Optical vortices in fiber

1 Introduction

In the last decade, perhaps the most extensively studied complex beam-shape of light is the class of vortex beams, characterized by a dark hollow center, which possess phase or polarization singularities. These beams have found a variety of applications in fields as disparate as high-field science and quantum optics, to nanoscale imaging. The bulk of the work with vortex beams has been conducted with free-space beams and elements. Recently, fiber means of generating and propagating these beams has become feasible, and this article reviews advances in fiber manipulation, control and applications of vortex beams. Section 2 is a brief overview of vortex beams – this is, by no means, an exhaustive review of vortex beams themselves, since the intent of this article is to describe developments in the realm of optical fibers in this regard, but this section elucidates the key properties of vortices that will be useful to understanding related fiber modes thereafter. For more comprehensive reviews on vortex beams themselves, the reader is directed to several exhaustive reviews in this subject, including [1, 2].

Following the brief introduction of vortices, Section 3 describes fiber/waveguide modes and how they are related to vortices. In Section 4, we discuss a new class of optical fibers that have made the generation and propagation of vortices possible. Sections 5 and 6 describe detailed optical characteristics of these fibers, and Section 7 reviews recent experiments and applications with them. We summarize the review in Section 8, and offer some thoughts on how the development of fibers supporting vortices could impact the multitude of areas in which optical vortices are finding applications.

2 Vortex beams

2.1 Polarization vortices (vector beams)

Plane waves and Gaussian beams are characterized by spatially uniform polarization orientations. This is in contrast to cylindrical vector beams of the kind shown in Figure 1A and B. Such beams are called polarization vortices, because their polarization is completely undetermined at the center of the beam, leading to a dark central region. Some key differences between plane polarized Gaussian beams and polarization vortices can be visualized in a physically intuitive fashion by considering their focussing properties when passing through a high-NA lens. Figure 1C and D schematically illustrate the focussing of a Gaussian beam and radially polarized beam by a high-NA lens, respectively. A simple ray-tracing schematic shows that, at focus, the polarization orientation of the Gaussian beam is highly complex, leading to self-aperturing effects. In contrast, the contributions at focus from the different spatial portions of the radially polarized beam constructively interfere to create a very unique distribution of light at focus [3]. The electric field vector actually points in the direction of beam flow, thus leading to a singular solution at focus, where there is a very high energy density, but no flow (Poynting vector S_z) associated with it in the beam propagation direction. These beams have found several uses, such as realizing super-resolved spots [4], plasmonic nano-focussing [5], single-molecule spectroscopy [6], metal machining [7], and they are considered to be potentially useful for laser-based electron and particle acceleration [8].

Figure 1

Images depicting the intensity profile as well as polarization orientations (arrows) of (A) radially and (B) azimuthally polarized beams; (C, D) Focusing properties with high-NA lenses for different beam shapes; (C) Gaussian beam results in complex polarization, while (D) a radially polarized beam yields an intense, well-defined electric field along the optic axis.

2.2 Phase vortices (OAM beams)

It has been well known for about a century, that paraxial light beams carry spin (polarization) as well as orbital angular momentum (OAM). Allen and coworkers [9] demonstrated in 1992 that commonly encountered Laguerre-Gaussian beams with helical phase given by e^iLφ, where L is topological charge and ϕ is the azimuthal angle, carry OAM of L•ħ per photon, and this has spurred widespread interest across a range of scientific and technological disciplines. These beams have been applied to optical tweezers [10, 11], atom manipulation [1, 12], nano-scale microscopy [13], as well as communications. Several classical [14] and quantum [15] communications experiments have exploited the inherent orthogonality of OAM modes in free-space by multiplexing information in this additional degree of freedom, thereby increasing the capacity of free-space communications links.

Figure 2 shows a schematic representation of the helical wavefront propagation of an OAM beam. Beams with non-zero OAM necessarily have a dark central spot since the phase is undefined at the center – this has led to their additional nomenclature as phase vortices. Common methods used for creating them include spiral phase plates [16], computer-generated fork holograms [17], and most recently, free-space mode sorters [18].

Figure 2

Intensity (white image in black background), (helical) phase (green) and spin (or polarization state, shown as curved arrows) of beams carrying OAM of L and spin of S: (A) a spin and OAM aligned beam; (B) a spin and OAM anti-aligned beam; (C, D) Gaussian beams with L=0 and with two degrees of freedom for spin.

Fiber generation and propagation of phase or polarization vortices has recently become possible, and the rest of this review will describe recent developments in this regard.

3 Fiber generation and mode stability

In its simplest form, ignoring polarization, the wave equation for the transverse electric field e_t, in a step-index fiber is given by

where n is the refractive index of the waveguide, k is the free-space wavevector (given by 2π/λ, λ is the wavelength), and

Figure 3

Modes of a step index fiber in the (A) scalar approximation, and (B) full vector solutions. Vertically stacked lines and associated arrows show relative n_eff for each solution: (C, D and E) Vector mode solutions, as in (B), but showing mode degeneracies and polarization orientations.

where β is the propagation constant of each individual vector mode solution, and all other terms were previously defined.

Figure 3B shows the corresponding intensity distributions for

Inspection of the allowed solutions shown in Figure 3D makes it immediately apparent that fiber means of generating polarization vortex states, such as the azimuthally or radially polarized beams using the TE₀₁ or TM₀₁ fiber modes, respectively, appears feasible. However, the near degeneracy of a multitude of modes other than the desired mode also implies that fiber means of generating cylindrical vector beams requires robust mode selection techniques. Indeed, various mode control techniques have been applied to few-mode fibers (i.e., fibers supporting a few – 6 to 10 – modes) with great success. A free-space Laguerre-Gaussian beam has been used to excite a mixture of vortex states in a 35-cm long fiber held rigid and straight, following which free-space polarization transformations with a combination of waveplates yields a pure polarization vortex [20]. A variant of this approach [21] uses pressure on the fiber itself to induce wave rotation such that the output interference pattern (resulting from the coherent combination of multiple modes) yields the desired cylindrical vector beam. Another approach to producing azimuthally polarized light is to use a fiber grating in conjunction with a free-space etalon to provide cavity feedback only to the desired mode [22]. Alternatively, mode conversion before coupling the desired beam into fibers has resulted in the propagation of high-power fs pulses of the radially polarized mode in short lengths (~20 cm) of hollow-core fibers [23]. Some of the lowest-loss and compact means to obtain vortices with high modal purity directly from fibers have involved inducing the required mode transformations in the fiber itself. This has been demonstrated with fiber gratings [24] or adiabatic mode couplers [25]. In both cases, immediately after the mode-converter (i.e., within <3 cm of the rigidly held fiber), a radially or azimuthally polarized beam is obtained. In addition to the aforementioned passive generation techniques, few-mode fibers that have been doped with rare-earth elements (e.g., Yb, Er) to provide gain have been used to achieve high mode selectivity in the desired mode – forcing stimulated emission in the desired mode (via the use of mode selective elements) further enhances the mode purity of the output, while also providing for high-power lasers with vector beam outputs [26–28] (as mentioned in Section 2, high-power vector beams have several applications in metal machining and, potentially, electron and particle acceleration). The aforementioned reports are, by no means, an exhaustive list of techniques published on fiber generation of cylindrical vector beams, but are illustrative of the most common techniques employed.

