Multipole Born series approach to light scattering by Mie-resonant nanoparticle structures

Exciting optical effects such as polarization control, imaging, and holography were demonstrated at the nanoscale using the complex and irregular structures of nanoparticles with the multipole Mie-resonances in the optical range. The optical response of such particles can be simulated either by full wave numerical simulations or by the widely used analytical coupled multipole method (CMM), however, an analytical solution in the framework of CMM can be obtained only in a limited number of cases. In this paper, a modification of the CMM in the framework of the Born series and its applicability for simulation of light scattering by finite nanosphere structures, maintaining both dipole and quadrupole resonances, are investigated. The Born approximation simplifies an analytical consideration of various systems and helps shed light on physical processes ongoing in that systems. Using Mie theory and Green’s functions approach, we analytically formulate the rigorous coupled dipole-quadrupole equations and their solution in the different-order Born approximations. We analyze in detail the resonant scattering by dielectric nanosphere structures such as dimer and ring to obtain the convergence conditions of the Born series and investigate how the physical characteristics such as absorption in particles, type of multipole resonance, and geometry of ensemble influence the convergence of Born series and its accuracy.

Application of Born series for modeling of Mie-resonant nanostructures

Born series formalism is a widely-used approach to solve a scattering problem in quantum mechanics and optics, including a problem of electromagnetic scattering on the ensembles of Mie-resonant nanoparticles. In the latter case, the Born series formalism can be used when the electromagnetic coupling between nanoparticles is weak. This can be violated near the multipole Mie-resonance of the nanoparticle. In this work, we analyze the applicability of the Born series approach for modeling the resonant optical response of Mie-nanoparticle ensembles and formulate quantitative criteria of Born series convergence and, subsequently, the applicability of this approach.

Symmetries and Selection Rules of the Spectra of Photoelectrons and High-Order Harmonics Generated by Field-Driven Atoms and Molecules

Using the strong-field approximation we systematically investigate the selection rules for high-order harmonic generation and the symmetry properties of the angle-resolved photoelectron spectra for various atomic and molecular targets exposed to one-component and two-component laser fields. These include bicircular fields and orthogonally polarized two-color fields. The selection rules are derived directly from the dynamical symmetries of the driving field. Alternatively, we demonstrate that they can be obtained using the conservation of the projection of the total angular momentum on the quantization axis. We discuss how the harmonic spectra of atomic targets depend on the type of the ground state or, for molecular targets, on the pertinent molecular orbital. In addition, we briefly discuss some properties of the high-order harmonic spectra generated by a few-cycle laser field. The symmetry properties of the angle-resolved photoelectron momentum distribution are also determined by the dynamical symmetry of the driving field. We consider the first two terms in a Born series expansion of the T matrix, which describe the direct and the rescattered electrons. Dynamical symmetries involving time translation generate rotational symmetries obeyed by both terms. However, those that involve time reversal generate reflection symmetries that are only observed by the direct electrons. Finally, we explain how the symmetry properties, imposed by the dynamical symmetry of the driving field, are altered for molecular targets.

Algorithms for Non-Relativistic Quantum Integral Equation

In low energy scattering in Non-Relativistic Quantum Mechanics, the Schödinger equation in integral form is used. In quantum scattering theory the wave self-function is divided into two parts, one for the free wave associated with the particle incident to a scattering center, and the emerging wave that comes out after the particle collides with the scattering center. Assuming that the scattering center contains a position-dependent potential, the usual solution of the integral equation for the scattered wave is obtained via the Born approximation. Assuming that the scattering center contains a position-dependent potential, the usual solution of the integral equation for the scattered wave is obtained via the Born approximation. The methods used here are arbitrary kernels and the Neumann-Born series. The result, with the help of computational codes, shows that both techniques are good compared to the traditional method. The advantage is that they are finite solutions, which does not require Podolsky-type regularization.

Iterative treatment of the Coulomb potential in laser–atom interactions

We discuss a Faddeev-like iterative approach which allows one to consistently include the Coulomb potential in strong field phenomena through a Born series. To assess the validity of this approach, we calculate the probability of excitation to given states of atomic hydrogen exposed to radiation pulses of various frequencies, durations and peak intensities and compare our results with those obtained by solving numerically the time-dependent Schrödinger equation. We obtain excellent agreement for a range of frequencies. As the frequency decreases, many high-order terms have to be included in order to get convergence of the Born series. Our results indicate that this Faddeev-like method is particularly suitable to treat the interaction of atoms with attosecond pulses. For the lowest frequency considered ($$\omega = 0.057$$ ω = 0.057 a.u.), we study in more detail the re-collision-based frustrated tunneling process in atomic hydrogen and compare our results with those existing in the literature. Graphic Abstract

Born approximation and sequence for hyperbolic equations

The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are C ∞ , and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations.