i
− ⃗μ · Ēpol ⃗ E⃗ pol R
h
i;
GΩ;
1 ⃗
1
d
Ω
exp
−
⃗μ⋅
Ē
pol
−1
kBT
exp
1
kBT
(119)
⃗ represents the
where ⃗μ is the dipole moment of each molecule in the ensemble, Ω
three Euler angles, Ēpol the applied poling field, and k B is Boltzmann’s constant.
Using Eqs. (111) and (119) and taking the result to first order in ⃗μ⋅E⃗ pol ∕k B T
yields
χ 2
ijk −2ω; ω; ω
N βs
IJ K −2ω; ω; ω
8π 2
Z
⃗
dΩ
⃗μ · E⃗ pol
⃗ jJ Ωa
⃗ kK Ω;
⃗
a Ωa
k B T iI
(120)
⃗
is the Euler rotation matrix discussed previously. Assuming that
where aiI Ω…
the molecule is one-dimensional, that is, the only nonvanishing component of
s
⃗μ and βs
IJ K −2ω; ω; ω are μZ and βZZZ −2ω; ω; ω, respectively, then the two
independent tensor components of the bulk response are
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60
μEpol
;
5k B T
(121)
μEpol
:
15k B T
(122)
s
x2
zzz −2ω; ω; ω N βZZZ −2ω; ω; ω
s
χ 2
xxz −2ω; ω; ω N βZZZ −2ω; ω; ω
This thermodynamic model for poling was first applied to Disperse Red azo 1
dye (DR1) in poly(methyl methacrylate) (PMMA), and the theory—using the
measured value of βs
ZZZ −2ω; ω; ω for this dye in solution, correctly predicted
the bulk second-harmonic coefficient of DR1/PMMA within experimental
uncertainties for a wide range of poling field and number density [43].
While the above model is useful for a poled isotropic polymer, many materials
are anisotropic. Examples of anisotropic materials include liquid crystals and
stretched polymers. For stretched polymers and nonferroelectric materials, the
orientational order is uniaxial, in contrast to a poled polymer, which has polar
order. The additional force can be added to the distribution function according to
h
i
exp k B1T − ⃗μ · E⃗ pol U fcosΘg
⃗ E⃗ pol R
h
i;
GΩ;
(123)
1 ⃗
1
⃗ pol U fcosΘg
d
Ω
exp
−
⃗μ
·
E
−1
kBT
where U fcosΘg is the axial ordering potential. With no poling field applied,
the order parameter of the dyes is given by
h
i
R 1
U cosθ
d
cosθP
cosθ
exp
−
i
−1
k T
h
i B
hPi i ;
(124)
R 1
U cosθ
dΩ
exp
−
−1
kBT
where Pi cos θ is the ith Legendre polynomial. Because the axial forces represented by U cos Θ are centrosymmetric, only even-order order parameters will
be nonzero. Equations (123) and (124), with the help of Eq. (111) to first order in
μEpol ∕k B T , then lead to
μEpol 1 4
8
2
s
hP i hP4 i ; (125)
xzzz −2ω; ω; ω N βZZZ −2ω; ω; ω
k BT 5 7 2
35
x2
xxz −2ω; ω; ω
μEpol
s
N βZZZ
−2ω; ω; ω
k BT
1
1
4
hP i − hP i : (126)
15 21 2 35 4
We stress that the above equations relate the orientational order of the material
before poling as quantified by hP2 i and hP4 i to the second-order
susceptibility
after the material is poled. Without a poling field E⃗ pol 0 , all tensor
components of βijk vanish. Errors in this general theory were later corrected
by Ghebremichael and associates [133].
This formalism was applied to the study of dye-doped polymers with applied uniaxial stress during the poling process. It was shown that, by adjusting the poling
2
field and the applied stress, the tensor ratio χ 2
xxz −2ω; ω; ω∕χ zzz −2ω; ω; ω could
be controlled to make the ratio near unity, making these materials useful in
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electro-optic devices whose operation is independent of the polarization of the
light beam [134].
