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Abstract
We numerically study the structure of polarization singularity lines in a near-field of the sub-wavelength dielectric particle when it is irradiated by a monochromatic elliptically polarized plane wave. For the various values of the ellipticity degree of the incident radiation, we trace the CT and LT-lines on which the polarization ellipse turns into a circle or a line, respectively. The continuous variation of the isotropy parameters of the singularities is visualized along the lines. The main rules of CT and LT-lines deformation, emergence, and vanishing are revealed.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When the electromagnetic radiation propagates through a medium, its initial polarization state often has a major impact on its propagation dynamics [1–8]. The usage of specially designed inhomogeneously polarized light beams allows one to overcome the traditional limitations implied by the symmetry of the medium on the character of nonlinear optical interaction [9]. The opposite is also true: significantly complicated distribution of light intensity and polarization may arise during its propagation in the nonlinear medium [10,11], which is of high academic interest, at least. Modern optics more and more often considers fully three-dimensional harmonic light fields, which can appear even in linear light-matter interaction. For example, such fields are formed in non-collinear mixing of paraxial beams or in the tight focusing of them [12,13]. They are also present near the nanoscale objects of various shapes [14–17]. Thorough research of the finest features of these kind of fields is nowadays of high demand, because of their applicability in nanoparticle trapping [18], quantum information [19] and nanoscopic sensing and imaging [20].
Within a paraxial approximation, the ellipticity and the orientation of polarization ellipse in each point of the cross-section of a beam can be characterized by two quantities: $M_0 = (|E_+|^{2}-|E_-|^{2})/(|E_+|^{2}+|E_-|^{2})$ and $\Psi _0 = \arg ({E}_+{E}_-^{*})/2$, where ${E}_\pm =E_x\pm i E_y$ are the complex slowly varying amplitudes of circularly polarized components of the transverse electric field. The first quantity varies from $-1$ to $1$ and is called the ellipticity degree of polarization ellipse, the second one is the angle between its main axis and the $x$-axis in the cross-section. These geometrical characteristics are related to commonly used normalized Stokes parameters as follows: $s_1+\mathrm {i} s_2 = \mathrm {e}^{2\mathrm {i}\Psi }\sqrt {1-M^{2}}$, $s_3 = M$. The longitudinal component of electric field strength vector $E_z$ is present in paraxial beams as well because of its transverse finiteness, however, its magnitude is relatively small. Polarization singularity points may appear in the propagating paraxial beam and these are C-points [21–24], in which the tip of the transverse electric field vector draws out a circle ($M_0=\pm 1$) and L-points, in which the tip of this vector draws out a line ($M_0=0$). The same mathematical description of polarization singularities is also applicable when the third component of the field, being comparable with the transverse ones, is for some reason not taken into account [14,25]. Polarization singularities are resistant to small disturbances of the electromagnetic field and form isolated C-lines and L-surfaces in the three-dimensional space in which the beam propagates.
In nonparaxial fields the shape, the orientation and the sizes of the polarization ellipses in the points of space $\mathbf {r}=(x,y,z)$ are given by a scalar $M(\mathbf {r}) =|[\mathbf {E} \times \mathbf {E}^{*}]|/|\mathbf {E}|^{2}$, which is analogous to the absolute value of ellipticity degree $M_0$, by a unit vector $\mathbf {n}(\mathbf {r})=\mathrm {i}[\mathbf {E} \times \mathbf {E}^{*}]/|[\mathbf {E} \times \mathbf {E}^{*}]|$ orthogonal to the plane of the ellipse and a pair of antiparallel vectors $\mathbf {a}$ and $-\mathbf {a}$, which is used for indicating the direction and length of ellipse major axis [17,26–28]. Here $\mathbf {E}(\mathbf {r})$ is the complex amplitude of electric field, $\mathbf {a}(\mathbf {r})=\rm {Re}\left (\mathbf {E}^{*}\sqrt {(\mathbf {E}\cdot \mathbf {E})/|(\mathbf {E}\cdot \mathbf {E})|}\right )$ and the direction of normal is related to the direction of the electric field vector rotation by the right-hand rule. The polarization ellipses of a three-dimensional light field can have various sizes and shapes in different points of space and, unlike as it was in the paraxial approximation, the orientation of their planes can be different as well.
