Abstract:The optical properties of nanoparticle clusters vary with the spatial arrangement of the constituent particles, but also the overall orientation of the cluster with respect to the incident light. This is particularly important in the context of nanoscale chirality and associated chiroptical responses, such as circular dichroism or differential scattering of circularly polarised light in the far-field, or local degree of optical chirality in the near-field. We explore the angular dependence of such quantities f… Show more
“…In contrast, the cubature methods used in this work have predetermined nodes and weights, regardless of the integrand's angular profile. The advantage is expected to be highest for sharply-defined angular properties, as exhibited by large elongated nanostructures supporting collective modes with well-defined momentum 40 . We tested briefly some adaptive cubature methods 61,62 , and did not observe a better performance on the above examples compared to Levedev, spherical t-design, or Gauß-Legendre cubatures.…”
Section: Discussionmentioning
confidence: 99%
“…Indeed, averaging the response along x, y, z axes is a commonly-used procedure for orientation-averaging in this regime 10 . Larger particles and/or clusters of particles may however show very directional responses, such as the photonic jet effect in large Mie scatterers 41 , or waveguide modes along chains of particles 40 or dielectric particles with a high aspect ratio 42 .…”
Section: Spherical Cubaturementioning
confidence: 99%
“…Our first illustration considers light scattering by a 'fingers crossed' dimer of gold nanorods 43 , which has attracted considerable interest in recent years as a prototypal structure for chiral plasmonics [44][45][46][47][48][49][50][51] . We recently revisited this geometry with a focus on the angular dependence of the chiroptical far-field properties, namely the circular dichroism in extinction, scattering and absorption 40 . Despite its sim- plicity, involving only two particles, the structure reveals an interesting response arising from the hybridisation of localised plasmon resonances, and a balance between absorption and scattering contributions as the size of the structure is varied 43 .…”
Section: Dimermentioning
confidence: 99%
“…For very small dimers this angular response is relatively smooth nonetheless, and as a result the orientation-averaged response may be reasonably approximated by the average over 3 orthogonal incidence directions. As the scale of the dimer increases, however, this is no longer an accurate approximation 40 . We confirm here this observation with a more complete characterisation of numerical angular averaging, using the previouslydefined seven cubature methods (Fig.…”
The optical properties of nanoparticles can be substantially affected by their assembly in compact aggregates. This is a common situation notably for nanoparticles synthesised and self-assembled into rigid clusters in colloidal form, where they may be further characterised or used in spectroscopic applications. The theoretical description of such experiments generally requires averaging the optical response over all possible cluster orientations, as they randomly orient themselves over the course of a measurement. This averaging is often done numerically by simulating the optical response for several directions of incidence, using a spherical cubature method. The simulation time increases with the number of directions and can become prohibitive, yet few studies have examined the trade-off between averaging accuracy and computational cost. We benchmark seven commonly-used spherical cubature methods for both far-field and near-field optical responses for a few paradigmatic cluster geometries: dimers of nanospheres and of nanorods, and a helix. The relative error is rigorously evaluated in comparison to analytical results obtained with the superposition T -matrix method. Accurate orientation averaging is especially important for quantities relating to optical activity, the differential response to left and right circularly polarised light, and our example calculations include in particular far-field circular dichroism and near-field local degree of optical chirality.
“…In contrast, the cubature methods used in this work have predetermined nodes and weights, regardless of the integrand's angular profile. The advantage is expected to be highest for sharply-defined angular properties, as exhibited by large elongated nanostructures supporting collective modes with well-defined momentum 40 . We tested briefly some adaptive cubature methods 61,62 , and did not observe a better performance on the above examples compared to Levedev, spherical t-design, or Gauß-Legendre cubatures.…”
Section: Discussionmentioning
confidence: 99%
“…Indeed, averaging the response along x, y, z axes is a commonly-used procedure for orientation-averaging in this regime 10 . Larger particles and/or clusters of particles may however show very directional responses, such as the photonic jet effect in large Mie scatterers 41 , or waveguide modes along chains of particles 40 or dielectric particles with a high aspect ratio 42 .…”
Section: Spherical Cubaturementioning
confidence: 99%
“…Our first illustration considers light scattering by a 'fingers crossed' dimer of gold nanorods 43 , which has attracted considerable interest in recent years as a prototypal structure for chiral plasmonics [44][45][46][47][48][49][50][51] . We recently revisited this geometry with a focus on the angular dependence of the chiroptical far-field properties, namely the circular dichroism in extinction, scattering and absorption 40 . Despite its sim- plicity, involving only two particles, the structure reveals an interesting response arising from the hybridisation of localised plasmon resonances, and a balance between absorption and scattering contributions as the size of the structure is varied 43 .…”
Section: Dimermentioning
confidence: 99%
“…For very small dimers this angular response is relatively smooth nonetheless, and as a result the orientation-averaged response may be reasonably approximated by the average over 3 orthogonal incidence directions. As the scale of the dimer increases, however, this is no longer an accurate approximation 40 . We confirm here this observation with a more complete characterisation of numerical angular averaging, using the previouslydefined seven cubature methods (Fig.…”
The optical properties of nanoparticles can be substantially affected by their assembly in compact aggregates. This is a common situation notably for nanoparticles synthesised and self-assembled into rigid clusters in colloidal form, where they may be further characterised or used in spectroscopic applications. The theoretical description of such experiments generally requires averaging the optical response over all possible cluster orientations, as they randomly orient themselves over the course of a measurement. This averaging is often done numerically by simulating the optical response for several directions of incidence, using a spherical cubature method. The simulation time increases with the number of directions and can become prohibitive, yet few studies have examined the trade-off between averaging accuracy and computational cost. We benchmark seven commonly-used spherical cubature methods for both far-field and near-field optical responses for a few paradigmatic cluster geometries: dimers of nanospheres and of nanorods, and a helix. The relative error is rigorously evaluated in comparison to analytical results obtained with the superposition T -matrix method. Accurate orientation averaging is especially important for quantities relating to optical activity, the differential response to left and right circularly polarised light, and our example calculations include in particular far-field circular dichroism and near-field local degree of optical chirality.
“…• Orientation-averaged far-field properties can be obtained at very little computational cost, with analytical formulae 21,[36][37][38][39] . Orientationaveraged near-field quantities can also be computed 15,22 , albeit with some computational overhead, providing analytical benchmark results 40 .…”
We introduce terms, an open-source Fortran program to simulate near-field and far-field optical properties of clusters of particles. The program solves rigorously the Maxwell equations via the superposition T -matrix method, where incident and scattered fields are decomposed into series of vector spherical waves.terms implements several algorithms to solve the coupled system of multiple scattering equations that describes the electromagnetic interaction between neighbouring scatterers. From this formal solution, the program can compute a number of physically-relevant optical properties, such as far-field cross-sections for extinction, absorption, scattering and their corresponding circular dichroism, as well as local field intensities and degree of optical chirality. By describing the incident and scattered fields in a basis of spherical waves the T -matrix framework lends itself to analytical formulas for orientation-averaged quantities, corresponding to systems of particles in random orientation; terms offers such computations for both far-field and nearfield quantities of interest. This user guide introduces the program, summarises the relevant theory, and is supplemented by a comprehensive suite of stand-alone examples in the website accompanying the code.
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