Exact Mie Research Articles

Comparison of rigorous scattering models to accurately replicate the behaviour of scattered electromagnetic waves in optical surface metrology

Rigorous scattering models are based on Maxwell's equations and can provide high-accuracy solutions to model electromagnetic wave scattering from objects. Being able to calculate the scattered field from any surface geometry and considering the effect of the polarisation of the incident light, make rigorous models the most promising tools for complex light-matter interaction problems. The total intensity of the electric near-field scattering from a silicon cylinder illuminated by the transverse electric and transverse magnetic polarisation of the incident light is obtained using various rigorous models including, the local field Fourier modal method, boundary element method and finite element method. The intensity of the total electric near-field obtained by these rigorous models is compared using the Mie solution as a reference for both polarisation modes of the incident light. Additionally, the intensity of the total electric near-field scattered from a silicon sinusoid profile using the same rigorous models is analysed. The results are discussed in detail, and for the cylinder, the deviations in the intensity of the total electric field from the exact Mie solution are investigated.

Intuitive understanding of extinction of small particles in absorbing and active host media within the MLWA

In an absorbing or an active host medium characterized by a complex refractive index n2=n2′+in2′′, our previously developed modified dipole long-wave approximation (MLWA) is shown to essentially overlie with the exact Mie theory results for localized surface plasmon resonance of spherical nanoparticles with radius a≲25nm (a≲20nm) in the case of Ag and Au (Al and Mg) nanoparticles. The agreement for Au and Ag (Al and Mg) nanoparticles, slightly better in the case of Au than Ag, continues to be acceptable up to a∼50nm (a∼40nm), and can be used, at least qualitatively, up to a∼70nm (a∼50nm) correspondingly. A first order analytic perturbation theory (PT) in a normalized extinction coefficient, κ¯=n2′′/n2′, around a nonabsorbing host is developed within the dipole MLWA and its properties are investigated. It is shown that, in a suitable parameter range, the PT can reliably isolate and capture the effect of host absorption or host gain on the overall extinction efficiency of various plasmonic nanoparticles.

Hollow titanium nitride nanoshells for enhanced plasmon-driven hot electron generation and improved photocatalytic and photovoltaic applications.

Nowadays, plasmonic titanium nitride (TiN) is widely employed as a potential alternative to noble metals in semiconductor-metal hybrid nanoparticles (S-M HNPs) for improving the utilization efficiency of solar energy in photocatalytic and photovoltaic systems. In semiconductor-TiN nanosystems, TiN NPs convert solar energy into highly energetic (hot) electrons that can be transmitted to the attached semiconductor for enhanced applications. In this paper, we propose TiN nanoshells with a nonabsorbing dielectric core as an improved energy conversion component in S-M HNPs, compared to homogenous TiN nanospheres, with higher geometrical optimization flexibility, wider absorption range tuneability, and effective hot electron generation and utilization due to the reduced plasmonic-shell size. For understanding the impact of the core material on the functionality of the nanoshells, we assume three core materials with different refractive indices (air, silica (SiO2), and magnesium oxide (MgO)). The exact Mie theory is utilized to calculate the absorption coefficient and the plasmon field of the proposed TiN nanoshells. To quantify the absorbance effectiveness on the solar spectrum, we calculate a relevant figure of merit (FoM) that depends on the spectral features of the absorption coefficient. By optimizing the geometrical parameters of nanoshells, it is found that hollow TiN nanoshells with the lowest core refractive index exhibit the highest FoM of solar energy absorption. Also, the plasmon field intensity of hollow TiN nanoshells is higher and more concentrated in a smaller volume of TiN material in comparison to the field intensity of other nanoshells (SiO2-TiN and MgO-TiN nanoshells) and TiN nanospheres. Factors affecting the utilization of the generated hot electrons, including the radiative damping of plasmons and the spreading of the plasmon field inside the nanoparticles, have been investigated. In view of the temporal dynamics of hot electrons, it is shown that using the hollow TiN nanoshells with thin shells greatly enhances the effectiveness of the generated hot electrons to reach the attached semiconductor. In fact, the reduced plasmonic-shell thickness results in a trade-off between a longer radiative relaxation time and less solar energy absorption with regard to the selected core material.