While Figure 3 and the associated discussion described the fundamental fiber modes in the weakly guiding approximation, none of the modes illustrated thus far have phase vortices which, as mentioned in Section 2, have a characteristic helical phase and thus carry OAM. However, this is just an artifact of the coordinate system in which we solved the eigenvalue equation. For instance, in the coordinate system that we chose, the HE_L+1,m and EH_L-1,m (L>1) modes have the following transverse electric field distributions [19]:

where F_L,m(r) is the radial field distribution of the corresponding scalar mode (LP) solution, and ϕ is the azimuthal coordinate. Since the even and odd solutions of each of these modes form a degenerate pair, we may equivalently represent the eigenmode as a linear combination of the degenerate pairs in another coordinate system of our choice. Specifically, for one choice of coordinate systems, we obtain:

where

Equations 3 and 4 describe only the L>1 modes, which, as mentioned earlier, exist in pairs of two degenerate states. The fundamental L=0 mode may also be described with these equations by noting that there are no EH modes associated with the L=0 state (and thus only the HE_1,m solutions of equations 3 and 4 are relevant). Nevertheless, we again see that they may be written in linear combinations that yield SAM, S=±1 states with OAM, L=0, which are uniformly polarized, much like the fundamental Gaussian mode solutions in free space.

Finally, we turn our attention to the L=1 modes, which are uniquely different in fibers from all other modes. The analytic form of these vector modes are adequately described by equation 3 if one substitutes the designation even EH_0,m with TM_0,m, and odd EH_0,m with TE_0,m. However, crucially, unlike the L>1 case, these two modes (TE_0,m and TM_0,m) are not degenerate. Thus, while one may use the linear combinations of the HE modes in the L=1 set to yield OAM modes (as was done for the L>1 modes), the same cannot be done with the TE and TM modes. So, we explicitly write the solutions for the L=1 mode in the OAM basis set as follows:

Crucially, in analogy to the polarization vortices in a fiber, OAM states are formed from the same LP_L,m mode groups, albeit with different true vector modes and mode combinations. Likewise, several of the fiber generation techniques mentioned earlier can also, in principle, be used to generate OAM states in fiber. In addition, given the helical symmetry of these modes, helecoidal gratings in photonic crystal fibers [29], and helically phased inputs into multicore fibers [30] have had success in producing states that resemble the first non-zero OAM state in short (<1 m) fibers. One of the lowest loss, high purity techniques to generate OAM in fiber remains acousto-optic gratings [24] mentioned earlier. Akin to the use of doped fibers to achieve radially polarized outputs, mJ energy level lasers with OAM output have been realized using pressure induced mode transformations in a laser cavity [31].

A common feature of all these techniques is that they work with fibers in which the desired mode is almost degenerate with the other modes within the same mode group. Thus, careful alignment of mode selective elements is required in the case of intra-cavity realizations of the vector beams. For the cases where extra-cavity mode conversion is used (fiber gratings [24], nonadiabatic couplers [25]), it is important that any fiber after mode conversion be of limited length. The reason such care must be exercised is that, in conventional fibers, the n_eff splitting between modes within a mode group is much too small to isolate the modes from one another, and the elements of each LP set randomly share power due to longitudinal inhomogeneities caused by fiber bends, geometry imperfections during manufacture, and stress-induced index perturbations. Examples of the resulting unstable combinations are illustrated in Figure 4, in which the top row shows true vector mode solutions for a step index waveguide, and the bottom row shows experimentally recorded non-azimuthally symmetric intensity distributions corresponding to various linear combinations of these vector modes. Since this pattern is formed by interference of two modes with slightly different n_eff, this pattern is itself not stable and rotates if the fiber is perturbed, much like a speckle pattern in a multimode fiber would change due to fiber handling. Note the striking similarity between the obtained interference pattern and the LP₁₁ mode obtained from the scalar mode solutions (see Figure 3A). Experimental observations of this pattern are often claimed to be observations of the LP₁₁ mode, but care must be exercised when using the nomenclature of eigenmodes when describing these states. A true eigenmode in a fiber is lengthwise invariant – that is, it would not change in size or shape as it propagates in a fiber. The LP₁₁ state, on the other hand, would switch its orientation as it propagates, even in a perfectly unperturbed, straight fiber. This is because constructive and destructive interference (see the “+” and “–” signs in Figure 4) lead to an LP₁₁ pattern with 90° rotated intensity patterns, and light propagation in the constituent true (vector) modes in a fiber will necessarily lead to such beating with a beat length of λ/Δn_eff, where λ is the wavelength of operation and Δn_eff is the difference in n_eff between the vector modes. Although the mathematical description above is only for the circularly symmetric fiber case, experimental observations of the vortex states in the case of non-circularly symmetric fibers (such are photonic crystal and multicore fibers) [29, 30] confirm the same intermodal coupling phenomenon.

Figure 4

Modal intensity patterns for the first higher order mode group; (top row) simulated true vector representation of the eigenmodes; (bottom row) the resultant, experimentally measured unstable intensity patterns due to mode mixing between the top row of eigenmodes.

Finally, we note that fiber generation techniques for realizing OAM beams or polarization vortices include a subset of free-space generation techniques that are implemented at the facet of a single mode fiber (SMF). This is feasible because the output of SMF is almost Gaussian shaped, and hence shares many characteristics with conventional free-space beams. By exploiting advanced lithographic techniques, axicons [32] and nano-structured metal films [33] have been incorporated on fiber facets to yield vortex beam outputs from a fiber.