A large applied electric field can affect the order parameters, making the calculations more complex because the small field approximation is no longer valid. Then, higher powers in E⃗ pol 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 mean
field approximation must be applied.
Van der Vorst and Picken extended the thermodynamic model of poling to the
high-field regime [135]. They included in their potential function, U , the effect of
poling to second order in the electric field and an effective single particle potential:
1
1
U Θ − ⃗μ · E⃗ pol − αE 2pol − ΔαE 2 P2 cos Θ − εhP2 iP2 cos Θ; (127)
2
3
where the first term corresponds to dipolar poling and the second term to the
energy shift of an isotropic material in response to the field (here α represents
the isotropic average of the polarizability). The third term represents poling of the
induced dipole moment (Δα is the difference in the polarizability between the long
and short axes of the cylindrical molecule), and P2 is the second-order Legendre
polynomial. The fourth term is due to internal liquid crystalline forces as originally modeled by Maier and Saupe [136–138]. Here ε represents the strength of the
mean field single particle potential and hP2 i the second-order order parameter,
which depends on the strength of the poling field and the single particle potential.
But, because hP2 i is a parameter in the single particle potential, the order parameters must be calculated self-consistently. The consequence of such a selfconsistent calculation is that the poling field affects the order parameter hP2 i
which can induce a phase transition between the isotropic and nematic phase
of a liquid crystal.
Poled ferroelectric materials have odd-order order parameters that are predetermined: for example, a ferroelectric liquid crystal or a Langmuir Blodgett film
[139–142]. Because an even-order response of order n depends only on the
material’s odd-order order parameters less than n 2, the second-order response
will depend only on hP1 i and hP3 i. For the one-dimensional molecule, the two
independent tensor components are
3
2
2
s
(128)
xzzz −2ω; ω; ω N βZZZ −2ω; ω; ω hP1 i hP3 i ;
5
5
x2
xxz −2ω; ω; ω
N βs
ZZZ −2ω; ω; ω
1
1
hP i − hP i :
5 1 5 3
(129)
All the models above relate the molecular hyperpolarizabilities to the bulk nonlinearity. By choosing appropriate molecules and polymers and poling them
under well-defined conditions, stretching them, and using naturally present
internal forces, a wide range of nonlinear optical properties result. Aligned
χ 2 -active films have also been produced by electric field poling of molecules
with octupolar symmetry or by using Langmuir–Blodgett techniques to deposit
multiple monolayers [91,139–142]
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5.5. Additional Contributions to Third-Order
Nonlinearities
In the preceding sections only the ultrafast electronic hyperpolarizability leading
to χ 3
ijkl −ωp ωq ωr ; ωp ; ωq ; ωr was considered. However, there are many
other slower effects that lead to nonlinear refractive index and absorption
changes in molecular media [11,110]. They mostly arise from collective
phenomena and include
1.
2.
3.
4.
5.
6.
7.
nuclear (vibrational) contributions to n2 ,
single reorientation of molecules with anisotropic polarizabilities,
collective reorientation of molecules with anisotropic polarizabilities,
photorefractive effects,
electrostriction,
thermal nonlinearities, and
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, vibration
and rotation. With the exception of the first one, the others have been discussed
in detail in a recent review paper [110]. Single molecule rotational nonlinearities
are well understood, and little progress has been made in this field over the last
decade.
Vibrational contributions were discussed first in the 1970s for glasses, and they
were found to contribute up to 20% of the Kerr nonlinearity [110]. As a result
there was only limited interest in these contributions. In the 1990s large vibronic
contributions were observed and discussed theoretically in the linear absorption
spectrum of linear molecules and conjugated polymers [143,144]. They have
also been observed in the nonlinear optics of such materials [115,144–146].
