1. INTRODUCTION

The large nonlinear-optical susceptibilities of certain organic molecular and polymeric materials have stimulated considerable research interest.[1]–[3] It has been shown that these large susceptibilities can be traced to the electronic properties of the constituent molecules.[4] Much of this research has been on molecular and polymeric crystals, where, for the second-order nonlinear-optical response, the relationship between the molecular and the crystal susceptibilities has been reported for various crystal classes.[5] Molecular studies have been carried out to optimize the molecular nonlinear susceptibility. This has been followed by further molecular and crystal engineering such that the crystals grow with the constituent molecules oriented favorably for second-order nonlinear-optical processes.[3] Studies of third-order nonlinear-optical susceptibilities have been carried out on liquid crystals, and the relationship between molecular and bulk susceptibilities has been enumerated for some susceptibility components.[6]–[13] These relationships have also been calculated in the quadrupole approximation for second-harmonic generation in unpoled liquid crystals.[14] Molecular orientations that allow second-order processes have been demonstrated in noncrystalline materials, such as liquid crystals, and more recently in polymer glasses.[15]–[18]

Orientationally ordered materials can be formed by incorporating dipolar nonlinear-optical molecules in a material that possesses a phase transition above room temperature in which molecular motion is greatly enhanced. In this state, the nonlinear moieties can be oriented by an external field. The material is then cooled through the phase transition, and, when the field is then removed, the molecular orientation remains. The residual orientation allows the material to exhibit second-order nonlinear-optical properties. The magnitude of the nonlinear-optical response is determined not only by the poling field, but also by molecular interactions and disordering thermal processes.

In this paper, the relationship between the second-order nonlinear-optical susceptibility of the oriented bulk material and molecular constituents is derived for orientationally ordered materials belonging to the point group ∞mm. The susceptibility is shown to depend on microscopic order parameters, with the results discussed in limits of interest for liquid-crystalline and isotropic systems as well as for neat and guest–host materials. The application of the theory to poled polymer glasses is described, including experimental results on second-harmonic generation and the linear electro-optic effect. The results of electro-optic measurements are compared with those of second-harmonic generation, in order to assess the origin of the electro-optic effect.

2. THEORY

The origin of nonlinear-optical effects can be found in the expansion of a material polarization density in powers of the electric field, which, in the electric-dipole approximation and instantaneous response, is given by

where summation notation is implied. The first two terms correspond to the spontaneous polarization and to linear-optical effects, respectively. The terms in higher orders of the electric field yield the various phenomena of nonlinear optics. The lowest-order nonlinear term leads to three-wave mixing and the linear electro-optic effect, which are considered for orientationally ordered materials in this paper. For monochromatic fields, the polarization can be expressed in terms of the relevant Fourier components and is described by the frequency-dependent second-order susceptibility, χ_ijk⁽²⁾(−ω₃; ω₁, ω₂), where ω₃ is the frequency obtained by mixing the incident frequencies ω₁ and ω₂ with energy conserved (ω₃ = ω₁ + ω₂). Since the electric polarization and electric fields in the dipole approximation are polar vectors, the inversion operation requires that all even-order susceptibilities vanish if the material possesses a center of inversion. For crystalline materials, 32 point groups describe the possible crystal classes, and 21 of these are noncentrosymmetric.[19]

The bulk second-order susceptibilities in many organic crystals are traceable to the nonlinear-optical properties of the individual molecular units.[4] The molecular polarization in analogy to Eq. (1) is given by

where μ_I⁰ is the molecular ground-state dipole moment, α_IJ(t) the linear polarizability, and β_IJK(t) and γ_IJKL(t) the two lowest-order nonlinear-optical susceptibilities or hyperpolarizabilities. Crystalline ordering arises from intermolecular interactions in which the crystal binding energy is much greater than kT, and the molecules are fixed within the unit cell. For organic van der Waals crystals, although the molecules are bound in the lattice, the crystal susceptibility can still be expressed as a sum over the individual molecules, taking into account local fields and the molecular orientation within the unit cell. For these crystals, the second-order nonlinear susceptibility is related to the molecular hyperpolarizability by[5]

where the sum is over the n molecules in the unit cell (indexed by s), N is the density of unit cells, and f_i^ω3, f_j^ω1, and f_k^ω2 are local field factors. The cosines represent the transformation from the molecular to the crystal unit-cell frames. The relationship, then, mainly involves geometrical considerations and has been applied to a series of organic crystalline materials.[20]

In addition to the crystalline symmetry, additional point groups exist for materials that do not possess long-range three-dimensional positional order. Of particular interest here is the noncentrosymmetric point group, ∞mm. This group is applicable to positionally disordered systems, which contain a unique axis. Such materials exhibit an infinite-fold rotation and an infinity of mirror planes about the unique axis. They exhibit pyroelectricity and are optically uniaxial. This point group constrains the properties of materials such as oriented glasses and nematic liquid crystals and liquid-crystal polymers. Smectic A liquid crystals and liquid-crystal polymers, when composed of poled achiral molecules or racemic mixtures, also belong to this point group. Other smectic and ferroelectric liquid crystals possess a higher degree of order and are not considered here. However, in any liquid crystal, intermolecular forces are comparable to kT, so the thermal energy effects considered here can be extended to more ordered liquid crystals. It is assumed that the materials are homogeneous so that, for instance, the liquid crystals exist in a single domain.