In the problems of light-matter interaction the polarization singularity points of a three-dimensional light field, namely, L$^{T}$ and C$^{T}$-points, are of great interest. The light is linearly polarized in the former ($M=0$) and one cannot uniquely determine the direction of the normal $\mathbf {n}$ in them. In the latter case, polarization ellipse degenerates into a circle ($M=1$) and it is the vector $\mathbf {a}$ that cannot be uniquely determined. These points behave differently with respect to the paraxial approximation, as both L$^{T}$ and C$^{T}$-points form lines in three-dimensional space, with exception to rare cases when the symmetry of the field is high. Then these points may form two-dimensional surfaces or even occupy a continuous volumes in three-dimensional space. However, such non-generic pictures are destroyed when a minor disturbance is added to the electric field [26].
Near C$^{T}$-points (L$^{T}$-points) all polarization ellipses (all normal vectors $\mathbf {n}$) lay nearly in the same plane and can be projected onto it. To make a connection to the paraxial theory and to avoid the artifacts of projection we will, in the case of C$^{T}$-points, choose this plane to be the plane of the electric field rotation in the C$^{T}$-point itself, and in the case of L$^{T}$-points we will choose this plane to be at all times perpendicular to the direction of an oscillating electric field in the L$^{T}$-point. One of the quantities, characterizing main features of a pattern formed by the projections of ellipses (the projections of vectors $\mathbf {n}$) onto the corresponding plane in the vicinity of a singular points is the isotropy parameter $\Upsilon _C$ ($\Upsilon _L$), which is dependent on the complex amplitude of electric field and its first derivatives, calculated in the singular point itself [17]:
The isotropy parameters contain extremely important information on polarization singularities. The topological index of a C$^{T}$-point is equal to $0.5\operatorname {sgn}\Upsilon _C$ and the topological index of L$^{T}$-point is $\operatorname {sgn}\Upsilon _L$. The absolute values of $\Upsilon _C$ and $\Upsilon _L$ qualitatively show the degree of the monotony of the angle $\Psi _C$, giving the orientation of the main axis of the ellipse near the C$^{T}$-point, and the angle $\Psi _L$, giving the orientation of vector $\mathbf {n}$ near the L$^{T}$-point on the angle $\varphi$ of the polar coordinate system in the chosen plane of projection with the center in the singular point. For example, when $|\Upsilon _{C,L}|=1$ the angle $\Psi _{C,L}$ uniformly increases or decreases with the increase of $\varphi$ and when $\Upsilon _{C,L}=0$ the functions $\Phi _{C,L}(\varphi )$ have two break points in which the corresponding angle is shifted by $\pi /2$ for C$^{T}$-points and by $\pi$ for L$^{T}$-points.
The present paper is devoted to the polarization singularities of a light field near the surface of dielectric spherical nanoparticle exposed to a homogeneously polarized wide laser beam. In linear optical approximation, basic topological features of L$^{T}$ and C$^{T}$-lines are revealed. The dynamics, emergence, and disappearance of the lines during the smooth variation of incident light and particle parameters are studied. The algorithm of L$^{T}$ and C$^{T}$-lines reconstruction was designed in [17] together with the visualization scheme for the changes of isotropy parameters along these lines. However, in [17] it was applied to an analytic solution of an optical problem, while in the present paper it is extended to visualize the solution obtained by numerical simulations.
2. Formulation of the problem and computation methods
Consider the electromagnetic plane wave propagating along the $z$-axis of the Cartesian coordinate system. The polarization ellipse of the wave is stretched along the $x$-axis (the angle $\Psi _0=0$) and has the ellipticity degree $M_0$. The complex amplitude of the electric field of the wave can be written as $\mathbf {E}_0=E_0(\mu _+ \mathbf {e}_x + \mathrm {i}\mu _-\mathbf {e}_y)$, where $\mu _\pm = (1-M_0)^{1/2} \pm (1+M_0)^{1/2}$, $\mathbf {e}_{x,y}$ are the unit vectors of the coordinate system and $E_0$ is a constant. The center of the dielectric sphere lies on the $z$-axis, while its south pole coincides with the origin point of the coordinate system.
