Nanoscale control over optical singularities

1. INTRODUCTION

Optical vortices (OVs) [1] are phase singularities in electromagnetic fields which, with proper circular symmetry, can carry an orbital angular momentum (OAM). This OAM can be converted into mechanical torque [2] and be used to trap and manipulate nanometer-sized particles [3,4]; transfer information via optical communication [5]; enable high-capacity wireless [6], fiber-optical [7], and quantum communication [8]; generate spatial resolution beyond the diffraction limit [9,10]; and contribute in small scale optical communication between chips [11]. Vector electromagnetic fields can also have an undefined orientation of their polarization ellipse, giving rise to polarization singularities called C-points. Recent developments have shown that polarization singularities in nano-photonic structures can be used, for example, to control the directionality [12,13] and intrinsic properties [14] of quantum dot emission.

Such applications typically require high field amplitude as close as possible to the optical singularity. Creating the optical singularities in two dimensions, e.g., using polaritonic [15–18] vortices gives rise to smaller OVs and allows their generation by specifically designed coupling slits, paving the way to new schemes for OAM light–matter interactions [19,20] and Fano resonance phenomena [21]. Creation of such optical singularity points is typically achieved using structurally generated OVs or polarization singularities controlled by nanofabrication and scattering [22]. However, optimal light–matter interaction of this kind requires precise control of the singularity properties, especially its location.

Interestingly, such degree of control can be achieved by utilizing the inherent instability of high-order OVs, which are sensitive to imperfections and tend to break under perturbations into a series of low-order vortices [23]. While this property is often considered a drawback to be contended with (see, e.g., [24]), a specifically designed perturbation may lead to precise control of the newly created singularities.

Here, we demonstrate continuous nanoscale control of optical singularities by engineering the splitting of high-order plasmonic Bessel modes. We vary the polarization state of a plane wave coupled to surface plasmon polaritons (SPPs) through a spiral slit which, in turn, results in controlled interplay between two Bessel modes of different orders. These two degrees of freedom stem from the two-dimensional (2D) polarization state (best expressed by the Poincaré sphere, as portrayed in Fig. 1), generating phase singularities in the out-of-plane field and polarization singularities in the in-plane field, while determining their exact position with deep subwavelength precision. Such precise positioning of optical singularities can be the key to enabling new on-chip light–matter interactions [25], with potential contributions in quantum information processing and communications; and can also greatly increase the range of abilities in particle nano-manipulation [26,27], with important consequences in biology [28] and chemistry [29].

 

Fig. 1. Control over phase singularities. Coupling plane waves into surface plasmon polaritons by a spiral slit allows the mapping of any polarization state, represented by the Poincaré sphere, into the location of plasmonic phase singularities in a 2D plane. The amplitude of the out-of-plane electric field resulting from coupling plane waves of different polarization states is shown along with their position on the Poincaré sphere. The black spots correspond to the phase singularities, with their separation and rotation angle controlled by varying only the polarization state of the incident light. Polarization variation along the longitudinal lines of the Poincaré sphere results in a control over the separation of the OVs, while along the latitudinal lines it results in a rotation of their shared axis. The super-oscillatory nature of OVs enables such control in precision much higher than allowed by the diffraction limit.

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2. METHODS

A. Sample Fabrication

The sample is a thin gold layer, 200 nm thick, deposited atop a 1 mm thick glass substrate using an e-gun (EVATEC ltd). As adhesion coating, we use a 3 nm thick titanium layer deposited below the gold layer. The coupling slit was fabricated using focused ion beam milling (FEI Strata 400s) from the metal side. For example, Fig. 2(b) features a coupling slit with topological charge of $l=2$, where ${\lambda }_{\mathrm{spp}}=633\mathrm{nm}$ is the wavelength of SPPs in the gold–air interface and the diameter of the slit is 16 μm.