In summary, the fiber generation techniques mentioned in this section provide for a versatile and powerful means of obtaining a variety of vortex beams. However, they do not allow propagation of the generated mode over reasonable lengths. In the following section, we will consider fiber designs that would enable vortex propagation in addition to generation.

4 Vortex fiber design

The vortex mode instability problem in fibers arises from the near-degeneracy of the constituent vector modes described in Section 3. For the purposes of the remainder of this review, we will restrict our analysis to the instability within the LP₁₁ mode group, which comprises the TM₀₁ (radially polarized), TE₀₁ (azimuthally polarized) and HE₂₁ (OAM) modes. We note, in passing, that it is conceivable that the same design principles may be applied to modes with higher OAM (L>1). It is well known that coupling between co-propagating modes decreases as Δn_eff, the difference in n_eff between modes, increases [34]. There have been multiple attempts over the years to understand the relationship between Δn_eff and mode coupling, especially in the context of understanding microbend losses in SMF (where the loss mechanism is a cascaded process starting with the guided LP₀₁ mode coherently coupling to a quasi-guided higher order mode (HOM), which is then radiated at the silica-cladding – polymer-jacket interface) [34]. However, since this problem intimately depends on the electric field overlap between the modes of interest, and the form, symmetry and strength of perturbations on a fiber, only rough phenomenological rules have been developed for a limited number of cases. One such rule, which has had ample experimental confirmation, providing for a degree of confidence, is that Δn_eff>10^-4 yields polarization maintaining (PM) fibers in which the orthogonal polarizations of the LP₀₁ modes remain stable for lengths scales exceeding 100 m. This rule has been successfully implemented in the context of designing large mode area fibers – a counterintuitive but universal phenomenon, discovered recently [35], reveals that Δn_eff between LP_0m and LP_1m mode increases with mode order m, and thus higher order HOMs are actually more stable than lower order modes, yielding a scalable pathway for increasing the mode area of fibers, of utility in high-power applications. In these HOM experiments, it was experimentally confirmed that Δn_eff>10^-4 is a reasonable proxy for mode stability [36].

The design problem, therefore, is to find the means to achieve Δn_eff>10^-4 between distinct vector modes within the LP₁₁ mode group by choice of fiber index profile n(r). The key physical intuition behind the solution is the elementary observation that, during total internal reflection, phase shifts at index discontinuities (index steps) critically depend on an incident wave’s polarization orientation, and the propagation constant of a mode represents phase accumulation. The translation of this physical reasoning into mathematical expressions amenable to optimization can be achieved with a full vectorial solution of the Maxwell equations. However, more analytical insight is obtained by means of a first-order perturbative analysis. First, we calculate the scalar propagation constants of the LP₁₁ mode group (identical in the scalar approximation) following which, the real propagation constants are obtained through a vector correction, given by [37]:

where r is the radial coordinate, F₁₁(r) is the radial wavefunction for the scalar LP₁₁ mode, β₁₁ is its unperturbed (scalar) propagation constant, a is the size of the waveguiding core, n_co is the core refractive index, Δn(r) is its refractive index profile relative to the index of the infinite cladding, and Δn_max is the maximum index of the waveguide (see schematic in Figure 5). Equation (6) embodies the physical intuition of the relationship between the field profile of a mode and its propagation constant. The search for a solution that substantially separates the propagation constants of the TE₀₁, TM₀₁, and HE₂₁ modes boils down to a search for a waveguide that yields high fields (F₁₁(r)) and field gradients (∂F₁₁(r)/∂r) at index steps. Furthermore, this separation, and hence mode stability, grows with the magnitude of the index step Δn_max.

Figure 5

Normalized refractive index profile (Gray background), and corresponding mode intensity for the scalar LP₁₁ mode (red) for (A) a conventional step-index fiber, and (B) the novel ring design. Intensity |E(r)|², rather than electric field E(r), plotted for visual clarity – field reverses sign at r/λ=0; (C) Measured refractive index profile (relative to silica index) for fabricated fiber, and corresponding LP₁₁ mode intensity profile; (D) Effective index for the 3 vector components of the scalar LP₁₁ mode for fiber shown in (C). n_eff of radially polarized (TM₀₁) mode separated by 1.8×10^-4 from other modes.

We now describe a fiber that was designed and fabricated to achieve a large vector separation, and hence stable propagation of vortex modes. We start by first considering a conventional fiber (index profile shown in Figure 5A). In light of equation (6), the problem with obtaining stable vortex modes from such fibers becomes immediately apparent – increasing Δn_max to increase mode separation does not help because the mode becomes increasingly confined and the field amplitudes dramatically decrease at waveguide boundaries. Using the intuition gathered from (6), we conclude that a waveguide whose profile mirrors that of the mode itself – i.e., an annular waveguide resembling an anti-guide – would be more suitable for maximizing Δn_max while also maximizing field-gradients at index steps. This is schematically illustrated in Figure 5B.

For this class of designs, increasing Δn_max does not automatically lead to reductions of F(r) and ∂F(r)/∂r at index steps. The refractive-index profiles shown in Figure 5A and B are plotted in normalized units to elucidate the fundamental differences between the two profiles. The simulations assumed an identical Δn_max of 0.025 for both cases, and their lateral dimensions were adjusted to obtain similar cutoff wavelengths (~2600 nm) for the 1st higher-order antisymmetric modes. This ensures that at the simulated wavelength (1550 nm), the modes (intensity profiles shown as red curves) in both waveguides are similarly well-confined or stable, and hence propagate with roughly similar losses. Calculations indicated that the n_eff of the radially polarized (TM₀₁) mode is separated from its nearest neighbor (the mixed-polarization HE₂₁ mode) by ~10^-5 for the step index fiber and by ~1.6×10^-4 for the new design. This satisfies the design goal to have Δn_eff >10^-4.

Figure 5C shows the measured refractive index profile of the fiber preform that was fabricated to test this concept. Note that this fiber possesses the high-index ring as demanded by the design depicted in Figure 5B. However, it also has a step-index central core, as do conventional fibers. This core does not sufficiently perturb the spatial profile of the LP₁₁ mode of interest, and hence does not detract from the design philosophy illustrated in Figure 5B. Instead, it allows for the fundamental mode to be Gaussian-shaped, enabling low-loss coupling.

