## Abstract

We report a method of generating exotic optical polarization Möbius strips through tightly focusing an arbitrary vector beam. A heart-shaped Möbius strip, an “8”-shaped twin Möbius strip, and a circular Möbius strip with varying polarization twisting rate are demonstrated. The ability of tailoring three-dimensional optical polarization topologies may spur novel studies of optics and physics and find their applications in sensing, light coupling to nanostructures, light-matter interaction, and metamaterial fabrication.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The Möbius strip, featuring a one-sided surface, can be easily obtained artificially by taking a rectangular strip of paper, twisting one end through an odd number of π radians, and then joining the ends [1,2]. However, the Möbius strip structure is rarely observable in optics until recently focused optical fields emerging from a q-plate are measured through three-dimensional nanotomography and the polarization topology structure is unveiled to be a Möbius strip [3]. Alternatively, the Möbius strip polarization topology can also be realized through the interference of two noncoaxial circularly polarized beams of opposite handedness with different scalar topological charges [4–6], through scattering from high-index dielectric nanoparticles [7], or even through tightly focusing a linearly polarized Gaussian beam [8]. The abovementioned Möbius strips are all calculated or measured along a circular path on which the major axes of the three-dimensional polarization ellipses of all points form the topology structure.

Since the Möbius strip is rarely seen in optics, it is intriguing to know if it is possible to even tailor the Möbius strip topology as required. In the current paper, we propose and demonstrate a method to generate exotic optical polarization Möbius strips through tightly focusing an arbitrary vector beam. Instead of twisting along a circular path, these Möbius strips wander along a prescribed path. A heart-shaped Möbius strip and an “8”-shaped twin Möbius strip are considered as examples. The Möbius strips in nature are usually twisted at an ordinary rate. We demonstrate, in the third example, a Möbius strip with varying polarization twisting rate. These exotic optical polarization Möbius strips can be physically realized through focusing vector optical fields under high numerical aperture (NA) conditions. With the advancement in liquid crystal spatial light modulators, digital micromirror devices and three-dimensional nanotomography, it is now possible to create arbitrary vector beams [9] and measure exotic optical polarization Möbius strips in the focal region. The ability of generating exotic optical polarization Möbius strips and other polarization topologies may spur novel studies of optics and physics and find their applications in sensing, light coupling to nanostructures, light-matter interaction, metamaterial fabrication, and developing novel nanophotonics devices.

## 2. Tight focusing of arbitrary vector beams

Focusing a light beam under high NA conditions creates a non-negligible longitudinal focal component. An arbitrary vector beam has four spatially varying degrees of freedom: the amplitude and phase of the x- and y-polarized components, respectively [9]. The versatility of arbitrary vector beams enables the generation of a variety of intensity distributions in the focal region such as the optical needle [10], the optical tube [11] and the optical chain [12,13]. Figure 1 shows the schematic of the optical configuration. An arbitrary vector beam is focused by a high-NA objective lens. The radius of the aperture stop is R, and the focal length is *f*.

The wave vector ${\overrightarrow{k}}_{\Omega}$ can be decomposed into three individual components in spherical coordinates:

_{0}= 2π/λ

_{0}is the wave number. The symbols

*θ*and

*ϕ*, as shown in Fig. 1, denote the elevation angle and azimuthal angle in spherical coordinates, respectively.

The Debye integral is commonly used to calculate the optical field distributions in the focal region under tightly focusing conditions [14–16]:

_{${\stackrel{\rightharpoonup}{E}}_{x\_\Omega}\left(\theta ,\varphi \right)$},${\stackrel{\rightharpoonup}{E}}_{y\_\Omega}\left(\theta ,\varphi \right)$ and ${\stackrel{\rightharpoonup}{E}}_{z\_\Omega}\left(\theta ,\varphi \right)$ represent the

*E*,

_{x}*E*and

_{y}*E*components of the optical fields in the wavefront surface Ω. ${\overrightarrow{E}}_{x\_f}\left({r}_{{}_{p}},\Psi ,{z}_{p}\right)$,${\overrightarrow{E}}_{y\_f}\left({r}_{{}_{p}},\Psi ,{z}_{p}\right)$ and ${\overrightarrow{E}}_{z\_f}\left({r}_{{}_{p}},\Psi ,{z}_{p}\right)$ represent the

_{z}*E*,

_{x}*E*and

_{y}*E*components of the optical fields in the focal region, where $\left({r}_{{}_{p}},\Psi ,{z}_{p}\right)$is the position vector near the focal point.