B. Near-Field Measurements

We used a scattering (aperture-less) near-field scanning optical microscope (s-NSOM, Neaspec ltd.) for the measurement of the near-field signal, both amplitude and phase [30]. The measurement setup is illustrated in Fig. 2. A 660-nm continuous wave (CW) semiconductor laser (Cobalt) beam was weakly focused to a 50-μm-diameter spot and thus illuminates the sample from the glass side (aka, transmission mode measurement). The sample was placed on a moving stage, where a silicon atomic force microscope (AFM) tip coated with platinum scattered predominantly the out-of-plane electric field component into a detector. The tip vibrates at a frequency of about 250 MHz. The incoming laser beam was split into two optical paths for an interferometric pseudo-heterodyne detection; a beam modulated by a vibrating mirror with a frequency of 600 Hz interfered with the signal scattered from the sample to reconstruct the full electric field information, including the phase. Therefore, the detected signal is demodulated at higher harmonics in order to suppress the background signal scattered from the tip [30]. The resolution ability of the s-NSOM is determined by the size of the tip apex, which is, in our case, 8–15 nm.

3. RESULTS

A. Concept Description

OVs can be created either at the focal plane of helical beams or by propagating surface or guided modes on a 2D plane [15,17,18,31]. OVs of the first type are usually regarded as Laguerre–Gaussian (LG) beams [2], which are a solution of the paraxial wave equation. The second type are 2D Bessel beams, decaying exponentially in the direction normal to the interface. Such 2D OVs were demonstrated in plasmonic structures [15,17,18,31] and silicon-plasmonic high-index waveguides [10,16].

The electric field of such 2D OVs is comprised of the component perpendicular to the surface: $E_z(\rho ,\theta )$ and the parallel to it: $E_{\rho }(\rho ,\theta )\widehat{\rho }+E_{\theta }(\rho ,\theta )\widehat{\theta }$ (also known as the out-of-plane field and in-plane field, respectively),

The surface modes can be excited by means of momentum matching between free propagating light and the guided mode (e.g., plasmons on a metal–dielectric interface [32]) or by either grating or slit coupling. When the slit width is much smaller than the wavelength of the incident light, it practically acts as a polarizer, which couples the electric field components normal to the slit/grating into the surface mode. This leads to an interplay between the polarization states of the incident light and the phase variations along the slit (“source” for the guided modes), which depends on both the shape of the slit and the illumination’s polarization, providing an important degree of freedom in determining the order of surface Bessel modes [15,17].

Consider a spiral slit carved in metal to couple incident radiation into SPP modes [Fig. 2(b)]. This slit is identified by the separation distance between successive turnings which, for plasmons, is $d=l{\lambda }_{\mathrm{spp}}$. Here ${\lambda }_{\mathrm{spp}}$ is the wavelength of the SPP and $l$ is an integer, which can be considered the topological charge of the slit. Coupling circularly polarized light (i.e., angular momentum of $\sigma =\pm 1$) into the guided modes by such a slit results in a Bessel mode whose order (“topological charge”) is the sum of the light’s OAM and $l$, respectively,

This property has been demonstrated in numerous works, from an $l=1$ slit resulting in Bessel modes of orders 0 and 2 [15] up to higher-order OAM modes [16–18]. These observations have indeed shown a ring whose radius increases with the Bessel order, but at the same time, did not provide evidence to a perfect high-order OV. They either lacked the phase information or showed that close to their center, high-order OVs tend to break into a series of single-unity OVs [18,23]. This breakdown is attributed to the lack of a pure state, namely, the high-order Bessel modes are “contaminated” by noise, far-field radiation from the slit, disorder [33], or unwantedly excited lower-order Bessel modes. In plasmonic Bessel modes, low-order modes may be excited due to fabrication imperfections but also because of an impure polarization state. As this phenomenon occurs close to the singularity where the intensity is low, it is typically overlooked by the intensity pattern and can mostly be observed only in the phase structure (see simulation in [15] and phase-resolved measurement in [18]).

 

Fig. 2. Experimental setup. (a) A linearly polarized laser beam propagates through $\lambda /2$ and $\lambda /4$ plates which provide complete control over the polarization state. The beam is then incident upon a metal–dielectric interface and is coupled to the interface via a spiral slit [for example, a scanning electron microscope image of a $l=2$ slit is presented in (b)]. The polarization state is translated to two Bessel modes of different orders, creating a controlled interference pattern. The full field distribution is mapped by phase-resolved s-NSOM showing both the amplitude and phase at 15 nm resolution.

