Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres

Mie scattering with 3D angular spectrum method

Mie theory is a powerful method to model electromagnetic scattering from a multilayered sphere. Usually, the incident beam is expanded to its vector spherical harmonic representation defined by beam shape coefficients, and the multilayer sphere scattering is obtained by the T-matrix method. However, obtaining the beam shape coefficients for arbitrarily shaped incident beams has limitations on source locations and requires different methods when the incident beam is defined inside or outside the computational domain or at the scatterer surface. We propose a 3D angular spectrum method for defining beam shape coefficients from arbitrary source field distributions. This method enables the placement of the sources freely within the computational domain without singularities, allowing flexibility in beam design. We demonstrate incident field synthesis and spherical scattering by comparing morphology-dependent resonances to known values, achieving excellent matching and high accuracy. The proposed method has significant benefits for optical systems and inverse beam design. It allows for the analysis of electromagnetic forward/backward propagation between optical elements and spherical targets using a single method. It is also valuable for optical force beam design and analysis.

Progressive algorithm for the scattering of electromagnetic waves by a multilayered eccentric sphere

This paper presents a general progressive algorithm for the computational study of electromagnetic wave scattering by a multilayered eccentric nanoparticle. The presented methodology is based on a combination of the vector addition theorem for spherical wave functions and an efficient progressive algorithm that matches the boundary conditions of every two adjacent shell layers from the outmost to the innermost layer. As a result, only a solution of small-sized matrices is required rather than solving a large set of system equations as reported in other works. With the developed approach, explicit expressions of the Mie scattering coefficients of the eccentric particle can be obtained. Moreover, the Mie coefficients of a specific inner layer could be calculated selectively, instead of having to compute those of all layers of the entire particle as required by other algorithms. The presented methodology can be used to study practically any type of spherical particle inclusions and the most widely studied cases such as scattering by solid particles, concentric particles, and inclusions with centers displaced along a straight line are just special cases of the algorithm presented. Computed results are also presented, illustrating that the eccentric structure allows extra freedom in the design of multilayered nanoparticles for optical applications.

Efficient progressive algorithm for light scattering of a multilayered concentric nanoparticle

An efficient progressive methodology is presented for the computation of multi-scattering of electromagnetic waves by a multilayered concentric nanoparticle. Instead of solving a large set of system equations as reported in other works, the proposed approach utilizes a progressive algorithm which considers two adjacent shell layers at a time, marching progressively from the innermost to the outmost layer, and requires only multiplication of 4×4 matrices. The progressive algorithm yields the analytical expression for the scattering parameter of the concentric particle. Moreover, the progressive algorithm allows the scattering coefficients of a specific internal layer to be computed selectively, rather than having to calculate those of all layers of the entire particle as required by other algorithms. We show that the presented progressive method has equivalent accuracy to the well-known recursive algorithm, but it is more attractive due to its lower complexity in implementation. It is shown that light scattering of both a single solid sphere and two-layered concentric shell are special cases of the proposed methodology. Case study demonstrates that the presented methodology is useful in assisting the design of a multilayered core/shell structure with maximum forward scattering feature, indicating it is applicable to the exploration of optical phenomena of nanoparticles with numerous layers. Moreover, the present progressive algorithm is further extended to the electromagnetic scattering by an eccentric multilayered particle with inner cores displaced along a line defined by the centers of the spheres, which provides extra freedoms for the design of optical core shell spherical particles.

Light propagation in layered media in a total reflection geometry: a transfer matrix method using virtually linear basis functions to handle critical conditions

For light propagation in a layered refractive index profile, critical conditions occur when the wave vector perpendicular to the layering becomes zero. Such conditions can occur in a total reflection geometry. Conventional transfer matrix methods become singular, and geometrical optics concepts break down at critical conditions. We introduce two new, to the best of our knowledge, basis systems using virtual linear functions that obey the Helmholtz equation and turn to linear functions required for critical conditions.

Approximated Backscattered Wave Models of a Lossy Concentric Dielectric Sphere for Fruit Characterization

Approximated models of electromagnetic waves scattered from a sphere with two different dielectric layers were developed and reported in this paper. We proposed that the dielectric properties of a concentric dielectric sphere object, for example, some types of fruit, could be estimated by this model, from some wave components of the backscattered wave. The models were suitable for lossy objects because only a single bounce of the wave was assumed. In terms of first bounce as well as total backscattered wave results, the reported values agreed well with the values calculated by a commercial software. The measurement results verified the calculated wave components. The dielectric properties determination of real fruits was performed and exhibited the potential in fruit characterization. The main advantage of these models is that they can provide the magnitude and phase information of each backscattered wave component, which makes quality monitoring of fruits to be possible.

