Abstract

Nonlinear optics is the study of all effects that can be described as multi-photon interactions in various material systems, including cases where the frequency of one or more photons tends to zero. This feature issue is dedicated to works on both second-order nonlinear optics (three-photon interactions) and third-order nonlinear optics (four-photon interactions) that focus on understanding the fundamental mechanisms of the nonlinear optical response when the nonlinearity is large and approaches the fundamental quantum limit—a regime required by applications and characterized by interesting physics. Slightly more than half of the papers focus on theoretical analysis, with a strong emphasis on understanding model systems in view of the fundamental limits to the nonlinear optical response, which are dictated by quantum mechanics, and with several discussions of new theoretical approaches to understand and model the nonlinear optical response. The remaining papers provide an experimental counterpoint that includes examples of nonlinear optical responses in a variety of systems, an overview of how the fundamental limits can be approached in real materials, and of how the nonlinear response scales with spatial dimensions. The latter is quantified by identifying scale-invariant intrinsic quantities such as the dimensionless ratios between experimental results and the quantum limits.

© 2016 Optical Society of America

Since the invention of the laser, the field of nonlinear optics has grown into a worldwide research enterprise that has led to the discovery and the design of novel materials, devices, measurement methods, and light sources. While there has been enormous success with many new materials and applications, one could argue that the development of practical nonlinear optical materials has stalled in recent years. We believe that this may be due in part to the fact that most of the design rules available to researchers reflect past empirical paradigms that have led to molecules and materials with large absolute nonlinearities, and these same systems are not necessarily efficient from the point of view of the intrinsic limitations faced by the materials used and the fundamental principles of quantum mechanics itself.

During the last two decades, an analysis based on fundamental limits and scale invariance—which removes the influence of molecular size—has spawned new ideas for molecular design and optimization, has led to new ways for understanding and evaluating experimental data based on the fundamental limits that can be derived from basic properties such as the size of a conjugated system in an organic molecule and the energy gap between ground and excited states and has led to deep puzzles and paradoxes about the nature of light–matter interaction. This area of research spans from basic science to applications, unified by the fundamental principles of quantum mechanics that dictate the scaling of material or molecular properties.

Organic molecules, because of their large conjugated electron systems that increase the cross-section for light–matter interaction, have always led to the largest nonlinear optical responses at the molecular level. This response, mostly quantified by second- and third-order optical polarizabilities—or first and second hyperpolarizabilities—has been the focus of much past and current research. While quantum chemistry calculations have been increasingly successful in predicting the optical response of specific molecules, an alternative approach has arisen whose goal is to develop a more fundamental understanding of the intrinsic origins of the nonlinear response and of how the laws of quantum mechanics set its upper limit.

Kuzyk’s development of the fundamental quantum limits of the nonlinear optical response provided an important theoretical insight as well as a new tool for experimentalists who can now express their results using dimensionless, scale-invariant quantities (the “intrinsic hyperpolarizabilities”; see the work of Perez-Moreno et al. and Erickson et al. in this feature issue) that inform us on the efficiency of a given molecular system. This allows a systematic comparison of different material systems to each other that is not intrinsically blinded by the fact that larger molecules have necessarily larger hyperpolarizabilities. In addition to fundamental quantum limits that put experimental work into a large context, the renewed focus toward studying the intrinsic origins of the nonlinear response has also taken aim at going beyond the standard analysis by considering the nonlinear optical response of electrons in exotic Hamiltonians. We expect that the increased fundamental understanding obtained in this way will lead to new paradigms for the development of new ideas and systems.

Motivated by the needs of a growing number of researchers working in this area, a new series of meetings on the topic of the quantum foundations of nonlinear optics has spawned organically. After the first meeting was held at Washington State University in 2014 (http://nlosource.com/ASRS-NLO.html), a second meeting took place at Lehigh University in 2015 (http://www.lehigh.edu/inlo/FoNLO), where it acquired the title “Foundations of Nonlinear Optics” (FoNLO). A third FoNLO meeting followed this year at Tufts University (http://sites.tufts.edu/fonlo2016), and a next meeting is planned for the Bahamas in 2017 (http://www.nathandawson.org/FoNLO2017/). Collective discoveries presented at these meetings, coupled with recent advances in synthetic chemistry, point toward the emergence of a coherent set of ideas for designing materials whose optical nonlinearities approach the fundamental limits.

This special feature of the Journal of the Optical Society of America B contains contributions from several participants in past FoNLO meetings and from others. These articles range from the fundamental, first principles analysis of the nonlinear response and its origins, to experimental work that uses scale-invariant quantities to put the strength of the nonlinear optical response in context, thus giving an overview of the field that is attentive to constraints for real materials and focuses on the efficiency of the nonlinear optical response. By offering a balance between theoretical and experimental papers, and between reviews and current state of the art, this feature issue connects new approaches to the modeling of the nonlinear optical response in various systems to the insights that can be gained by experimental data and the discussion of different material systems.

Mark Kuzyk, who first investigated the fundamental limits of the nonlinear optical response in 2000, presents a paper where he seeks to define a path to ultra-large nonlinear optical susceptibilities. Rick Lytel gives a theoretical perspective of the physics of the fundamental limits of nonlinear optics, followed by three papers by Mossman et al. that discuss various aspects of theoretical modeling, from quantum graphs, to Dalgarno–Lewis perturbation theory, to analyzing the more practical fundamental limit of the figure of merit of an electro-optic device in a molecular material. The electro-optic response and its optimization are also discussed by Tillack and Robinson. Theoretical work that tests the fundamental limits from a fundamental perspective is presented by Christopher Burke et al., who show the effect of multiple electrons on the second-order polarizabilities of 1D potentials. Nathan Dawson investigates the nonlinear optical response of a space-fractional quantum mechanical system subjected to a static electric field, and Dawson and Kuzyk then attack the inverse problem of extracting the potentials for systems near the fundamental limit from their energy spectra and dipole moment matrices by using polynomial potential approximations. The boundary between experimental results and the fundamental limits is investigated by Javier Perez-Moreno, who provides an analysis of a three-state system and how it determines the second-order nonlinear optical response, and, in collaboration with Shoresh Shafei and Mark Kuzyk, presents an extensive review of how experimental second- and third-order polarizabilities scale with the size of a molecule, thus identifying which real systems meet the super-scaling requirement. Further experimental data and analysis on this topic are provided by Erickson et al., who show how the spectra of the third-order polarizability and the related two-photon absorption cross sections and zero-frequency values vary with the conjugation length in donor–acceptor substituted molecules while remaining close to the fundamental limit. Finally, an overview of experimental work on several new materials is provided by several authors: Barbano et al. contribute experimental work on the tensorial nature of the third-order susceptibility, Rashid Ganeev discusses frequency conversion in laser-produced graphite plasma, Andrews et al. investigate nonlinear optical effects in resonant cavities, Rawal et al. present results on new organic chromophores based on cross-conjugated bridges, and Sullivan et al. analyze hybrid metal–organic structures that can give enhanced nonlinear optical susceptibilities.

It is our hope that this wide range of topics will provide an overview of a diverse field and that this feature issue will help push the envelope for the development of exotic materials, the re-engineering of existing materials, and the control of quantum effects to manipulate the macroscopic nonlinear-optical response and other desirable bulk properties.

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