Numerical analysis of tiny-focal-spot generation by focusing linearly, circularly, and radially polarized beams through a micro/nanoparticle.

Obtaining a tiny focal spot is desired for super resolution. We do a vectorial numerical analysis of the linearly, circularly, and radidally polarized electromagnetic fields being focused through a dielectric micro/nanoparticle of size comparable to the wavelength. We find tiny focal spots (up to ∼0.

Generalized far-field integral.

The propagation of light in homogeneous media is a crucial technology in optical modeling and design as it constitutes a part of the vast majority of optical systems. Any improvements in accuracy and speed are therefore helpful. The far-field integral is one of the most widely used tools to calculate diffraction patterns.

Generalized Debye integral.

The Debye integral is an essential technique in physical optics, commonly used to efficiently tackle the problem of focusing light in lens design. However, this approximate method is only valid for systems that are well designed and with high enough Fresnel numbers. Beyond this assumption, the integral formula fails to provide accurate results.

Theory and algorithm of the homeomorphic Fourier transform for optical simulations.

The introduction of the fast Fourier transform (FFT) constituted a crucial step towards a faster and more efficient physio-optics modeling and design, since it is a faster version of the Discrete Fourier transform. However, the numerical effort of the operation explodes in the case of field components presenting strong wavefront phases-very typical occurrences in optics- due to the requirement of the FFT that the wrapped phase be well sampled. In this paper, we propose an approximated algorithm to compute the Fourier transform in such a situation.

Isolating the Gouy phase shift in a full physical-optics solution to the propagation problem.

The Gouy phase shift has remained an object of fascination since its discovery by the eponymous scientist at the end of the nineteenth century. The reason behind this uninterrupted interest resides, at least in part, in the fact that the Gouy effect is to be found in the borderland between geometrical optics and diffractive behavior. Using purely mathematical arguments in a full electromagnetic solution to the propagation problem, it is possible to derive a formula where all the physical effects that we know must appear are laid bare, including the Gouy phase.

Application of the semi-analytical Fourier transform to electromagnetic modeling.

The Fast Fourier Transform (FFT) algorithm makes up the backbone of fast physical optics modeling. Its numerical effort, approximately linear on the sample number of the function to be transformed, already constitutes a huge improvement on the original Discrete Fourier Transform. However, even this orders-of-magnitude improvement in the number of operations required can fall short in optics, where the tendency is to work with field components that present strong wavefront phases: this translates, as per the Nyquist-Shannon sampling theorem, into a huge sample number.