Theoretical prediction of photophoretic force on a dielectric sphere illuminated by a circularly symmetric high-order Bessel beam: on-axis case

Compared to the experimental progresses made in the optical trapping of aerosol particles in gaseous media by means of photophoretic forces, the theoretical analysis of photophoretic forces is less developed, the underlying mechanisms being yet not fully understood. In this paper, theoretical derivations of photopheresis of a dielectric sphere in gaseous media illuminated by a circularly symmetric Bessel beam of arbitrary order is presented within the framework of generalized Lorenz-Mie theory. An analytic and closed-form formula for the asymmetry factor, which ultimately determines the sense of direction of photophoretic force, is provided. The influences of particle size, absorptivity of the particle, half-cone angle, beam order of the Bessel beam on the asymmetry factor are explored in detail. The method proposed in this paper can be applied to a wider class of axisymmetric beams carrying nonzero topological charges.

Fluorescence Emission Triggered by Radioactive β decay in Optimized Hyperbolic Cavities

Luminescence arising from β -decay of radiotracers has garnered much interest recently as a viable in-vivo imaging technique. The emitted Cerenkov radiation can be directly detected by high sensitivity cameras or used to excite highly efficient fluorescent dyes. Here, we investigate the enhancement of visible and infrared emission driven by β -decay of radioisotopes in the presence of a hyperbolic nanocavity. By means of a transfer matrix approach, we obtain quasi-analytic expressions for the fluorescence enhancement factor at the dielectric core of the metalodielectric cavity, reporting a hundred-fold amplification in periodic structures. A particle swarm optimization of the layered shell geometry reveals that up to a ten-thousand-fold enhancement is possible thanks to the hybridization and spectral overlapping of whispering-gallery and localized-plasmon modes. Our findings may find application in nuclear-optical medical imaging, as they provide a strategy for the exploitation of highly energetic gamma rays, Cerenkov luminescence, and visible and near-infrared fluorescence through the same nanotracer.

Characteristics of photonic jets generated by a dielectric sphere illuminated by a Gaussian beam

Photonic jets (PJs) formed on the shadow side of micro-sized dielectric spheres excited by focused Gaussian beams are investigated within the framework of the generalized Lorenz-Mie theory (GLMT). The intrinsic advantages of rapidity and high accuracy of the GLMT in calculations enable us to conduct a systematic study of PJs at a low cost and a high reliability. To reveal the influence of beam parameters on the properties of PJs, numerical results concerning variations of key parameters of PJs, including the maximal intensity, the focal distance, which is linked to the position of maximal intensity, and longitudinal and transversal dimensions are presented with the change of the beam waist radius and the focal center location of the Gaussian beam. The results show that as the beam waist radius approaches the radius of the particle, the energy stream of the Gaussian beam contributes more efficiently to the formation of PJs. By properly tuning the location of the beam focal center, the PJ pattern can be efficiently engineered to a large extent.

Electromagnetic energy in multilayered spherical particles

We obtain exact analytic expressions for (i) the electromagnetic energy radial density within and outside a multilayered sphere and (ii) the total electromagnetic energy stored within its core and each of its shells. Explicit expressions for the special cases of lossless core and shell are also provided. The general solution is based on the compact recursive transfer-matrix method, and its validity includes also magnetic media. The theory is illustrated on examples of electric field enhancement within various metallo-dielectric silica-gold multilayered spheres. The user-friendly MATLAB code, which includes the theoretical treatment, is available as a supplement to the paper.

Iterative design of multilayered dielectric microspheres with tunable transparency windows

Suspensions of microparticles dispersed in air or liquids are useful for designing media with desirable optical extinction properties within the visible or infrared spectrum. We describe here a numerical iterative optimization algorithm used to design multilayered concentric dielectric spheres with prescribed optical scattering properties. Our method integrates a computationally efficient rigorous electromagnetic solver, based on Mie theory, within an optimization loop to determine specific particle configurations that best meet a desired optical response. In particular, we show that this method can be used to design all-dielectric spherical particles that possess narrow tunable transparency windows while removing any angular dependency on the optical response.

Analytical method for analysis of electromagnetic scattering from inhomogeneous spherical structures using duality principles

In this article, an analytical approach is presented for the analysis of electromagnetic (EM) scattering from radially inhomogeneous spherical structures (RISSs) based on the duality principle. According to the spherical symmetry, similar angular dependencies in all the regions are considered using spherical harmonics. To extract the radial dependency, the system of differential equations of wave propagation toward the inhomogeneity direction is equated with the dual planar ones. A general duality between electromagnetic fields and parameters and scattering parameters of the two structures is introduced. The validity of the proposed approach is verified through a comprehensive example. The presented approach substitutes a complicated problem in spherical coordinate to an easy, well posed, and previously solved problem in planar geometry. This approach is valid for all continuously varying inhomogeneity profiles. One of the major advantages of the proposed method is the capability of studying two general and applicable types of RISSs. As an interesting application, a class of lens antenna based on the physical concept of the gradient refractive index material is introduced. The approach is used to analyze the EM scattering from the structure and validate strong performance of the lens.

Scattering of aerosol particles by a Hermite-Gaussian beam in marine atmosphere

Based on the complex-source-point method and the generalized Lorenz-Mie theory, the scattering properties and polarization of aerosol particles by a Hermite-Gaussian (HG) beam in marine atmosphere is investigated. The influences of beam mode, beam width, and humidity on the scattered field are analyzed numerically. Results indicate that when the number of HG beam modes u (v) increase, the radar cross section of aerosol particles alternating appears at maximum and minimum values in the forward and backward scattering, respectively, because of the special petal-shaped distribution of the HG beam. The forward and backward scattering of aerosol particles decreases with the increase in beam waist. When beam waist is less than the radius of the aerosol particle, a minimum value is observed in the forward direction. The scattering properties of aerosol particles by the HG beam are more sensitive to the change in relative humidity compared with those by the plane wave and the Gaussian beam (GB). The HG beam shows superiority over the plane wave and the GB in detecting changes in the relative humidity of marine atmosphere aerosol. The effects of relative humidity on the polarization of the HG beam have been numerically analyzed in detail.

BIM-Sim: Interactive Simulation of Broadband Imaging Using Mie Theory

Understanding the structure of a scattered electromagnetic (EM) field is critical to improving the imaging process. Mechanisms such as diffraction, scattering, and interference affect an image, limiting the resolution, and potentially introducing artifacts. Simulation and visualization of scattered fields thus plays an important role in imaging science. However, EM fields are high-dimensional, making them time-consuming to simulate, and difficult to visualize. In this paper, we present a framework for interactively computing and visualizing EM fields scattered by micro and nano-particles. Our software uses graphics hardware for evaluating the field both inside and outside of these particles. We then use Monte-Carlo sampling to reconstruct and visualize the three-dimensional structure of the field, spectral profiles at individual points, the structure of the field at the surface of the particle, and the resulting image produced by an optical system.

Analysis of rainbow scattering by a chiral sphere

Based on the scattering theory of a chiral sphere, rainbow phenomenon of a chiral sphere is numerically analyzed in this paper. For chiral spheres illuminated by a linearly polarized wave, there are three first-order rainbows, with whose rainbow angles varying with the chirality parameter. The spectrum of each rainbow structure is presented and the ripple frequencies are found associated with the size and refractive indices of the chiral sphere. Only two rainbow structures remain when the chiral sphere is illuminated by a circularly polarized plane wave. Finally, the rainbows of chiral spheres with slight chirality parameters are found appearing alternately in E-plane and H-plane with the variation of the chirality.

List of problems for future research in generalized Lorenz-Mie theories and related topics, review and prospectus [invited]

The expression "generalized Lorenz-Mie theories" generically denotes a class of light-scattering theories describing the interaction between an illuminating electromagnetic arbitrary-shaped beam and a particle possessing a high degree of symmetry. This allows one to use the method of separation of variables in which the illuminating beam is expressed as an expansion over a set of basis functions. Such theories have been derived and applied over the past 35 years. Although, as a whole, these theories are now well developed, there remains a list of problems to be solved, some of which are described in this paper.

Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam

Using the theory of electromagnetic scattering of a uniaxial anisotropic sphere, we derive the analytical expressions of the radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam. The beam's propagation direction is parallel to the primary optical axis of the anisotropic sphere. The effects of the permittivity tensor elements ε(t) and ε(z) on the axial radiation forces are numerically analyzed in detail. The two transverse components of radiation forces exerted on a uniaxial anisotropic sphere, which is distinct from that exerted on an isotropic sphere due to the two eigen waves in the uniaxial anisotropic sphere, are numerically studied as well. The characteristics of the axial and transverse radiation forces are discussed for different radii of the sphere, beam waist width, and distances from the sphere center to the beam center of an off-axis Gaussian beam. The theoretical predictions of radiation forces exerted on a uniaxial anisotropic sphere are hoped to provide effective ways to achieve the improvement of optical tweezers as well as the capture, suspension, and high-precision delivery of anisotropic particles.