Physical optics approximation of electromagnetic field scattered from large PEC objects covered with chiral metamaterials

Electromagnetic scattering analysis from conducting objects comparable in size with the incident wavelength, using exact numerical approaches, always suffers from high computational complexity, especially when covered with composite penetrable materials. In this paper the asymptotic method of the physical optics (PO) is developed to predict the scattering patterns from conducting objects coated by chiral metamaterials (CMMs) as an absorber. In first stage, the backward ray-tracing is used to determine the lit region on the outer surface of the objects represented by flat elementary facets. Simple closed-form expressions are derived for the co- and cross-polarized reflection coefficients from a metal-backed CMM layer to approximate the equivalent currents on lit facets in terms of the geometrical-optics fields. Then, for stealth applications, a CMM layer with appropriate electromagnetic properties is suggested with reflected power lower than −20 dB in a wide aspect angle of incident ray from 0 to 57 degrees. Finally, the scattering patterns are calculated via the PO approximation and the equivalent currents. In order to check the accuracy of the proposed solutions, the simulation results of several objects will be compared with the experimental data and the results of the exact Mie theory and method of moment (MoM).

Exact Solution of New Magnetic Current Based Surface-Volume-Surface EFIE and Analysis of Its Spectral Properties

A novel magnetic current based Surface-Volume-Surface Electric Field Integral Equation (SVS-EFIE-M) is presented for the problem of scattering on homogeneous non-magnetic dielectric objects. The exact Galerkin Method of Moments (MoM) utilizing both the rotational and irrotational vector spherical harmonics as orthogonal basis and test functions according to the Helmholtz decomposition is implemented to solve SVS-EFIE-M analytically for the case of dielectric sphere excited by an electric dipole. The field throughout the sphere is evaluated and compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. This exact solution validates the rigorous nature of the new SVS-EFIE-M formulation. It also reveals the spectral properties of its individual operators, their products and their linear combination. The spectrum of the MoM impedance matrix is also obtained. It is shown that upon choosing basis and test functions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}(S)$</tex-math></inline-formula> space and evaluating testing inner products in the same space, the MoM impedance matrix features bounded condition number with increasing order of discretization and/or at low frequencies. This makes the proposed SVS-EFIE-M formulation free of oversampling and low-frequency breakdowns giving it advantage both over its SVS-EFIE-J predecessor and classical double-source integral equations such as PMCHWT, Muller, and others suffering from this type of numerical instabilities inherent to their inferior spectral properties.

Analytic Solution of Surface–Volume–Surface Electric Field Integral Equation on Dielectric Sphere and Analysis of Its Spectral Properties

The exact solution of the surface–volume–surface electric field integral equation (SVS-EFIE) is presented for the problem of radiation in the vicinity of the homogeneous dielectric sphere. The solution is obtained via the Galerkin method of moments (MoM) utilizing rotational and irrotational complete sets of orthogonal vector spherical harmonics as basis and test functions according to the Helmholtz decomposition. In the case of radial or tangential electric dipole radiation, the electric field throughout the sphere evaluated via the analytic MoM solution of the SVS-EFIE is compared against the exact classical Mie series solution. The two are shown to agree to 12 digits of accuracy upon a sufficient number of basis/test functions taken in the MoM solution and the Mie series expansion. The exact solution confirms and validates the rigorous nature of the SVS-EFIE formulation. It also reveals the spectral properties of its individual operators, their products, and their linear combination, as well as the spectrum of the MoM impedance matrix. It is shown that, upon choosing basis and test functions in the Sobolev space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{1/2}(S)$ </tex-math></inline-formula> and performing testing inner products evaluation in the space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{div}^{-1/2}(S)$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> being the surface of the sphere, the MoM impedance matrix features bounded condition number with increasing order of discretization similar with analogous exact MoM solution of the surface EFIE (S-EFIE) on the perfectly electrically conducting (PEC) sphere.

Unified treatment of nonlinear optical force in laser trapping of dielectric particles of varying sizes

Optical trapping using laser tweezer has revolutionized the field of force spectroscopy having enormous applications in biological manipulation. While a number of theories were developed for particles of different sizes to estimate trapping force under continuous-wave excitation, they were not under short pulsed excitation which leads to nonlinear optical force. Here, we present a comparative study of various theories and provide a unified description for laser trapping under femtosecond pulsed excitation. Numerical results show that exact Mie theory (EMT) can provide a precise qualitative and quantitative prediction of trapping force when optical Kerr effect is included. Moreover, we also show how Mie interference phenomena, leading to observation of Fano resonance, are naturally captured within EMT. Thus, our findings pave the way for potential far-reaching applications in the accurate numerical estimation of nonlinear optical force on arbitrary-sized spherical dielectric particles.

Discrete dipole approximation method for electromagnetic scattering by particles in an absorbing host medium.