The relevant symmetry operations can be applied to the third-rank tensor describing second-order nonlinear-optical processes. The resulting nonzero tensor components contributing to the nonlinear polarization are[19]

Thus the nonzero tensor components are ${\chi }_{333}^{(2)},{\chi }_{113}^{(2)}={\chi }_{223}^{(2)},{\chi }_{131}^{(2)}={\chi }_{232}^{(2)}$, and ${\chi }_{311}^{(2)}={\chi }_{322}^{(2)}$. For clarity, the superscript (2) denoting second-order susceptibilities will be suppressed for the remainder of this paper.

The bulk susceptibility of oriented materials is calculated by considering the molecular ensemble in the high-temperature state at which molecular motion is enhanced. For isotropic materials, this state corresponds to a liquidlike ensemble of molecules. For liquid crystals, this state is the appropriate mesophase exhibiting liquid-crystalline behavior. The room-temperature phase consists of the molecules locked into the orientation attained in the high-temperature phase. The room-temperature nonlinear-optical susceptibility is then identical to the orientational contribution calculated in the high-temperature phase at which molecular motion is enhanced and the poling field is present.

The aligning energy for orientationally ordered materials originates in both the short-range intermolecular interactions and the externally imposed field and is comparable to kT. The relationship between the molecular and bulk susceptibility is calculated under the assumption that the molecular nonlinear-optical properties are not severely perturbed by neighboring molecules. The bulk susceptibility is, then, the statistical average of the molecular susceptibilities,[21]

where N is the number density of the nonlinear-optical molecules, ${\beta }_{IJK}^{*}$ the molecular susceptibility including local field effects, and the angle brackets denote the statistical average. If $\overleftrightarrow{a}$ is the transformation between the molecular and laboratory frame, as given in Appendix A, then Eq. (5) can be rewritten as

When the orientational order is imposed with an electric field in the phase exhibiting enhanced motion, then the ensemble average is given by

where the normalized Gibbs distribution function is

with m* the molecular dipole moment including corrections arising from the local poling field, E_p. The term U is the short-range interaction potential between molecules. Although U is very small in isotropic materials, it is not in liquid crystals. In general, it depends on all Euler angles, as defined in Appendix A. However, it is assumed here that U is independent of the angle ϕ, that is, the molecular interaction potential possesses the uniaxial symmetry associated with the mesophase.[22] Further, if the molecular interactions responsible for ordering depend only on a unique axis, U is independent of ψ.

The constituent molecules are approximated to be rigid aromatic moieties. In the absence of an external aligning field, the molecules align so that the bulk material, while possessing a unique axis, does not exhibit a spontaneous polarization. Thus the molecular potential does not distinguish +n from −n, where n is the director associated with the mesophase. This implies that, in the absence of an external field, the mesophase will not exhibit second-order nonlinear-optical processes in the electric-dipole approximation. Under these assumptions, G(Ω, E_p) depends only on θ. These arguments are also valid for the orientational interactions in smectic A phases. Again, for isotropic phases, U is zero.

The function G(Ω, E_p) can then be expanded in terms of Legendre polynomials,[22]

with

The A_l are the ensemble average of the P_l, 〈P_l〉, and are defined as the microscopic order parameters associated with an axially symmetric liquid crystal. The 〈P_l〉 may be evaluated by using an appropriate model, such as mean-field theory,[23],[22] but, for our purposes here, it is sufficient that the 〈P_l〉 can be independently measured.[6],[11],[12] Since no spontaneous polarization is present, only 〈P_l〉’s with even l are nonzero. It should be noted that ferroelectric liquid crystals are characterized by a spontaneous polarization that arises from the nonaxial nature of the molecules. The intermolecular potential responsible for the ferroelectric behavior arises from ϕ- or ψ-dependent U, so the expansion in Legendre polynomials, as done in Appendix A, is not valid. In this case, values of 〈P_l〈 for odd l are nonzero.

