Lorentz Beams

A new kind of tridimensional scalar optical beams is introduced. These beams are called Lorentz beams because the form of their transverse pattern in the source plane is the product of two independent Lorentz functions. Closed-form expression of free-space propagation under paraxial limit is derived and pseudo non-diffracting features pointed out. Moreover, as the slowly varying part of these fields fulfils the scalar paraxial wave equation, it follows that there exist also Lorentz-Gauss beams, i.e. beams obtained by multipying the original Lorentz beam to a Gaussian apodization function. Although the existence of Lorentz-Gauss beams can be shown by using two different and independent ways obtained recently from Kiselev [Opt. Spectr. 96, 4 (2004)] and Gutierrez-Vega et al. [JOSA A 22, 289-298, (2005)], here we have followed a third different approach, which makes use of Lie's group theory, and which possesses the merit to put into evidence the symmetries present in paraxial Optics.Comment: 11 pages, 1 figure, submitted to Journal of Optics

The angular momentum of vectorial non-paraxial fields and the role of radial charges in orbit-spin coupling

Electromagnetic fields carry a linear and an angular momentum, the first being responsible for the existence of the radiation pressure and the second for the transfer of torque from electromagnetic radiation to matter. The angular momentum is considered to have two components, one due to the polarization state of the field, usually called Spin Angular Momentum (SAM), and one due to existence of topological azimuthal charges in the field phase profile, which leads to the Orbital Angular Momentum (OAM). For non-paraxial fields these two contributions are not independent of each other, something which is described as spin-orbit coupling. It has been recently proved that electromagnetic fields necessarily carry also invariant radial charges that, as discussed in this work, play a key role in the angular momentum. Here we show that the total angular momentum consists in fact of three components: one component only dependent on the spin of the field, another dependent on the azimuthal charges carried by the field and a third component dependent on the spin and the radial charges contained in the field. By properly controlling the number and coupling among these radial charges it is possible to design electromagnetic fields with a desired total angular momentum. In this way it is also possible to discover fields with no orbital angular momentum and a spin angular momentum typical of spin-3/2 objects, irrespective of the fact that photons are spin-1 particles

On the nonparaxial corrections of Bessel–Gauss beams

The nonparaxial corrections for Bessel–Gauss beams were derived recently using two different approaches [Borghi et al., J. Opt. Soc. Am. A 18, 1618 (2001) and Vaveliuk et al., J. Opt. Soc. Am. A 24, 3297 (2007)]. However, the two obtained results do not agree, so it is necessary to determine which method is correct. In the most recent of those papers, Vaveliuk et al. claimed that their method is correct while the method described by Borghi et al. is incorrect. In the present work, just by solving the rigorous propagation problem, we show that exactly the converse is true.Imaging Science and TechnologyApplied Science

Electromagnetic scattering beyond the weak regime: Solving the problem of divergent Born perturbation series by Padé approximants

Electromagnetic scattering is the main phenomenon behind all optical measurement methods where one aims to retrieve the shape or physical properties of an unknown object by measuring how it scatters an incident optical field. Such an inverse problem is often approached by solving, several times, the corresponding direct scattering problem and trying to find the best estimate of the object which is compatible with a set of measurements. In the direct scattering problem, two regimes can be distinguished depending on the size of the object and the permittivity contrast: the weak-scattering regime and the strong-scattering regime. Generally, the presence of the scatterer alters the form of the incident field inside the scatterer. If that effect is neglected in the physical model, then one speaks of the so-called single-scattering regime or, more often, the Born approximation. The regime in which this approximation is valid is the weak-scattering regime. The corresponding inverse problem, that aims to retrieve the object from scattering data, becomes linear in this case. Linearizing the problem simplifies the method to solve it, but also introduces limitations to the maximum spatial resolution achievable in the reconstruction of the object. In the strong-scattering regime, multiple-scattering effects are not neglected and the inverse problem is treated in its full non-linear nature, which makes finding its solution a far more challenging task. Despite the existence of numerical methods, a powerful way to solve those direct problems would be to use a perturbation approach where the field is expressed as a series, known as the Born series. The advantage of a perturbation approach stems from the fact that each term of the series has a clear physical meaning and can unveil much more about the scattering process than a purely numerical approach can offer. Unfortunately, the series solution turns out to be strongly divergent in the strong-scattering regime, making it an unpractical approach for problems under these strong-scattering conditions. Thus, despite the fact that multiple scattering could, in principle, allow resolving sub-wavelength details of the unknown object, this possibility is in practice hampered by the divergent nature of the higher-order terms of the Born series. In this work, we show how to solve this problem by employing Padé approximants and how to treat electromagnetic problems well beyond the weak-scattering regime and provide an accurate evaluation of the scattered field even under strong-scattering conditions. Padé approximants are rational functions that can offer improvements in two ways, namely series acceleration of converging series and analytic continuation of a series outside its region of convergence. In the case of a symmetric approximant of order N, the approximant is calculated from 2N + 1 terms in the Born series, therefore incorporating multiple-scattering effects to which these higher-order corrections in the Born series correspond. We apply the method to two scalar scattering problems: that of a one-dimensional slab and that of an infinitely long cylinder, which reduces to a two-dimensional problem under normal incidence. In particular, we treat cases in the strong-scattering regime where the Born series diverges, but where Padé approximation retrieves a valuable result. In Fig. 1 the case of a cylinder is shown which is well beyond the weak-scattering regime, but where the most accurate Padé approximant gives a good result for the field. The presented approach incorporates multiple-scattering effects and can therefore represent an important building block to the application of the Born series to direct and inverse problems, with potential applications in superresolution, optical metrology, and phase retrieval

Padé resummation of divergent Born series and its motivation by analysis of poles

The Born series is in principle a powerful way to solve electromagnetic scattering problems. Higherorder terms can be computed recurrently until the desired accuracy is obtained. In practice, however, the series solution often diverges, which severely limits its use. We discuss how Padé approximation can be applied to the Born series to tame its divergence. We apply it to the scalar problem of scattering by a cylinder, which has an analytical solution that we use for comparison. Furthermore, we improve our understanding of the divergence problem by analyzing the poles in the analytical solution. This helps build the case for the use of Padé approximation in electromagnetic scattering problems. Additionally, the poles reveal the region of convergence of the Born series for this problem, which agrees with actual calculations of the Born series

Scaling symmetry and conserved charge for shape-invariant optical fields

In this work we present an extensive study of the scaling symmetry typical of a paraxial wave theory. In particular, by means of a Lagrangian approach we derive the conservation law and the corresponding generalized charge associated with the scale invariance symmetry. In general, such a conserved charge, qs say, can take any value that remains constant during propagation. However, it is explicitly proven that for the whole class of physically realizable shape-invariant fields, that is, fields whose intensity distribution maintains its shape on propagation, qs must necessarily vanish. Finally, an interesting relation between such charge qs and the effective radius of a beam, as introduced by Siegman some years ago, is derived.IST/Imaging Science and TechnologyApplied Science