Rick Lytel

The influence of geometry and topology of quantum graphs on their nonlinear-optical properties

We analyze the nonlinear optics of quasi one-dimensional quantum graphs and manipulate their topology and geometry to generate for the first time nonlinearities in a simple system approaching the fundamental limits of the first and second hyperpolarizabilities. Changes in geometry result in smooth variations of the nonlinearities. Topological changes between geometrically-similar systems cause profound changes in the nonlinear susceptibilities that include a discontinuity due to abrupt changes in the boundary conditions. This work may inform the design of new molecules or nano- scale structures for nonlinear optics and hints at the same universal behavior for quantum graph models in nonlinear optics that is observed in other systems.

Geometry-controlled nonlinear optical response of quantum graphs

We study for the first time the effect of the geometry of quantum wire networks on their nonlinear optical properties and show that for some geometries, the first hyperpolarizability is largely enhanced and the second hyperpolarizability is always negative or zero. We use a one-electron model with tight transverse confinement. In the limit of infinite transverse confinement, the transverse wavefunctions drop out of the hyperpolarizabilities, but their residual effects are essential to include in the sum rules. The effects of geometry are manifested in the projections of the transition moments of each wire segment onto the 2D lab frame. Numerical optimization of the geometry of a loop leads to hyperpolarizabilities that rival the best chromophores. We suggest that a combination of geometry and quantum-confinement effects can lead to systems with ultralarge nonlinear response.

DRESSED QUANTUM GRAPHS WITH OPTICAL NONLINEARITIES APPROACHING THE FUNDAMENTAL LIMIT

In this paper, we dress bare quantum graphs with finite delta function potentials and calculate optical nonlinearities that are found to match the fundamental limits set by potential optimization. We show that structures whose first hyperpolarizability is near the maximum are well described by only three states, the so-called three-level Ansatz, while structures with the largest second hyperpolarizability require four states. We analyze a very large set of configurations for graphs with quasi-quadratic energy spectra and show how they exhibit better response than bare graphs through exquisite optimization of the shape of the eigenfunctions enabled by the existence of the finite potentials. We also discover an exception to the universal scaling properties of the three-level model parameters and trace it to the observation that a greater number of levels are required to satisfy the sum rules even when the three-level Ansatz is satisfied and the first hyperpolarizability is at its maximum value, as specified by potential optimization. This exception in the universal scaling properties of nonlinear optical structures at the limit is traced to the discontinuity in the gradient of the eigenfunctions at the location of the delta potential. This is the first time that dressed quantum graphs have been devised and solved for their nonlinear response, and it is the first analytical model of a confined dynamic system with a simple potential energy that achieves the fundamental limits.

Nonlinear optics of quantum graphs

Quantum graphs are graphical networks comprised of edges supporting Hamiltonian dynamics and vertices conserving probability flux. Lateral confinement of particle motion on every edge results in a quasi one-dimensional quantum-confined system for which nonlinear optical effects may be calculated. Our ongoing research program is the first to investigate the nonlinear optical properties of quantum graphs. We seek to discover configurations with intrinsic first and second hyperpolarizabilities approaching their respective fundamental limits, to explore the NLO variation with the geometry and topology of the graphs, and to develop scaling laws for more complex graphs with self-similar properties. This paper describes a new methodology for calculating the hyperpolarizabilities of a class of graphs comprised of sequentially-connected edges. Such graphs include closed-loop topologies and their geometrically-similar but topologically-different open loop cousins, as well as other bent wire graphs and their combinations.

Phase disruption as a new design paradigm for optimizing the nonlinear-optical response

The intrinsic optical nonlinearities of linear structures, including conjugated chain polymers and nanowires, are shown to be dramatically enhanced by the judicious placement of a charge-diverting path sufficiently short to create a large phase disruption in the dominant eigenfunctions along the main path of the probability current. Phase disruption is proposed as a new general principle for the design of molecules, nanowires, and any quasi-1D quantum system with large intrinsic response and does not require charge donor-acceptors at the ends.