Figure 5D shows the n_eff for the TM₀₁, HE₂₁, and TE₀₁ modes in this fiber, measured by recording grating resonance wavelengths for a variety of grating periods (for more details, see [37]). The n_eff for the desired TM₀₁ mode (radially polarized mode) is separated by at least 1.8×10^-4 from any other guided mode of this fiber. Note that this value is actually larger than that of the theoretical schematic in Figure 5B, primarily because the fabricated fiber effectively had a higher Δn_max owing to the down-doped region between the core and the ring. For conventional fibers, the 3 curves would be indistinguishable in the scale of this plot.

Since the first demonstration, in 2009, of “ring” shaped cores for stable propagation of optical vortices in fibers [37], several fiber designs with enhanced mode separation, broadband characteristics, and designs in which the dispersion as well as mode separation can be controlled, have been revealed [38–41]. Although the designs are very efficient in generating large splittings of the effective index they require large index contrasts, field and field gradients, which may lead to high scattering losses [42]. Given significant current interest in vortex mode supporting fibers, it is conceivable that new design classes of fiber may provide further improvements, though, as of this writing, most of the designs proposed to date follow the high-index ring design architecture described in this section.

5 Fiber propagation of vortices

Figure 6A shows the experimental setup used to generate vector beams in this fiber. The fiber-input is spliced to SMF so that the light entering it is in the conventional fundamental mode. Thereafter, a fiber grating, comprising periodic microbends, converts the fundamental mode to the desired higher order vector mode. Fiber gratings are periodic perturbation introduced in the core of a fiber that are well known to resonantly provide mode conversion between propagating modes of a fiber [43]. When the grating wavevector is matched with the difference in propagation constants between different modes, one obtains mode conversion with experimentally measured efficiencies [44] exceeding 99.99%, and by proper fiber design, these gratings may be made broadband [45] with bandwidths exceeding 500 nm. For a review of the physics and applications of fiber gratings and how their mode-converting properties are related to the propagation characteristics of modes, see [43–46]. Figure 6B shows the mode conversion spectra when the grating period Λ, is 800 μm – the measured conversion efficiency exceeds 99.8% at the respective resonance wavelengths λ_res, for each vector mode. The HE₂₁ resonance can be excited with any state of polarization (SOP) of light entering the microbend grating, but for the TE₀₁ or TM₀₁ modes excitation is maximized only when the input SOP is parallel or perpendicular, respectively, to the plane of microbend perturbation. Thus, we can use input SOP as well as grating period to choose the mode we excite, as well as the wavelength at which we do so.

Figure 6

(A) Experimental setup – the ring fiber is spliced to SMF (bottom branch) for spectral measurements shown in (B), or cleaved and imaged on camera (top branch) for measurements shown in (C); (B) Measured grating resonance spectra for coupling from fundamental LP₀₁ mode to desired antisymmetric mode – efficiency >99.8%; (C) Experimentally recorded near-field images for the radially polarized (TM₀₁) mode (top) and azimuthally polarized (TE₀₁) mode (bottom). Clean annular intensity profile for both. Image rotation with polarizer in beam path consistent with expected polarization orientation for the two modes, and confirms the polarization state of the two beams.

Figure 6C shows experimentally recorded near-field images of the fiber output when the grating is tuned either to the radially polarized (TM₀₁), or the azimuthally polarized (TE₀₁) resonance. All images were obtained after mode propagation over fiber lengths exceeding 20 m. Both beams have annular shapes and appear to be remarkably pure and stable – a condition that is maintained as we perturb the fiber (with bends and twists of radii as small as 5 mm). With the polarizer in the beam path, only the projections of the mode that are aligned with polarizer are transmitted, leading to the LP₁₁-like intensity profile that rotates as the polarizer is rotated. The insertion loss, measured with a power meter, is <5% (~0.2 dB), and intensity measurements along the azimuth of the annular intensity profiles revealed variations of <1% (0.05 dB).

We now turn our attention to the generation of OAM modes in this fiber, which, based on the discussion in Section 3 (equation 3 and 4, and associated discussion), is feasible by exciting a pair of (degenerate) HE₂₁ modes. Note that OAM generation in a fiber requires that the two HE₂₁ modes be generated with the appropriate phase shift – with microbend gratings, this is readily achieved with a circularly polarized fundamental (LP₀₁) mode input at the grating (each polarization automatically excites the two orthogonal HE₂₁ modes of the fiber with the correct (π/2) phase shift, as demanded by equation 4 for OAM generation. A schematic of the experimental setup is shown in Figure 7A, and B shows the near field intensity at the output fiber facet after the OAM mode has propagated through 20 m of the vortex fiber. A clear donut shaped intensity profile is indicative of a pure mode. However, to confirm that it is an OAM mode, we also need a measurement of the (helical) phase. This is achieved by observing the interference of this beam with a uniformly polarized reference beam, obtained from an SMF whose path length is matched to ensure that the two beams coherently interfere. When the reference beam is slightly diverging (i.e., it has the form

Figure 7

(A) Experimental setup used to excite and characterize OAM states; (B) Mode intensity image at output of 20-m long vortex fiber indicating stable transmission of a vortex states; (C–F) image of the vortex fiber output when interfered with an expanded (C and D) or tilted (E and F) Gaussian-reference beam; Spiral rotation (C and D) or fork orientation (E and F) depend on OAM state at output.

Next, we show that, once an OAM state is generated in this fiber, we can control its topological charge (L), by simple, off-the-shelf, commercial fiber components. It can be shown that a linear combination of two L=±1 OAM modes will have a total OAM of topological charge that lies between -1≤L≤1 [47]. By adjusting paddles on a commercial polarization controller mounted on the vortex fiber (Polcon2 of Figure 7A), we are able to tune the output OAM state from L=-1 to L=+1. This is possible because Polcon2 serves to control the relative phase between the

6 Mode purity – measurement and characterization

The previous Section (5) described the vortex fiber and its output characteristics qualitatively. Methods for estimating the purity of these states in a fiber are necessary for characterizing both the (de)multiplexing devices and the fibers themselves. In this section, we describe a quantitative method to ascertain mode purities, and then apply it to characterize the length dependent purity of OAM modes in these fibers.

