## Abstract

Topological properties of light attract tremendous attention in the
optics communities and beyond. For instance, light beams gain
robustness against certain deformations when carrying topological
features, enabling intriguing applications. We report on the
observation of a topological structure contained in an optical beam,
i.e., a twisted ribbon formed by the electric field vector *per se*, in stark contrast to recently reported
studies dealing with topological structures based on the distribution
of the time averaged polarization ellipse. Moreover, our ribbons are
spinning in time at a frequency given by the optical frequency divided
by the total angular momentum of the incoming beam. The number of full
twists of the ribbon is equal to the orbital angular momentum of the
longitudinal component of the employed light beam upon tight focusing,
which is a direct consequence of spin-to-orbit coupling. We study this
angular-momentum-transfer-assisted generation of the twisted ribbon
structures theoretically and experimentally for tightly focused
circularly polarized beams of different vorticity, paving the way to
tailored topologically robust excitations of novel coherent
light–matter states.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The topological structure of light beams is enriched by spin–orbit interactions
(SOI) or the interplay of spin angular momentum (SAM), associated with the
polarization of light, and the orbital angular momentum (OAM), which is
related to the spatial structure of the beam’s wavefront [1,2].
While complex topological structures such as optical vortices, polarization
knots [3,4], and optical polarization Möbius strips [5,6] were extensively
studied and even observed experimentally [3,4,7–10], surprisingly, the
experimental observation of an optical twisted ribbon was investigated less
actively and reported only very recently [11]. This is despite the fact that an optical ribbon is a ubiquitous
structure [12]. It is important to note
here that polarization topological structures reported so far have solely been
discussed in the context of the polarization ellipse and its major and minor
semiaxes [5–8,11,12] or higher-order Lissajous figures [13,14]. In the present
study, it is the electric field vector itself, which twists and turns to form
a ribbon. Additionally, quite elaborate schemes including interference of
several beams or the focusing of multiple co-propagating modes are often
required to observe topologies such as optical polarization Möbius strips or
ribbons [7,10,11]. In the present
work, we show that focusing a circularly polarized light beam with or even
without a central phase vortex generates a twisted ribbon of the focal
electric field *per se,* spinning in time around
the optical axis. Importantly, our optical ribbon can be a “pure” or “true”
ribbon in the sense that it manifests itself around a point of zero field
intensity and exists by itself while not being accompanied/surrounded by other
topological structures. The number of full ribbon twists corresponds to the
total angular momentum (AM) in accordance with the index theorem
(and equal to the OAM of the longitudinal component of the focused beam)
[15]. The theoretical predictions are
fully corroborated by experimental results recorded utilizing a
nano-interferometric amplitude and phase reconstruction
technique [16].

In order to prove the proposed concept of optical twisted ribbons in the time-instantaneous electromagnetic field, we study topological structures created by tight focusing of fundamental Gaussian and first-order Laguerre–Gaussian beams resulting in electric (and magnetic) field ribbons of low twist numbers. For comparison with the full-field experimental data retrieved for the aforementioned tightly focused light beams, we use theoretical data calculated based on vectorial diffraction theory [17]. However, we start our discussion by representing the general structure of highly confined light beams via an approximate analytical model, which will allow us to retrieve a fundamental connection between the AM of the input beam and the twist number of the resulting field ribbons. In this model, we use at the input a narrow ring aperture (see also [18,19]), simplifying the integral equations of vectorial diffraction theory to analytic expressions. For the limiting case of $\text{NA} \to 1$, one obtains the field components of a circularly polarized annular beam with azimuthal index $m$ in a cylindrical [20] as well as Cartesian basis [19],

Next, we discuss the time evolution of the aforementioned ribbon structure. The components of the electric field in cylindrical coordinates [see Eq. (1)] have their azimuthal coordinate dependence solely given by the common phase factor $\text{exp}[{i({m \pm 1})\varphi - i \omega t}]$. The temporal evolution of the field is thus uniquely linked to an azimuthal coordinate change by $\frac{{\text{d}\varphi}}{{\text{d}t}} = \frac{\omega}{{m \pm 1}}$. Consequently, the topological structure of the instantaneous electric field vector generated upon tight focusing rotates on a circular trace in the focal plane at a $\frac{1}{{m \pm 1}}$ fraction of the optical frequency [25], or inversely proportional to the total AM of the full field, which is equal to $m \pm 1$.

To demonstrate experimentally the appearance of twisted ribbons in the time-instantaneous electric field distribution, we use the experimental setup shown in Fig. 1, which consists of a homebuilt confocal-like microscope [7,16]. The chosen near-field probe is a single gold nanosphere with a diameter of approximately 80 nm (see scanning electron micrograph in Fig. 1) that is immobilized on a glass substrate. This probe can be precisely scanned through the investigated field distribution using a piezo stage, resulting in scattering of the local electric field at the point of the probe. An oil-immersion microscope objective ($\text{NA} = {1.3}$) is collecting the forward-scattered and transmitted light from the substrate side, leading to the angularly resolved detection of their interference by imaging the back focal plane of the microscope objective onto a CCD camera. This Fourier-microscopy-based interference approach is equivalent to observing the scattering process from various directions. Thus, it allows for retrieval of the relative phase information between all three electric field components in the fully vectorial distribution under study from the far field. This technique, labeled Mie-scattering nano-interferometry [16], was proven to achieve deep sub-wavelength spatial resolution in the experimental study of vectorial focal fields and enabled the experimental verification of optical polarization Möbius strips and twisted ribbons in the distribution of the polarization ellipse (major axis) resulting from tailored light fields [7,8,11]. Details regarding this technique are discussed in [16].