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Surprisingly, this drawback can be utilized to control the splitting of the higher-order Bessel modes, i.e., the location of the newly generated single-unity OVs, by merely controlling the polarization state of the excitation beam, which can be achieved using half- and quarter-wave plates.

A general polarization state can be expressed as a superposition of the two circular polarizations,

B. Experimental Setup

The experimental system is described in Fig. 2. Light at polarization state determined by a quarter- ($\lambda /4$) and half- ($\lambda /2$) wave plates is coupled through an Archimedean spiral slit to plasmonic Bessel modes with relative amplitude and phase

C. Control over Phase Singularities

We map the plasmonic waves using a scattering near-field scanning optical microscope (s-NSOM) that enables phase-resolved mapping of the out-of-plane near field on the metal surface via pseudo-heterodyne interferometric detection [30]. We demonstrate control over the distance between the singularities (radial position) by changing the relative amplitude between circular polarizations, i.e., mainly changing the orientation of the $\lambda /4$ plate. Control over the rotation axis (azimuthal position) is achieved by changing the relative phase between circular polarizations, i.e., mainly changing the orientation of the $\lambda /2$ plate.

Our main results, summarized in Figs. 3 and 4, show the plasmonic field generated by coupling through slits of $l=1,2$, where $l=1$ generates plasmonic Bessel modes of orders 0 and 2 while $l=2$ creates orders 1 and 3. Both amplitude and phase are shown for different ratios between the high- and low-order Bessel beams, obtained for different orientations of the wave plates. The phase singularities are marked by black circles and correspond to the dark spots in the amplitude. Evidently, the polarization state controlled by the wave plates is directly translated to a two-dimensional position of the optical (plasmonic) dislocations with nanoscale precision. Irrespective of the total number of singularities, the two polarization degrees of freedom allow control of only two optical singularities at a time, where the central vortices [e.g., in Figs. 3(g)–3(l) and 4(g)–4(l)] are stationary.

 

Fig. 3. Controlling the distance between OVs. Amplitude (upper row) and phase (lower row) of near-field mapping for $l=1(\text{left columns})$ and $l=2(\text{right columns})$. The orientations of the $\lambda /4$ and $\lambda /2$ plates are indicated above the amplitude images. The singularities are marked by black circles in the phase maps. The distance between the singularities is indicated above the phase images.

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Fig. 4. Controlling the rotation angle of the OVs. Amplitude (upper row) and phase (lower row) of near-field mapping for $l=1$ (left columns) and $l=2$ (right columns). The orientations of the $\lambda /4$ and $\lambda /2$ plates are indicated above the amplitude images. The singularities are marked by black circles in the phase maps. The rotation angle of the singularities is indicated above the phase images.

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The minimal distance between two singularities is limited by the purity of the circular polarization, the quality of fabrication, and an intrinsic separation ($\sim \lambda /17$) caused by the circular asymmetry of the spiral slit (which we verified using simulations–see Supplement 1). The maximal distance between singularities, however, is only limited by the first node of the low-order Bessel mode. Even though Figs. 3 shows specific distances between the optical singularities, it is crucial to emphasize that the control is continuous, with the precision determined by the resolution of the waveplates’ rotation. Namely, the precision can potentially be far better than the $\lambda /12$ demonstrated here and is, in fact, not limited by the wavelength. The underlying reason for this virtually unlimited resolution is that optical vortices are a super-oscillation phenomenon [34,35] and as such, can possess features much smaller than the diffraction limit. The rotational resolution, depicted in Fig. 4, is also limited only by the $\lambda /2$ plate precision.

Experimental results of continuous control over OVs of order $l=1$ and $l=2$ are attached as Visualization 1 and Visualization 2, respectively.

