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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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 1943-8206/13/010004-79$15/0$15.00 © OSA . . . . . . . . . . . . . . . . 6 10 11 11 12 4 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 5 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 6 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 7 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 8 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 9 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 10 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 11 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 12 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 13 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, Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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) Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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) Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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; Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 (43) 18 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, Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 22 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 27 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 28 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 30 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 31 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 32 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) Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 33 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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., Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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]. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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]. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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]. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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]. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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]. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 53 ⃗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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 56 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 57 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 58 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 59 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 61 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] Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 62 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 63 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 64 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 65 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): Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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 Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 70 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 72 References 1. J. Kerr, “A new relation between electricity and light: dielectrified media birefringent,” Philos. Mag. 4th Series 50(332), 337–348 (1875). 2. J. Kerr, “Electro-optic observations on various liquids,” Philos. Mag. 5th Series 8(47), 85–102, 202–245 (1879). 3. J. Kerr, “Electro-optic observations on various liquids,” J. Phys. Theor. Appl. 8, 414–418 (1879). 4. T. H. Maiman, “Stimulated optical radiation in ruby,” Nature 187(4736), 493–494 (1960). 5. F. J. McClung and R. W. Hellwarth, “Giant optical pulsations from ruby,” J. Appl. Phys. 33(3), 828–829 (1962). 6. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7(4), 118–119 (1961). 