Study on Focusing Mechanism of Radial Polarization with Immersion Objective

Abstract

We describe a procedure for determining the generalised scaling functions f n (g) at all the values of the coupling constant. These functions describe the high spin contribution to the anomalous dimension of large twist operators (in the sl(2) sector) of N = 4 SYM. At fixed n, f n (g) can be obtained by solving a linear integral equation (or, equivalently, a linear system with an infinite number of equations), whose inhomogeneous term only depends on the solutions at smaller n. In other words, the solution can be written in a recursive form and then explicitly worked out in the strong coupling regime. In this regime, we also emphasise the peculiar convergence of different quantities ('masses', related to the f n (g)) to the unique mass gap of the O(6) nonlinear sigma model and analyse the first next-to-leading order corrections.ArXiv ePrint: 0808.1886 Furthermore, the same linear integral equation (2.11) still controls this next-to-leading order (nlo), f (0) (g, j). Now, similarly, we may imagine that the dots should initially be inverse integer powers of ln s, with coefficients, at each power, depending on g and j. Afterwards, inverse integer powers of s should also enter the stage, but they are determined by the complete non-linear integral equation (NLIE) of [18]. 1 However, in this paper we will constrain ourselves to the leading Sudakov factor f (g, j), leaving the analysis of its corrections for future publications.Actually, in [19] we have initiated the study of the strong coupling regime of the first generalised scaling function f 1 (g) and have shown the proportionality of its leading order to the mass gap m(g) (see (3.13) below) of the O(6) nonlinear sigma model (NLSM). This gives a first positive test, in the strong coupling regime j ≪ m(g) of the NLSM, for the Alday-Maldacena proposal [11]. This claims that as long as g ≫ j the quantity f (g, j) + j should coincide with the O(6) NLSM energy density. The latter was expanded and checked for the first orders in the perturbative regime j ≫ m(g) of the NLSM by [11]. Hence, our test was a first indication in another valuable region of the NLSM, i.e. j ≪ m(g), where the free energy series is, besides, convergent [22]. Afterwards, the embedding of the O(6) NLSM into N = 4 SYM at large g was brilliantly shown in a formal way by [20], where the leading strong coupling contribution of f 3 (g) was computed too. In a contemporaneous paper [21], starting from the our linear integral equation [19], we have set down the initial ideas for a systematic study of all the f n (g) and confined our study to the first four f 1 (g), f 2 (g), f 3 (g) and f 4 (g), by finding for them some analytic relations and expressions. These have been then evaluated numerically with additional analytic results for large g, finding agreement with the suitable results from the O(6) NLSM [22]. Furthermore, the agreement on f 4 (g) is highly nontrivial, since it contains the details of the specific interaction in the O(6) NLSM. For completeness sake, all these results will be reported in the f...

Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam

“…The vectorial beam at the back aperture of the objective can be written as the superposition of radial and azimuthal beams E = P ( α , )( γ · e r + δ · e φ ), where P ( α , ) is the pupil apodization function of the apodizer, e r and e φ are the unit vectors in the radial and azimuthal directions, respectively. The focal field of the superposed vectorial beam can be derived by the vectorial diffraction theory as 13 16 where A is a constant, u is the convergence angle, a is the maximum angle of convergence determined by the numerical aperture of the objective, J n ( x ) is the n th order Bessel function of its first kind and φ is the azimuthal angle in the image plane. The ratio of the averaged longitudinal to transverse polarization intensity components within the focal plane defined by the full width at half maximum (FWHM) can be expressed as Although the ratio of the polarization field components is generally used to define the polarization angle at individual points in free space, the ratio of the energy between different polarization components averaged within the focal plane is more relevant when the threshold effect of material response is considered during the polarized light interaction with materials, for example, in the polarization encryption with nanorods.…”

“…A way to break this limit is to bend the wavefront of light through vectorial diffraction by a high numerical aperture (NA) objective lens 12 13 . Although a longitudinal polarization state can be created by focussing a radially polarized light beam 14 15 16 , there has been a key obstacle of generating arbitrary 3D polarization orientation in the focus.…”

Quasi-linearly polarized hybrid modes in tapered and metal-coated tips with circular apertures: understanding the functionality of aperture tips

“…This nature is akin to focusing of a Gaussian beam with a lens. As the Gaussian beam is focused with a high numerical aperture lens, the longitudinal component gets stronger at the focal plane [41][42][43]. However, by guiding a mode through a taper with a metal cladding, we can achieve a ratio (|E z | 2 /|E x | 2 >1) that is not achievable for a Gaussian beam focused by any physical lens (|E z | 2 /|E x | 2 <1 always) [44].…”

Quantification of the NA dependent change of shape in the image formation of a z‐polarized fluorescent molecule using vectorial diffraction simulations

“…The field distribution is, for the case depicted here of a dipole aligned co-linear to the optical axis, purely radially polarized; this distribution is akin to the field distribution of a vector beam. In the image plane, two alternatives are commonly seen: (1) the point emitter is imaged with a dip in the intensity at the Centre (a "donut") as known from single molecule microscopy, or (2) the emitter appears as a tight spot, as used in applications such as direct laser writing with radially polarized beams in photolithography, machining, or optical data storage While several groups have contributed to an increased understanding of image formation in high-NA vector beam focusing (Brown, 2011;Lan & Tien, 2008;Quabis et al, 2000;Sheppard & Choudhury, 2004;Sheppard & Török, 1997;Youngworth & Brown, 2000), none has yet answered a key question: where exactly does the donut turn into a spot? This transition point can be accurately determined by calculating the image of a dipole but will require several steps as discussed below.…”

“…While several groups have contributed to an increased understanding of image formation in high‐NA vector beam focusing (Brown, 2011 ; Lan & Tien, 2008 ; Quabis et al, 2000 ; Sheppard & Choudhury, 2004 ; Sheppard & Török, 1997 ; Youngworth & Brown, 2000 ), none has yet answered a key question: where exactly does the donut turn into a spot? This transition point can be accurately determined by calculating the image of a dipole but will require several steps as discussed below.…”