MieLab: A Software Tool to Perform Calculations on the Scattering of Electromagnetic Waves by Multilayered Spheres

In this paper, we present MieLab, a free computational package for simulating the scattering of electromagnetic radiation by multilayered spheres or an ensemble of particles with normal size distribution. It has been designed as a virtual laboratory, including a friendly graphical user interface (GUI), an optimization algorithm (to fit the simulations to experimental results) and scripting capabilities. The paper is structured in five different sections: the introduction is a perspective on the importance of the software for the study of scattering of light scattering. In the second section, various approaches used for modeling the scattering of electromagnetic radiation by small particles are discussed. The third and fourth sections are devoted to provide an overview of MieLab and to describe the main features of its architectural model and functional behavior, respectively. Finally, several examples are provided to illustrate the main characteristics of the software.

Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam

An analytical solution to the scattering of an off-axis Gaussian beam obliquely incident on a uniaxial anisotropic sphere is obtained in the particle-centered system. Based on the local approximation to the off-axis beam shape coefficients and the coordinate rotation theory, the off-axis obliquely incident Gaussian beam is expanded with the spherical vector wave functions in the primary coordinate of the uniaxial anisotropic sphere. The internal fields of the uniaxial anisotropic sphere are proposed in the integrating form of the spherical vector wave functions by introducing the Fourier transform. By matching the fields on the boundary and solving matrix equations, the expansion coefficients are analytically derived. The influences of the beam waist center positioning and the obliquely incident angles, as well as the permittivity tensors on the far scattered field distributions, are numerically presented. The correctness of the theory is verified by comparing our numerical results in special cases with results from the references and with calculations by other algorithms.

Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere

Scattering of an on-axis Gaussian beam by a uniaxial anisotropic sphere is studied. The incident on-axis Gaussian beam is expanded in terms of spherical vector wave functions, and the beam shape coefficients are obtained by applying the local approximation. The internal fields of a uniaxial anisotropic sphere are proposed in the integrating form of the spherical vector wave functions by introducing the Fourier transform. Utilizing the continuous tangential boundary conditions, both the scattered and the internal field coefficients are derived analytically. Numerical calculations are presented. The effects of the beam width, beam waist center positioning, and anisotropy on scattering properties are analyzed. The internal and near-surface field distributions are also discussed, and the two eigenmodes are characterized. The continuity on the surface of a uniaxial anisotropic sphere is well confirmed.

Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence

The Debye series expansion expresses the Mie scattering coefficients into a series of Fresnel coefficients and gives physical interpretation of different scattering modes, but when an infinite multilayered cylinder is obliquely illuminated by electromagnetic plane waves, the scattering process becomes very complicated because of cross polarization. Based on the relation of boundary conditions between global scattering process and local scattering processes, the generalized Debye series expansion of plane wave scattering by an infinite multilayered cylinder at oblique incidence is derived in this paper. The formula and the code are verified by the comparison of the results with that of Lorenz-Mie theory in special cases and those presented in the literatures.

Scattering of an electromagnetic plane wave by a Luneburg lens. III. Finely stratified sphere model

The parallel iteration procedure for computing scattering by a multilayer sphere is described. The procedure uses a successive doubling strategy applied to four sets of multiple-scattering amplitudes, which is reminiscent of the fast Fourier transform (FFT) algorithm. The procedure is then used to calculate scattering of a plane wave by a modified Luneburg lens. The evolution of the transmission rainbow for the Luneburg lens parameter f>1 into an orbiting ray for f=1 and into a series of morphology-dependent resonances for f<1 is studied, and various features of the scattered intensity as a function of scattering angle are commented on. It is found that some resonances are formed without the presence of an exterior centrifugal barrier to confine them.

The expansion coefficients of arbitrary shaped beam in oblique illumination

In generalized Lorenz-Mie theories, (GLMTs), the most difficult task concerns the description of the illuminating beam. We provide an approach for expansions of the incident arbitrary shaped beam in spherical and spheroidal coordinates in the general case of oblique illumination. The representations for shaped beam coefficients are derived by using addition theorem for spherical vector wave functions under coordinate rotations. For the special cases of the particle center located on shaped beam axis and plane wave, the simplified expressions are given. The convergence of the beam shape coefficients is discussed.

Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder

We derive the formula of the Debye-series decomposition for normally incident plane-wave scattering by an infinite multilayered cylinder. A comparison of the scattering diagrams calculated by the Debye series and Mie theory for a graded-index polymer optical fiber is given and the agreement is found to be satisfied. This approach permits us to simulate the rainbow intensity distribution of any single order and the interference of several orders, which is of good use to the study of the scattering characteristics of an inhomogeneous cylinder and to the measurement of the refractive index profile of an inhomogeneous cylinder.

