Light has proved superior to electricity in technologies from memory storage media, such as DVDs, to fiber optics for information transmission. Optical switching and imaging, 3D photolithography, and photodynamic cancer therapy are each similarly dependent upon light-matter interactions. It is natural to ask whether these interactions have fundamental limits in terms of efficiency, in what ways they might affect the pace of innovation, and how we might use knowledge about such constraints to produce better devices with improved materials.
Almost a decade ago, one of us (MGK) determined that light-matter interactions are theoretically constrained by a fundamental limit in terms of strength, which is quantified by the function known as the hyperpolarizability.1 All molecules measured until then fell short of the ‘Kuzyk limit’ by a factor of 30.2 This work made it clear for the first time that molecules do not achieve anywhere near full potential and that the then current paradigms of molecular design were fundamentally flawed. At the time, efforts to increase the hyperpolarizability focused on fabricating larger molecules. Zhou and co-workers used these theoretical observations as the basis for computer-modeled design of the ultimate molecule. Their concept of modulated conjugation3 led to the synthesis of a molecule with record-high hyperpolarizability.4
Calculations of the fundamental limit are based on general principles of quantum mechanics and must therefore be obeyed in any system. Although this approach has helped develop insights into what makes molecules efficient, many questions remain. The record-breaking molecules still fall 20-fold short of the fundamental limit, indicating room for dramatic improvements, while computer-optimized numerical models range at about 30% below the limit. Although this shortfall may be inconsequential for most applications, it leads to deeper questions about the underlying physics of light-matter interactions.
To investigate the shortfall found in Zhou's numerical optimization studies, which were carried out using the potential energy method, we employed a Monte Carlo approach.5 Both techniques use transition dipole moments and energies to calculate the hyperpolarizability. Zhou had optimized this value by varying the shape of the potential energy function from which these moments and energies were calculated. We randomly assigned moments and energies under the constraint of sum rules, which are equivalent to the Schrödinger equation. Use of the Monte Carlo method generated a broad distribution of hyperpolarizabilities that tails off at the fundamental limit (see Figure 1).
Figure 1. Distribution of calculated hyperpolarizabilities (points with error bars) for various energy weightings. βINT=±1,corresponding to the fundamental limit. In the inset, n is the fractional power that determines the shape of the curve, and χ2, R2, and A are fit parameters.
Figure 2. Summary of the intrinsic hyperpolarizability of a series of molecules, labeled TMC-2, TM-2, and TMC-3. The horizonal blue line represents the long-standing factor-of-30 shortfall.
These results suggest the intriguing possibility that nature allows for systems that are not described by typical Hamiltonians (i.e., those with kinetic and potential energy terms that have electromagnetic scalar and vector potentials). The implication is that exotic systems are required if the fundamental limits are to be obtained. Unfortunately, the Monte Carlo techniques, as used in our studies, do not provide guidelines for such systems, nor do they elucidate their properties. Though consistent with the laws of physics, such a system might not exist at present but could perhaps be deliberately fabricated. In either event, these simulations may define new areas of study.
Recently, Zhou and M. G. Kuzyk have shown that the intrinsic hyperpolarizability (the ratio of the hyperpolarizability to the fundamental limit) serves as a good figure of merit for evaluating electro-optic materials.6 Figure 2 shows a summary of the best molecules as reported by Kang and co-workers. The molecules TMC-2, TM-2, and TMC-37 have intrinsic hyperpolarizabilities inside 30% of the fundamental limit, thus living up to their full potential. The researchers identify the twisting of the rings relative to each other as the source of the the enhancement. Substantial improvement to electro-optic materials through the use of these molecules would considerably enhance the efficiency of modulators.
Theoretical studies of fundamental limits provide significant insight into the strength of light-matter interactions, and they can foster the design of better materials and novel devices. The phenomena of light interaction with a material, as studied with limit theory, can impact a broad range of applications and technologies, including enhanced efficacy in cancer therapies, increased speed of the Internet, and the development of ultra-smart materials. Accordingly, future efforts will focus on building a deeper understanding of how light can be harnessed by matter.
I thank the National Science Foundation (ECS-0756936) and Wright Paterson Air Force Base for generously supporting this work.
Mark G. Kuzyk, Mark C. Kuzyk
Washington State University
7. H. Kang, A. Facchetti, H. Jiang, E. Cariati, S. Righetto, R. Ugo, C. Zuccaccia, A. Macchioni, C. L. Stern, Z. F. Liu, Ultralarge hyperpolarizability twisted pi-electron system electro-optic chromophores: synthesis, solid-state, and solution-phase structural characteristics, electronic structures, linear and nonlinear optical properties, and computational studies, J. Am. Chem. Soc. 129, pp. 3267, 2007.