From the full amplitude and phase information of the complex vectorial light field reconstructed within the plane of observation, we also retrieve information about the temporal evolution of the field and, thus, the rotation behavior of this topological structure during an optical cycle. Given the common time-dependent phase factor of $\exp [{- i\omega t}]$ for a harmonically oscillating field, the instantaneous vectorial electric field distribution can be inferred at every time step within the optical cycle by a global phase shift of $\Delta \varphi = \omega \Delta t$.

In contrast to the analytical considerations discussed above, the highly confined focal field distribution containing a twisted ribbon in its time-instantaneous electric field vector is experimentally generated by focusing a circularly polarized Laguerre–Gaussian beam of order $m$ with the full aperture of a microscope objective with an NA of 0.9 (no annular aperture). As stated, this affects only the strength of the sidelobes in the focal field distribution but not the comprised topological structure, while simplifying the experimental setting. The input beam for $m \ne 0$ is generated by transmitting a circularly polarized Gaussian beam through an optimally tuned $\boldsymbol{q}$-plate [26] of order $q = \frac{m}{2}$. The resulting field with an on-axis phase vortex of charge $m$ and circular polarization of opposite handedness is filtered spatially with a pinhole to obtain the lowest radial order of the created Laguerre–Gaussian modes [27] before being focused by the objective.

For the experimental reconstruction, we scan the nano-probe across the highly confined focal field and apply the reconstruction algorithm detailed in [16] to the collected far-field intensity information. As a result, we retrieve the experimentally reconstructed focal electric energy density and phase distributions shown in Figs. 2(a) and 2(b) for two different input light beams $\text{LG}_0^{\text{circ}-}$ and $\text{LG}_1^{\text{circ}+}$, respectively. The excitation wavelength in this case was chosen as $\lambda = 530\;\text{nm} $, with the nano-probe exhibiting an experimentally determined relative permittivity of $\varepsilon = - 3.1 + 2.5i$ at this wavelength.

It can be seen that the total electric energy density (depicted on the left of Fig. 2) strongly resembles the cylindrically symmetric field distributions predicted by Eq. (1) as well as numerical calculations via vectorial diffraction theory [17] for both cases (not shown). While the energy density distributions of the individual electric field components (right side of Fig. 2) show minor deviations from the expected symmetries given by Eq. (1), the resulting phase distributions (depicted as insets) exhibit the expected on-axis phase vortices with charge $m$ and $m \pm 1$ for the transverse and longitudinal field components, respectively, in the case of the first-order Laguerre–Gaussian beam [Fig. 2(b)].

From this experimentally determined fully vectorial complex electric field distribution, we now trace the real-valued electric field vector on a circular path with radius $r = 150\;\text{nm} $ around the optical axis (shown in Fig. 2 as white or black dashed lines) for a fixed time $t$ within an optical period ${T}$. For a fundamental Gaussian input beam $\text{LG}_0^{\text{circ}-}$, the electric field for ${t} = {0}$ and ${t} = {T}/{4}$ traced along the closed circle features the topological structure of a twisted ribbon as depicted in Fig. 3(a). Dark and light blue arrows correspond to the two times indicated above. When following the trace in a ccw direction, the electric field vector rotates once in a cw manner around the trace, corresponding to a topological charge or twist index of the ribbon of ${-}{1}$ [15]. This index can also be inferred from the projection of the electric field vector onto the transverse plane, as shown in Fig. 3(a). To verify the experimentally obtained twist index, we additionally calculated the focal field distribution of a $\text{LG}_0^{\text{circ} -}$ beam with the same parameters used in Fig. 3(a) using vectorial diffraction theory [17], resulting in the twisted ribbon shown in Fig. 3(b). The excellent agreement of the experimentally determined and numerically calculated topological structures confirms their robustness against aberrations and experimental noise. Furthermore, our results show that such intriguing structures are even present in the most fundamental and widely used light beams such as a circularly polarized fundamental Gaussian beam.

Comparing the resulting twisted ribbons for both time steps ${t} = {0}$ and ${t} = {T}/{4}$ (light and dark blue arrows in Fig. 3), we can also follow the instantaneous rotation of the topological structure upon time evolution, confirming the analytically expected rotation at the optical frequency $\omega$ for the shown case of $m = 0$ (see also Visualization 1).

To verify the analytically determined twist index for higher-order light beams and probe a “pure” twisted ribbon around a dark spot of the electric field, we also plot the corresponding data for a Laguerre–Gaussian beam $\text{LG}_1^{\text{circ} +}$ [see Fig. 2(b)]. The resulting twisted ribbon with twist index 2 is depicted in Fig. 4(a) and agrees very well with the numerically determined twisted ribbon [Fig. 4(b)]. Here, a single optical cycle will lead to a rotation of the twisted ribbon topology by only 180° when traced around the optical axis (see Visualization 2), confirming the inverse relation between the rotation frequency of the focal field and its total AM.

In conclusion, we have analytically predicted and experimentally and
numerically observed twisted ribbons formed by the time-instantaneous electric
field vector *per se* with the number of full
twists depending on the total AM of the underlying tightly focused light beam, $m \pm 1$, spinning in time around the optical axis at
a ${1}/({m \pm 1})$ fraction of the optical frequency. Our
findings show that beside the distribution of polarization (ellipse) and
phase, the electromagnetic field itself can also feature an interesting
topological structure. We envision that the underlying robustness of these
entities might allow for novel concepts in shaping coherent multi-particle
light–matter states such as hybrid polaritons.

## Funding

Nederlandse Organisatie voor Wetenschappelijk Onderzoek; FP7 Ideas: European Research Council (340438); Russian Foundation for Basic Research (20-07-00505).

## Acknowledgment

T. B., P. B., and G. L. thank S. Orlov for the initial theoretical implementation of the employed field reconstruction algorithm.

## Disclosures

The authors declare no conflicts of interest.

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