7. N. Bloembergen, Nonlinear Optics (Addison-Wesley, 1965) and references therein. 8. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8(1), 21–22 (1962). 9. W. N. Herman and L. M. Hayden, “Maker fringes revisited: secondharmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). 10. J. Giordmaine, “Mixing of light beams in crystals,” Phys. Rev. Lett. 8(1), 19–20 (1962). 11. G. I. Stegeman and R. A. Stegeman, Nonlinear Optics: Phenomena, Materials and Devices (Wiley, 2012). 12. G. Valentin, G. Dmitriev, G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 2010). 13. M. Di Domenico, “Calculation of the nonlinear optical tensor coefficients in oxygen-octahedra ferroelectrics,” Appl. Phys. Lett. 12(10), 352–355 (1968). 14. M. Di Domenico, “Oxygen-octahedra ferroelectrics. I. Theory of electrooptical and nonlinear optical effects,” J. Appl. Phys. 40(2), 720–734 (1969). 15. B. F. Levine, “Bond-charge calculation of nonlinear optical susceptibilities for various crystal structures,” Phys. Rev. B 7(6), 2600–2626 (1973). 16. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5(1), 17 (1964). 17. S. K. Kurtz and T. T. Perry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39(8), 3798–3813 (1968). 18. M. Bass, D. Bua, and R. Mozzi, “Optical second-harmonic generation in crystals of organic dyes,” Appl. Phys. Lett. 15(12), 393–396 (1969). 19. P. D. Southgate and D. S. Hall, “Second harmonic generation and Miller’s delta parameter in a series of benzene derivatives,” J. Appl. Phys. 43(6), 2765–2770 (1972). 20. A. F. Garito and K. D. Singer, “Organic crystals and polymers—a new class of nonlinear optical materials,” Laser Focus 18(2), 59–64 (1982). 21. D. D. Eley, “Phthalocyanines as semiconductors,” Nature 162(4125), 819 (1948). 22. A. Pochettino, “Sul comportamento foto-elettrico dell’antracene,” Accad. Lincei Rend. 15, 355 (1906). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 73 23. M. Pope and C. E. Swenberg, Electronic Processes in Organic Crystals and Polymers, 2nd ed. (Oxford, 1999). 24. H. Kuhn, “Free electron model for absorption spectra of organic dyes,” J. Chem. Phys. 16(8), 840–841 (1948). 25. H. Kuhn, “A quantum-mechanical theory of light absorption of organic dyes and similar compounds,” J. Chem. Phys. 17(12), 1198–1212 (1949). 26. B. L. Davydov, L. D. Derkacheva, V. V. Dunina, M. E. Zhabotinskii, V. F. Zolin, L. G. Koreneva, and M. A. Samokhina, “Connection between charge transfer and laser second harmonic generation,” Eksp. Teor. Fiz. 12, 24–26 (1970) [JETP Lett. 12, 16–18 (1970)]. 27. J. L. Oudar and D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66(6), 2664–2668 (1977). 28. S. J. Lalama and A. F. Garito, “Origin of the nonlinear second-order optical susceptibilities of organic systems,” Phys. Rev. A 20(3), 1179–1194 (1979). 29. B. J. Orr and J. F. Ward, “Perturbation theory of the non-linear optical polarization of an isolated system,” Mol. Phys. 20(3), 513–526 (1971). 30. J. F. Ward, “Calculation of nonlinear optical susceptibility using diagrammatic perturbation theory,” Phys. Rev. 37, 1–18 (1965). 31. B. F. Levine and C. G. Bethea, “Molecular hyperpolarizabilities determined from conjugated and nonconjugated organic liquids,” Appl. Phys. Lett. 24(9), 445–447 (1974). 32. K. D. Singer and A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic-generation,” J. Chem. Phys. 75(7), 3572–3580 (1981). 33. B. F. Levine and C. G. Bethea, “Second and third order hyperpolarizabilities of organic molecules,” J. Chem. Phys. 63(6), 2666–2682 (1975). 34. J. L. Oudar, “Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds,” J. Chem. Phys. 67(2), 446–457 (1977). 35. J. L. Oudar, D. S. Chemla, and E. Batifol, “Optical nonlinearities of various substituted benzene molecules in the liquid state and comparison with solid state nonlinear susceptibilities,” J. Chem. Phys. 67(4), 1626–1635 (1977). 36. K. Clays and A. Persoons, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66(23), 2980–2983 (1991). 