A systematic study of the dependence of the band gap, the linear optical susceptibility, the polarizability α, and the second hyperpolarizability γ, on the number of repeat unit is conducted for the thiophene series from monomer to hexamer. The linear optical susceptibilities for oligomers have been determined from the refractive index measurements on vacuum deposited films using the m lines technique. The orientationally averaged polarizabilities 〈α〉 have been measured from refractive index measurements of THF solutions. The orientationally averaged second hyperpolarizabilities 〈γ〉 have been measured by degenerate four-wave mixing studies of THF solutions. The validity of the Lorentz–Lorenz approximation is tested and found to be satisfactory. The experimental values of 〈α〉 and 〈γ〉 for thiophene and 〈α〉 for bithiophene are found to be in qualitative agreement with those obtained by a recent ab initio calculation which used the finite field method and included diffuse polarization functions. The experimentally observed dependence of 〈α〉 and 〈γ〉 on the number N of the thiophene repeat unit is compared with that predicted by a free electron model, PPP methods, and the ab initio calculations. For the case of polarizability, the repeat unit dependence is in good agreement with that predicted by the ab initio calculations but for the case of second hyperpolarizability, the agreement is not as good.

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In this paper, we show the existence and qualitative properties of traveling wave solutions to the Allen-Cahn equation with fractional Laplacians. A key ingredient is the estimation of the traveling speed of traveling wave solutions.

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We study the set S = {(x, y) ∈ + × Z n : x + B j y j ≥ b j , j = 1, . . . , n}, where B j , b j ∈ + −{0}, j = 1, . . ., n, and B 1 | · · · |B n . The set S generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and the mixing-MIR set of Günlük and Pochet. In addition, it arises as a substructure in general mixed-integer programming (MIP), such as in lot-sizing. Despite its importance, a number of basic questions about S remain unanswered, including the computational complexity of optimization over S and how to efficiently find a most violated cutting plane valid for P = conv(S).One popular approach is to establish and analyze the full inequality description of P , or at least interesting families of facets valid for it. Unfortunately, the inequality description of P turns out to be tremendously complicated, and establishing it either fully or partially is an extraordinary task. Fortunately, the extreme points and extreme rays of P have a simple and elegant structure that proves insightful in deriving important properties of S. We give all extreme points and extreme rays of P . In the worst case, the number of extreme points grows exponentially with n. However, we show that, in some interesting cases, it is bounded by a polynomial of n. Finally, we use our results on the extreme points of P to give a polynomial-time separation oracle for the polar set of P , which, in turn, gives a polynomial-time separation oracle for P and establishes the tractability of optimization over S.

We have investigated the influence of two-photon absorption on the third-order nonlinear optical properties of model organic molecules using the technique of degenerate four-wave mixing (DFWM). A theoretical formulation developed here shows that the presence of two-photon absorption, which is related to the imaginary part of the third-order susceptibility χ(3), leads to an enhancement of the effective third-order nonlinearity and to the appearance of effects caused by the formation of two-photon generated excited states. The dynamic behavior of the nonlinearity is then governed by the properties of excited molecules. The nonlinear effects also involve contributions which depend on the fifth power of the electric field. We have performed a systematic study of third-order nonlinear optical properties of alkoxy (–C10H21OCH) substituted p-polyphenyl oligomers using the technique of time-resolved degenerate four-wave mixing with subpicosecond pulses at 602 nm. Experimentally determined values of the second-order hyperpolarizability γ for the oligomers increase smoothly from the monomer to the trimer, with a more rapid increase to the pentamer and to the heptamer. In addition, the hyperpolarizabilities for the pentamer and the heptamer appear to be complex. A smooth increase of the γ value is expected from an increase of the π conjugation from a shorter chain oligomer to a longer chain oligomer. The more rapid increase of the γ value in the pentamer, and especially in the heptamer, however, cannot be explained satisfactorily by only taking into account the π-conjugation length. Two-photon absorption for the pentamer and the heptamer at the measurement wavelength of 602 nm is suggested to be important as the observed dynamic behavior is satisfactorily explained by the predictions of the theoretical model presented here. It is shown that the effective γ value for a two-photon absorbing material is a function of optical intensity, pulse width, and sample length if one uses the conventional degenerate four-wave mixing description.

As part of a continuing study, a series of highly conjugated aromatic benzobisthiazole, benzobisoxazole, and N,N-diphenylbenzbisimidazole model compounds were synthesized, and their third-order nonlinear optical properties investigated by using subpicosecond degenerate four-wave mixing. Measurementa were made at 602 nm on THF solutions, vacuum-deposited films, or melt-quenched films. In all but one material, the third-order effect was determined to be instantaneous within the employed temporal resolution, suggesting dominance of electronic nonlinearity. From the experimental data it was possible to formulate structure-nonlinear optical property correlations and demonstrate that the molecular second hyperpolarizability can be increased almost 3 orders of magnitude by simple structural modifications. IX: #I= L4((NIylr + N~Y,,)~ + (N1yli)2)'/2 (15) Shand, M. L.; Chance, R. R. In Nonlinear Optical Propertiesof Organic and Polymeric Materiale; Williams, D. J., Ed.; ACS Symp. Ser. No. 233; American Chemical Society: Washington, DC, 1983.