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 . 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 Sz) associated with it in the beam propagation direction. These beams have found several uses, such as realizing super-resolved spots , plasmonic nano-focussing , single-molecule spectroscopy , metal machining , and they are considered to be potentially useful for laser-based electron and particle acceleration .
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  demonstrated in 1992 that commonly encountered Laguerre-Gaussian beams with helical phase given by eiLφ, 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 , as well as communications. Several classical  and quantum  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 , computer-generated fork holograms , and most recently, free-space mode sorters .
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 et, 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
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 . A variant of this approach  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 . 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 . 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  or adiabatic mode couplers . 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 HEL+1,m and EHL-1,m (L>1) modes have the following transverse electric field distributions :
where FL,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:
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 HE1,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 EH0,m with TM0,m, and odd EH0,m with TE0,m. However, crucially, unlike the L>1 case, these two modes (TE0,m and TM0,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 LPL,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 , and helically phased inputs into multicore fibers  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  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 .
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 , nonadiabatic couplers ), 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 neff 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 neff, 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/Δneff, where λ is the wavelength of operation and Δneff is the difference in neff 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.
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  and nano-structured metal films  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 Δneff, the difference in neff between modes, increases . There have been multiple attempts over the years to understand the relationship between Δneff 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) . 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 Δneff>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 , reveals that Δneff between LP0m and LP1m 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 Δneff>10-4 is a reasonable proxy for mode stability .
The design problem, therefore, is to find the means to achieve Δneff>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 :
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, nco is the core refractive index, Δn(r) is its refractive index profile relative to the index of the infinite cladding, and Δnmax 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 Δnmax.
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 Δnmax 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 Δnmax while also maximizing field-gradients at index steps. This is schematically illustrated in Figure 5B.
For this class of designs, increasing Δnmax 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 Δnmax 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 neff 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 Δneff >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 neff 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 ). The neff 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 Δnmax 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 , 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 . 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 . When the grating wavevector is matched with the difference in propagation constants between different modes, one obtains mode conversion with experimentally measured efficiencies  exceeding 99.99%, and by proper fiber design, these gratings may be made broadband  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 λres, 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 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
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 . 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
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 |γi|2, in the expansion of the transverse electric field Et(r,ϕ) at the fiber output
where the sum is over the mode index i. The six fields
Figure 8 shows the experimental setup that we used to control and analyse the OAM states of a vortex fiber . 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).
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 Δ1,2 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 |γi|2, from which mode purity, quantified by the multipath interference (MPI) level [defined as MPI=10 log10(|γi|2/Ptotal)] [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 Δ1,2 parameters, and Figure 9C illustrates the powers of the extracted modes. Note that mode purities exceeding 20 dB may be measured with this technique.
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.
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
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 neff 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 Δneff>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 . 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%.
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  – 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  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 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  as Pth=16‧Aeff/gR‧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 . 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.
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.
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 .
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 .
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 , was 14.2 dB.
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  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.
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 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 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).
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  or free-space optics , 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  to facilitate spots as small as 0.14λ2.
We thank several students, postdoctoral scientists and colleagues from BU, OFS, Aarhus Univ., Univ. Southern California and MIT Lincoln Lab for collaborations that led to the successful experiments reviewed in this article. We gratefully acknowledge M. Yan (OFS) for several early interactions that led to the development of the vortex fiber. We thank P. Balling and C. Smith (Univ. Aarhus) for the collaborations on nonlinear measurements. N. Bozinovic (BU), whose PhD thesis encompassed several of the OAM related experiments described here, played a significant role in the development of this technology. We thank P. Gregg (BU) for insightful discussions as well as his help with several of the illustrations. We also thank S. Golowich (MIT-LL) for interactions that led to the theoretical formulation of the fiber characterization problem. Last, but not least, we would like to thank A. Willner, Y. Yue and Y. Ren (USC) whose contributions were critical for the realization of the OAM-MDM experiments. The work on OAM described in this manuscript was funded, in part, by the DARPA InPho program.
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