Scientists at Imperial College London in London, England have devised a novel way of analyzing tricky 3D plasmonic systems affected by nonlocality – through transformation optics.
Though van der Waals forces are relatively easy to calculate in parallel surfaces farther apart than 10 nm, they become very difficult to calculate in tiny distances of less than 5 nm. Furthermore, at such a miniscule scale, nonlocality – the direct interaction of objects separated in space with no apparent intermediate mechanism – also works to obscure analysis.
However, in a paper recently published in Proceedings of the National Academy of Sciences, Imperial College London scientists Prof. Sir John Pendry, Dr. Yu Luo and Dr. Rongkuo Zhao detail how transformation optics can be used to analyze nonlocality in 3D plasmonic systems.
The paper marks the first time that transformation optics have been used to calculate van der Waals forces, managing to solve the dilemma of nonlocal effects resulting from nanoscale bodies in plasmonic systems. Plasmons are quasiparticles that result from the quantization of plasma oscillations at optical frequencies; thus, transformation optics decide the direction that electromagnetic radiation will propagate by arranging electromagnetic fields in specific ways.
“Nonlocality introduces computational complexity which makes doing the calculations difficult,” Pendry told Phys.org. “We’ve found a workaround that greatly simplifies the calculations by replacing the nonlocal system with a local system that reproduces the results to a high degree of accuracy.”
“The key to successfully exploiting transformation optics is to choose the right transformation. In our case we were able to transform the problem of two nearly-touching spheres into the much more symmetric problem of two concentric spheres,” Pendry said.
However, the researchers were faced with the challenge of juggling several scales; specifically, the length of the spheres themselves (about 10 nm) and the spacing between the spheres, which the scientists attempted to push to the limit of atomic spacing (about 0.2 nm). Additionally, the scientists were also challenged by the fact that the forces they were studying were constituted by a host of different frequencies, ranging almost 100 eV in scope.
According to Pendry, scientists are just beginning to probe the effects of nonlocality in nonoscale surface phenomena, and are also just beginning to build reliable models. Unlike other methods, however, Pendry’s focus on transformation optics is able to take multiple electromagnetic fields into account, helping to field the specific difficulties of nonlocality. “The nanoscale forces in our paper are just one instance of where it’s important to treat nonlocality,” Pendry said, “where the main complication is that the response of a system at a given point depends not just on the electromagnetic fields at that point, but on the fields in the surrounding region as well – a problem that many traditional approaches fail to address.”
With the help of his colleagues, Pendry found that nonlocality significantly weakened the field enhancement, and thus the van der Waals force, between the spheres. “Van der Waals forces – although long range relative to standard chemical bonds – are only significant when surfaces are quite close to one another,” Pendry explained. “The standard local theory predicts infinite force in the limit that surfaces touch – but of course this is nonsense. Therefore, predictions that make sense and can be compared to experiments need to take nonlocality into account.”
Pendry hopes that his research of van der Waals forces will eventually lead to further research on chemical bonding. Though currently peripheral to the main focus of his paper, Pendry asserts that bonding will be integral to the final approach that immediately precedes contact between the two surfaces, in which direct contact of the charges will occur through electron tunneling.
“The forces we consider are complementary to chemical bonding, in that the current theoretical approach to chemical bonds exploits the local density approximation,” Pendry said. “In other words, just as a study of pure van der Waals forces omits chemical bonding, so a pure local density study of bonds has nothing to say about the longer range dispersion forces that we calculate. Of course, at some stage the two have to come together…but for that to happen we need experimental input – and theoretical studies of the van der Waals forces are the first steps in making this happen.”
Through transformation optics, the new approach detailed in the PNAS paper is able to analyze 3D nonlocal problems while shedding light on a more general understanding of nonlocal effects in plasmonic nanostructures. “Calculations are always difficult when treating singular structures – by which we mean situations such as the nearly touching spheres considered in our paper – but also the interaction of needle-sharp points with surfaces,” Pendry said. “Using transformations to unravel the singularity reveals how the forces work in each of these situations, and in fact often enables us to show a common origin.” Indeed, according to Pendry, “any nanomechanical system must consider the effects of van der Waals forces – and our paper is an attempt to further our understanding of these problems.”
Pendry has high hopes for the implications of, and further investigations inspired by, his research; specifically, in areas of electromagnetic heating and cooling. “On the near horizon is heat transfer between surfaces that are close but not in physical contact,” Pendry said. “Electromagnetic fluctuations responsible for the van der Waals force also enable heat to leap across the gap – an effect different from, and much stronger than, radiative cooling. In the longer term, we’ll try to generalize our theory of quantum friction, whereby surfaces which are close but not in physical contact can experience frictional drag. Nonlocality is also an important issue in the effects.”
Because transformation optics is used in a wide range of electromagnetic applications and theories, Pendry points towards other ways that his method could be used in other, EM-focused research. “The present study is just one in a whole series of applications,” he said. “We’ve already seen many studies of its application to invisibility, and we have used it extensively to study intense field enhancements in plasmonic structures, such as surface enhanced Raman spectroscopy. In fact,” he continued, “virtually any problem that has electromagnetic radiation interacting with a physical structure could potentially benefit from transformation optics – and in the case of plasmonic systems, nonlocality will always be an important issue whenever surface in close proximity are considered.”
– Melanie Abeygunawardana