D. Control over Polarization Singularities

The results above were obtained by measuring the evanescent tail of the electric field component normal to the interface, $E_z$, comprised of two different order plasmonic Bessel modes. Being transverse-magnetic (TM) vectorial waves, these modes also contain two field components parallel to the interface, best represented by the radial and tangential in-plane field components $E_{\rho },E_{\theta }$, which can be derived directly from Maxwell’s equations using the out-of-plane field,

The three TM electric field components $E_z$ (measured) and $E_{\rho },E_{\theta }$ (derived) are shown in Fig. 5 for a representative separation between singularity points in $E_z$. Evidently, the singularity points in $E_z$ are at the same positions as the maximal intensity in $E_{\rho }$. Namely, the control over the OVs is translated to controlling 2D hot spots via polarization control. These results are in quite a good agreement with the simulations shown in Figs. 6(a)–6(f) and 6(i)–6(n), from which one can extract the polarization singularities in the in-plane field, known as C-points (see Visualization 3 for simulation of the in-plane field).

 

Fig. 5. Experimental in-plane field. In-plane field components $E_{\rho }$, $E_{\theta }$ (c)–(f), (i)–(l) derived out of the measured out-of-plane field component $E_z$ (a), (b), (g), (h) for $l=1$. Two polarization states examined: ${\vartheta }_{\lambda /2}=160$, ${\vartheta }_{\lambda /4}=0$ (a)–(f) and ${\vartheta }_{\lambda /2}=175$, ${\vartheta }_{\lambda /4}=0$ (g)–(l). Amplitude (upper row) and phase (bottom row) presented.

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Fig. 6. Analytical in-plane field and control over C-points location. Numerical simulations fitted to two polarization states depicted in Fig. 5 respectively. Amplitude (upper row) and phase (bottom row) of the out-of-plane field component $E_z$ and the in-plain field component $E_{\rho }$, $E_{\theta }$ are presented. The polarization states expressed by the polarization ellipse orientation $\alpha$ and handedness $ϵ$ of the two states have also been calculated [(g), (h) and (o), (p), respectively]. The distance between the C-points, marked by black circles, is indicated above the $\alpha$ image.

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The polarization state of a three-dimensional vectorial field is defined on a plane, and therefore can be described by two parameters, creating, in the general case, an ellipse. One may characterize this polarization ellipse using its handedness $ϵ$ (the eccentricity of the ellipse) and its orientation angle $\alpha$, both of which can be expressed using the in-plane electric fields [36] (see Supplement 1). C-points are created in the special case where the orientation of the polarization ellipse is undefined (circular polarization, also called a singularity in polarization space).

Similarly to phase singularities, C-points also carry a topological charge, defined as the times it takes the orientation angle $\alpha$ to accumulate $2\pi$ in a closed path around the C-point in a counterclockwise manner. The allowed values are at multiples of $\frac{1}{2}$ [37].

Figures 6(g), 6(h), 6(o), 6(p) exemplify the control over the C-points’ positions by deriving the polarization ellipse orientation $\alpha$ and handedness $ϵ$ from the simulations fitted to experimental results. Points of undefined orientation $\alpha$ can be seen in Figs. 6(h) and 6(p), which feature polarization handedness $ϵ=-1$ (that is, circular polarization) and TC of $\frac{1}{2}$. The two measurements differ by a slight shift in rotation axis and a $\lambda /5$ shift in separation. The resolution with which the position of C-points can be controlled is the same as the out-of-plane field singularities.

4. CONCLUSIONS

In conclusion, we demonstrated continuous control over the location of both phase and polarization optical singularities in 2D space at deep subwavelength resolution by simply changing the polarization state of light incident on a fairly common plasmonic structure. Depending on the sensitivity of the setup (precision of wave plate rotation), our method can allow control with sub-10-nm resolution. This method is relatively simple, since it does not employ wavefront shaping techniques [38] or require multiple wavelengths [26].

The concept demonstrated here can find applications in many fields, e.g., super-resolution and localization microscopy (by scanning the optical singularities over emitting objects) and particle nano-manipulation (defining specific trajectories by generating and moving optical vortices in different locations [27] and controlling the velocity of trapped particles as a means to create optically driven nanomotors).

Coupled with tight light confinement (for example, via short-wavelength hybrid Si-plasmonic waveguides [18]), it could become a useful tool for manipulation of sub-100-nm particles and even for engineering new light–matter interactions on a chip, potentially beyond the dipole approximation [25]. Another advantage of using such a platform is the large in-plane field (relative to that of surface plasmons), enabling a much stronger influence of the C-points in controlling radiation from quantum emitters.