Although no quantitative estimate of the vibronic contribution to the two-photon
absorption spectrum of the polydiacetylene PTS was reported in 1996, it is clear
that the integrated intensity of the vibronic subbands with picosecond pulses was
larger than for the contribution of the nonvibronic peak [115].
In 2000 Chernyak and associates showed that in second and third harmonic generation experiments, when the wavelengths are tuned below the lowest-energy,
excited electronic state, the purely electronic hyperpolarizabilities account for
90%–95% of the total [146]. A theoretical analysis of the Chernyak experimental results came to the same conclusion [147]. From the early 1990s, when the
first theoretical papers began to appear, most of the theoretical calculations have
been in the static limit (non-resonant limit) where the one- and two-photon contributions to the nonlinear refractive index interfere destructively, as discussed in
the preceding two sections [148,149]. Bishop pointed out that, in this case, the
vibronic component could dominate the Kerr component.
The most recent experiments on carbon disulphide with pulses ranging in width
from 30 fs to ∼10 ps at frequencies close to the non-resonant regime have shown
that the vibrational and rotational contributions can indeed be dominant, as
shown in Fig. 23 [150]. Additional experiments showed a lack of dispersion with
wavelength in the nonlinear data and confirm that indeed the conditions approximate the non-resonant limit. The vibrational contribution appears at pulse
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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].
widths comparable to the vibrational frequencies of the molecule. Note that the
vibrational contribution is three times the Kerr, and the rotational contribution is
three times the vibrational one.
Clearly there is much still to be learned from further experiments in the
non-resonant regime.
6. Conclusions
In molecular materials, the nonlinear optical properties of the bulk medium are
determined from a sum over the nonlinear optical properties of the molecules by
virtue of the weak interactions between them. This is in contrast to some inorganic crystals (for example, semiconductors) where strong interactions lead
to delocalization of the wave functions, yielding band structures. Molecular
materials made of organic molecules offer a vast choice of molecular structures
that can be custom designed through organic synthesis. This combination of
custom tailoring molecular structure and molecular assembly into a bulk structure 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 have
a large nonlinear optical response because the molecules can be engineered to
individually have large nonlinearities, the molecules cannot always be arranged
in a unit cell in a way to optimize the bulk response. This has led to a different
way to use molecules, namely doped and functionalized into host polymers in
which molecular alignment is achieved by electric field poling to optimize the
second-order susceptibility [51]. Furthermore, such polymers can be made into
thin films, fibers, and moldable components, making them amenable to highvolume manufacturing.
Since the strength of the nonlinearity in dipolar molecules derives from the same
physics as the dipole moment, i.e., the charge transfer mechanism, the tendency of
strong dipoles is to be counteraligned in the crystal phase. Only 20%–30% of
organic crystals containing highly nonlinear molecules exhibit any bulk nonlinearity at all. Furthermore, the stronger the charge transfer mechanism, the
narrower the transmission window in the visible and near-infrared, another characteristic of the charge transfer. Inorganic crystals have also been shown to be up to
the task for frequency conversion, while this is not the case for integrated
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electro-optic devices. As a result, the major application of molecular nonlinear
optics is in poled polymer devices used for electro-optics for operation in the
near-infrared.
There has been steady progress to understand the nonlinear mechanism through
quantum modeling, for example the SOS approach with feedback from experiments, as first applied by Garito and associates [151]. Given the complexity
of quantum modeling, attempts were made to connect structures with nonlinear
optical properties in an effort to identify motifs for synthetic chemists to follow.
These included work on length scaling [152–156], bond-length alternation in
conjugated molecules [86,87], and symmetry [157], which found that the best
third-order materials are made from centrosymmetric molecules [157]. Describing the molecules in terms of the irreducible tensors associated with a molecule’s
spatial symmetry has been proven to be a very powerful tool for understanding
their microscopic and macroscopic optical properties. More recently, a broader
understanding is being developed using scaling arguments [158–160], which are
based on fundamental limits of the nonlinear optical response [161].
Once good molecules are identified and built into materials, models of the local
electric fields need to be used to predict the bulk response. Conversely, determination of molecular properties relies on local field models.
In this article, we have reviewed this bottom-up approach, and shown how limited state models can be used to describe the generic dispersion of the nonlinear
optical response and how the dispersion can be used as a probe of the symmetry
of the system [162]. Such symmetry arguments can be used to determine which
class of states contribute. As a result, for example, given just the sign of the offresonant response of a centrosymmetric system, one can show that a nonlinear
Miller’s delta approach does not apply.
The magnitude and sign of the nonlinearities change with frequency. Armed
with knowledge of the locations of the excited states relative to the ground state
and the electric transition dipole moments, it is possible to predict the magnitude
and sign of the nonlinearity. However, this detailed knowledge is usually restricted to just a few states and transition moments, those which dominate
the linear and nonlinear absorption spectrum. For asymmetric molecules,
primarily useful for electro-optics, a two-level model (the ground state and
one excited state, both measurable by linear absorption spectroscopy) provides
useful information about the nonlinear dispersion, sign, and magnitude of the
nonlinearity. A key result is that, for χ 3 , there is a cancellation between
one- and two-photon transitions in the region of lower loss.
Symmetric molecules require a three-level model. The symmetry separates out
the one from the two-photon transitions. The three levels consist of an evensymmetry ground state and two excited states, one with even-symmetry wave
functions and one with odd-symmetry wave functions. Also required are twodipole transition moments, one from the ground state to the odd-symmetry excited 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 explaining the frequency dispersion and sign of the nonlinearity, and most important the sign of the non-resonant nonlinearity. If the molecule exhibits more than
one strong peak in the linear and nonlinear absorption spectrum, multiple excited
states contribute significantly to the nonlinearity requiring “brute force” numerical calculations to understand the nonlinearity.
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Molecular nonlinear optics has come a long way over the last three decades in
the design of better materials and in better understanding the physics behind
light–matter interactions important for nonlinear optical imaging, interfacial studies, and other applications of nonlinear optics. Given the richness of materials
available through organic synthesis and materials processing, and the complexity of nonlinear interactions, new phenomena and applications surely await
discovery.
Appendix A: Cartesian Tensor Decomposition
In this appendix, we will describe the decomposition of the hyperpolarizability
into its Cartesian form. We will also express the components of various weights
in terms of the elements of the permutation operations. In decomposing a tensor
into its irreducible components in Cartesian form, we will need to have a method
for reducing the rank of the tensor since the irreducible components have a rank
either the same as or lower than the tensor being decomposed. Following reduction, we can recover the rank n 3 tensor by embedding the irreducible tensors
into this higher rank, arriving at a description where the nonlinear optical tensor
is a sum of its embedded irreducible components. Both of these operations derive from the two rotationally invariant Cartesian tensor forms: (1) the symmetric second-rank form δij , the Kronecker delta whose components are
unity when i j and zero otherwise, and (2) the antisymmetric Levi–Civitas
tensor εijk whose components are 1 (−1) if ijk is an even (odd) permutation
of 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 and
also lowers its rank. Thus, since irreducible forms are extracted with these δ and
ε tensors, the irreducible form of n J , known as its natural form, must be
traceless and symmetric, and appear as appropriate linear combinations of various permutations and combinations of the reduction products with the δ and ε
tensors. Returning to our example in Eqs. (61) and (62), we see that the first term
in Eq. (62) follows from contracting T with δ as T ij δij , the second term as T ij εijk ,
and so forth.
