Abstract:
The off-resonant hyperpolarizability is calculated using the dipole-free sum-over-stats expression from a randomly chosen set of energies and transition dipole moments that are forced to be consistent with the sum rules. The process is repeated so that the distribution of hyperpolarizabilities can be determined. We find this distribution to be a cycloid-like function. In contrast to variational techniques that when applied to the potential energy function yield an intrinsic hyperpolarizability less than 0.71, our Monte Carlo method yields values that approach unity. While many transition dipole moments are large when the calculated hyperpolarizability is near the fundamental limit, only two excited states dominate the hyperpolarizability - consistent with the three-level ansatz.

Abstract:
The Schr\"{o}dinger equation has the property that when changing the length scale by $\vec{r} \to \epsilon \vec{r}$ and the energy scale by $E \to E / \epsilon^2$, the shape of the wavefunction remains unchanged. The same re-scaling leaves the intrinsic hyperpolarizability (as well as higher-order hyperpolarizabilities) unchanged. As such, the intrinsic hyperpolarizability is the best quantity for comparing molecules since it re-normalizes for trivial differences that are due to molecular size and energy gap. Similarly, the intrinsic hyperpolarizability is invariant to changes in the number of electrons. In this paper, we review the concept of scale invariance and how it can be applied to better understand the nonlinear optical response, which can be used to develop new paradigms for it's optimization.

Abstract:
Using sum rules and a new dipole-free sum-over-states expression, we calculate the fundamental limits of the dispersion of the real and imaginary parts of all electronic nonlinear-optical susceptibilities. As such, these general results can be used to study any nonlinear optical phenomena at any wavelength, making it possible to push both applications and our understanding of such processes to the limits. These results reveal the ultimate constraints imposed by nature on our ability to control and use light.

Abstract:
Truncated sum rules have been used to calculate the fundamental limits of the nonlinear susceptibilities; and, the results have been consistent with all measured molecules. However, given that finite-state models result in inconsistencies in the sum rules, it is not clear why the method works. In this paper, the assumptions inherent in the truncation process are discussed and arguments based on physical grounds are presented in support of using truncated sum rules in calculating fundamental limits. The clipped harmonic oscillator is used as an illustration of how the validity of truncation can be tested; and, several limiting cases are discussed as examples of the nuances inherent in the method.

Abstract:
Using sum rules, the dipolar terms can be eliminated from the commonly-used sum-over-states (SOS) expression for nonlinear susceptibilities. This new dipole-free expression is more compact, converges to the same results as the common SOS equation, and is more appropriate for analyzing certain systems such as octupolar molecules. The dipole-free theory can be used as a tool for analyzing the uncertainties in quantum calculations of susceptibilities, can be applied to a broader set of quantum systems in the three-level model where the standard SOS expression fails, and more naturally leads to fundamental limits of the nonlinear susceptibilities.

Abstract:
The Thomas Kuhn Reich sum rules and the sum-over-states (SOS) expression for the hyperpolarizabilities are truncated when calculating the fundamental limits of nonlinear susceptibilities. Truncation of the SOS expression can lead to an accurate approximation of the first and second hyperpolarizabilities due to energy denominators, which can make the truncated series converge to within 10% of the full series after only a few excited states are included in the sum. The terms in the sum rule series, however, are weighted by the state energies, so convergence of the series requires that the position matrix elements scale at most in inverse proportion to the square root of the energy. Even if the convergence condition is met, serious pathologies arise, including self inconsistent sum rules and equations that contradict reality. As a result, using the truncated sum rules alone leads to pathologies that make any rigorous calculations impossible, let alone yielding even good approximations. This paper discusses conditions under which pathologies can be swept under the rug and how the theory of limits, when properly culled and extrapolated using heuristic arguments, can lead to a semi-rigorous theory that successfully predicts the behavior of all known quantum systems, both when tested against exact calculations or measurements of broad classes of molecules.

Abstract:
We propose the scale-invariant intrinsic hyperpolarizability as a measure of the figure of merit for electrooptic molecules. By applying our analysis to the best second-order nonlinear-optical molecules that are made using the present paradigms, we conclude that it should be possible to make dye-doped polymers with electrooptic coefficients of several thousand picometers per volt.

Abstract:
This work focuses on understanding the nonlinear-optical response of a 1-D quantum wire embedded in 2-D space when quantum-size effects in the transverse direction are minimized using an extremely weighted delta function potential. Our aim is to establish the fundamental basis for understanding the effect of geometry on the nonlinear-optical response of quantum loops that are formed into a network of quantum wires. Using the concept of leaky quantum wires, it is shown that in the limit of full confinement, the sum rules are obeyed when the transverse infinite-energy continuum states are included. While the continuum states associated with the transverse wavefunction do not contribute to the nonlinear optical response, they are essential to preserving the validity of the sum rules. This work is a building block for future studies of nonlinear-optical enhancement of quantum graphs (which include loops and bent wires) based on their geometry. These properties are important in quantum mechanical modeling of any response function of quantum-confined systems, including the nonlinear-optical response of any system in which there is confinement in at leat one dimension, such as nanowires, which provide confinement in two dimensions.

Abstract:
All observations of photodegradation and self healing follow the predictions of the correlated chromophore domain model. [Ramini et.al. Polym. Chem., 2013, 4, 4948.] In the present work, we generalize the domain model to describe the effects of an electric field by including induced dipole interactions between molecules in a domain by means of a self-consistent field approach. This electric field correction is added to the statistical mechanical model to calculate the distribution of domains that are central to healing. Also included in the model are the dynamics due to the formation of an irreversibly damaged species. As in previous studies, the model with a one-dimensional domain best explains all experimental data of the population as a function of time, temperature, intensity, concentration, and now applied electric field. Though the nature of a domain is yet to be determined, the fact that only one-dimensional domain models are consistent with observations suggests that they might be made of correlated dye molecules along polymer chains.

Abstract:
We demonstrate optical limiting using the self-lensing effect of a higher-order Laguerre-Gaussian beam in a thin dye-doped polymer sample, which we find is consistent with our model using Gaussian decomposition. The peak phase shift in the sample required for limiting is smaller than for a fundamental Gaussian beam with the added flexibility that the nonlinear medium can be placed either in front of or behind the beam focus.