It can be assumed that exp(m · E_p/kT) can be expanded to first order in m · E_p/kT at even reasonably large poling fields (m · E_p < kT) since ensemble averages of odd-ranked tensors require that the next term contributing to the macroscopic susceptibility be cubic in the field. The products of transformation components and G(Ω, E_p) can be expanded in terms of Legendre polynomials, which can then be integrated by using their orthogonality relationships. After expanding in terms of the P_l, integrating over ϕ and ψ, and summing over I, J, and K, the bulk susceptibility in Eq. (6) is given by

where frequency dependence is suppressed, and $u_{ijk}^{(l)}$ is given in Appendix A for the nonzero components of χ_ijk, with the molecular dipole directed arbitrarily in the molecular frame and E_p directed in the laboratory 3-direction. In this case, the permutation symmetries of the bulk reflect that of the contributing molecular components.

Under the above assumption that the molecular system possesses axial symmetry in the ground state, insight can be gained by examining the expressions in Appendix A in the limit of a one-dimensional molecule, i.e., only ${\beta }_{zzz}^{*}$ and $m_z^{*}$ are nonzero. This symmetry is reflected in p-disubstituted aromatic molecules and is consistent with the axial nature assumed.[24] With these assumptions, Eq. (11) can be rewritten for the nonzero components as

and

In the isotropic phase of liquid crystals or in polymer glasses, the order parameters 〈P₂〉 and 〈P₄〉 are zero, so that only the first term contributes to χ₃₃₃ and the other components of χ_ijk are exactly ⅓ of χ₃₃₃. Here, the local field effects are reasonably well characterized, and the local field effects factor out as

The local fields at optical frequencies can be described with Lorenz–Lorentz-type expressions

and that of the dipole in the presence of the local static poling field by the Onsager expression

where ∊ is the static dielectric constant, μ_z the molecular dipole moment, and n_∞ and n_ω optical indices of refraction.[25]

In isotropic materials, the integral in Eq. (7) can be evaluated exactly without expanding the dipolar energy in a Taylor series. It is found that

where L₃(p) is the third-order Langevin function whose series expansion for a one-dimensional molecule is given by[21]

and where

The first term in Eq. (18) is identical to the first term in Eq. (12), and Eq. (18) shows that the next term in the Langevin function is small if μ_zE_p < kT.

For liquid crystals, all terms in Eqs. (12) and (13) contribute, and the local fields are more complicated.[6],[10],[12] In the limit when the order parameters are unity, the term in parentheses in expression (12) becomes one, so that, assuming similar local field effects, a susceptibility five times that of isotropic materials is obtained. In addition, the term in parentheses in Eq. (13) becomes zero, and that tensor component vanishes. The order parameters of liquid crystals are found to be less than one. For instance, far from the phase transition of a typical nematic, N-(p′-methoxybenzylidine)-p-cyanoaniline (MBBA), 〈P₂〉 ~ 0.6 and 〈P₄〉 ~ 0.25,[12],[6] χ₃₃₃ is about three times the isotropic case and is about equal to that of isotropic materials for the other components. For certain smectic liquid crystals and liquid-crystal polymers, the order parameters can be as high as 〈P₂〉 ~ 0.9 and 〈P₄〉 ~ 0.8.[12] Here, χ₃₃₃ and χ₃₁₁ are about 4.5 and 0.3 that of the isotropic case, respectively.

For a guest–host composite system, Eq. (8) can be rewritten as

where G_u is the distribution function of species u, U_uv is the interaction energy between the species in the system, ${\mathbf{m}}_u^{*}$ is the local field-corrected dipole moment of species u, and v runs over all the species in the system. The susceptibility is then given by

For isotropic materials, U_uv is taken to be zero, and the total susceptibility is the sum over independent species. For a liquid-crystal host, the host–host interactions that lead to the liquid-crystal ordering may be larger than guest–host interactions.[12],[26] This implies that the order parameter describing the orientation of the nonliquid-crystalline guest will be less than that of the neat host. The nonlinear-optical susceptibility calculated above is responsible for a variety of nonlinear-optical effects, including second-harmonic generation and the linear electro-optic effect. The relationship between the susceptibility and these phenomena along with a comparison between experimental results and the theory for poled guest–host glasses is now described.