Electromagnetic (EM) scattering by particles in an absorbing host medium is frequently encountered in practical applications, which makes the conventional EM scattering theory controversial and most of the theoretical methods for EM scattering inapplicable. Most of the relevant works in literature are confined to spherical particles. In this work, we develop the discrete dipole approximation (DDA) method for EM scattering by an arbitrary particle immersed in an absorbing host medium. We elaborate how the near- and far-field scattering quantities can be calculated by DDA. The accuracy of DDA is validated by comparison with the apparent and inherent scattering quantities of spherical particles computed by exact Mie theory. Then EM extinction by non-absorbing spheroids in absorbing host medium is studied by DDA. We find that particles that are prolonged in the incident direction are more likely to produce a negative apparent extinction, which is also supported by the near-field electric field distribution. The DDA method we develop will be useful and flexible in the study of EM scattering by particles in absorbing host medium.

Comparison of equations used to estimate soot agglomerate absorption efficiency with the Rayleigh-Debye-Gans approximation

The contribution of soot aerosols to radiative forcing and climate change depends on their optical properties. Since these aerosols are agglomerates composed of quasi-spherical primary particles (usually small with respect to the radiation wavelength), their properties can be calculated following the Rayleigh-Debye-Gans approximation, which considers primary particles as independent absorbers and scatterers, with corrections made to account for multiple scattering and absorption within the agglomerate. A variety of equations can be used for particles small compared to the wavelength. In this study, results of the Rayleigh, Bohren and Huffman, simplified Bohren and Huffman, and Dobbins and Megaridis equations have been compared to the results from the exact Mie equations for the absorption efficiency. This comparison has been made for individual particles with different diameters and for an ensemble of primary particles composing an agglomerate. It has been shown that some of the largest primary particles of a typical agglomerate may not behave as Rayleigh absorbers at short wavelengths, contributing to errors in the estimation of the absorption efficiency. Both Bohren and Huffman and Dobbins and Megaridis equations are useful to extend the application range of the Rayleigh equation, thus reducing the accumulated error in the estimation of the agglomerate absorption efficiency. This analysis has been performed for several complex refractive index spectra. To choose the best equation for a particular application, the wavelength spectrum of the radiation and the range of primary particle sizes should be considered, aiming for a compromise between simplicity and accuracy.

Modeling of light scattering and haze in semicrystalline polymers

AbstractThis article reports a new model approach for the description of light scattering in semicrystalline polymers, to describe more precisely the influence of supermolecular structure on the optical properties. This is the first study in which light scattering of polymer films has been modeled using exact Mie scattering theory of radially anisotropic spheres. As a model material a well‐known polymer, isotactic polypropylene (iPP) was used. Samples were prepared with different sample thicknesses and crystalline structures in order to identify the key parameters of light scattering in polycrystalline polymeric systems. Validation haze measurements were carried out with a spectrophotometer equipped with a 150 mm snap‐in integrating sphere. It was found that the optical properties of the polycrystalline sample can be described using multiple light scattering on these scattering centers. Good agreement was found between the simulated and experimentally measured haze values which proves the reliability and applicability of our new approach.

Landau broadening of plasmonic resonances in the Mie theory.

Landau damping in the metal nanosphere is considered beyond the quasistatic approximation with the use of the exact Mie theory when an incident plane wave can excite not only the dipole mode but also higher-order modes. In resonance approximation, when one considers excitation of a single mode, the analytical formula for the Landau damping coefficient for various modes has been derived. It was demonstrated that the simultaneous excitation of several eigenmodes, which are overlapped in the frequency domain, can lead to substantial correction of the Landau damping coefficients for the modes.

Microwave absorption by small dielectric and semi-conductor coated metal particles

The presence of the oxide coatings on conducting metal particles is often ignored in the calculations of microwave absorption. However, we find that the optical properties of the coatings play a significant role in enhancing or suppressing the absorption of electromagnetic energy. Here, we solve the Mie scattering equations numerically to separate and quantify the role of electric and magnetic field absorption from microwaves at 2.45 GHz by small metal spheres coated with dielectric or semi-conducting materials. The range of size and conductivities of the metal particles and the optical properties of the coatings are chosen for their practical importance. We also provide simple approximate expressions for the absorption per unit volume by coated spheres in the small particle limit which agrees very well with the exact Mie solution. We have demonstrated that for highly conducting particles coated with a material of low conductivity, the electric field absorption depends only on the optical properties, and the volume fraction of the coating. In contrast, the magnetic field absorption depends only on the properties of the core which is the same as the bare metal. We find that the power absorbed by coated particles via the electric field is maximized when loss tangent, tanδ~1. A key result of this study is that a thin layer of lossy coating (~15% of the particle size) on highly conductive particles will significantly enhance (up to a factor of 105) both the power absorbed from the E-field and the total power absorbed.