37. J. Zyss and I. Ledoux, “Nonlinear optics in multipolar media: theory and experiments,” Chem. Rev. 94(1), 77–105 (1994). 38. T. Verbiest, K. Clays, C. Samyn, J. Wolff, D. Reinhoudt, and A. Persoons, “Investigations of the hyperpolarizability in organic molecules from dipolar to octopolar systems,” J. Am. Chem. Soc. 116(20), 9320–9323 (1994). 39. S. F. Hubbard, R. G. Petschek, K. D. Singer, N. D’Sidocky, C. Hudson, L. C. Chien, and P. A. Cahill, “Measurements of Kleinman-disallowed hyperpolarizability in conjugated chiral molecules,” J. Opt. Soc. Am. B 15(1), 289–301 (1998). 40. V. Ostroverkhov, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien, “Measurements of the hyperpolarizability tensor using hyper-Rayleigh scattering,” J. Opt. Soc. Am. B 17(9), 1531–1542 (2000). 41. J. Oudar and J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A 26(4), 2016–2027 (1982). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 74 42. J. Zyss and J. Oudar, “Relations between microscopic and macroscopic lowest-order optical nonlinearities of molecular crystals with one-or twodimensional units,” Phys. Rev. A 26(4), 2028–2048 (1982). 43. K. D. Singer, J. E. Sohn, and S. J. Lalama, “Second harmonic generation in poled polymer films,” Appl. Phys. Lett. 49(5), 248–250 (1986). 44. M. G. Kuzyk, K. D. Singer, and R. J. Twieg, eds., feature issue on “Organic and Polymeric Nonlinear Optical Materials,” J. Opt. Soc. Am. B 15(1–2) 1–932 (1998). 45. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, “Second-order nonlinear optical processes in orientationally ordered materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 4(6), 968–976 (1987). 46. K. D. Singer, M. G. Kuzyk, W. R. Holland, J. E. Sohn, S. J. Lalama, R. B. Comizzoli, H. E. Katz, and M. L. Schilling, “Electro-optic phase modulation and optical second-harmonic generation in corona-poled polymer films,” Appl. Phys. Lett. 53(19), 1800–1801 (1988). 47. M. G. Kuzyk, U. C. Paek, and C. W. Dirk, “Guest-host polymer fibers for nonlinear optics,” Appl. Phys. Lett. 59(8), 902–903 (1991). 48. D. J. Welker, J. Tostenrude, D. W. Garvey, B. K. Canfield, and M. G. Kuzyk, “Fabrication and characterization of single-mode electro-optic polymer optical fiber,” Opt. Lett. 23(23), 1826–1828 (1998). 49. J. I. Thackara, G. F. Lipscomb, M. A. Stiller, A. J. Ticknor, and R. Lytel, “Poled electro-optic waveguide formation in thin-film organic media,” Appl. Phys. Lett. 52(13), 1031–1033 (1988). 50. G. F. Lipscomb, A. F. Garito, and R. S. Narang, “An exceptionally large linear electro-optic effect in the organic-solid MNA,” J. Chem. Phys. 75(3), 1509–1516 (1981). 51. L. R. Dalton, P. A. Sullivan, and D. H. Bale, “Electric field poled organic electro-optic materials: state of the art and future prospects,” Chem. Rev. 110(1), 25–55 (2010). 52. C. Sauteret, J. P. Hermann, R. Frey, F. Pradere, J. Ducuing, R. H. Baughman, and R. R. Chance, “Optical nonlinearities in one-dimensional-conjugated polymer crystals,” Phys. Rev. Lett. 36(16), 956–959 (1976). 53. J. M. Hales, J. Matichak, S. Barlow, S. Ohira, K. Yesudas, J.-L. Brédas, J. W. Perry, and S. R. Marder, “Design of polymethine dyes with large third-order optical nonlinearities and loss figures of merit,” Science 327(5972), 1485–1488 (2010). 54. P.-J. Kim, J.-H. Jeong, M. Jazbinsek, S.-B. Choi, I.-H. Baek, J.-T. Kim, F. Rotermund, H. Yun, Y. S. Lee, P. Günter, and O.-P. Kwon, “Highly efficient organic THz generator pumped at near-infrared: quinolinium single crystals,” Adv. Funct. Mater. 22(1), 200–209 (2012). 55. P. D. Cunningham, N. N. Valdes, F. Vallejo, L. M. Hayden, B. Polishak, X.-H. Zhou, J. Luo, A. K.-Y. Jen, J. C. Williams, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys. 109(4), 043505 (2011). 56. T. F. Heinz, H. W. K. Tom, and Y. R. Shen, “Determination of molecularorientation of monolayer adsorbates by optical second-harmonic generation,” Phys. Rev. A 28(3), 1883–1885 (1983). 