Design of highly focused fields that remain unpolarized on axis

Abstract

Research on the properties of highly focused fields mainly involved fully polarized light, whereas partially polarized waves received less attention. The aim of this Letter is to provide an appropriate framework, for designing some features of the focused field, when dealing with incoming partially polarized beams. In particular, in this Letter, we describe how to get an unpolarized field on the axis of a high numerical aperture objective lens. The study of field distribution, in the focal region of a high numerical aperture (NA) focusing system, is attracting increasing attention because of their possible applications in many fields, e.g., electron acceleration, nonlinear optics, or particle trapping and manipulation. In general, these techniques require three-dimensional (3D) focused electromagnetic fields, with special characteristics: shape, polarization coherence, and so on. Obtaining these specific features involves the suitable design of the input field [1][2][3][4][5][6]. Research on the polarization properties of highly focused fields has been mainly devoted on fully polarized light, whereas partially polarized waves have received less attention [7][8][9]. Moreover, the use of partially coherent fields has been recently proposed as a suitable light source for optical trapping systems [10]; this kind of fields are also useful in tomography [11], plasmonics spectroscopy [12], or invisibility cloaking [13]. Accordingly, the aim of this Letter is to provide an appropriate framework for designing some features of the focused field, when dealing with incoming partially polarized beams. First, we relate the circular components of the transverse incident field with the circular and longitudinal content of the focused field; then, this formalism is extended to quasi-monochromatic statistically stationary incident beams. Finally, we focus on getting an unpolarized field on the axis of an imaging system with a high NA. Moreover, the polarization over the focal plane is also analyzed.The electric field distribution, at any point in the focal region of a high NA focusing system, is given by the well- where A is a constant, related to the focal length and the wavelength; k is the wave number, r and ϕ denote in this Letter the polar coordinates at the focal plane; and angles θ and θ 0 are represented in Fig. 1. Pθ denotes the so-called apodization function. obtained from energy conservation and geometric considerations; and E 0 is the so-called vector angular spectrum; this angular spectrum is usually written aswhere f 1 and f 2 are, respectively, the azimuthal and radial components of incident field (which we assume transversal). The unitary vectors e 1 and e 2 are given by e 1 ϕ − sin ϕ; cos ϕ; 0;e 2 θ; ϕ cos θ cos ϕ; cos θ sin ϕ; sin θ:Frequently, it is useful to write the angular spectrum using a convenient change of basis [15][16][17]. Instead of using the conventional radial and azimuthal description, it is advisable to describe the angular spectrum in terms of the circular content of the beam (see below). Accor...

Experimental implementation of tightly focused beams with unpolarized transversal component at any plane

“…The use of this circular basis is particularly useful in the analysis of the angular momentum and when dealing with vortex beams [34][35][36]. Moreover, this basis is very suitable for designing the polarization characteristics of the transverse component of a highly focused beam [18,32]. We rewrite the field at the vicinity of the focus plane E as follows:…”

“…This method relates the circular content of the incident beam with the circular and longitudinal components of the focused field [32]. Using this formalism, in this paper we derive analytical expressions that link the coherence-polarization properties of a quasi-monochromatic statistically stationary incident beam and the transversal part of a focused field.…”

Effect of linear polarizers on highly focused spirally polarized fields

“…Nowadays, highly focused beams are present in a variety of scenarios and multiples studies and potential applications have been proposed in the literature [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For this reason, we have recently published a study on how linear polarizers modify a tightly focused field when these devices are placed at the focal plane of a high numerical aperture objective lens [18].…”

On the behavior of linear polarizers on highly focused radially polarized beams

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] In this context, an important issue is to develop the theoretical framework to determine the behavior of a polarizer on a focused beam. [15][16][17] We have recently published a study on how linear polarizers modify a tightly focused field.…”