To this end, the three-wave mixing tensor yields one scalar,
β0 βijk εijk βijk εjkl δil ;
(A1)
three vectors (J 1) as traces
β1;1
βijk δjk ;
i
β1;2
βjik δjk
i
β1;3
βjki δjk ;
i
(A2)
two traceless symmetric second-rank tensors (J 2),
1
1
β2;1
εikl βklj εjkl βkli − β0 δij ;
ij
2
3
1
1
βikl εklj βjkl εkli − β0 δij ;
β2;2
ij
2
3
(A3)
and a single third-rank tensor (J 3):
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66
1
β3
ijk βijk βikj remaining permutations − traces:
6
(A4)
Note where symmetrizing sums and subtracted traces appear in these equations.
Equations (A2)–(A4) illustrate the reduction procedure yielding the irreducible
representations in Cartesian coordinates using the δij and εijk tensors. We can
then write the hyperpolarizability tensor in terms of these irreducible representations by embedding them in a third-rank tensor using the same two tensors yielding a sum with terms related to the irreducible tensors:
2;1
2;2
1;1
1;2
1;3
0
βijk β3
ijk βijk βijk βijk βijk βijk βijk ;
(A5)
where
1 0
β0
ijk β εijk ;
6
1
β1;1
δjk − β1;1
δik − β1;1
δij ;
4β1;1
i
j
ijk k
10
1
−β1;2
δjk 4β1;2
δik − β1;2
δij ;
β1;2
i
j
ijk k
10
1
−β1;3
δjk − β1;3
δik 4β1;3
δij ;
β1;3
i
j
ijk k
10
1
2;1
β2;1
εljk ;
β2;2
ijk 2εijl βlk
il
3
1
2;2
β2;1
2β2;2
εljk :
ijk εijl βlk
il
3
(A6)
In deriving these irreducible representations, we merely labeled the tensors of
J ;m
common rank by m, the second index in the superscript of βijk
indicated in
Eqs. (A2)–(A6). However, when proceeding to consider the second-harmonic
and Kleinman symmetric cases, it is more useful to label these m in terms of
their behavior under Cartesian index permutation symmetry. This can be done
by considering the irreducible representation of the permutation group of three
objects. This group has three irreducible representations (two 1D and one 2D)
yielding four possible permutation projection operators:
1
Ps 1 1↔2 1↔3 2↔3 1 → 2 → 3 1 → 3 → 2;
6
1
Pa 1 1 → 2 → 3 1 → 3 → 2 − 1↔2 − 1↔3 − 2↔3;
6
1
Pm 2 22↔3 − 1↔2 − 1↔3 − 1 → 2 → 3 − 1 → 3 → 2;
6
1
Pm0 2 22↔3 1↔2 1↔3 − 1 → 2 → 3 − 1 → 3 → 2: (A7)
6
The two-digit sequences denote the exchange of the two indices, while the threedigit sequences are cyclic index permutations, where, for example, 1↔2βijk βjik and 1↔2↔3βijk βkij . These are orthogonal operators which extract
tensors that are fully symmetric (s), fully antisymmetric (a), both being onedimensional operators, and two of mixed symmetry (m), (m0 ) corresponding
to the two-dimensional operator. The factors ensure that Pi Pj Pi δij . Since they
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67
belong to the same representation, they are chosen so that (m) is symmetric with
respect 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 permutation group is especially convenient since, for second-harmonic generation and
the linear electro-optic effect, both are symmetric in one pair of indices, thus
yielding (m0 ) and (a) components that are identically zero. Then, the (a) component is applicable to parametric processes (three waves of different frequency), but does not contribute to second-harmonic generation, and the (s)
projection contributes to the fully Kleinman symmetric case. This, then, explains
the 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 literature [68]. The dielectric surrounding a molecule will also affect the molecule’s
static dipole moment. For a molecular vacuum moment ⃗μ in a cavity made of a
dielectric of relative dielectric constant ε1r , a charge is induced on the cavity
wall, as shown in Fig. 24. The surface charge will modify the electric field inside
and outside the cavity. The electric potentials inside the cavity, φin , and outside
the cavity, φout , are given by
1
μ
2ε1r − 1 μ
cos Θ −
r cos Θ ;
(B1)
φin 4πε0 r2
2ε1r 1 a3
φout 1
μ
2ε1r − 1 μ
1
3
μ
cos
Θ
−
r
cos
Θ
cos Θ;
4πε0 r2
2ε1r 1 r2
4πε0 2ε1r 1 r2
(B2)
where Θ is the angle between the field and the z axis, r is the distance to the field
point, and a is the cavity radius. The first term in Eq. (B1) is the dipole due to ⃗μ
and the second term is the uniform reaction field due to the induced surface
Figure 24
z
(a)
r
(b)
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 in
the cavity. (c) Molecule represented as a point dipole and in a dielectric of permittivity, ε1r (d) represented as a dielectric sphere of permittivity ε2r and polarization P⃗ 0 . No electric fields or induced polarization are shown.