3. SECOND-HARMONIC GENERATION

Second-harmonic generation is a special case of the general phenomena of optical three-wave mixing. For second-order nonlinear-optical processes, a convenient definition of the electric field is[27]

where the e_j(ω) are complex field amplitudes at the incident frequencies, ω₁ and ω₂ and c.c. denotes the complex conjugate. This definition provides for the transformation of the time-dependent polarization of Eq. (1) into the frequency-dependent polarization given in Eqs. (4). It is seen for second-harmonic generation that the energy-conservation requirement

is fulfilled and that additional permutation symmetries arising from E_i(ω₁) = E_i(ω₂) in Eqs. (4) result in the degeneracy of the j and k indices in χ_ijk given in Eq. (1). This leads to the definition of the second-harmonic coefficient d_ijk through

Finally, owing to the degeneracy of j and k, d_ijk can be rewritten in the standard contracted notation,[19] d_iu(−2ω;, ω), where u runs from 1 to 6, so that, for second-harmonic generation, Eqs. (4) are rewritten as

These definitions of the electric field can also be applied to the molecular polarization given in Eq. (2), which leads to

The molecular second-harmonic coefficients, ${\beta }_{IJK}^{\prime }(-2\omega ;\omega ,\omega )$, are those normally measured with electric-field-induced second-harmonic generation.[25]

Second-harmonic-generation measurements were carried out on poled molecule-doped polymer films.[16] The films were composed of the azo dye, Disperse Red 1, whose molecular structure is shown in Fig. 1, dissolved in poly(methyl methacrylate) (PMMA). Thin films were prepared by spin coating onto indium tin oxide coated glass. The films were poled by raising the composite film above its glass–rubber transition temperature in the presence of an intense electric field and then cooling to room temperature with the field still applied.

Second-harmonic generation in transmission was measured with p-polarized light at 1.58-μm wavelength. The value of d₃₃(−2ω; ω, ω) was determined by analyzing the interference function for p-polarized light under the assumption that d₃₁ = ⅓d₃₃, as predicted by expressions (12) and (13) and confirmed by an independent measurement of d₃₁ using s-polarized incident light (the results are shown in Table 2 below). The value of the product ${\beta }_{zzz}^{\prime }{\mu }_z$ [defined in Eqs. (14) and (26)] as measured independently, using electric-field-induced second-harmonic generation of liquid solutions of Disperse Red 1, is given in Table 1 along with other relevant properties of the film. It is assumed that the azo dye is a one-dimensional molecule, as has been seen in other p-disubstituted aromatic compounds.[5] Using expressions (12)–(16), d₃₃ was calculated by using values given in Table 1. The results are also listed in Table 2. The range in the calculated d₃₃ reflects the uncertainty in ${\beta }_{zzz}^{\prime }{\mu }_z$. The measurements and theory are in reasonable agreement, confirming the theory leading to expressions (12) and (13), where 〈P₂〉 and 〈P₄〉 are zero. The validity of taking the dipolar energy to first order has been confirmed by the plot of d₃₃ versus the poling field E_p (Fig. 2), and thus the first term in Eq. (18) dominates.[16]

Since the films represent a composite system, the susceptibility should be given by Eq. (21). That is, where U_vu = 0, the measured susceptibility is simply the sum of the susceptibilities of the polymer and the dye. The second-harmonic coefficient is linear in the number density and does not extrapolate to d₃₃ = 0 at zero concentration, as shown previously and here in Fig. 3.[16] The nonzero intercept represents the contribution to d₃₃ of PMMA, which is expected since PMMA is somewhat polar. The linearity and intercept in Fig. 3 do suggest that Eq. (21) adequately describes this material.

Since Fig. 2 is not corrected for the contribution of PMMA, those data and those depicted in Fig. 3 indicate that the measured values of d₃₃ generally fall in the lower range of the error bars when compared with the calculated value. This is probably due to several factors. First, the third-order susceptibility contributes to the value of ${\beta }_{zzz}^{\prime }{\mu }_z$, as determined by electric-field-induced second-harmonic generation, but not to the film susceptibility; a 10% contribution is not unusual.[28] Injected charge may be screening the poling field, thus reducing d₃₃ in the films. Finally, polymer glasses are not in thermodynamic equilibrium, so molecular relaxation occurs following poling. In fact, some relaxation has been observed and contributes to the lower values of d₃₃.

Thus we see the validity of Eq. (11) and expressions (12) and (13) for a poled glass, where the microscopic order parameters vanish. The applicability of the theory to the molecular-macroscopic relationship in liquid crystals has yet to be demonstrated, though appropriate liquid-crystal structures have been fabricated.[17] The applicability of the theory for polymer glasses for the linear electro-optic effect is now discussed.