The scattering of a plane electromagnetic wave on a spherical particle is a classical problem in optics. In fact, it is analytically solvable for any ratio of the incident light wavelength $\lambda$ and particle radius $r$. The most interesting interaction pattern occurs for $r \sim 0.1\lambda$, as the dipole momenta of the particle, excited in its various parts by the incident wave, oscillate at almost its frequency (see e.g. [29,30]). Due to the high complexity of analytical solution, in this paper we solve the problem numerically in the COMSOL Multiphysics package by the finite element method. Firstly, we set the values of the ellipticity degree $M_0$, the wavelength of the incident radiation, the radius of the sphere and the properties of its material. Periodic boundary conditions are used at the boundaries of the computational space. It is surrounded by a model medium able to absorb all the incident radiation (Perfectly Matched Layers) [31]. This allows one to avoid possible “reflections” of the wave from the boundaries of the computational space. The computational space is split into $50000$ finite elements.
Fig. 1. The examples of vector $\mathbf {n}$ distribution near an L$^{T}$-point (in the center, colored blue) and polarization ellipses distribution near a C$^{T}$-point (in the center, colored red) for different values of the isotropy parameters $\Upsilon _C$ and $\Upsilon _L$. The colorbars depict the color scheme that is used in Fig. 2 for mapping $\Upsilon _{C,L}$ on the C$^{T}$ and L$^{T}$-lines.
Numerically solving the system of Maxwell equations, we obtained the values of the complex amplitude $\mathbf {E}$ of the superposition of incident and scattered electric fields in the nodes of a three-dimensional cubic grid. Tricubic interpolation was used to obtain the values of the amplitude in other points of space. The points in which scalar $(\mathbf {E}\cdot \mathbf {E})$ (C$^{T}$-points) or vector $[\mathbf {E}\times \mathbf {E}^{*}]$ (L$^{T}$-points) is zero were found in three sets of planes perpendicular to three Cartesian axes. The found points of polarization singularities were grouped in smooth lines by the nearest neighbor method and the lines were colored with respect to the values of the corresponding isotropy parameter Eqs. (1) and (2) (see Fig. 1).
Below we use the following representation of the results. C$^{T}$-lines are drawn thicker than L$^{T}$-lines and the former are colored from yellow to red, while the latter is colored from purple to green. Bright tones (yellow for C$^{T}$ and pale purple for L$^{T}$-lines) correspond to positive values of the isotropy parameters, and dark tones (dark red for C$^{T}$ and dark green for L$^{T}$-lines) correspond to the negative ones. Figure 1 illustrates some examples of distributions of vectors $\mathbf {n}$ near an L$^{T}$-point and of polarization ellipses near a C$^{T}$-point for different values of $\Upsilon _L$ and $\Upsilon _C$. Notice that pale yellow (bright and thick in grayscale) parts of lines correspond to the C$^{T}$-points with positive topological index and dark red (dark and thick in grayscale) correspond to the ones with negative topological index. Analogously, pale purple (bright and thin in grayscale) parts of lines correspond to L$^{T}$-points with positive topological index and dark green (dark and thin in grayscale) correspond to the ones with negative topological index. Special round markers show the points in which the sign of the topological index is reversed.
3. Results of numerical simulations
Our numerical simulations have shown that the value of the ellipticity degree of incident wave polarization has a significant impact on the number and geometry of the formed singular lines when the wavelength of the incident wave is one order of magnitude greater than the radius of the dielectric spherical nanoparticle. If the incident light is almost linearly polarized then two pairs of closed L$^{T}$-lines appear near the particle. They have the shapes of slightly deformed closely located rings and lie outside the particle (colored gray in Fig. 2(a)). Two of the four L$^{T}$-lines appear to be non-closed because they exist partially beyond the computational space. The lines are the closest to each other near four points of the equatorial plane of the sphere. As the $M_0$ gets bigger one pair of rings shrinks rapidly and completely disappears (Fig. 2(b)). The length of lines in the other pair also gets smaller but at a much slower pace. Up to the disappearance, each of the deformed rings has two points of topological index change.
Fig. 2. Configurations of L$^{T}$ and C$^{T}$-lines near a dielectric spherical particle with radius 90 nm for $\lambda = 710$ nm and $M_0=0.02$ (a), $0.1$ (b), $0.7$ (c), $0.96$ (d). The lines are colored according to the values of isotropy parameters of singularities. The direction of incident wave propagation and its polarization state are shown in the corners of each subfigure.