57. C. Anceau, S. Brasselet, and J. Zyss, “Local orientational distribution of molecular monolayers probed by nonlinear microscopy,” Chem. Phys. Lett. 411, 98–102 (2005). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 75 58. F. Zaera, “Probing liquid/solid interfaces at the molecular level,” Chem. Rev. 112(5), 2920–2986 (2012). 59. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999). 60. S. Yue, M. M. N. Slipchenko, and J.-X. Cheng, “Multimodal nonlinear optical microscopy,” Laser Photon. Rev. 5(4), 496–512 (2011). 61. W. Min, C. W. Freudiger, S. Lu, and X. S. Xie, “Coherent nonlinear optical imaging: beyond fluorescence microscopy,” Annu. Rev. Phys. Chem. 62(1), 507–530 (2011). 62. L. Loew, A. Millard, and P. Campagnola, “Second harmonic imaging microscopy,” Microsc. Microanal. 9(Suppl. S02), 170–171 (2003). 63. K. L. Wustholz, D. R. B. Sluss, B. Kahr, and P. J. Reid, “Applications of single-molecule microscopy to problems in dyed composite materials,” Int. Rev. Phys. Chem. 27(2), 167–200 (2008). 64. R. Carriles, D. N. Schafer, K. E. Sheetz, J. J. Field, R. Cisek, V. Barzda, A. W. Sylvester, and J. A. Squier, “Invited review article: Imaging techniques for harmonic and multiphoton absorption fluorescence microscopy,” Rev. Sci. Instrum. 80(8), 081101 (2009). 65. A. T. Yeh, H. Gibbs, J.-J. Hu, and A. M. Larson, “Advances in nonlinear optical microscopy for visualizing dynamic tissue properties in culture,” Tissue Eng. Part B Rev. 14(1), 119–131 (2008). 66. G. C. R. Ellis-Davies, “Two-photon microscopy for chemical neuroscience,” ACS Chem. Neurosci. 2(4), 185–197 (2011). 67. S.-H. Park, D.-Y. Yang, and K.-S. Lee, “Two-photon stereolithography for realizing ultraprecise three-dimensional nano/microdevices,” Laser Photon. Rev. 3(1–2), 1–11 (2009). 68. M. G. Kuzyk and C. W. Dirk, Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials (Marcel Dekker, 1998). 69. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2009). 70. W. Thomas, “Über die zahl der dispersionselektronen, die einem station aren zustande zugeordnet sind (vorlaufige mitteilung),” Naturwissenschaften 13(28), 627 (1925). 71. W. Kuhn, “Über die gesamtstarke der von einem zustande ausgehenden absorptionslinien,” Z. Phys. A Hadrons Nuclei 33, 408–412 (1925). 72. F. Reiche and U. W. Thomas, “Über die zahl der dispersionselektronen, die einem stationären Zustand zugeordnet sind,” Z. Phys. 34(1), 510–525 (1925). 73. M. G. Kuzyk, “Quantum limits of the hyper-Rayleigh scattering susceptibility,” IEEE J. Sel. Top. Quantum Electron. 7(5), 774–780 (2001). 74. J. Zhou, U. B. Szafruga, D. S. Watkins, and M. G. Kuzyk, “Optimizing potential energy functions for maximal intrinsic hyperpolarizability,” Phys. Rev. A 76(5), 053831 (2007). 75. J. Pérez-Moreno, K. Clays, and M. G. Kuzyk, “A new dipole-free sum-over-states expression for the second hyperpolarizability,” J. Chem. Phys. 128(8), 084109 (2008). 76. M. G. Kuzyk, “Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 85(6), 1218–1221 (2000). 77. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities,” Opt. Lett. 25(16), 1183–1185 (2000). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 76 78. M. G. Kuzyk, “Erratum: Physical limits on electronic nonlinear molecular susceptibilities,” Phys. Rev. Lett. 90(3), 039902 (2003). 79. M. G. Kuzyk, “Fundamental limits on third-order molecular susceptibilities: erratum,” Opt. Lett. 28(2), 135 (2003). 80. Y. Liao, B. E. Eichinger, K. A. Firestone, M. Haller, J. Luo, W. Kaminsky, J. B. Benedict, P. J. Reid, A. K. Jen, L. R. Dalton, and B. H. Robinson, “Systematic study of the structure-property relationship of a series of ferrocenyl nonlinear optical chromophores,” J. Am. Chem. Soc. 127(8), 2758–2766 (2005). 81. J. Zhou, M. G. Kuzyk, and D. S. Watkins, “Pushing the hyperpolarizability to the limit,” Opt. Lett. 31(19), 2891–2893 (2006). 