_{z}To expedite the numerical calculation of the Debye integral, Eq. (2) can be rewritten in terms of *k _{x}* and

*k*[17]:

_{y}_{0}plane, respectively.

The optical polarization topology is defined as the geometry formed by the major axes of three-dimensional polarization ellipses of all points lying on a prescribed path. The approach introduced in [18] can be used to trace out the 3D orientation of the polarization ellipses. In this notation, the major axes of the polarization ellipses in the complex representation of the electric fields are given by:

The regular optical Möbius strip topology as in [3] wanders along a circular path at an ordinary twisting rate. Here we intend to construct exotic optical Möbius strip topologies that twist along a more complex path with a varying polarization twisting rate. Since the optical polarization topology is regulated by the three-dimensional polarization ellipses of all points on the path, the three individual optical field components E_{x}, E_{y} and E_{z} must be shaped simultaneously to form the desired topology.

As illustrated in Fig. 2, the algorithm starts with initial optical fields in the pupil plane. For simplicity, we have used a circularly polarized Gaussian beam as the initial optical fields. By applying the projection function of the objective lens, the optical fields in the wavefront surface are obtained. These optical fields divided by cos*θ*, as demonstrated in Eq. (3), are Fourier transform related to the optical fields in the focal plane. The constraints in the focal region require that the three individual optical field components E_{x}, E_{y} and E_{z} along the prescribed path form the desired topology structure whereas the optical fields off the path can choose arbitrary spatial amplitude, phase and polarization distributions. The Möbius strip is a one-sided surface that can be characterized by parametric equations. Specifying the constraints of three individual optical field components in the focal region based on parametric equations is plausible. However, it is not a simple procedure to follow because the convergence of the algorithm requires that the three individual iterative processes satisfy their constraints simultaneously. To handle the convergence problem and simplify the numerical process, we nonuniformly sample and interpolate the data of optical fields of a regular optical Möbius strip generated through focusing a light beam emerging from a q-plate [3] and utilize the interpolated data as the specified constraints in the focal region. The modified fields are then inversely Fourier transformed and times cos*θ*. The constraints in the wavefront surface require that the optical fields beyond the NA are zero whereas the optical fields within the NA are allowed to have arbitrary spatial amplitude, phase and polarization distributions. The algorithm iterates back and forth between the wavefront surface and the focal region until the merit function is satisfied and the desired polarization topology is formed [19,20].

## 3. Numerical simulation

Three examples are considered to demonstrate that the method is capable of generating exotic optical polarization Möbius strips in the focal region. The first example is to generate a heart-shaped Möbius strip polarization topology. The numerical aperture of the objective lens is set to 0.9. The constraint condition in the focal region is that three individual optical components form a Möbius strip polarization topology and the Möbius strip wanders along a heart-shaped trajectory of around 0.27λ in size in the z = 0 plane. The constraint condition in the wavefront surface is that the optical fields outside the NA are zero. The algorithm is performed based on Eq. (3). When the iterations are converged, the intensity and polarization distributions of the required pupil fields are obtained as shown in Fig. 3(a). The left-handed polarization and right-handed polarization are denoted in red and white, respectively. The pupil fields have a star-shaped polarization topology [21]. Figures 3(b) and 3(c) display the intensity distributions of E_{x} and E_{y} components in the pupil plane, respectively. Figures 3(d) and 3(e) display the phase distributions of E_{x} and E_{y} components in the pupil plane, respectively. Evidently, all the four degrees of freedom of an arbitrary vector beam are utilized to form the desired polarization topology.