4. LINEAR ELECTRO-OPTIC EFFECT

The electro-optic effect is defined through the change in the electric impermeability induced by an applied electric field[27] and, to first order in the applied field, is given by

where B_ij is the impermeability tensor, ∊ the dielectric tensor, E_k the applied electric field, and r_ij,k the linear electro-optic or Pockels coefficient.[27] r_ij,k is symmetric with respect to the first two indices and thus can be written in reduced tensor notation, r_uk.[19] Kleinman symmetry does not apply to the linear electro-optic effect.[29]

The linear electro-optic effect can also be defined through the change in the optical susceptibility[27]

for Δ∊_ij ≪∊_ii and ∊_jj, where χ_ij is the linear susceptibility and ∊_ij the dielectric tensor. The induced polarization is then given by

The electric fields are defined as

and

where e_l(ω) are the Fourier amplitudes of the electric fields including the overall phase. Taking ω₁ = ω as the optical frequency and ω₂ = 0, the polarization at the optical frequency is then

If a similar analysis is carried out in the notation of a second-order nonlinear optical susceptibility defined through

with the electric fields given by Eq. (22), then the following relationship between the nonlinear susceptibility and the electro-optic coefficient is obtained:

where ∊_ll(ω) = n_l², the principal indices of refraction. For typical doped poled polymer films, all principal indices are nearly identical. The electro-optic coefficient is then related to the second-order susceptibility by

At optical frequencies, the second-harmonic susceptibility is dominated by virtual electronic processes if both fundamental and second-harmonic fields are far from the electronic states of the material. Since modulating waveforms are much lower in frequency, acoustic and optical phonon modes can contribute to the electro-optic coefficient.[30],[31] In this case, the electro-optic coefficient is the sum of three contributions

where $r_{uk}^{el}$ is the electronic contribution, $r_{uk}^B$ the acoustic phonon contribution, and $r_{uk}^R$ the optical phonon contribution.[31] Since the second-harmonic generation is only electronic in origin, the second-harmonic coefficient can be used to deduce the electronic contribution to the electro-optic coefficient.[31]

When the molecular second-order susceptibility β_ijk is dominated by the change in dipole moment between the ground and first excited electronic states along the z axis, it can be described by a two-level model[24],[28]:

where the z_ij’s are the transition dipole moments between states i and j, ω₀ is the angular frequency of the first excited state, and (ω₁, ω₂, ω₃) are the applied and radiated fields.

The macroscopic second-order susceptibility of poled, doped polymer systems can be related to the microscopic susceptibility through the frequency-dependent local field factors [expression (17)]. Using the two-level model [Eq. (37)], the bulk electronic electro-optic coefficient, $r_{ij,k}^{el}$, can be related to the second-harmonic coefficient d_kij by accounting for dispersion and local field effects through[31]

where ω′ is the frequency of the fundamental used in measuring d_kij and the electro-optic coefficient is evaluated at frequency ω. Figures 4 and 5 depict the dispersion of d₃₃ and $r_{33}^{el}$ of poled azo-dye-doped polymer films per poling field and the number density, as calculated using Eqs. (37) and (38). The dispersion of β for substituted benzene molecules agrees with a two-level model.[24],[32] For an extended molecule such as the azo dye, the two-level model gives a reasonable approximation to the shape of the dispersion over a limited wavelength range.[33]

Electro-optic properties of poled polymer glasses were determined from the dependence of the optical phase on voltage in a Mach–Zehnder interferometer.[34] The r₁₃ component was measured by applying the modulating field normal to the film and parallel to the laser beam in one arm of the interferometer. Using the symmetry of a poled film, r₃₃ is equal to 3r₁₃, as can be derived from expressions (12) and (13).

Films of Disperse Red 1 dye in PMMA were used as the model system. Properties of the film are given in Table 3. The electro-optic measurements were performed at a wavelength of 633 nm, a modulating frequency of 35 kHz, and a modulating voltage of 50 V rms. The measured value of the electro-optic coefficient is given in Table 4. As observed in second-harmonic generation, the dependence of the electro-optic coefficient is linear in the poling field, so the nonlinear optical coefficient per unit poling field can be used to compare different films.

The electronic contribution to the electro-optic coefficient can be estimated from second-harmonic-generation measurements using Eq. (38). The results of this calculation are given in Table 4. The calculated value of r^el/E_p is somewhat larger than the measured field-independent electro-optic coefficient at the measured frequency. The sign and the percentage difference between r^el/E_p and r/E_p observed in the poled polymer film are comparable with those found in organic crystals such as 3-methyl-4-nitropyridine-1-oxide.[35] It has generally been found that the electro-optic coefficient is mainly electronic in origin in such crystals, and the difference between r^el and r in crystals has been attributed to phonon contributions.[31] This may be responsible for the difference here, since molecular motion is present, and the phonon structure of PMMA will contribute. The difference between r^el/E_p, as determined from second-harmonic generation, and r/E_p may also be due to screening of the modulating field by trapped charge and to inadequacies in the two-level model. Once these considerations have been addressed, the relationship between the molecular and macroscopic electro-optic properties will be more fully understood.