Optimum topology of quasi-one-dimensional nonlinear optical quantum systems

In this paper, we determine the optimum topology of quasi-one-dimensional nonlinear optical structures using generalized quantum graph models. Quantum graphs are relational graphs endowed with a metric and a multiparticle Hamiltonian acting on the edges, and have a long application history in aromatic compounds, mesoscopic and artificial materials, and quantum chaos. Quantum graphs have recently emerged as models of quasi-one-dimensional electron motion for simulating quantum-confined nonlinear optical systems. This paper derives the nonlinear optical properties of quantum graphs containing the basic star vertex and compares their responses across topological and geometrical classes. We show that such graphs have exactly the right topological properties to generate energy spectra required to achieve large, intrinsic optical nonlinearities. The graphs have the exquisite geometrical sensitivity required to tune wave function overlap in a way that optimizes the transition moments. We show that this class of graphs consistently produces intrinsic optical nonlinearities near the fundamental limits. We discuss the application of the models to the prediction and development of new nonlinear optical structures.

Optimization of eigenstates and spectra for quasi-linear nonlinear optical systems

Quasi-one-dimensional quantum structures with spectra scaling faster than the square of the eigenmode number (superscaling) can generate intrinsic, off-resonant optical nonlinearities near the fundamental physical limits, independent of the details of the potential energy along the structure. The scaling of spectra is determined by the topology of the structure, while the magnitudes of the transition moments are set by the geometry of the structure. This paper presents a comprehensive study of the geometrical optimization of superscaling quasi-one-dimensional structures and provides heuristics for designing molecules to maximize intrinsic response. A main result is that designers of conjugated structures should attach short side groups at least a third of the way along the bridge, not near its end as is conventionally done. A second result is that once a side group is properly placed, additional side groups do not further enhance the response.

Physics of the fundamental limits of nonlinear optics: a theoretical perspective [Invited]

The theory of the fundamental limits (TFL) of nonlinear optics is a powerful tool for experimentalists seeking to create molecules and materials with large responses and for theorists who are seeking to understand how the basic elements of quantum theory delineate the boundaries within which these searches should be conducted. On a practical level, the TFL provides a metric for measuring the performance or “goodness” of new molecules, relative to what is possible. Explorations of large sets of structures within the theory provide insight into new design rules for creating more active molecules. This paper is a review of the TFL, starting with a history of its development and its first use in discovering that all molecules as of the year 2000 fell a factor of 30 below the limits, and continuing to the present day, where the theory continues to provide research opportunities and challenges. The review focuses on off-resonant nonlinear optics in order to sharply focus on the key elements of the TFL, but pointers are provided to the literature for near- and on-resonance applications.

Fundamental limits on the electro-optic device figure of merit

Device figures of merit are commonly employed to assess bulk material properties for a particular device class, yet these properties ultimately originate in the linear and nonlinear susceptibilities of the material, which are not independent of each other. In this work, we calculate the electro-optic device figure of merit based on the half-wave voltage and linear loss, which is important for phase modulators and serves as the simplest example of the approach. This figure of merit is then related back to the microscopic properties in the context of a dye-doped polymer, and its fundamental limits are obtained to provide a target. Surprisingly, the largest figure of merit is not always associated with a large nonlinear optical response, the quantity that is most often the focus of optimization. An important lesson for materials design is that the figure of merit alone should be optimized. The best device materials can have low nonlinearity provided that the loss is low, or near resonance high loss may be desirable because it is accompanied by a resonantly enhanced, ultralarge nonlinear response, so device lengths are short. Our work shows which frequency range of operation is most promising for optimizing the material figure of merit for electro-optic devices.

Topological optimization of nonlinear optical quantum wire networks

Spatially extended molecular structures, modeled as quantum graphs with one-dimensional electron dynamics, exhibit optical responses that can approach the fundamental limits. We present the results of a comprehensive study of the topological dependence of the nonlinearities of quantum graphs and show exactly how the first and second hyperpolarizability of a graph depend upon its topological class and how the hyperpolarizability tensors vary with graph geometry. We show how graphs with star motifs share universal scaling behavior near the maximum nonlinear responses and articulate design rules for quantum-confined, quasi-one dimensional systems that may be realized using molecular elements and nanowires.