82. H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. Liu, S. T. Ho, E. C. Brown, M. A. Ratner, and T. J. Marks, “Ultralarge hyperpolarizability twisted pielectron system electro-optic chromophores: synthesis, solid-state and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies,” J. Am. Chem. Soc. 129(11), 3267–3286 (2007). 83. A. D. Slepkov, F. A. Hegmann, S. Eisler, E. Elliott, and R. R. Tykwinski, “The surprising nonlinear optical properties of conjugated polyyne oligomers,” J. Chem. Phys. 120(15), 6807–6810 (2004). 84. J. C. May, J. H. Lim, I. Biaggio, N. N. P. Moonen, T. Michinobu, and F. Diederich, “Highly efficient third-order optical nonlinearities in donorsubstituted cyanoethynylethene molecules,” Opt. Lett. 30(22), 3057–3059 (2005). 85. J. C. May, I. Biaggio, F. Bures, and F. Diederich, “Extended conjugation and donor-acceptor substitution to improve the third-order optical nonlinearity of small molecules,” Appl. Phys. Lett. 90(25), 251106 (2007). 86. S. R. Marder, C. B. Gorman, B. G. Tiemann, J. W. Perry, G. Bourhill, and K. Mansour, “Relation between bond-length alternation and second electronic hyperpolarizability of conjugated organic molecules,” Science 261 (5118), 186–189 (1993). 87. F. Meyers, S. R. Marder, B. M. Pierce, and J. L. Bredas, “Electric field modulated nonlinear optical properties of donor-acceptor polyenes: sum-over-states investigation of the relationship between molecular polarizabilities (α, β, and γ) and bond length alteration,” J. Am. Chem. Soc. 116(23), 10703–10714 (1994). 88. K. C. Rustagi and J. Ducuing, “Third-order optical polarizability of conjugated organic molecules,” Opt. Commun. 10(3), 258–261 (1974). 89. B. I. Greene, J. Orenstein, R. R. Millard, and L. R. Williams, “Nonlinear optical response of excitons confined to one dimension,” Phys. Rev. Lett. 58(26), 2750–2753 (1987). 90. N. J. Dawson, B. R. Anderson, J. L. Schei, and M. G. Kuzyk, “Classical model of the upper bounds of the cascading contribution to the second hyperpolarizability,” Phys. Rev. A 84(4), 043406 (2011). 91. M. Joffre, D. Yaron, J. Silbey, and J. Zyss, “Second order optical nonlinearity in octupolar aromatic systems,” J. Chem. Phys. 97(8), 5607–5615 (1992). 92. For an introduction to the subject including examples, see: R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals (Springer, 2010). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 77 93. J. Jerphagnon, D. S. Chemla, and R. Bonneville, “The description of the physical properties of condensed matter using irreducible tensors,” Adv. Phys. 27(4), 609–650 (1978). 94. V. Ostroverkhov, O. Ostroverkhova, R. G. Petschek, K. D. Singer, L. Sukhomlinova, R. J. Twieg, S.-X. Wang, and L. C. Chien, “Optimization of the molecular hyperpolarizability for second harmonic generation in chiral media,” Chem. Phys. 257(2–3), 263–274 (2000). 95. V. P. Ostroverkhov, “Chiral second order nonlinear optics,” Ph.D. dissertation (Case Western Reserve University, 2001). 96. K. D. Singer, R. G. Petschek, V. Ostroverkhov, R. J. Twieg, and L. Sukhomlinova, “Non-polar second-order nonlinear and electro-optic materials: axially ordered chiral polymers and liquid crystals,” J. Polym. Sci. B Polym. Phys. 41(21), 2744–2754 (2003). 97. V. Ostroverkhov, O. Ostroverkhova, R. G. Petschek, K. D. Singer, L. Sukhomlinova, and R. J. Twieg, “Prospects for chiral nonlinear optical media,” IEEE J. Sel. Top. Quantum Electron. 7(5), 781–792 (2001). 98. J. F. Nye, Physical Properties of Crystals (Oxford University, 1985). 99. G. Heesink, A. Ruiter, N. van Hulst, and B. Bölger, “Determination of hyperpolarizability tensor components by depolarized hyper Rayleigh scattering,” Phys. Rev. Lett. 71(7), 999–1002 (1993). 100. Y. Wu, G. Mao, H. Li, R. G. Petschek, and K. D. Singer, “Control of multiphoton excited emission and phase retardation in Kleinman-disallowed hyper-Rayleigh scattering,” J. Opt. Soc. Am. B 25(4), 495–503 (2008). 