5. CONCLUSIONS

Orientationally ordered materials have been shown to possess second-order nonlinear optical properties. The orientational order is obtained by the application of an external orienting force whose energy is comparable with kT so that the relationship between the properties of the constituent molecules and the bulk reflect the effects of the thermal energy. The relationship between the molecular and bulk properties has been derived and is found to depend on microscopic order parameters. The theory is applicable to liquid crystals and isotropic materials belonging to the point group ∞mm. All nonzero tensor components of χ_ijk have been related to components of the molecular hyperpolarizability and the microscopic order parameters corresponding to the statistical average of the first three even-order Legendre polynomials, 〈P₀〉, 〈P₂〉, and 〈P₄〉.

In the limit of a one-dimensional molecule, the largest poled liquid-crystal susceptibility component is found to be between one and five times greater than that of a poled isotropic material. Measurements of the second-harmonic and electro-optic coefficients for a guest–host isotropic material consisting of Disperse Red 1 dye dissolved in PMMA have been found to be in reasonable agreement with the theory, and discrepancies can be attributed to molecular relaxation and Coulombic screening of the poling field. In addition, the electro-optic measurements indicate that the origin of the electro-optic effect in the dye–PMMA material is mostly electronic in origin.

APPENDIX A

The macroscopic second-order susceptibility, χ_ijk, can be expressed in terms of the molecular second-order susceptibility, β_IJK, as

(A1)$${\chi }_{ijk}(-{\omega }_3;{\omega }_1,{\omega }_2)=N{〈{\beta }_{IJK}^{*}(-{\omega }_3;{\omega }_1,{\omega }_2)〉}_{ijk},$$

where N is the number density and 〈 ${\beta }_{IJK}^{*}$〉 is an orientational average of the local field-corrected molecular second-order susceptibility. The indices I, J, and K define the coordinate system of the molecule, while indices i, j, and k define those of the macroscopic material. The two coordinate systems are related by the Euler angles, ϕ, θ, and ψ.[36]

The orientational average in the presence of a poling field E_p is of the form

(A2)$${〈{\beta }_{IJK}^{*}〉}_{ijk}=\int \text{d}\mathrm{\Omega }{a}_{iI}{a}_{iJ}{a}_{kK}G(\mathrm{\Omega },E_p){\beta }_{IJK}^{*},$$

where the volume element is given by

(A3)$$\int \text{d}\mathrm{\Omega }={\int }_0^{2\pi }\text{d}\varphi {\int }_0^{2\pi }\text{d}\psi {\int }_{-1}^1\text{d}(\text{cos}\hspace{0.17em}\theta ),$$

$\overleftrightarrow{a}$ is the transformation matrix:

(A4)$$\overleftrightarrow{a}=\left(\begin{array}{lll}+\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}\psi -\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}\psi \hfill & +\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}\psi +\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}\psi \hfill & -\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\psi \hfill \\ -\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}\psi -\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}\psi \hfill & -\text{cos}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi \hspace{0.17em}\text{sin}\hspace{0.17em}\psi +\text{cos}\hspace{0.17em}\varphi \hspace{0.17em}\text{cos}\hspace{0.17em}\psi \hfill & +\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\psi \hfill \\ +\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{cos}\hspace{0.17em}\varphi \hfill & +\text{sin}\hspace{0.17em}\theta \hspace{0.17em}\text{sin}\hspace{0.17em}\varphi \hfill & +\text{cos}\hspace{0.17em}\theta \hfill \end{array}\right),$$

and G(Ω, E_p) a distribution function.[37] Summation notation is used unless otherwise stated. If the poling field couples to the local field-corrected molecular dipole moment, m*, and the molecules interact through a potential U, the thermodynamic distribution function takes the form

(A5)$$G(\mathrm{\Omega },E_p)=\frac{\text{exp}[-(U-{\mathbf{m}}^{*}·{\mathbf{E}}_p)/kT]}{\int \text{exp}[-(U-{\mathbf{m}}^{*}·{\mathbf{E}}_p)/kT]\text{d}\mathrm{\Omega }},$$

where 1/kT is the Boltzmann factor.

In the limit in which the interaction energy between the poling field and the molecular dipole is smaller than the thermal energy, the exponential distribution factor can be approximated as

(A6)$$\text{exp}[-(U-{\mathbf{m}}^{*}·{\mathbf{E}}_p)/kT]\approx \left(1+\frac{{\mathbf{m}}^{*}+{\mathbf{E}}_p}{kT}\right)\hspace{0.17em}\text{exp}(-U/kT).$$

If the poling field is applied in the laboratory 3-direction and the components of the dipole moment in the molecular frame are given by m*l(l = x, y, or z), the dipole energy becomes

(A7)$$-{\mathbf{m}}^{*}·{\mathbf{E}}_p=-a_{3l}{m^{*}}_l{E}_3.$$

Using expressions (A7) and (A6), we can rewrite the distribution function [Eq. (A5)] as