6.1 The “ring” technique – Spatial domain characterization

Equation 4 and the associated discussion in Section 3 showed that these vortex states are linear combinations of degenerate pairs of true guided modes, and therefore, are themselves true guided modes, or eigenstates of the fiber. An important feature of the solutions (4) is that the orbital and spin degrees of freedom are decoupled: the phase terms exp(±iLϕ) and polarization terms

always appear in product form. Physically, this means that the phase fronts have the spiral angular dependence of free-space Laguerre-Gaussian beams, which gives rise to values of OAM of ±Lħ, and the fields have spatially uniform circular polarization, which give rise to spin angular momentum (SAM) of ±ħ. Practically, this decoupling of spin and orbital degrees of freedom means that existing mode-sorter technology, for example based on spatial light modulators or specially designed phase masks, that was developed for free-space applications may also be applied to (de)multiplexing with optical fibers.

In order to analyse the state present at the fiber output, we may project it onto various polarization states, as well as interfere it with a reference beam. The goal is to use the recorded output spatial intensity patterns to uniquely determine the mode content, as given by the amplitudes |γ_i|², in the expansion of the transverse electric field E_t(r,ϕ) at the fiber output

where the sum is over the mode index i. The six fields

represent the vector modes either in the standard basis

TE_0,1, and TM_0,1, or alternatively, the vortex basis given in Eq. (5). From this representation, it is evident that a spatially uniform circular polarizer of positive (resp. negative) helicity will project onto the “+” (resp. “-”) vortex states dramatically reducing the number of interference terms that must be analysed to deduce mode purity.

Figure 8 shows the experimental setup that we used to control and analyse the OAM states of a vortex fiber [48]. Using standard single mode fiber (SMF), a 50-nm-wide 1550-nm LED, and a narrowband CW tunable laser (Agilent 8168F) were multiplexed into a 20-m-long vortex fiber. Thereafter, using a microbend grating (40-mm length, 475-μm period), with only the LED source turned on, we obtained 18-dB of mode conversion from the input LP₀₁ mode to the desired HE₂₁ modes (see the transmission spectrum in Figure 8B). Next, we switched the source to the laser, tuned to the resonant mode-conversion wavelength (1527 nm). The vortex fiber output was imaged onto a camera (VDS, NIR-300, InGaAs).

Figure 8

(A) Experimental setup for characterizing mode purity using the “ring” technique; (B) Grating resonance spectrum used to deduce HE₂₁ mode conversion level; (C) Camera image showing L=1; S=1 state; simultaneous projection of all spin polarized projections enables mode-purity measurement with a single camera shot.

In order to observe the state present at the fiber output, and to analyze its purity, we projected the fiber output onto the left circular (LC) and right circular (RC) polarization states (Figure 8C). In addition, to observe the phase of the beam, the interference of the vertical (V) polarization projection with a reference beam is recorded. The “V+ref” image in Figure 8C shows a clear spiral interference pattern indicative of the output mode carrying OAM. Using a combination of non-polarizing beam splitters (NPBS), quarter wave plates (QWP) and polarizing beam displacing prisms (PBDP), we devised a setup capable of recording these projections in one camera shot.

When most of the light is confined to one of the modes (as is the case in this experiment, where a single OAM state was excited by means of a fiber grating) evaluating the positive and negative SAM states’ intensity projections on the ring of peak radial intensity r₀ reveals valuable information about the modal content. The result of the projection onto the, say, positive SAM state, can be obtained by combining Eqs. 5 and 7:

The detected power then becomes

where DC is a constant offset, and Δ_1,2 are coefficients of a Fourier series of azimuthally varying intensity patterns, and Δϕ₁ and Δϕ₂ are arbitrary constants. The form of this interference is consistent with the fact that the interference between the three modes of the same SAM can, at most, yield cos(2ϕ) type azimuthal variations (due to the interference of, say, the S=+1; L=+1 (spin aligned HE) modes and the S=+1; L=-1 (spin anti-aligned, circularly polarized linear combination of the TE and TM modes). Deducing these Fourier coefficients is sufficient to recover the mode powers |γ_i|², from which mode purity, quantified by the multipath interference (MPI) level [defined as MPI=10 log₁₀(|γ_i|²/P_total)] [49, 50], can be ascertained. Figure 9A shows an example of a measurement of |P₊E(r₀,ϕ)|²; the azimuthal intensity variation of the image that occurs due to interference between the vortex states. Figure 9B shows an example of the Fourier series analysis used to recover the DC and Δ_1,2 parameters, and Figure 9C illustrates the powers of the extracted modes. Note that mode purities exceeding 20 dB may be measured with this technique.

Figure 9

(A) Azimuthal intensity profile of P₊ (i.e., left circularly polarized) projection for radius r₀; (B) Fourier series coefficients for profile in (A); (C) Extracted modal power contribution, indicating the ability to measure ~20-dB level mode purities.

6.2 Long-length propagation

We now use this ring technique to study lengthwise mode purity in vortex fibers. The setup, which is similar to that shown in Figure 8, comprises a tunable narrowband laser that is spliced to the input of the vortex fiber and conversion from the fundamental mode into the OAM mode is achieved using a microbend fiber grating, as before. Using polarization controllers before the microbend grating, mode conversion efficiencies of ~97% to the L=±1 modes was obtained. At the other fiber end, after 1.1 km propagation, the mode was imaged or interfered with a reference beam, in order to perform the modal purity analysis described in the previous section. The measurement is repeated for multiple wavelengths over a narrow spectral range, not to deduce wavelength dependent mode coupling behavior (which is neither expected nor was it observed, over small bandwidths), but to use wavelength as a proxy for randomizing intermodal phase differences. Thus, we obtain average MPI values as well as error bounds for the measurement. Finally, a cutback measurement was implemented in order to study the length dependent mode coupling phenomena in these fibers.

Figure 10A shows MPI measured by the ring technique over a bandwidth of 0.6 nm, for a 360-meter long vortex fiber. Figure 10B shows a cutback study of the mode purity. The experiments reveal that, even after starting with imperfect input coupling (with mode purities of ~97%), OAM purity decreases only by ~10% over a km of propagation.

Figure 10

(A) MPI measurement using ring technique for 360-m long propagation of OAM states in vortex fiber; (B) Cutback measurement to show the lengthwise evolution of mode purity of the OAM states.

7 Applications of vortex fibers

The vortex fibers described in the previous sections enable the generation and propagation of OAM as well as polarization vortex beams over long lengths of fiber. In this section, we discuss some experiments that have been conducted with these fibers.

7.1 Polarization-maintaining fibers

Polarization maintaining (PM) fibers are critical elements for several applications, particularly in the realm of fiber lasers, since they allow single polarization outputs that do not drift with time, in the presence of perturbations. The fundamental (HE₁₁, or equivalently, LP₀₁) mode in an optical fiber is two-fold degenerate, with the two modes being orthogonally polarized (equivalently, with SAM=± ħ, or circular polarization

Just as with the degenerate OAM-carrying HE₂₁ modes, this means that propagation in fiber with bends and twists results in coupling between the two modes, resulting in a linear combination of the

modes at the output (also called an elliptically polarized state).

PM fibers lift this degeneracy sufficiently to enable mode coupling free propagation in only one of the two orthogonal polarizations of the HE₁₁ mode. The only means of doing this is by breaking the circular symmetry of the fiber waveguide itself. This is typically achieved either by realizing oval cores or by adding stress rods in the cladding, which leads to birefringence – the so-called PANDA fibers. As a rule of thumb, the separation in n_eff between the two orthogonally polarized HE₁₁ modes must be >10^-4 in order to achieve polarization maintaining operation. Indeed, as mentioned in Section 3, the Δn_eff>10^-4 metric for separation of vortex modes had its genesis in the success PM fibers have had with this metric.

The fact that distinct cylindrical vector beams in the vortex fiber – namely, the TE₀₁ and TM₀₁ modes – are not degenerate with any other mode brings up the intriguing prospect of using them to obtain PM operation from fibers. We test this hypothesis with the following experiment. Figure 11A illustrates the setup used to quantify the polarization extinction afforded by these modes [37]. The distinction from earlier setups is that the vortex fiber has a microbend grating at both the input as well as output. Since gratings are reciprocal devices, light enters the vortex fiber, gets converted to the desired radially polarized (TM₀₁) mode by the first grating, and after propagation over 20 m in this mode, gets reconverted to a conventional, Gaussian-like, fundamental mode of an SMF by the second grating. This facilitates using conventional fiber optic measurement equipment to measure polarization extinction ratios (PER) and SOPs. We record the SOP of this setup on a Poincare sphere (Figure 11B) with a commercial polarization analyser while perturbing the fiber with multiple twists and bends with radii-of-curvature down to 1 cm. The top part of Figure 11B shows a reference measurement on a conventional SMF – as expected, perturbations on SMF result in an output that traverses virtually every point on the surface of the Poincare sphere. In contrast, a similar measurement with the vortex fiber shows only small changes in the SOP, as represented by the limited “smearing” of the blue trace on the bottom sphere in Figure 11B. The PER, measured by recording the minimum and maximum transmitted power with a rotating polarizer at the SMF output of the bottom setup of Figure 11A, is ~28.7 dB, corresponding to a modal purity level of 99.8%.

Figure 11

(A) Setup to compare PM characteristics of the polarization vortex modes in the ring fiber and conventional SMF. Input and output gratings on the fiber ensure that light entering or exiting the setup is conventionally polarized (i.e., spatially uniform, as in Gaussian beams), thus facilitating measurements with conventional fiber-optic test sets (such as polarization analyzers); (B) Poincare sphere representation of output SOP, to measure polarization-state variations as both fibers are perturbed. Red traces represent evolution of SOP on the front surface of the sphere (as viewed in the fig.), while blue traces show SOP states on back surface of the sphere.

Unlike conventional PM fibers, using the TM₀₁ mode of this fiber does not require breaking the cylindrical symmetry of the fiber. This, in turn, potentially makes connections and splices with these fibers more robust, since the fiber need not be “keyed” for PM operation. Another potential advantage to using this schematic arises from the fact that higher order modes typically have larger mode areas than does the fundamental mode [35] – so constructing high-power PM fiber lasers with this fiber may be naturally beneficial. However, the polarization extinction ratio (PER) measured with this fiber (28.7 dB) is not as good as the industry standard PANDA fiber which was also measured in the same setup (PER~35 dB; measurement not shown in Figure 11), and so it remains to be seen if vortex fibers can be improved in this regard.

7.2 Nonlinear wavelength conversion

A natural question that arises, once fiber propagation of vortices become feasible, is what are the nonlinear-optical properties of vortex modes in fibers? Knowledge of the nonlinear response is useful, for instance, for realizing nonlinear optical device effects (e.g., frequency conversion), and for studying the impact of using such modes for long-length data transmission. Since dispersion was not designed or tailored in the first generation of vortex fibers, testing the interplay between dispersion and nonlinearity, of critical importance in several nonlinear-optical phenomena, will not be feasible. Instead, here we focus on studying the response with respect to one ubiquitous nonlinearity that does not require phase matching and hence dispersion tailoring – stimulated Raman scattering (SRS).

The measurement setup used in the experiment [51] is shown in Figure 12. The output of a Q-switched Nd:YAG (Yttrium-Aluminum Garnet) laser with a pulse duration of ~10 ns at a wavelength of 1064 nm and a 10-Hz repetition rate is coupled into a standard single-mode fiber (SMF). This fiber is spliced onto the vortex fiber. A microbend grating of 560-µm period excited the radially polarized (TM₀₁) mode with ~99% efficiency. After 100 m of fiber propagation, the output beam’s power, field profile (at multiple discrete wavelengths) and spectrum are recorded.

Figure 12

Experimental setup for the generation and characterization of the Raman-scattered vortex mode. Light at a wavelength of 1064 nm excites the polarization-vortex mode by means of a mechanical micro-bend grating and is sent through 100 m of the vortex fiber. A silicon CCD, an optical spectrum analyzer (OSA), and a power meter are used to characterize the radially polarized beams at 1064 nm, 1115 nm and 1175 nm, respectively.

Figure 13 shows clear evidence of SRS, since the spacing between peaks is ~13 THz, which corresponds to the Stokes shift of Raman scattering in Silica. With peak power levels of ~300 W, up to the 4th order of Stokes shift is clearly seen. With a transmission of ~90% through the vortex fiber, only fairly limited amounts of power were lost during the non-linear wavelength conversion. The threshold power for SRS can be estimated from the effective mode area, the fiber length and the Raman gain coefficient [52] as P_th=16‧A_eff/g_R‧L, which yields a value of 93 W in excellent agreement with the experimentally observed threshold. The Raman gain is dependent on whether the polarizations of the pump and signal are copolarized or orthogonal, with copolarized pumping being far more efficient [52]. One usually requires a polarization maintaining fiber to get copolarized SRS in a fiber, but the same can be achieved by the stable polarization state of the radially-polarized mode, since, as we discussed in the previous Section (7.1), the radially polarized mode in vortex fibers behaves like a PM fiber.