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68
charge. Similarly, the first term in Eq. (B2) is the dipole potential of the bare
dipole ⃗μ and the second term is the dipole potential from the induced charge on
the cavity surface, with the sum of the two defining the dipole field outside the
cavity.
Here we follow the approach of Kuzyk and Dirk [68]. The electric field is
calculated from the potential according to E⃗ ε1r ∇φ1 , so, from Eq. (B2), the
effective dipole moment is given by
⃗μe 3ε1r
⃗μ:
2ε1r 1
(B3)
The electric field inside the cavity is a superposition of a dipole field and a uniform electric field due to the surface charge that is induced by the dipole and is
⃗ If the molecule is polarizable, the reaction field can
called the reaction field R.
act on the molecule and change its dipole moment, which in turn can change the
reaction field. The total dipole moment, ⃗μ0 , is then the sum of the permanent
dipole moment and the induced dipole moment:
⃗μ0 ⃗μ ε0 α2 R⃗ 1−
⃗μ
α2
3
2ε1r −1
2ε1r 1
.
(B4)
Applying the Onsager approximation 4πN a3 ∕3 1 and the Clausius–Mossotti
equation for the polarizability α2 , the effective internal dipole moment is given
by [68]
⃗μ0 1−
⃗μ
n22 0−1
n22 02
2ε1r 0−1
2ε1r 01
⃗μ
2ε1r 0 1n22 0 2
;
32ε1r 0 n22 0
(B5)
pffiffiffiffi
where n εr , and n0 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 analogy to Eq. (B4), the effective dipole moment measured by an observer whose
perspective is external to the cavity is
⃗μ0e ⃗μ0
3ε1r 0
n2 0 2ε1r 0
⃗μ 22
:
2ε1r 0 1
n2 0 2ε1r 0
(B6)
B.1. Local Field Model of a Two-Component Dipolar
Composite
The dipole moment of an orientationally fixed molecule can be written as a series expansion in the electric field, where the first two terms for a dipole in a
vacuum are
pi μi 1
α F:
ε0 ij j
(B7)
If the molecule freely rotates, the orientational average of the permanent dipole
moment vanishes at nonzero temperatures. An applied electric field will induce
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69
an electronic polarization due to electron cloud deformation and a reorientational
polarization due to partial alignment of the permanent dipole moments will
occur, leading to [69]
j ⃗μj2 F⃗
⃗
⃗
⃗p ε0
⃗α · F ;
3k B T
(B8)
where α is the orientationally averaged polarizability. (For a spherical molecule,
α is a scalar.) We can then rewrite Eq. (B8) as
⃗
⃗p ε0 αeff F⃗ ε0 αor αF;
(B9)
where the effective polarizability ᾱeff is the sum of the orientational and electronic parts.