101. C. A. Dailey, B. J. Burke, and G. J. Simpson, “The general failure of Kleinman symmetry in practical nonlinear optical applications,” Chem. Phys. Lett. 390(1–3), 8–13 (2004). 102. M. M. Ayhan, A. Singh, C. Hirel, A. G. Gürek, V. Ahsen, E. Jeanneau, I. Ledoux-Rak, J. Zyss, C. Andraud, and Y. Bretonnière, “ABAB homoleptic bis(phthalocyaninato)lutetium(III) complex: toward the real octupolar cube and giant quadratic hyperpolarizability,” J. Am. Chem. Soc. 134(8), 3655–3658 (2012). 103. V. Ostroverkhov, R. G. Petschek, K. D. Singer, and R. J. Twieg, “Λ-like chromophores for chiral non-linear optical materials,” Chem. Phys. Lett. 340(1–2), 109–115 (2001). 104. L. Sanguinet, J. C. Williams, R. J. Twieg, G. Mao, G. Wiggers, R. G. Petschek, and K. D. Singer, “Synthesis and HRS NLO characterization of new triarylmethyl cations,” Nonlinear Opt. Quantum Opt. 34, 41–44 (2005). 105. L. Sanguinet, R. J. Twieg, G. Wiggers, G. Mao, K. D. Singer, and R. G. Petschek, “Synthesis and spectral characterization of bisnaphthylmethyl and trinaphthylmethyl cations,” Tetrahedron Lett. 46(31), 5121–5125 (2005). 106. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phasematched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). 107. C. W. Dirk, L. T. Cheng, and M. G. Kuzyk, “A simplified three-level model for describing the molecular third-order nonlinear optical susceptibility,” Int. J. Quantum Chem. 43(1), 27–36 (1992). 108. G. I. Stegeman, M. G. Kuzyk, D. G. Papazoglou, and S. Tzortzakis, “Off-resonance and non-resonant dispersion of Kerr nonlinearity for symmetric molecules [Invited],” Opt. Express 19(23), 22486–22495 (2011). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 78 109. M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, “Mechanisms of quadratic electrooptic modulation of dye-doped polymer systems,” J. Opt. Soc. Am. B 7(5), 842–858 (1990). 110. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudess,” Adv. Opt. Photon. 2(1), 60–200 (2010). 111. G. Stegeman and H. Hu, “Refractive nonlinearity of linear symmetric molecules and polymers revisited,” Photon. Lett. Poland 1, 148–150 (2009). 112. G. I. Stegeman, “Nonlinear optics of conjugated polymers and linear molecules,” Nonlinear Opt. Quantum Opt. 43(1), 143158 (2012). 113. D. Jacquemin, B. Champagne, and B. Kirtman, “Ab initio static polarizability and first hyperpolarizability of model polymethineimine chains. II. Effects of conformation and of substitution by donor/acceptor end groups,” J. Chem. Phys. 107(13), 5076–5087 (1997). 114. J. H. Andrews, J. D. V. Khaydarov, K. D. Singer, D. L. Hull, and K. C. Chuang, “Characterization of excited states of centrosymmetric and noncentrosymmetric squaraines by third-harmonic spectral dispersion,” J. Opt. Soc. Am. B 12(12), 2360–2371 (1995). 115. W. E. Torruellas, B. L. Lawrence, G. I. Stegeman, and G. Baker, “Twophoton saturation in the band gap of a molecular quantum wire,” Opt. Lett. 21(21), 1777–1779 (1996). 116. D. M. Bishop, B. Kirtman, and B. Champagne, “Differences between the exact sum-over-states and the canonical approximation for the calculation of static and dynamic hyperpolarizabilities,” J. Chem. Phys. 107(15), 5780–5784 (1997). 117. P. McWilliams, P. Hayden, and Z. Soos, “Theory of even-parity state and two-photon spectra of conjugated polymers,” Phys. Rev. B 43(12), 9777–9791 (1991). 118. For example, J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1996). 119. V. Ostroverkhov, K. D. Singer, and R. G. Petschek, “Second-harmonic generation in nonpolar chiral materials: relationship between molecular and macroscopic properties,” J. Opt. Soc. Am. B 18(12), 1858–1865 (2001). 120. D. Wanapun, V. J. Hall, N. J. Begue, J. G. Grote, and G. J. Simpson, “DNA-based polymers as chiral templates for second-order nonlinear optical materials,” Chem. Phys. Chem. 10(15), 2674–2678 (2009). 121. M. G. Kuzyk, “Third order nonlinear optical processes in organic liquids,” Ph.D. dissertation (University of Pennsylvania, 1985). 