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Abstract
Topologies associated with polarization point and line singularities can provide tools for controlling light propagation. By using the Stokes parameter, we demonstrate the emergence of polarization singularities hidden in deep subwavelength confined electromagnetic fields with angular momentum. We show that when the incoming orbital angular momentum is appropriately chosen, highly confined electromagnetic fields with super-diffraction-limited spatial dimensions can be obtained. At the same time, a conversion of orbital to spin angular momentum occurs, leading to a non-trivial topology. Our method provides a platform for developing topological photonics and studying the behavior of polarization singularities under strong focusing.
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1. Introduction
Phase singularities—also called vortices or dislocations—are points or lines in space where the phase of an optical field is undefined and the corresponding intensity vanishes [1–15]. An optical vortex is characterized by a spiral phase and is topological in nature. In 1992, Allen and coworkers demonstrated that a light beam possessing a spiral phase can carry an optical orbital angular momentum (OAM) of lћ per photon, where ћ is the reduced Planck constant, h divided by 2π. [2]. New families of phase singularities in the form of closed, knotted, and linked dislocation lines have also been reported and demonstrated [16–21]. Because of their nontrivial spatial features, optical fields possessing phase singularities have formed the basis of a rapidly growing number of unique applications, including super-resolution imaging [22–24], optical micromanipulation [25,26], quantum information processing [27,28], and optical communication [29–32].
In addition to scalar phase singularities, the free space optical field can also support singularities in its polarization vector that have attracted growing attention [33–51]. For the two-dimensional paraxial fields, the polarization singularities in these fields are isolated points, at which at least one of the parameters defining the state of polarization (SoP) is indeterminate. The generic singularities in an ellipse field are called C-points. The C-point is circularly polarized when the orientation is undefined and is surrounded by an elliptical SoP. Similar to C-point singularities, V-point singularities also occur in a vector field and are characterized by integer values of the Poincare–Hopf index. In difference, the V-point is a vector-point singularity at which the direction of the electric vector of a linearly polarized field is undefined. Moreover, in the neighborhood of the V-point, the SoPs are all linearly polarized and undergo changes only in the azimuth. These singularities are structurally stable when perturbed and propagating. In this sense, they are the natural counterpart to phase singularities in scalar optics. Polarization singularities usually appear in random fields, multi-beam interference, and vector beams. Most recently, polarization singularities with torus knot and link topologies as well as topologically more complicated figure-eight knots have been generated and observed [52,53]. In addition, a nonparaxial 3D polarization topology in the form of Möbius strips was found to appear in an appropriately chosen observation plane near the focus of a tightly focused Gaussian beam [54,55].
In this Letter, based on the general integrated analytical formula for tight focusing of arbitrary locally linearly polarized fields, the focusing properties of azimuthally polarized (AP) Laguerre-Gaussian (LG) beams are numerically investigated and discussed in detail. Calculations show that the intrinsic optical degree of freedom of the OAM does provide a simple but powerful solution for tailoring the highly confined field distributions. By manipulating the topological charge, we obtain a superdiffraction-limited electromagnetic field without additional phase or amplitude modulation. Meanwhile, orbital-to-spin angular momentum conversion occurs, resulting in nontrivial polarization topological structures. In particular, by exploiting the two-dimensional Stokes parameters, the topological behavior of polarization hidden in complex polarization distributions is revealed.
2. Deep subwavelength confined electromagnetic field with angular momentum
Polarization, as an intrinsic optical degree of freedom, is one of the salient features of light. From theory, a monochromatic light beam with arbitrary locally linear SoPs may be written in polar coordinates (ρ, ϕ) as [56–62]
Highly confined electromagnetic fields, which can be generated, for example, in high numerical-aperture (NA) objective lens focusing conditions shown in Fig. 1, are widely used as excellent tools to study detailed nano-optics. Strongly focused light fields have been widely analyzed using the generalized vector Debye integral. Richards and Wolf considered the contributions of input polarization to the focused field and built an analytical model for a uniform linearly polarized beam [63]. Similarly, the electric field E at each point r near the focus can also be derived when the incident polarized light beam is embodied as in Eq. (1).
Fig. 1. Schematic of the high NA objective lens focusing system and coordinate system followed in our calculations. The focal plane is located at z = 0.
Here, we employ the LG complex amplitude distribution $\textrm{LG}_p^l$, for which l and p are integers associated with intertwined helices and relate to the topological charge and the number of additional concentric rings. With p = 0, the electromagnetic field in the image space becomes
Fig. 2. Calculated total electric field intensity distributions in the focal (left column) and through-focus (right column) planes for the input AP $\textrm{LG}_0^0$, $\textrm{LG}_0^1$ and $\textrm{LG}_0^2$ beams (from the first row to the third row, respectively) All intensities have been normalized by the peak intensity of the total field in the focal volume for each input light.