Figures 4(a)-4(c) show the intensity distributions of the E_{x}, E_{y} and E_{z} components in the z = 0 plane, respectively. Figures 4(d)-4(f) show the phase distributions of the E_{x}, E_{y} and E_{z} components in the z = 0 plane, respectively. The total intensity distributions and the heart-shaped path are shown in Fig. 4(g). Figure 4(h) displays the major axes of the three-dimensional polarization ellipses along the red heart-shaped path. The major axes of the polarization ellipses are denoted in blue and green to indicate the orientation. Apparently the polarization topology is a heart-shaped Möbius strip.

The second example is to generate an “8”-shaped twin Möbius strip in the focal region. The constraint condition in the focal region is that three individual optical components form an “8”-shaped twin Möbius strip polarization topology. The “8” trajectory in the z = 0 plane consists of two circles positioning in contact side by side. Each circle has a radius of 0.15λ. Figure 5(a) shows the intensity and polarization distributions in the pupil plane. Figures 5(b) and 5(c) show the intensity distributions of E_{x} and E_{y} components in the pupil plane, respectively. Figures 5(d) and 5(e) show their corresponding phase distributions in the pupil plane, respectively.

Figures 6(a)-6(c) display the intensity distributions of the E_{x}, E_{y} and E_{z} components in the z = 0 plane, respectively. Figures 6(d)-6(f) display the phase distributions of the E_{x}, E_{y} and E_{z} components in the z = 0 plane, respectively. The total intensity distributions are shown in Fig. 6(g). The “8”-shaped path is denoted in white and located in the central region of the z = 0 plane. The major axes of the three-dimensional polarization ellipses along the red “8”-shaped path are plotted in Fig. 6(h). The major axes of the polarization ellipses are denoted in blue and green to indicate the orientation. Each circular path of the “8” shape has a Möbius strip polarization topology and their adjacent part share the same three-dimensional polarization states. It is worthy of noting that the “8”-shaped path appears in the area of low optical intensity.

The third example is to generate a circular Möbius strip with varying polarization twisting rate in the focal region. The constraint condition in the focal region is that three individual optical components form a circular Möbius strip polarization topology. More importantly, the Möbius strip twists at different rates along the two halves of the circular path of 0.27λ in size. The required intensity and polarization distributions in the pupil plane are shown in Fig. 7(a). Figures 7(b) and 7(c) display the intensity distributions of E_{x} and E_{y} components in the pupil plane, respectively. Figures 7(d) and 7(e) display the phase distributions of E_{x} and E_{y} components in the pupil plane, respectively.

Figures 8(a)-8(c) display the intensity distributions of the E_{x}, E_{y} and E_{z} components in the z = 0 plane, respectively. Figures 8(d)-8(f) display the corresponding phase distributions of the E_{x}, E_{y} and E_{z} components, respectively. The total intensity distributions are shown in Fig. 8(g). The circular path is divided to two halves with one half denoted in solid white and the other half in dashed white. These two halves are plotted in red and black respectively in Fig. 8(h). The major axes of the three-dimensional polarization ellipses along the circular path are denoted in blue and green to indicate the orientation. From Fig. 8(h), it is noticeable that the Möbius strip surrounding the black semicircle has a polarization twisting rate approximately twice that surrounding the red semicircle. Thus a Möbius strip polarization topology of varying twisting rate in the focal region has been demonstrated.

## 4. Discussions and conclusions

In summary, we demonstrate a method of generating exotic optical polarization Möbius strips through tightly focusing an arbitrary vector beam. A heart-shaped, “8”-shaped and circular Möbius strip with varying polarization twisting rate are designed with the method. The generation of exotic optical polarization topologies may find their way to the disciplines of optics and physics and spur potential applications in sensing, light-matter interaction, metamaterial fabrication, and developing novel optical devices.

## Funding

National Natural Science Foundation of China (NSFC) (61505062, 61875245).

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