122. J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Pair correlations, cascading, and local-field effects in nonlinear optical susceptibilities,” Phys. Rev. A 46(7), 4172–4184 (1992). 123. J. H. Andrews, K. L. Kowalski, and K. D. Singer, “Molecular orientation, pair correlations and cascading in nonlinear optical susceptibilties,” Mol. Cryst. Liq. Cryst. 223(1), 143–150 (1992). 124. A. Baev, J. Autschbach, R. W. Boyd, and P. N. Prasad, “Microscopic cascading of second-order molecular nonlinearity: new design principles for enhancing third-order nonlinearity,” Opt. Express 18(8), 8713–8721 (2010). 125. G. R. Meredith, “Local field cascading in third-order non-linear optical phenomena of liquids,” Chem. Phys. Lett. 92(2), 165–171 (1982). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 79 126. G. R. Meredith, “Second-order cascading in third-order nonlinear optical processes,” J. Chem. Phys. 77(12), 5863–5871 (1982). 127. N. J. Dawson, B. R. Anderson, J. L. Schei, and M. G. Kuzyk, “Quantum mechanical model of the upper bounds of the cascading contribution to the second hyperpolarizability,” Phys. Rev. A 84(4), 043407 (2011). 128. G. R. Meredith, “Cascading in optical third-harmonic generation by crystalline quartz,” Phys. Rev. B 24(10), 5522–5532 (1981). 129. G. I. Stegeman, D. J. Hagan, and L. Torner, “Cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28(12), 1691–1740 (1996). 130. M. Asobe, I. Yokohama, H. Itoh, and T. Kaino, “All-optical switching by use of cascading of phase-matched sum-frequency-generation and difference-frequency-generation processes in periodically poled LiNbO3 ,” Opt. Lett. 22(5), 274–276 (1997). 131. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970). 132. M. Canva and G. I. Stegeman, “Parametric interactions in organic waveguides,” Adv. Polym. Sci. 158, 87–121 (2002). 133. F. Ghebremichael, M. G. Kuzyk, K. D. Singer, and J. H. Andrews, “Relationship between the second-order microscopic and macroscopic nonlinear optical susceptibilities of poled dye-doped polymers,” J. Opt. Soc. Am. B 15(8), 2294–2297 (1998). 134. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second order nonlinear optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6(4), 742–752 (1989). 135. C. P. J. M. van der Vorst and S. J. Picken, “Electric field poling of acceptor– donor molecules,” J. Opt. Soc. Am. B 7(3), 320–325 (1990). 136. W. Maier and A. Saupe, “Eine einfache molekulare theorie des nematischen kristallinflussigen zustandes,” Z. Naturforsch. A 13, 564–566 (1958). 137. W. Maier and A. Saupe, “Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase 1,” Z. Naturforsch. A 14, 882–889 (1959). 138. W. Maier and A. Saupe, “Eine einfache molekular-statistische theorie der nematischen kristallinflussigen phase 2,” Z. Naturforsch. A 15, 287–292 (1960). 139. I. R. Girling, N. A. Cade, P. V. Kolinsky, and C. M. Montgomery, “Observation of second-harmonic generation from a Langmuir-Blodgett monolayer of merocyanine dye,” Electron. Lett. 21(5), 169–170 (1985). 140. I. R. Girling, P. V. Kolinsky, N. A. Cade, J. D. Earls, and I. R. Peterson, “Second harmonic generation from alternating Langmuir-Blodgett films,” Opt. Commun. 55(4), 289–292 (1985). 141. G. J. Ashwell, T. Handa, and R. Ranjan, “Improved second-harmonic generation from homomolecular Langmuir-Blodgett films of a transparent dye,” J. Opt. Soc. Am. B 15(1), 466–470 (1998). 142. I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, R. A. Hann, P. F. Gordon, B. D. Bothwell, S. K. Gupta, S. Allen, P. Robin, E. Chastaing, and J. C. Dubois, “Second harmonic generation by Langmuir-Blodgett multilayers of an organic azo dye,” Europhys. Lett. 3, 803–809 (1987). 143. A. Painelli, “Vibronic contribution to static NLO properties: exact results for the DA dimer,” Chem. Phys. Lett. 285(5–6), 352–358 (1998). Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 80 144. S. Polyakov, F. Yoshino, M. Liu, and G. I. Stegeman, “Nonlinear refraction and multi-photon absorption in polydiacetylenes from 1200 to 2200 nm,” Phys. Rev. B 69(11), 115421 (2004). 