Numerous optical instruments and equipment require the use of sharply focused light fields, ranging from lithography, data storage, as well as microscopy to optical trapping. The vector properties of light have been utilized to obtain the smallest possible spot. It has been demonstrated that a RP input plane wave can lead to a spot size significantly smaller than for linear and circular polarizations [66–68]. To be better understanding of the on-axis circularly symmetric hot spot shown in Figs. 2(c) and 2(d), Fig. 3 depicts its corresponding normalized focal field intensity profiles along x and z axes. For the sake of comparison, the results for RP $\textrm{LG}_0^0$ input beam under the same focusing conditions are also given. The RP $\textrm{LG}_0^0$ beam can have a narrow central peak due to the appearance of the strong longitudinal field component that is sharply centered on the optical axis as shown in Fig. 3(a). The beam size contributed by the longitudinal field component has a full width at half-maximum (FWHM) of 0.506λ in the transverse direction. However, the beam size for the total field is as large as 0.778λ, which is larger than the diffraction limit for this focusing lens λ/(2NA) = 0.526λ. Nevertheless, the FWHM value for the total fields of input AP $\textrm{LG}_0^1$ beam is only 0.522λ, which is beyond diffraction limit. From Fig. 3(b), the beam size along optical axis is calculated to be about 1.458λ and 1.416λ for RP $\textrm{LG}_0^0$ and AP $\textrm{LG}_0^1$ beams, respectively. So, an AP $\textrm{LG}_0^1$ input beam can generate a tighter hot spot than RP $\textrm{LG}_0^0$ beam, not only in the transverse direction but also in the longitudinal direction. Most importantly, we believe that this transversely polarized ultra-diffraction-limited spot is an important complement to its longitudinally polarized counterpart, since polarization plays a crucial and even decisive role in light-matter interactions.
Fig. 3. (a) Calculated normalized focal electric intensity profiles in the x axis direction for the longitudinal component and the total fields of input RP $\textrm{LG}_0^0$ beam as well as total fields of input AP $\textrm{LG}_0^1$ beam. (b) The corresponding intensity profiles along z axis for the total fields of the above two beams.
For the input paraxial AP $\textrm{LG}_0^1$ beam, it carries optical OAM directed along the propagation axis and no spin angular momentum (SAM). For its strongly focused non-paraxial field, the corresponding angular momentum properties can be studied using SAM and OAM densities, which can be expressed, respectively, as [69–71]
Figure 4 illustrates the normalized distributions of the SAM and OAM densities in the focal and through-focus planes for the input AP $\textrm{LG}_0^1$ beam. From calculations, only a longitudinal component of the SAM density [Figs. 4(a) and 4(b)] emerges, arising from a missing out-of-plane electric field component [see Figs. 2(c) and 2(d)], and thus results in the absence of a transverse component. Here, the positive and negative values mean that its direction is along the forward and reverse directions of the optical axis, respectively. And the appearance of longitudinal SAM also shows an interesting optical process referred to as orbital-to-spin conversion, in contrast to the well-known spin-to-orbital angular momentum conversion. There are on-axis densities null and annular distributions for both transverse and longitudinal components of optical OAM [see Figs. 4(c)–4(f)]. Additionally, compared with the longitudinal component, the transverse component is much stronger, which is much different from its SAM distribution.
Fig. 4. Optical SAM and OAM density distributions in the focal (left column) and through-focus (right column) planes for the input AP $\textrm{LG}_0^1$ beam. The three rows from top to the bottom are, respectively, the normalized longitudinal SAM, transverse and longitudinal OAM density distributions.
3. Polarization singularities hidden in the complex polarization distributions
Polarization plays a vital role in the light-matter interaction. Figures 5(a) and 5(b) illustrates the polarization distributions on the propagation and focal planes for the input AP $\textrm{LG}_0^1$ beam, respectively. The polarization projections in both the x-z and y-z planes are only linearly polarized [see Fig. 5(a)], indicating the spinning axis of the electric field is parallel to the z axis; i.e., the polarization ellipses all lie in the plane parallel to the x-y plane, arising from the absence of longitudinal electric field component. However, the polarization projection on the x-y plane is extremely complicated because the polarization ellipse becomes a solid ellipse in some regions away from the optical axis, indicating that the direction of the long axis of the polarization ellipse rotates as the light propagates. Overall, the polarization distribution on the focal plane exhibits perfect cylindrical symmetry, and linear, circular and elliptical SoPs mixed in the beam cross section [see Fig. 5(b)]. Meanwhile, the local polarization ellipse changes not only in its handedness but also in its ellipticity and orientation in the radial direction, whereas it only changes in the orientation in the azimuthal direction.
