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

is the propagation constant of the propagating mode (this is equivalently characterized by an effective index of the mode group given by

The resultant intensity distributions for

*e*_{t}, along with schematic representations of

*n*_{eff} for the first few modes are shown in

Figure 3A. Inspection of these field profiles appears to suggest that an optical fiber, in spite of its strictly cylindrically symmetric nature, contains no solutions that resemble vortices of the kind described in Section 2. This fallacy arises from the scalar approximation of Eq. 1, which neglects index gradients. Thus, obtaining accurate solutions would require using the full vector wave equation given by [19]

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

along with schematic representations of

*n*_{eff}, for the full vector solution of the same step index waveguide. It becomes immediately apparent that, though their intensity profiles resemble free-space vortices of Section 2, the true modes of an optical fiber have more complex degeneracies, as illustrated in

Figure 3C–E. For the azimuthally symmetric modes, designated as LP

_{0m} modes in the scalar approximation and HE

_{1m} in the vector solutions, there exist two-fold degeneracies, accounting for the two orthogonal polarization orientations (

Figure 3C). These modes are uniformly polarized and carry no OAM, and their degeneracies arise out of the two polarization states that are allowed. These two states are either horizontal, vertical, or their linear combinations, as depicted in this figure, or, equivalently, left, or right circular (or their linear combinations – elliptical), which may also be denoted by the spin angular momentum (SAM) that each photon in the beam can carry, given by

*S*=±1, representing a SAM of ±1ħ per photon. For the first higher order solutions designated as LP

_{11} modes in the scalar approximation, there exist four modal solutions – TE

_{01},

and TM

_{01} (see

Figure 3B or 3D). The two HE

_{21} modes are strictly degenerate, and have an

*n*_{eff} distinct from the TE

_{01} and TM

_{01} modes. Thereafter, all higher order solutions (with higher azimuthal index – i.e., LP

_{L,}_{m} modes with

*L*>1) comprise two distinct modal solutions that are doubly degenerate, called HE

_{L+1,}_{m} and EH

_{L}_{-1,}_{m} modes, respectively (only one of each degenerate pair’s field distribution is shown in

Figure 3E). In general, modes with almost similar

*n*_{eff} also have similar group velocities, and comprise a single mode group. In this context, the LP

_{1m} mode group is the scalar representation of 4 vector modes – the TE

_{0m},

and TM

_{0m}, and all other LP

_{L}_{,}_{m} (

*L*>1) mode groups comprise HE and EH true mode combinations.

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 TE_{01} or TM_{01} 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

representing left or right handed circular polarization, respectively, also denoted by SAM with S=±1 carried by each photon in this state. Note the uniform polarization as well as explicit helical phase for these mode solutions. This illustrates that OAM=

*Lħ* states can be represented as π/2-phase-shifted linear combinations of the vector modes HE

_{L}_{+1,}_{m}, with their spin aligned with the sign of

*L*, or the vector combinations of EH

_{L}_{-1,}_{m} states with spin anti-aligned with the sign of

*L*.

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 LP_{11} mode obtained from the scalar mode solutions (see Figure 3A). Experimental observations of this pattern are often claimed to be observations of the LP_{11} 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 LP_{11} 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 LP_{11} 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.

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