The van de Hulst approximation for light scattering and its use for transmittance predictions in transparent ceramics

The in-line transmittances of transparent spinel and YAG ceramics with a low volume fraction of monosized spherical pores calculated via the exact Mie solution and the so-called van de Hulst approximation are compared. It is shown that the transmittances calculated via the van de Hulst approximation are in surprisingly good qualitative agreement with the exact Mie solutions. In particular, all relevant trends and features of transmittance-versus-wavelength-and-pore-size plots are correctly predicted, so that the van de Hulst approximation may be considered as a perfect tool for material design and process optimization purposes. It avoids the “unphysical” results that can occur in the Rayleigh and Fraunhofer approximations for pores sizes beyond their validity and is much simpler than the Rayleigh-Gans-Debye approximation. Tables listed in the Appendix to this paper allow the reader reconstruct the exact Mie solution results from the easily calculated van de Hulst approximation, if required.

Reproducing the morphology-dependent resonances of spheres with the discrete dipole approximation.

During the development and application of a scattering algorithm, its accuracy is normally validated by comparing with results of spherical particles given by the exact Mie theory. Being the simplest shape, sphere supports morphology-dependent resonances (MDRs), which cause sharp variations of the scattering properties in narrow size ranges. We show that MDRs may mislead the validation of any volume- or surface-discretization methods, including the discrete dipole approximation (DDA) and, thus, should be explicitly avoided. However, the brute-force DDA simulations can actually capture the narrow peaks in the extinction efficiency over the size parameter, but only if a dipole size parameter is smaller than twice the MDR width. That is much more computationally intensive than typical DDA simulations. We find that a single Lorentzian MDR peak may be split into two due to the symmetry breaking by the DDA discretization. Furthermore, instead of time-consuming high-resolution DDA simulations for reproducing MDR, we developed and validated a significantly more computationally efficient method. It is based, first, on fitting simulated data with one or two Lorentzian peaks combined with a cubic baseline. Second, we use Richardson extrapolation of peak parameters to zero dipole size, exploiting the smooth convergence of these parameters towards the reference Mie values. When applied to two MDRs with relative widths 2 × 10-3 and 9 × 10-4, the developed workflow, powered by intensive simulations, reproduces the peak positions with unprecedented accuracy - errors less than 0.07% and 0.4% of their widths, respectively. This extends the way for studying the evolution of the MDR under non-axisymmetric deformations of a sphere or a spheroid.

Extended Multiplicative Signal Correction for Infrared Microspectroscopy of Heterogeneous Samples with Cylindrical Domains.

Optical scattering corrections are invoked to computationally distinguish between scattering and absorption contributions to recorded data in infrared (IR) microscopy, with a goal to obtain an absorption spectrum that is relatively free of the effects of sample morphology. Here, we present a modification of the extended multiplicative signal correction (EMSC) approach that allows for spectral recovery from fibers and cylindrical domains in heterogeneous samples. The developed theoretical approach is based on exact Mie theory for infinite cylinders. Although rigorous Mie theory implies utilization of comprehensive and time-consuming calculations, we propose to change the workflow of the original EMSC algorithm to minimize extensive calculations for each recorded spectrum at each iteration step. This makes the modified EMSC approach practical for routine use. First, we tested our approach using synthetic data derived from a rigorous model of scattering from cylinders in an IR microscope. Second, we applied the approach to Fourier transform IR (FT-IR) microspectroscopy data recorded from filamentous fungal and cellulose samples with pronounced fiber-like shapes. While the corrected spectra show greatly reduced baseline offsets and consistency, strongly absorbing regions of the spectrum require further refinement. The modified EMSC algorithm broadly mitigates the effects of scattering, offering a practical approach to more consistent and accurate spectra from cylindrical objects or heterogeneous samples with cylindrical domains.

Single scattering albedo of homogeneous, spherical particles in the transition regime

Aerosol single scattering albedo (SSA) is the most important intensive particle parameter controlling aerosol direct radiative forcing. For homogeneous, spherical particles and a complex refractive index independent of wavelength, the SSA is solely dependent on size parameter (ratio of particle circumference and wavelength) and complex refractive index of the particle and can be accurately calculated with Mie theory. Here, we explore this dependency for particles of intermediate size in the transition or peak regime between the Rayleigh scattering and geometric optics regimes. We show that in the transition regime, low-frequency oscillations (the interference structure) of SSA as function of size parameter occur for relatively small imaginary parts of the refractive index and are caused by interference between transmitted and diffracted components of the scattered light. Anomalous diffraction theory (ADT) semi-quantitatively describes this behavior of the SSA as function of size parameter and complex refractive index. While ADT accurately gives the size parameters of the interference peaks, ADT amplitudes only approximate exact Mie results. A significant improvement in agreement with Mie theory can be obtained with modified ADT (MADT) that adds a parameterization for the physics neglected in ADT, namely internal reflection/refraction, photon tunneling, and edge diffraction.