Figure 13

Spectra measured with the OSA at input peak-powers of 190 W, 360 W, and 470 W, showing generation of up to the 4th order cascade of Stokes shift.

To confirm that the generated wavelengths are produced in pure vortex modes with shape and polarization properties inherited from the pump mode, a tunable bandpass filter (interference filter), is added to the setup to view the beam at the different wavelengths separately. The filter allows transmission up to a wavelength of 1200 nm. Figure 14 shows the spectrum, as well as the mode intensity image, measured with different settings of the bandpass filter. While the resultant doughnut intensity profiles (insets in Figure 14A) suggest that all SRS peaks are in the radially polarized mode, we confirm this by taking multiple intensity profiles (Figure 14B) after projecting the beam through a polarizer at different angular positions – as was done when initially characterizing the fiber (see Section 5 and Figure 6). The power can be measured at each wavelength, and we observe that at a pump intensity of 200 W, the pump, the 1st, and the 2nd order Stokes-shifted components all carry the same amount of power, signifying high power-conversion efficiency.

Figure 14

(A) Images of the collimated radially polarized mode at the fiber output, at pump, 1st-order Stokes shift and 2nd-order Stokes shift. Spectra of the light transmitted through a bandpass filter selecting 1064 nm, 1115 nm and 1175 nm, showing good extinction of out-of-band light. Inset shows images of the output mode at these wavelengths; (B) Images of the mode shown in inset of (A) as an output polarizer is rotated, confirming radial polarization of the output.

In summary, (a) vortex modes in this fiber are robust under nonlinear transformations; (b) this represents a simple way to create multi-color vortices for several applications, and (c) all threshold values and observations indicate that the nonlinear response is no different from conventional fiber modes. While the last implication sounds trivial or obvious, it indicates that the complex polarization patterns do not alter the nonlinear response in any way. As we noted earlier, the present fiber does not offer dispersion control. Designs proposed to take dispersion into account predict the ability to achieve supercontinuum generation to achieve as much as an octave of bandwidth over which radially polarized beams may be obtained [40].

7.3 Quantum entanglement

One of the interesting prospects of OAM-based communications is to provide a higher-dimensional basis for quantum encryption applications. As a step in this direction, we have studied the ability of OAM modes in the vortex fiber to preserve entanglement [53].

A schematic of the entanglement transmission system is shown in Figure 15. To probe the reach of successful entanglement transport when one photon propagates in an OAM mode, we replace one of the channels with the vortex fiber. As before, we use fiber gratings to couple in and out of the OAM mode at the input and output, respectively, of the vortex fiber, so as to facilitate a conventional Gaussian beam input and output – this enables using commercial equipment to study entanglement. The loss of this 1-km-long vortex fiber “module” was 7 dB, arising from imperfect coupling, avoidable grating losses, as well as ~1.5 dB/km of OAM mode propagation loss. MPI, estimated from the spectral ripples in the transmission module [49], was 14.2 dB.

Figure 15

(A) Schematic of the experimental setup to measure preservation of entanglement with fiber OAM states. Polarization-entangled photons (1550 nm and 1558 nm) are sent in two different channels A (5 km SMF) and B (1.1 km vortex fiber). In channel B, two gratings are used to convert between fundamental and OAM mode with >80% efficiency; Time delay was tuned to target coincidence of channel A photons (travelling in an OAM mode) and channel B photons (travelling in SMF). (B) Density matrix of the entangled state measured at the output, indicating violation of Bell’s inequality.

Two polarization-entangled photon pulses were produced (1550 nm and 1558 nm, 50 MHz repetition rate) and launched in the two different channels (channel A: 1.1 km vortex fiber, channel B: 5 km single mode fiber). At the receiving end a complete quantum state tomography is performed. By measuring the density matrix of the photon pair at the output, a concurrence value of C=0.76 and a maximally possible S-parameter of S=2.47 was extracted. This clearly shows that the photon pair preserves entanglement even after one of the photons has been converted to, and back from, an OAM carrying state.

7.4 Mode-division multiplexing

While the OAM entangled photon experiments described in the previous section suggests that the mode purity of the OAM modes in vortex fibers may be sufficient for data transmission even at the single photon level, the real power of using multiple modes is in encoding information in all the modes, thereby scaling capacity by mode division multiplexing (MDM).

The setup for a recent OAM-MDM experiment [54] is shown in Figure 16. A CW-ECL operating at 1550 nm was modulated using a 50 GBaud QPSK signal, and subsequently split into four arms that were delayed sufficiently to obtain four de-correlated data channels. Two of the four channels were converted into the L=±1 OAM modes (modes A and B) using fork-holograms of topological charge ±1, created with a spatial light modulator (recall the discussion related to Figure 7 in Section 4, where we had shown the creation of fork patterns when interfering a plane wave with an OAM beam – from reciprocity, it follows that a plane wave (or collimated LP₀₁ mode from SMF) falling on a fork hologram would create an OAM mode with the desired topological charge). The other two LP₀₁ modes (C and D) were left unchanged. Fiber coupling losses were 0.7 dB for the LP₀₁ mode and 1.1 dB for the OAM modes. These modes are then propagated through 1.1 km of vortex fiber.

Figure 16

Systems experiment setup: signal from the laser or WDM source is modulated, amplified using an erbium doped fiber amplifier (EDFA), filtered using a band pass filter (BPF) and split into four individual fiber arms (two in the case of the WDM experiment). Two of the arms were converted into OAM modes using the input SLM. Two fundamental modes were also collinearly aligned with the two OAM modes using a beam-splitter, and all four modes were coupled into the fiber. After propagation, the modes are demultiplexed sequentially and sent for coherent detection and offline digital signal processing (DSP). Acronyms: ADC, analog digital convertor; Att, attenuator; FM, flip mirror; LO, local oscillator; OC, optical coupler; (N)PBS, (non)-polarizing beam-splitter; PBC, polarization beam combiner; PC, polarization controllers; PC-SMF, polarization controller on SMF; PC-VF, polarization controller on vortex fiber.

