Maxwell Garnett approximation (advanced topics): tutorial

A High-Throughput Microfluidic Cell Sorter Using a Three-Dimensional Coupled Hydrodynamic-Dielectrophoretic Pre-Focusing Module

Dielectrophoresis (DEP) is a powerful tool for label-free sorting of cells, even those with subtle differences in morphological and dielectric properties. Nevertheless, a major limitation is that most existing DEP techniques can efficiently sort cells only at low throughputs (<1 mL h-1). Here, we demonstrate that the integration of a three-dimensional (3D) coupled hydrodynamic-DEP cell pre-focusing module upstream of the main DEP sorting region enables cell sorting with a 10-fold increase in throughput compared to conventional DEP approaches. To better understand the key principles and requirements for high-throughput cell separation, we present a comprehensive theoretical model to study the scaling of hydrodynamic and electrostatic forces on cells at high flow rate regimes. Based on the model, we show that the critical cell-to-electrode distance needs to be ≤10 µm for efficient cell sorting in our proposed microfluidic platform, especially at flow rates ≥ 1 mL h-1. Based on those findings, a computational fluid dynamics model and particle tracking analysis were developed to find optimum operation parameters (e.g., flow rate ratios and electric fields) of the coupled hydrodynamic-DEP 3D focusing module. Using these optimum parameters, we experimentally demonstrate live/dead K562 cell sorting at rates as high as 10 mL h-1 (>150,000 cells min-1) with 90% separation purity, 85% cell recovery, and no negative impact on cell viability.

Effective Optical Properties of Laterally Coalescing Monolayer MoS2

Two-dimensional (2D) transition metal dichalcogenides (TMDCs) exhibit compelling dimension-dependent exciton-dominated optical behavior influenced by thickness and lateral quantum confinement effects. Thickness quantum confinement effects have been observed; however, experimental optical property assessment of nanoscale lateral dimension monolayer TMDCs is lacking. Here, we employ ex situ spectroscopic ellipsometry to evaluate laterally coalescing monolayer metalorganic chemical vapor deposited MoS₂. A multisample analysis is used to constrain Bruggeman and Maxwell-Garnett effective medium approximations and the effective dielectric functions are derived for laterally coalesced and uncoalesced MoS₂ films (∼10-94% surface coverage for ∼10-140 nm lateral grain sizes). This analysis demonstrates the ability to probe MoS₂ optical exciton behavior at growth-relevant grain sizes in relation to chemical vapor nucleation density, ripening, and lateral growth conditions. Our analysis is pertinent toward expected in situ epitaxial 2D TMDC film growth metrology to enable the facile development of monolayer films with targeted process-dependent optical properties.

Maxwell Garnett approximation in random media: tutorial

In the third part of the tutorial, we develop a Maxwell Garnett approximation, which is applicable, in particular, to disordered random media with non-spherical inclusions and to multicomponent mixtures. We are especially interested in the case when the inclusions are non-spherical and randomly distributed and oriented in space, but the composite is isotropic on average. The effective medium formula applicable to such media cannot be obtained by direct generalization of the results that were derived in the first two parts of the tutorial. We show that the Maxwell Garnett effective medium approximation can be stated in a very general form. The results derived by us previously as well as a result applicable to the medium described above can be obtained as special cases of this general result.

Collective Plasmon Coupling in Gold Nanoparticle Clusters for Highly Efficient Photothermal Therapy

Plasmonic nanomaterials with strong absorption at near-infrared frequencies are promising photothermal therapy agents (PTAs). The pursuit of high photothermal conversion efficiency has been the central focus of this research field. Here, we report the development of plasmonic nanoparticle clusters (PNCs) as highly efficient PTAs and provide a semiquantitative approach for calculating their resonant frequency and absorption efficiency by combining the effective medium approximation (EMA) theory and full-wave electrodynamic simulations. Guided by the theoretical prediction, we further develop a universal strategy of space-confined seeded growth to prepare various PNCs. Under optimized growth conditions, we achieve a record photothermal conversion efficiency of up to ∼84% for gold-based PNCs, which is attributed to the collective plasmon-coupling-induced near-unity absorption efficiency. We further demonstrate the extraordinary photothermal therapy performance of the optimized PNCs in in vivo application. Our work demonstrates the high feasibility and efficacy of PNCs as nanoscale PTAs.

Computational Electromagnetics: A Miscellany

The paper presents a miscellany of unorthodox and, in some cases, paradoxical or controversial items related to computational and applied electromagnetics. The topics include a definition of the magnetic source field via a line integral, losses in electric power transmission vs. losses in photonics, homogenization of periodic electromagnetic structures, spurious modes, models of plasmonic media, and more. It is hoped that this assortment of subjects will be of interest to a broad audience of scientists and engineers.

Equivalence between positive and negative refractive index materials in electrostatic cloaks

We investigate, both theoretically and numerically, the equivalence relationship between the positive and negative refraction index dielectric materials in electrostatic invisibility cloak. We have derived an analytical formula that enables fast calculate the corresponding positive dielectric constant from the negative refraction index material. The numerical results show that the negative refraction index material can be replaced by the positive refractive index materials in the static field cloak. This offers some new viewpoints for designing new sensing systems and devices in physics, colloid science, and engineering applications.