Assessing the uncertainties of the discrete dipole approximation in case of melting ice particles

We studied the applicability of the Discrete Dipole Approximation (DDA) for scattering by partially melted snowflakes in the microwave region. The DDA accuracy in the case of a liquid water coated ice sphere was tested at various particle sizes, water layer thicknesses, frequencies and DDA grid resolutions. We found that the backscattering and absorption cross section show the largest biases with respect to the Mie reference. The highest discrepancies were found for the thinnest water coating and the lowest frequency. We applied a previously published method to separately analyze the errors due to the representation of particle shape and discretization. The accuracy of the DDA seems to decrease particularly when only a single dipole is used to represent some structural element of the target with large refractive index. A similar behavior was also found by testing the variation of the scattering properties of complex shaped melting snowflakes at different grid resolutions. From both experiments we conclude that single isolated water dipoles should be avoided when modeling the scattering properties of melting snowflakes in the microwave using DDA. Although we found that a stability criterion which is commonly used for pure ice particles is not sufficient for melting particles, the DDA results in general converge to the exact Mie solution when shape and discretization errors are reduced by using a refined grid.

Alternate analytic formulation of optical force on a dielectric sphere in the ray optics limit

In the past, the optical force on a micrometer-sized dielectric sphere in a single-beam gradient laser trap was formulated by considering 2D distribution of rays for a plane-wave excitation. However, laser beams usually have a Gaussian transverse intensity profile, which, upon tight focusing, leads to a 3D optical trap. Here, we systematically formulate a generalized ray/geometric optics formalism for estimating force (and potential) for both flat-top and Gaussian laser beams using 2D distribution of light rays as well as 3D distribution of light cones. We also compare our method with the exact Mie theory. In addition, we present a detailed discussion on the nature of force (and potential) considering the optical Kerr effect under high-repetition-rate ultrafast pulsed excitation.

FORCE AND HIDDEN MOMENTUM FOR CLASSICAL MICROSCOPIC DIPOLES

The concept of hidden momentum is reviewed and the first rigorous derivation from Maxwell's equations is provided for the electromagnetic force on electrically small perfect electric conductors of arbitrary shape in bandlimited but otherwise arbitrarily time-varying fields. It is proven for the Amperian magnetic dipoles of these perfect conductors that a electromagnetic force exists that makes the force on these time varying Amperian magnetic dipoles equal to the force on magnetic-charge magnetic dipoles with the same time varying magnetic dipole moment in the same time varying externally applied fields. The exact Mie solution to the perfectly conducting sphere under plane-wave illumination is used to prove that the expressions for the total and hidden-momentum forces on the arbitrarily shaped electrically small perfect conductors correctly predict the forces on perfectly conducting spheres. Remarkably, it is found that the quadrupolar fields at the surface of the sphere are required to obtain the correct total force on the sphere even though the quadrupolar moments are negligible compared to the dipole moments as the electrical size of the sphere approaches zero.

Optical absorption of carbon-gold core-shell nanoparticles

In order to enhance the solar thermal energy conversion efficiency, we propose to use carbon-gold core-shell nanoparticles dispersed in liquid water. This work demonstrates theoretically that an absorbing carbon (C) core enclosed in a plasmonic gold (Au) nanoshell can enhance the absorption peak while broadening the absorption band; giving rise to a much higher solar absorption than most previously studied core-shell combinations. The exact Mie solution is used to evaluate the absorption efficiency factor of spherical nanoparticles in the wavelength region from 300 nm to 1100 nm as well as the electric field and power dissipation profiles inside the nanoparticles at specified wavelengths (mostly at the localized surface plasmon resonance wavelength). The field enhancement by the localized plasmons at the gold surfaces boosts the absorption of the carbon particle, resulting in a redshift of the absorption peak with increased peak height and bandwidth. In addition to spherical nanoparticles, we use the finite-difference time-domain method to calculate the absorption of cubic core-shell nanoparticles. Even stronger enhancement can be achieved with cubic C-Au core-shell structures due to the localized plasmonic resonances at the sharp edges of the Au shell. The solar absorption efficiency factor can exceed 1.5 in the spherical case and reach 2.3 in the cubic case with a shell thickness of 10 nm. Such broadband absorption enhancement is in great demand for solar thermal applications including steam generation.