With all four channels enabled simultaneously, the demuxing system sorts the modes according to their OAM (L) and SAM (S) values, using another SLM and a combination of a quarter-wave plate and a polarizer, respectively. The resulting output was mapped back into a conventional Gaussian-shaped beam with a planar phase, which was routed to a coherent receiver by coupling into an SMF. This enables a quantitative measure of mode purity by observing channel output power vs. time (Figure 17A and B), which reveals how much power leaked into other channels (i.e., cross-talk – Figure 17A), as well as how much power leaked out-and-back, during fiber propagation, into an individual channel (MPI, deduced from the amplitude Δ, of time-dependent, slow power fluctuations – Figure 17B) [49, 50]. We note that MPI is a fundamental property arising from fiber design, while cross-talk depends on both the fiber and the (de)multiplexing setup design. Both crosstalk and MPI increase bit-error rate (BER) in the absence of digital signal processing (DSP) based corrective algorithms (measured values for all the modes are shown in the table in Figure 17C).

Figure 17

(A) Output power vs. time measured at a specific channel’s receiver. Drop in power when that mode is turned off indicates level of crosstalk due, primarily, to imperfect mode projections in the free-space demultiplexing system; (B) Power fluctuations measured for a typical channel, suggesting the presence of MPI, which arises from mode-coupling within the fiber; (C) All values for cross-talk and MPI summarised in table.

Figure 18 summarises the results of the first data transmission tests done with the setup shown in Figure 16.Figure 18A shows constellation diagrams of the transmitted data, for each mode, when (i) only one channel is turned on at a time (top row), and (ii) when all channels are turned on simultaneously (bottom row). We see a slight degradation in data quality when all modes are transmitted though the existence of distinct constellations is indicative of successful data transmission – this is further quantified with the BER plots of Figure 18B. When only one channel is on at a time, the largest received power penalty for achieving a BER of 3.8×10^-3 (the threshold BER level at which forward-error-correction (FEC) algorithms ensure error-free data transmission) is 2.5 dB, mainly due to MPI. In the all-channel case, this largest power penalty increased to 4.1 dB, mainly due to cross-talk. In the latter case, a total transmission capacity at 400 Gbit/s below the FEC limit was achieved.

Figure 18

(A) Constellation diagrams in the case of 4-mode OAM-MDM, with 50 GBaud NRZ-QPSK, λ=1550 nm, for the single-channel, and all-channel cases. Note the larger distortion of the constellations in the all-channel case; (B) Corresponding BER vs. received power plots for the same cases as in (A) – power penalty is <2.5 dB when only one mode is used at a time, and 4.1 dB when data is transmitted in all modes together.

Figure 19 shows a variety of spiral interference patterns at different wavelengths, qualitatively indicative of broadband transmission of a pure OAM state obtained at the output of the 1.1-km vortex fiber in the setup shown in Figure 16. The images span a wavelength range of 30 nm, suggesting that the MDM technique may be combined with traditional WDM to further enhance capacity. While Figure 19A suggests that this system has the ability to operate over a bandwidth of at least 30 nm, several extraneous considerations related to imperfect free-space coupling of the modes constrained the experiment, and only two OAM modes and 10 WDM channels (from 1546.64 to 1553.88 nm) were chosen for the WDM experiment (Figure 19B). Reduction to two modes allowed us to choose a more complex modulation format (16-quadrature amplitude modulation) for data transmission, yielding higher spectral efficiency, albeit at a lower baud rate of 20 GBaud. Transmission of 20 channels (OAM-MDM and WDM), resulted in a total transmission capacity of 1.6 Tb/s under the FEC limit (Figure 19C shows typical constellation diagrams that were recorded).

Figure 19

(A) Spiral interference patterns, showing helical phase, of the L=+1; S=+1 OAM mode at multiple wavelengths across 30 nm from 1540 to 1570 nm, indicating that these vortices may be supported over at least the spectral range approximately covered by the C-band; (B) WDM experiments with 10 wavelength channels on top of modes: Spectrum of the modulated signal at the output of the WDM 16-QAM T_x, and spectrum of the L=+1; S=+1 OAM mode at the receiver after demultiplexing; (C) Constellation diagrams of 16-QAM modulation for the demultiplexed L=+1; S=+1 OAM mode at 1550.64 nm (channel A), for several cases (without and with crosstalk – XT).

The aforementioned experiments used conventional, free-space commercial components (SLMs, wave retarders, etc.) for muxing and demuxing, which are inherently lossy. Recent demonstrations of theoretically lossless OAM muxing devices, based on waveguides [55] or free-space optics [18], indicate that the required component technology is concurrently developing to enable realistic deployments of scalable networks based on OAM-MDM.

8 Summary and prospects for applications in nano-imaging

The use of optical fibers to generate or propagate cylindrical vector beams (polarization vortices) and OAM states dates back at least to the 1990s. In early implementations, the fiber served as a replacement to the generation device (such as the free-space phase plate or SLM), and this addressed key loss, alignment stability and reliability problems often encountered with free-space devices. However, this did not lead to widespread use of fibers in the field of singular optics, because the fiber had to be specially designed, and ultimately, it offered only an alternative to commercially available free-space components.

The advent of designs that enable propagation over long lengths (10–100 m, to km) opens the door to new applications. As described in Section 7.1, the polarization maintaining aspect of it enables building high-power fiber lasers that emit in desired vortex modes, while nonlinear control (Section 7.2) may lead to ultra-broadband coherent vortex beams and perhaps even the use of fiber for creating entangled OAM states. Finally, vortex stability in the fiber provides for new schemes for enhancing data capacity in fiber links via mode-division multiplexing. Current demonstrations for data transmission have been over 1-km links – this length scale represents an improvement of over three orders of magnitude with respect to lengths over which OAM transport had been previously attempted. However, these length scales fall far short of those needed for long-haul transmission links. On the other hand, these lengths are ideal for communications within data centers, where deployment of novel fibers is significantly easier than in long-haul networks.

Fiber stability of vortices also enhances several nano-imaging applications which currently use free-space elements. In analogy with the ubiquity of SMF in confocal microscopes today, it is conceivable that super-resolution imaging techniques that use vortices [4–6, 13] would greatly benefit from a fiber that could be bent, routed, or otherwise robustly handled in a microscope. More interestingly, this may open the door to envisage endoscopic implementations of several nanoscale microscopy techniques.

Finally, we note that only simple, ring-shaped index profiles have been demonstrated thus far. It is entirely conceivable that even more complex waveguide geometries could lead to much higher mode stability, or the ability to generate and propagate even more complex vortex beams that are theoretically postulated [56] to facilitate spots as small as 0.14λ².