For a dipole inside a dielectric, the local electric field F⃗ has two sources,
⃗
F⃗ E⃗ c R;
(B10)
⃗ The cavity field is the sum of
the cavity field E⃗ c , and the reaction field R.
the applied electric field and the field due to the induced charge on the surface
of the cavity, while the reaction field is due to the surface charge that is induced
by the dipole inside the cavity. The reaction field is always along the axis of the
dipole, 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 electron
cloud. The induced dipole moment of a molecule in a dielectric is thus of the
form
⃗
⃗p ε0 αor E⃗ c 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
E⃗ c 3ε1r ⃗
E.
2ε1r 1
(B12)
The induced dipole moment can thus be written in terms of the applied electric
field as
3ε1r
2ε1r − 1
3ε1r ⃗
⃗
⃗p ε0
⃗p ε0 αor
Eα
E:
(B13)
2ε1r 1
2ε1r 1
2ε1r 1
Solving this self-consistent expression for the dipole moment, ⃗p, we get
2
3
3ε1r
α
α
or
2
2ε 1
⃗
⃗p ε0 4 1r 2ε −1α 5E:
(B14)
1 − 2ε 1r1a23
1r
According to Eq. (100), the effective polarizability of the spherical molecule is
related to its dielectric function by
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αeff ε2r − 1 3
a:
ε2r 2
(B15)
The electronic polarizability is related to the “fast” part of the response and the
refractive index to the electronic part of the dielectric constant according to
n22 ε2r . With the understanding that n2 refers to the electronic part of εr ,
α2 n22 − 1 3
a.
n22 2
(B16)
Substituting Eqs. (B15) and (B16) into Eq. (B14) yields
⃗p ε0
ε1r ε2r − 1n22 2 3 ⃗
a E:
ε2r 2n22 2ε1r (B17)
Using Equation (104) with εr → ε2r and ⃗p → ⃗p2 and setting this equal to
Eq. (B17) gives
ε n2 2 ⃗
F⃗ 1r2 2
E:
n2 2ε1r
(B18)
This is the Onsager local field model. Note that this derivation is for the twocomponent system, such as a dye-doped polymer or liquid solution, so that ε1r is
the dielectric constant of the host and n2 is the refractive index of the guest. The
single component Onsager expression is obtained by removing the subscripts
from Eq. (B18).
The local field calculations above neglect the fact that the polarizability and
nonlinear susceptibility of a molecule change in the presence of an electric field.
The linear and nonlinear optical susceptibilities are peaked at the optical frequency corresponding to resonant excitations of the molecules. The fluctuations
in the local electric field can result in peak broadening while a static local electric
field can affect both the shape and positions of these peaks. The broadening is
commonly described by a phenomenological width (or excited state decay time
τ) parameter. The effect of the reaction field on the susceptibility through its
effect on the structure of the molecule is important.
The reaction acts on a molecule and changes its energy levels, normally by only
a small amount. The molecular susceptibilities are functions of the transition
frequencies ωvm ωv − ωm and the transition moments ⃗μvm of the molecule,
where m and v label the energy eigenstates. The energy levels of a molecule
in a dielectric material such as a polymer shift and the transition moments
change. The most pronounced effect on the linear absorption spectrum is a shift
in the wavelength of maximum absorbance. Such a shift also appears in the nonlinear optical spectrum. When the transition moments are not strongly affected
by the reaction field, the shifts can be formally introduced by an energy shift
^ m ; ω0m , which affects any arbitrary function, f ω0m , as follows:
operator, Oω
f ωm Ôωm ; ω0m f ω0m ;
(B19)
where ℏω0m is the vacuum energy of level m. Clearly, the shift operator depends
on the dielectric properties of the host matrix.