For any value of ellipticity degree $M_0$ there also exist two C$^{T}$-lines which also look like closely located deformed rings, crossing the surface of the particle. Outside of it they are noticeably curved and have a negative topological index. Inside the sphere, the parts of the C$^{T}$-lines are almost straight and have a positive index. The index change occurs in points near the intersections of the lines and the sphere. Near the points where $\Upsilon _C=0$ C$^{T}$-line turns by almost $90^{\circ }$ (Figs. 2(a) and (b)). With growth of $M_0$ the $\Upsilon _C=0$ points and the abrupt turning points move apart from each other (Figs. 2(c) and (d)).
As $M_0$ gets bigger, the parts of C$^{T}$-lines inside the sphere start to move away from each other. The points of topological index change are getting further from each other and move noticeably away from the surface of the particle. When $M_0 \approx 0.7$ C$^{T}$-lines become more and more curved and elongated in space (Fig. 2(d)). As $M_0$ tends to unity (Figs. 2(c) and (d), the length of C$^{T}$-lines grows, and the planes, associated with the orientation of the lines, turn in opposite directions. Eventually, when the incident wave is circularly polarized the C$^{T}$-lines almost lie in the plane $y=0$ (Fig. 2(d)). Herewith, additional points of topological index change appear close to the intersections of lines and the particle. The variation of the isotropy parameter along the C$^{T}$-lines becomes more complex. On the last diagram (Fig. 2(d)) the lines seem to brake because they partially exit the computational space.
For the wide range of the ellipticity degree of the incident wave (from 0.1 to 0.9 for the chosen wavelength and radius of the particle) closed C$^{T}$ and L$^{T}$-lines lie not further from the center of the sphere than four times its radius. As the polarization of the incident wave becomes linear (circular) the length of L$^{T}$-lines (C$^{T}$-lines) gets infinitely big. The L$^{T}$-lines exist only when the polarization of the incident wave is close to linear. In our case, they exist when $M_0<0.22$, which corresponds to the ratio of axes of the incident polarization ellipse being greater than $9$. On the other hand, ring-shaped C$^{T}$-lines exist for any value of the incident wave ellipticity degree. The shape, the degree of deformation, and the length of C$^{T}$-lines drastically change with the growth of $M_0$ in the range $0.7 < M_0 \leq 1$, which corresponds to the axes ratio being between $1$ and $2$.
The disappearance of the L$^{T}$-lines at certain $M_0$ is a typical feature of the singular lines dynamics, which can happen both in paraxial and non-paraxial optics. However, in non-paraxial optics it may not always be possible. For example, if closed C$^{T}$ and L$^{T}$-lines are entangled, they cannot be contracted to a point and thus remain stable with respect to the smooth change of the incident field parameters [17]. It is also worth mentioning that not only the direction of the C$^{T}$-lines changes near the surface of the sphere, but their topological charge as well. As it is in the paper [17], in our problem the topological index of the C$^{T}$-lines are mostly negative. At the same time, the parts of L$^{T}$-lines with positive and negative topological indices have roughly the same lengths.
4. Conclusion
We have investigated polarization singularities of three-dimensional electromagnetic field near a dielectric spherical nanoparticle irradiated by a plane monochromatic elliptically polarized wave. Assuming the optical response of the sphere is linear we had computed the components of three-dimensional electric field strength vector in area near the sphere using finite-element method and COMSOL Multiphysics package for various wavelengths and polarization of incident radiation. The obtained data was used to reconstruct the C$^{T}$ and L$^{T}$-lines in the near-field, in points of which the polarization of light is circular and linear respectively. Not only the topological index changes, but smooth variation of the isotropy parameters of the singular points along the lines were demonstrated.
The L$^{T}$-lines near the nanoparticle were found to be closed curves. The changes in their length and shape during the continuous variation of the polarization ellipticity of incident radiation were analyzed. It was shown that L$^{T}$-lines may contract to a point and disappear. Closed C$^{T}$-lines exist for any polarization of the incident wave. Their shape, degree of deformation and length vary significantly during a smooth transition from linear to weakly elliptical polarization of the incident radiation.
Funding
Foundation for the Advancement of Theoretical Physics and Mathematics (Basis); National University of Science and Technology (K2-2019-004); Russian Foundation for Basic Research (19-02-00069, 20-32-90123).
Disclosures
The authors declare no conflicts of interest.
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