Generalized Lorenz–Mie theory of photonic wheels
Introduction
In recent years structured light gathers even more attention due to the emergence of new methods and tools such as geometrical phase elements or spatial light modulators. Complex light structures are used widely from telecommunications [1], [2] to increase transmitted data quantity by employing orbital angular momentum to optical switching. The rapid pace has started from well known paraxial optical vortices [3], which posses a transverse angular momentum (AM) [4], and are applied for micro-manipulation of various objects [5]. Going from paraxial theory to solutions of Maxwell’s equations [6] has resulted in various research on polarization singularities [7], [8], [9], [10], vector vortices [11], [12], Poincare beams [13], [14], complexified optical beams [15], [16], [17], light spinning in ribbons [18], [19], [20] and photonic wheels [21], [22].
One of the exciting configurations here are optical beams with transverse angular momentum of light [23]. Usually, optical beams contain near plane wave components and can carry only longitudinal angular momentum of light [4]. However, when focusing systems approach high numerical apertures vector optical beams with transverse angular momentum are created. These beams are sometimes called photonic wheels, because electric and magnetic fields are spinning in the longitudinal plane [21], [22]. Photonic wheels are closely related to the spin-Hall effect of light [24] which was reported numerous times in various contexts [25], [26], [27], [28].
Perhaps, one of the well-known applications of optical vortices is optical trapping and spinning of various micro-objects [5]. In order to characterize such kind of interactions a proper theory has to be introduced. For example, for spherical single or clustered objects a generalized Lorenz–Mie theory (GLMT) is introduced [29], [30]. This theory enables prediction of forces and torques acting on a object [30], [31], [32], [33], [34] and describes electric and magnetic fields inside and outside of a single particle [35], [36] or a cluster of particles [37], [38], [39].
Development of generalized Lorenz–Mie theory requires an introduction of electromagnetic multipoles (vector spherical harmonics) [40] and a proper description in terms of multipoles for both an object and an incoming beam [29]. The multipolar response of a single spherical particle was introduced by Gustav Mie in 1908 [41]. Since then a number of techniques were developed. For arbitrary shaped particles a System Transfer Operator (also known as T-Matrix) approach was introduced [42], [43], [44], [45]. Usually, the T-Matrix is determined using an Extended Boundary Condition Method [45], [46]. We note here, that it is common in the literature to identify the T-matrix method and the Extended Boundary Condition Method, but this is misleading [30]. Indeed, a Multiple Scattering Method [37], [47], [48] can be introduced to deal with the scattering from multiple objects. This method is based on generalized Lorenz–Mie theories and it enables an alternative calculation of the T-matrix [49], [50]. Those advances have enabled simulation of non-spherically shaped particles [36] or various clusters of them [51], [52].
On the other hand, the incident beam in those approaches also has to be described in terms of vector spherical harmonics (VSH). The expansion of the plane wave into VSH is the simplest known expression in scattering theories [29], [41]. As the conventionally polarized optical fields become focused by a high numerical aperture system, this expansion is not valid anymore [17]. Therefore, a description of the interaction between an object and a highly focused field requires a calculation of expansion coefficients. This is done usually either by correction of the plane wave expansion coefficients via introduction of the beam shape coefficients [29], or by direct calculation of the correct expressions [35], [53]. The calculation can be done either semi-analytically by using a theory based on Richards–Wolf integrals [54], [55] or analytically by using complex source beams [35], [56].
In this work we develop a method to numerically calculate exact expansion coefficients for optical beams with transverse angular momentum [21], [57], [58]. Next, we introduce the generalized Lorenz–Mie theory of photonic wheels. We use this development to study interaction of these beams with a spherical nano-particle. Lastly, we build upon our previous study on chirality in three particle system [52] and use Multiple Scattering Method to investigate scattering of a photonic wheel from this cluster. We demonstrate not only internal and external electric fields, but also investigate the angular momentum of light.
Section snippets
Towards expansion of photonic wheels into vector spherical harmonics
A collimated monochromatic beam of light with spatially separated left- and right-handed circular polarization components and a TEM-like beam intensity profile is described bywhere is the beamwidth and are Cartesian coordinates. An appearance of transverse angular momentum can be explained within the framework of simple ray optics. The intensity profile of this spin-segmented beam is symmetric with respect to the axis, and the beam propagates along the
Comparison of the electric fields obtained using the expansion and direct integration
As we have arrived at the expansion of photonic wheel into VSHs, we compare here electric fields calculated with the help of this expansion to the fields calculated using semi-analytical solution given by Richards–Wolf integrals [55]. The numerical aperture is set to 0.9, the wavelength is nm and the beamwidth is mm [65]. At the entrance plane of the focusing system electric field components and are expressed by Eq. (2). Knowing all parameters we calculate expansion
Verification of the transversality of angular momentum in the VSH expansion of a photonic wheel
As we now are able to calculate both electric and magnetic fields, we will now verify, that the angular momentum in the expansion is still transversal. Therefore, we calculate individual components of the angular momentum density for the same situation as in the previous Section, except the beamwidth, which was taken to be infinitely large. We note that we simulate photonic wheels in the focus of an aplanatic system, so, the apodization function is present. For calculation of the angular
Application to the scattering problems. Multiple-scattering method
Now we come to an application of the expansion into VSHs presented above. We use semi-analytical (i.e. open form) expressions for expansion coefficients and develop a generalized Lorenz–Mie theory for photonic wheels interacting with a sphere located in the origin. The method is based on separation of variables. If the shape of a particle is well-defined, analytical solutions (electromagnetic multipoles) of the Helmholtz equation are found in spherical coordinates. The relation between incident
Discussion and conclusions
As a conclusion, we have introduced a semi-analytical (open form) expansion of highly confined vector optical beams with transverse angular momentum (photonic wheels) into electromagnetic multipoles. Our analysis has revealed a rather rich multipolar structure of photonic wheels - the expansion has infinite odd numbers of azimuthal indices and three even numbers . As the photonic wheel becomes more confined (i.e. the numerical aperture of the focusing system increases) the number of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This project has received funding from European Social Fund (project No. 09.3.3-LMT-K-712-01-0167) under grant agreement with the Research Council of Lithuania (LMTLT).
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