(A8)$$G(\mathrm{\Omega },E_p)\approx \frac{(1+a_{3l}{m^{*}}_l{E}_3/kT)\text{exp}(-U/kT)}{\int \text{exp}(-U/kT)\text{d}\mathrm{\Omega }}.$$

The only independent nonzero macroscopic second-order susceptibility tensor components that remain after performing the orientational averages, as dictated by the resulting system’s point symmetry ∞mm, are χ₃₃₃, χ₃₁₁, χ₁₁₃, and χ₁₃₁. We assume that the interaction potential U is independent of the angles ψ and ϕ. The only ordering forces among molecules will therefore be in the θ direction. Owing to the axial symmetry of the molecules, only terms with even powers of components of $\overleftrightarrow{a}$ will contribute to the integrals in Eq. (A2). The 〈P_i〉 are defined to be microscopic order parameters calculated as an orientational average of the ith Legendre polynomial, as defined in Eqs. (9) and (10) [P₀(x) = 1, P₂(x) = ½(3x² − 1), P₄(x) = ⅛(35x² − 30x² + 3)] and are given by[22]

(A9)$$〈P_i〉=\frac{{\int }_{-1}^1\text{d}(\text{cos}\hspace{0.17em}\theta )P_i(\text{cos}\hspace{0.17em}\theta )e^{-U/kT}}{{\int }_{-1}^1\text{d}(\text{cos}\hspace{0.17em}\theta )e^{-U/kT}}.$$

The product of components of $\overleftrightarrow{a}$ and G(Ω, E_p) can be expanded in terms of the Legendre polynomials. On integrating over ψ and ϕ, the macroscopic second-order susceptibility becomes

(A10)$${\chi }_{ijk}=\frac{N{E}_p}{kT}[u_{ijk}^{(0)}+u_{ijk}^{(2)}〈P_2〉+u_{ijk}^{(4)}〈P_4〉],$$

where the order parameters coefficients $u_{ijk}^{(n)}$ are expressed in terms of the molecular susceptibilities ${\beta }_{ijk}^{*}$ and the molecular dipole components m*_i. Defining the following matrices:

(A11)$${\overleftrightarrow{\gamma }}_0=\left(\begin{array}{lll}{\beta }_{xxx}^{*}\hfill & 0\hfill & 0\hfill \\ 0\hfill & {\beta }_{yyy}^{*}\hfill & 0\hfill \\ 0\hfill & 0\hfill & {\beta }_{zzz}^{*}\hfill \end{array}\right),$$

(A12)$${\overleftrightarrow{\gamma }}_1=\left(\begin{array}{lll}0\hfill & {\beta }_{xyy}^{*}\hfill & {\beta }_{xzz}^{*}\hfill \\ {\beta }_{yxx}^{*}\hfill & 0\hfill & {\beta }_{yzz}^{*}\hfill \\ {\beta }_{zxx}^{*}\hfill & {\beta }_{zyy}^{*}\hfill & 0\hfill \end{array}\right),$$

(A13)$${\overleftrightarrow{\gamma }}_2=\left(\begin{array}{lll}0\hfill & {\beta }_{yxy}^{*}\hfill & {\beta }_{zxz}^{*}\hfill \\ {\beta }_{xyx}^{*}\hfill & 0\hfill & {\beta }_{zyz}^{*}\hfill \\ {\beta }_{xzx}^{*}\hfill & {\beta }_{yzy}^{*}\hfill & 0\hfill \end{array}\right),$$

(A14)$${\overleftrightarrow{\gamma }}_3=\left(\begin{array}{lll}0\hfill & {\beta }_{yyx}^{*}\hfill & {\beta }_{zzx}^{*}\hfill \\ {\beta }_{xxy}^{*}\hfill & 0\hfill & {\beta }_{zzy}^{*}\hfill \\ {\beta }_{xxz}^{*}\hfill & {\beta }_{yyz}^{*}\hfill & 0\hfill \end{array}\right),$$

and

(A15)$${\overleftrightarrow{\mathbf{m}}}^{*}=\left(\begin{array}{lll}{m}_x^{*}\hfill & m_y^{*}\hfill & m_z^{*}\hfill \\ m_x^{*}\hfill & m_y^{*}\hfill & m_z^{*}\hfill \\ m_x^{*}\hfill & m_y^{*}\hfill & m_z^{*}\hfill \end{array}\right),$$

the resulting macroscopic susceptibility can be expressed in a relatively compact form. With the matrix product given by (ab)_ij = a_ikb_kj and γ = γ₁ + γ₂ + γ₃, the order parameter coefficients are (note that the ii subscripts denote the trace of the matrix) as follows:

• For χ₃₃₃

  (A16)$$u_{333}^{(0)}=¹/₁₅{[m^{*}(3{\gamma }_0+\gamma )]}_{ii},$$

  (A17)$$u_{333}^{(2)}=²/₇[3{(m^{*}{\gamma }_0)}_{zz}-{(m^{*}{\gamma }_0)}_{ii}]-¹/₂₁[2{(m^{*}\gamma )}_{ii}-3{(m^{*}\gamma +\gamma m^{*})}_{zz}],$$

  (A18)$$u_{333}^{(4)}=¹/₃₅[5{(m^{*}{\gamma }_0)}_{zz}+3{(m^{*}{\gamma }_0)}_{ii}-5{(m^{*}\gamma +\gamma m^{*})}_{zz}+{(m^{*}\gamma )}_{ii}];$$
• for χ₁₃₁

  (A19)$$u_{131}^{(0)}=¹/₁₅[{(m^{*}{\gamma }_0)}_{ii}+2{(m^{*}{\gamma }_2)}_{ii}],$$

  (A20)$$u_{131}^{(2)}=¹/₄₂[3{(m^{*}{\gamma }_0)}_{zz}-{(m^{*}{\gamma }_0)}_{ii}-5{(m^{*}{\gamma }_2)}_{ii}+18{({\gamma }_2{m}^{*})}_{zz}-3{(m^{*}{\gamma }_2)}_{zz}],$$

  (A21)$$u_{131}^{(4)}=-¹/₇₀[5{(m^{*}{\gamma }_0)}_{zz}+3{(m^{*}{\gamma }_0)}_{ii}+{(m^{*}{\gamma }_2)}_{ii}-5{({\gamma }_2{m}^{*}+m^{*}{\gamma }_2)}_{zz}];$$
• for χ₁₁₃

  (A22)$$u_{113}^{(0)}=¹/₁₅[{(m^{*}{\gamma }_0)}_{ii}+2{(m^{*}{\gamma }_3)}_{ii}],$$

  (A23)$$u_{113}^{(2)}=¹/₄₂[3{(m^{*}{\gamma }_0)}_{zz}-{(m^{*}{\gamma }_0)}_{ii}-5{(m^{*}{\gamma }_3)}_{ii}+18{({\gamma }_3{m}^{*})}_{zz}-3{(m^{*}{\gamma }_3)}_{zz}],$$

  (A24)$$u_{113}^{(4)}=-¹/₇₀[5{(m^{*}{\gamma }_0)}_{zz}+3{(m^{*}{\gamma }_0)}_{ii}+{(m^{*}{\gamma }_3)}_{ii}-5{({\gamma }_3{m}^{*}+m^{*}{\gamma }_3)}_{zz}];$$
• and for χ₃₁₁,

  (A25)$$u_{311}^{(0)}=¹/₁₅[{(m^{*}{\gamma }_0)}_{ii}+2{(m^{*}{\gamma }_1)}_{ii}],$$

  (A26)$$u_{311}^{(2)}=¹/₄₂[3{(m^{*}{\gamma }_0)}_{zz}-{(m^{*}{\gamma }_0)}_{ii}-5{(m^{*}{\gamma }_1)}_{ii}+18{({\gamma }_1{m}^{*})}_{zz}-3{(m^{*}{\gamma }_1)}_{zz}],$$

  (A27)$$u_{311}^{(4)}=-¹/₇₀[5{(m^{*}{\gamma }_0)}_{zz}+3{(m^{*}{\gamma }_0)}_{ii}+{(m^{*}{\gamma }_1)}_{ii}-5{({\gamma }_1{m}^{*}+m^{*}{\gamma }_1)}_{zz}].$$

ACKNOWLEDGMENTS

The authors wish to thank J. W. Goodby, S. J. Lalama, and R. D. Small for helpful discussions.

Figures and Tables

 

Fig. 1 Disperse Red 1, an azo dye.

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Fig. 2 d₃₃ versus poling field. Shaded area is calculated by using expressions (17)–(19). (Reference [16]; used with permission.)

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Fig. 3 d₃₃ versus number density. Shaded area is calculated by using expressions (17)–(19). (Reference [16]; used with permission.)

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Fig. 4 Dispersion of d₃₃/NE_p using a two-level model.

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Fig. 5 Dispersion of $r_{33}^{el}/N{E}_p$, using a two-level model.

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Table 1. Properties of Films Used in Second-Harmonic-Generation Measurements

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Table 2. Results of Second-Harmonic Generation at λ = 1.58 μm^a

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Table 3. Properties of Films Used in Electro-Optic Measurements

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Table 4. Results of Electro-Optic Measurements at λ = 0.633 μm

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