Fig. 5. (a) Polarization distributions in the x-z plane when y = 0 and their projections onto the three orthogonal planes. (b) Polarization ellipse fields in the focal plane for the input AP $\textrm{LG}_0^1$ beam.
Now, we focus our attention on the polarization topology. The light vibrations lie in the transverse plane for the aforementioned non-paraxial ultra-diffraction-limited light fields, analogous to the traditional 2D paraxial optical fields. Therefore, we can study the topological structures using the same analysis for traditional transverse fields. First, we recall the Stokes parameters, s0, s1, s2, and s3 used in 2D paraxial fields, but adapt them to the current field,
Detailed analysis of the polarization topology can be carried out with the help of the normalized Stokes parameters, namely, Si = si/s0, (i = 0, 1, 2, 3). For a spatially varying polarization distribution, the Stokes parameters are also functions of the position coordinates. The complex Stokes field S12 = S1 + iS2 can be used for probing polarization singularities. The phase distribution corresponding to this Stokes field is expressed as φ12 = arctan(S2/S1). The phase singularities of this complex Stokes field correspond to the polarization singularities in the actual field distribution.
Figure 6 depicts the phase variation of the Stokes field S12 in the focal plane. The phase variation of the Stokes field S12 exhibits a vortex of charge 2 in the center. In the outside of optical axis, four annular boundaries with different widths, which are π/2 phase shifted in the radial direction, separate the phase pattern into several concentric regions. Phase singularities appear on the inner and outer rings of the annulus boundaries. In total, there are about ten positions that exhibit phase singularities as can be found in the inset in Fig. 6.
Fig. 6. Phase variation of the Stokes field S12 on the focal plane for the input AP $\textrm{LG}_0^1$ beam.
To further explore the polarization topology hidden in the complex polarization distribution, the normalized stokes parameters S1, S2, and S3 are investigated. Figures 7(a)–7(c) shows, respectively, the contour lines of the normalized stokes parameters S1 = 0, S2 = 0, and S3 = 0. Two straight lines crossing each other on the axis and several circle can be found when S1 = 0 and S2 = 0. However, they are different, the two straight lines in Fig. 7(a) being directed, respectively, along the azimuthal angles of π/4 and 3π/4, whereas they are along the x- and y-axes in Fig. 7(b). For S3 = 0, however, only circles appear. The zero counters of all three normalized Stokes parameters S1, S2, and S3 are also plotted in a single map [see Fig. 7(d)]. It is known that at the C-points, S1 = S2 = 0, and for the L lines, S3 = 0. Interestingly, the straight lines in Fig. 7(a) and 7(b) cross each other on the axis in Fig. 7(d), resulting in a C point. And the circles in Fig. 7(a) and 7(b) are found to be coincident, resulting in the appearance of C lines. These locations should also be the positions where the annular boundaries emerge in the phase variation of the Stokes field S12 [see Fig. 6]. In contrast, the circles in Fig. 7(c) are L lines. For the innermost circle, the light vibrations should be all radially arranged and exhibit perfect circular symmetry [see Fig. 5(b)], and therefore is much different from the traditional L line. Furthermore, the location of this L line should be exactly where the SAM density is zero. On the two sides of this L line, the polarization ellipses have opposite handedness [see Figs. 4(a) and 4(b)].
Fig. 7. Theoretically calculated contour lines of the normalized Stokes parameters with (a) S1 = 0, (b) S2 = 0, (c) S3 = 0, and (d) the zero counters of all three normalized Stokes parameters S1, S2, and S3 in the focal plane for the input AP $\textrm{LG}_0^1$ beam.
4. Conclusions
In summary, we reveal the optical polarization topology hidden in a deep-subwavelength confined electromagnetic field with angular momentum. A general integrated analytical formula for calculating tightly focused fields of arbitrary locally linearly polarized beams based on Richards and Wolf’s vector diffraction theory is proposed. Calculations show that the intrinsic optical degree of freedom of the OAM provides a simple but powerful solution for tailoring the focal field distributions of the input AP beam. By manipulating the topological charge, an ultra-diffraction-limit electromagnetic field can be obtained without additional phase or amplitude modulations. Meanwhile, orbital-to-spin angular momentum conversion occurs, resulting in nontrivial polarization topological structures. In particular, by using the Stokes parameter, the topological behaviors of polarization hidden in the complex polarization distributions are revealed. These results may contribute to the research of complex highly confined optical fields.
Funding
National Natural Science Foundation of China (12074224, 61975128); Natural Science Foundation of Shandong Province (ZR2020MA087, ZR2021YQ02).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.
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