Debye series for light scattering by a multilayered sphere

We have derived the formula for the Debye-series decomposition for light scattering by a multilayered sphere. This formulism permits the mechanism of light scattering to be studied. An efficient algorithm is introduced that permits stable calculation for a large sphere with many layers. The formation of triple first-order rainbows by a three-layered sphere and single-order rainbows and the interference of different-order rainbows by a sphere with a gradient refractive index, are then studied by use of the Debye model and Mie calculation. The possibility of taking only one single mode or several modes for each layer is shown to be useful in the study of the scattering characteristics of a multilayered sphere and in the measurement of the sizes and refractive indices of particles.

Debye series analysis of scattering of a plane wave by a spherical Bragg grating

The Debye series decomposition of the partial-wave scattering amplitudes of a multilayer sphere is derived. The partial-wave transmission and reflection terms appearing in the Debye series are multiple-scattering amplitudes written in terms of four basic quantities and combined together layer by layer in an identical way. The resulting expressions are then used to calculate the scattered intensity of a spherical Bragg grating covering a dielectric core particle and to analyze a number of new structures appearing in the scattered intensity.

Electromagnetic computational method for the scattering of an axisymmetric laser beam by an inhomogeneous body of revolution

An original computational method for solving the two-dimensional problem of the scattering of an axisymmetric laser beam by an arbitrary-shaped inhomogeneous body of revolution is presented. This method relies on a domain decomposition of the scattering zone into concentric spherical radially homogeneous subdomains and on an expansion of the angular dependence of the fields on the Legendre functions. Numerical results for the fields obtained for various scatterer geometries are presented and analyzed.

Dimensionless parameters for the design of optical traps and laser guidance systems

Optical traps are routinely used for the manipulation of neutral particles. However, optical trap design is limited by the lack of an accurate theory. The generalized Lorenz-Mie theory (GLMT) solves the scattering problem for arbitrary particle size and predicts radial forces accurately. Here we show that the GLMT predicts the observed radial and axial forces in a variety of optical manipulators. We also present a dimensionless parameter beta for the prediction of axial forces. Coupled with our correlation for radial escape forces, we now have a set of two simple correlations for the practical design of radiation-force-based systems.

Mie-scattering calculation

The new Mie-scattering calculation is a robust and efficient algorithm used to compute light scattering from spheres. It calculates the ratio between Riccati-Bessel functions instead of the complicated logarithmic derivative. The Kapteyn inequality is used to estimate the number of significant digits of the calculated Riccati-Bessel functions and their ratio. This new algorithm is stable and accurate for both large and small particles. The implemented C++ code yields the same accurate results for both small and large particles compared with Wiscombe's MIEV0 code in double precision. Suggestions are provided for the porting of the MIEV0 code.

Improved recursive algorithm for light scattering by a multilayered sphere

An improved recurrence algorithm to calculate the scattering field of a multilayered sphere is developed. The internal and external electromagnetic fields are expressed as a superposition of inward and outward waves. The alternative yet equivalent expansions of fields are proposed by use of the first kind of Bessel function and the first kind of Hankel function instead of the first and the second kinds of Bessel function. The final recursive expressions are similar in form to those of Mie theory for a homogeneous sphere and are proved to be more concise and convenient than earlier forms. The new algorithm avoids the numerical difficulties, which give rise to significant errors encountered in practice by previous methods, especially for large, highly absorbing thin shells. Various calculations and tests show that this algorithm is efficient, numerically stable, and accurate for a large range of size parameters andrefractive indices.

Four-flux model for a multilayer, plane absorbing and scattering medium: application to the optical degradation of white paint in a space environment

We generalize the four-flux radiative transfer model to the case of a multilayer medium. A concrete application, that of the study of the optical degradation of white paint in a simulated space environment, is presented. The degraded material is decomposed in a damaged layer and in an unaffected layer, and we assume that the degradation is due to a variation Dkappa of the imaginary part of the refractive index in the damaged layer. Then we find an empirical law for variation Dkappa with dose, taking into account possible saturation.

Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference

We experimentally examine the primary rainbow created by the illumination of a coated cylinder. We present a simple technique for varying the coating thickness over a wide range of values, and we see evidence for two different scattering regimes. In one, where the coating thickness is large, twin rainbows are produced. In the second, where the coating is thin enough to act as a thin film, a single rainbow is produced whose intensity varies periodically as the coating thickness varies. We find good agreement with previous theoretical predictions.