145. D. M. Bishop, B. Champagne, and B. Kirtman, “Relationship between static vibrational and electronic hyperpolarizabilities of π-conjugated push-pull molecules within the two-state valence-bond charge-transfer model,” J. Chem. Phys. 109(22), 9987–9994 (1998). 146. V. Chernyak, S. Tretiak, and S. Mukamel, “Electronic versus vibrational optical nonlinearities of push-pull polymers,” Chem. Phys. Lett. 319(3–4), 261–264 (2000). 147. D. M. Bishop, B. Champagne, and B. Kirtman, “Comment on ‘Electronic versus vibrational optical nonlinearities of push–pull polymers,’” Chem. Phys. Lett. 329(3–4), 329–330 (2000). 148. G. P. Das, A. T. Yeates, and D. Dudis, “Vibronic contribution to static molecular hyperpolarizabilties,” Chem. Phys. Lett. 212(6), 671–676 (1993). 149. B. Kirtman and B. Champagne, “Nonlinear optical properties of quasilinear conjugated oligomers, polymers and organic molecules,” Int. Rev. Phys. Chem. 16(4), 389–420 (1997). 150. H. Hui, S. Webster, D. Hagan, and E. Van Stryland, CREOL, University of Central Florida, are working on a manuscript, title and journal to be determined. 151. S. J. Lalama, K. D. Singer, A. F. Garito, and K. N. Desai, “Exceptional second-order non-linear optical susceptibilities of quinoid systems,” Appl. Phys. Lett. 39(12), 940–942 (1981). 152. J. W. Wu, J. R. Heflin, R. A. Norwood, K. Y. Wong, O. Zamani-Khamiri, A. F. Garito, P. Kalyanaraman, and J. Sounik, “Nonlinear optical processes in lower-dimensional conjugated structures,” J. Opt. Soc. Am. B 6(4), 707–720 (1989). 153. J. R. Heflin, Y. M. Cai, andA.F. Garito,“Dispersion measurements of electricfield-induced second-harmonic generation and third-harmonic generation in conjugated linear chains,” J. Opt. Soc. Am. B 8(10), 2132–2147 (1991). 154. D. C. Rodenberger, J. R. Heflin, and A. F. Garito, “Excited-state enhancement of third-order nonlinear optical responses in conjugated organic chains,” Phys. Rev. A 51(4), 3234–3245 (1995). 155. J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, “Symmetry-controlled electron correlation mechanism for third order nonlinear optical properties of conjugated linear chains,” Mol. Cryst. Liq. Cryst. 160, 37–51 (1988). 156. J. R. Heflin, K. Y. Wong, O. Zamani-Khamiri, and A. F. Garito, “Nonlinear optical properties of linear chains and electron-correlation effects,” Phys. Rev. B 38(2), 1573–1576 (1988). 157. M. G. Kuzyk and C. W. Dirk, “Effects of centrosymmetry on the nonresonant electronic third-order nonlinear optical susceptibility,” Phys. Rev. A 41(9), 5098–5109 (1990). 158. S. Shafei and M. G. Kuzyk, “Critical role of the energy spectrum in determining the nonlinear optical response of a quantum system,” J. Opt. Soc. Am. B 28(4), 882–891 (2011). 159. M. G. Kuzyk, “A bird’s-eye view of nonlinear optical processes: unification through scale invariance,” Nonlinear Opt. Quantum Opt. 40, 1–13 (2010). 160. J. Pérez-Moreno and M. G. Kuzyk, “Comment on ‘Organometallic complexes for nonlinear optics. 45. Dispersion of the third-order nonlinear Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 81 optical properties of triphenylamine-cored alkynylruthenium dendrimers’ —Increasing the nonlinear optical response by two orders of magnitude,” Adv. Mater. 23(12), 1428–1432 (2011). 161. M. G. Kuzyk, “Using fundamental principles to understand and optimize nonlinear optical materials,” J. Mater. Chem. 19(40), 7444–7465 (2009). 162. J. Pérez-Moreno, S.-T. Hung, M. G. Kuzyk, J. Zhou, S. K. Ramini, and K. Clays, “Experimental verification of a self-consistent theory of the first-, second-, and third-order (non)linear optical response,” Phys. Rev. A 84(3), 033837 (2011). Mark G. Kuzyk received the B.A. (1979), M.S. (1981), 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. Advances in Optics and Photonics 5, 4–82 (2013) doi:10.1364/AOP.5.000004 82