Novel semi-analytical optoelectronic modeling based on homogenization theory for realistic plasmonic polymer solar cells

Numerical-based simulations of plasmonic polymer solar cells (PSCs) incorporating a disordered array of non-uniform sized plasmonic nanoparticles (NPs) impose a prohibitively long-time and complex computational demand. To surmount this limitation, we present a novel semi-analytical modeling, which dramatically reduces computational time and resource consumption and yet is acceptably accurate. For this purpose, the optical modeling of active layer-incorporated plasmonic metal NPs, which is described by a homogenization theory based on a modified Maxwell-Garnett-Mie theory, is inputted in the electrical modeling based on the coupled equations of Poisson, continuity, and drift-diffusion. Besides, our modeling considers the effects of absorption in the non-active layers, interference induced by electrodes, and scattered light escaping from the PSC. The modeling results satisfactorily reproduce a series of experimental data for photovoltaic parameters of plasmonic PSCs, demonstrating the validity of our modeling approach. According to this, we implement the semi-analytical modeling to propose a new high-efficiency plasmonic PSC based on the PM6:Y6 PSC, having the highest reported power conversion efficiency (PCE) to date. The results show that the incorporation of plasmonic NPs into PM6:Y6 active layer leads to the PCE over 18%.

Multifunctional composites for elastic and electromagnetic wave propagation

Composites are ideally suited to achieve desirable multifunctional effective properties since the best properties of different materials can be judiciously combined with designed microstructures. Here, we establish cross-property relations for two-phase composite media that link effective elastic and electromagnetic wave characteristics to one another, including the respective effective wave speeds and attenuation coefficients, which facilitate multifunctional material design. This is achieved by deriving accurate formulas for the effective electromagnetic and elastodynamic properties that depend on the wavelengths of the incident waves and the microstructure via the spectral density. Our formulas enable us to explore the wave characteristics of a broad class of disordered microstructures because they apply, unlike conventional formulas, to a wide range of incident wavelengths (i.e., well beyond the long-wavelength regime). This capability enables us to study the dynamic properties of exotic disordered "hyperuniform" composites that can have advantages over crystalline ones, such as nearly optimal, direction-independent properties and robustness against defects. We specifically show that disordered "stealthy" hyperuniform microstructures exhibit novel wave characteristics (e.g., low-pass filters that transmit waves "isotropically" up to a finite wavenumber). Our cross-property relations for the effective wave characteristics can be applied to design multifunctional composites via inverse techniques. Design examples include structural components that require high stiffness and electromagnetic absorption; heat sinks for central processing units and sound-absorbing housings for motors that have to efficiently emit thermal radiation and suppress mechanical vibrations; and nondestructive evaluation of the elastic moduli of materials from the effective dielectric response.

The non-stationary case of the Maxwell-Garnett theory: growth of nanomaterials (2D gold flakes) in solution

The solution-based growth mechanism is a common process for nanomaterials. The Maxwell-Garnett theory (for light-matter interactions) describes the solution growth in an effective medium, homogenized by a mean electromagnetic field, which applies when materials are in a stationary phase. However, the charge transitions (inter- and intra-transitions) during the growth of nanomaterials lead to a non-stationary phase and are associated with time-dependent permittivity constant transitions (for nanomaterials). Therefore, time-independence in the standard Maxwell-Garnett theory is lost, resulting in time dependence, ε i(t). This becomes important when the optical spectrum of a solution needs to be deconvoluted at different reaction times since each peak represents a specific charge/energy transfer with a specific permittivity constant. Based on this, we developed a time-resolved deconvolution approach, f(t) ∝ ε i(t), which led us to identify the transitions (inter- and intra-transitions) with their dominated growth regimes. Two gold ion peaks were precisely measured (322 nm and 367 nm) for the inter-transition, and three different polyaniline oxidation states (PAOS) for the intra-transition, including A (372 nm), B (680 nm), and C (530 nm). In the initial reaction time regime (0-90 min), the permittivity constant of gold was found to be highly dependent on time, i.e. f E ∝ ε i(t), since charge transfer takes place from the PAOS to gold ions (i.e. inter-transition leads to a reduction reaction). In the second time regime (90-180 min), the permittivity constant of gold changes as the material deforms from 3D to 2D (f S ∝ ε 3D-2D), i.e. intra-transition (combined with thermal reduction). Our approach provides a new framework for the time-dependent modelling of (an)isotropic solutions of other nanomaterials and their syntheses.

Omnidirectional optical spectra for a nanocomposite cholesteric elastomer

We perform a theoretical model for studying the propagation of electromagnetic waves, within a cholesteric elastomer slab, doped with metallic inclusions aleatorily distributed, whose corresponding dielectric properties can be expressed in terms of a resonant effective uniaxial tensor. We have obtained an omnidirectional narrow conducting band and two narrow reflection bands created in the optical spectrum for right and left circularly polarized light, as well as narrow reflection bands for right circularly polarized light. We have found that the transmission and reflection spectra depend on light incidence angle and the filling fraction of metallic inclusions.