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71
Accounting for the local fields using a local field tensor as described by
Eq. (112), Eq. (2) becomes
pn
i ω 1
ε0 Πm Ôωm ; ω0m ξn
ijk…ℓ −ω; ω1 ; ω2 …ωn Ljj0 ω1 E j0 ω1 2n−1
(B20)
× Lkk 0 ω2 E k 0 ω2 × × Lℓℓ0 ωn Eℓ0 ωn ;
where the index m spans all the energy eigenstates of the molecule and the
operator Πm represents the product over all energy eigenstates. Multiplying
Eq. (B20) by Lii0 ω, we get
p0n
i ω 1
ε ξ0n −ω; ω1 ; ω2 ; …; ωn E j ω1 E k ω2 n−1 0 ijk…ℓ
2
× × Eℓ ωn ;
(B21)
where the quantities with the primes (0) are called the “dressed” induced dipole
moment and susceptibility:
n
p0n
i ω pi ωLi0 i ω;
(B22)
n
0
ξ0n
ijk…ℓ −ω; ω1 ; ω2 ; …; ωn Πm ÔΩm ; Ωm ξijk…ℓ −ω; ω1 ; ω2 …ωn × Lii0 ωLjj0 ω1 × Lkk 0 ω2 × …Lℓℓ0 ωn : (B23)
Because p0n
i ω now contains the local field factor, it corresponds to a
“Maxwell” polarization when multiplied by N , the number of molecules per unit
volume. For a two-component isotropic system, the local field factors are given
by the Onsager local fields, which, for an isotropic medium associated with the
radiation field at frequency ω, are of the form [68]
Lii0 ω δii0
3ε1r ω
:
2ε1r ω 1
(B24)
The remaining local field factors are of the Onsager form, which, for an input
frequency 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 dependence 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: FA955010-1-0286), KDS acknowledges support by the NSF Center for Layered Polymeric Systems (DMR-0423914) and helpful discussions with Prof. Rolfe
Petschek, 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.
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(2011).
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144. S. Polyakov, F. Yoshino, M. Liu, and G. I. Stegeman, “Nonlinear refraction
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optical properties of triphenylamine-cored alkynylruthenium dendrimers’
—Increasing the nonlinear optical response by two orders of magnitude,”
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033837 (2011).
Mark G. Kuzyk received the B.A. (1979), M.S. (1981), and
Ph.D. (1985) degrees in physics from the University of
Pennsylvania. He was a Member of Technical Staff at
AT&T Bell Laboratories until 1990, then became a faculty
member at Washington State University, Pullman, where he
was also the Boeing Distinguished Professor of Physics and
Materials Science and is now Regents Professor. He is a Fellow of the Optical Society of America, the American Physical Society, and SPIE; was an Associate Chair of Physics and the Chair of
the Materials Science Program; and presented the 2005 WSU Distinguished Faculty Address. He served as topical editor for JOSA B and is one of the founders
of the ICONO conferences on organic nonlinear optics. In his spare time, he
plays ice hockey with The Geezers.
Kenneth Singer is Ambrose Swasey Professor of Physics
and Director of the Engineering Physics Program at Case
Western Reserve University. He received his B.S. summa
cum laude in physics from the Ohio State University in
1975 and Ph.D. in physics from the University of
Pennsylvania in 1981. He was a Member of Technical Staff
at Bell Laboratories from 1982 to 1989, and Distinguished
Member of Technical Staff from 1989 to 1990. From 1990 to
1993 he was the Warren E. Rupp Associate Professor of Physics at Case. Singer
is a Fellow of both the American Physical Society and the Optical Society of
America and has served as topical editor of JOSA B.
George I. Stegeman received his Ph.D. from the University
of Toronto and is the first recipient of the Cobb Family Chair
in Optical Sciences and Engineering. The principal interest
of Dr. Stegeman’s research group is the experimental study
of nonlinear optics in waveguide structures, especially the
properties of spatial solitons in various regions of the electromagnetic spectrum. Of particular interest are solitons in
photonic crystals, in semiconductor optical amplifiers, in
quasi-phase-matched doubling crystals, and in the discrete systems afforded
by coupled arrays of channel waveguides. He is a Fellow of the Optical Society
of America and has received the Hertzberg Medal for Achievement in Physics of
the Canadian Association of Physicists and the R. Woods Prize of the Optical
Society of America.
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