Extraordinarily transparent compact metallic metamaterials

The design of achromatic optical components requires materials with high transparency and low dispersion. We show that although metals are highly opaque, densely packed arrays of metallic nanoparticles can be more transparent to infrared radiation than dielectrics such as germanium, even when the arrays are over 75% metal by volume. Such arrays form effective dielectrics that are virtually dispersion-free over ultra-broadband ranges of wavelengths from microns up to millimeters or more. Furthermore, the local refractive indices may be tuned by altering the size, shape, and spacing of the nanoparticles, allowing the design of gradient-index lenses that guide and focus light on the microscale. The electric field is also strongly concentrated in the gaps between the metallic nanoparticles, and the simultaneous focusing and squeezing of the electric field produces strong 'doubly-enhanced' hotspots which could boost measurements made using infrared spectroscopy and other non-linear processes over a broad range of frequencies.

Interaction of Te and Se interlayers with Ag or Au nanofilms in sandwich structures

Noble metal nanolayers on flat substrates are often deposited with the use of semiconductor interlayers, which may strongly interact with the noble metal overlayer. We investigated the crystallinity, atomic concentration profile and optical parameters of ≈35 nm-thick silver and gold layers deposited on glass substrates with 2 nm-thick tellurium or selenium interlayers. Our study, based on X-ray reflectometry (XRR), X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS) and ellipsometric measurements, showed that using either of these interlayers introduces strain in nanocrystals of both plasmonic films. This, in turn, influences the migration of Se and Te into the metal layers. Selenium atoms migrate both in the silver and gold nanolayers, while tellurium atoms migrate only in silver. The Te concentration curve clearly suggests that this migration is an effect of the segregation of Te atoms in the silver structure. The differences in crystallinity, as well as the migration process, strongly influence the optical parameters of Ag and Au. In the permittivity of Ag deposited on either Te or Se, additional plasmonic bands originating from grain boundary segregation or diffusion occur, while for the Au layer, such resonances were not pronounced. In the permittivity of both materials, the intensity of the interband transition peaks is strongly altered, possibly due to the nano-alloy formation, but more likely due to the microstrain on metal grains.

External versus induced and free versus bound electric currents and related fundamental questions of the classical electrodynamics of continuous media: discussion

Standard textbooks on classical electrodynamics frequently operate with the notions of free and bound currents (charges). Alternative terminology of external and induced currents also exists. However, a clear physical definition of these physical objects is rarely given. The term "free current" can refer in some cases to the conductivity current, which is subject to constitutive relations in a material sample. In other cases, free current refers to the current that is completely extrinsic to a given material sample and is assumed to be known a priori or manipulated by the experimentalist at will. Although one can argue that all currents flowing in material media are subject to some constitutive relations, there is a clear distinction in the construction of the classical electrodynamics between the external and induced currents. The aim of this paper is to clarify this distinction while pointing out that the traditional distinction between free and bound currents is arbitrary and can be abandoned. In addition, the paper considers some relevant fundamental questions of classical electrodynamics, including the derivation of macroscopic Maxwell's equations, the properties of the external currents, and the physical interpretation of some auxiliary fields such as the field of polarization P.

Effective Optical Properties of Inhomogeneously Distributed Nanoobjects in Strong Field Gradients of Nanoplasmonic Sensors

Accurate and efficient modeling of discontinuous, randomly distributed entities is a computationally challenging task, especially in the presence of large and inhomogeneous electric near-fields of plasmons. Simultaneously, the anisotropy of sensed entities and their overlap with inhomogeneous fields means that typical effective medium approaches may fail at describing their optical properties. Here, we extend the Maxwell Garnett mixing formula to overcome this limitation by introducing a gradient within the effective medium description of inhomogeneous nanoparticle layers. The effective medium layer is divided into slices with a varying volume fraction of the inclusions and, consequently, a spatially varying effective permittivity. This preserves the interplay between an anisotropic particle distribution and an inhomogeneous electric field and enables more accurate predictions than with a single effective layer. We demonstrate the usefulness of the gradient effective medium in FDTD modeling of indirect plasmonic sensing of nanoparticle sintering. First of all, it yields accurate results significantly faster than with explicitly modeled nanoparticles. Moreover, by employing the gradient effective medium approach, we prove that the detected signal is proportional to not only the nanoparticle size but also its size dispersion and potentially shape. This implies that the simple volume fraction parameter is insufficient to properly homogenize these types of nanoparticle layers and that in order to quantify optically the state of the layer more than one independent measurement should be carried out. These findings extend beyond nanoparticle sintering and could be useful in analysis of average signals in both plasmonic and dielectric systems to unveil dynamic changes in exosomes or polymer brushes, phase changes of nanoparticles, or quantifying light absorption in plasmon assisted catalysis.

Two years of tutorials: editorial

Editor-in-Chief P. Scott Carney gives an update on JOSA A tutorials.