Nanophotonics 2021; 10(1): 209–224 Review Zelin Ma and Siddharth Ramachandran* Propagation stability in optical fibers: role of path memory and angular momentum https://doi.org/10.1515/nanoph-2020-0404 1 Introduction Received July 19, 2020; accepted September 30, 2020; published online October 21, 2020 Multimode fibers (MMFs) and their spatially diverse higher-order modes (HOMs) have experienced alternating Abstract: With growing interest in the spatial dimension levels of interest ever since the invention of optical fibers. of light, multimode fibers, which support eigenmodes Although one of the first applications of light propagation, with unique spatial and polarization attributes, have for image transport with flexible optical fiber waveguides, experienced resurgent attention. Exploiting this spatial utilized MMFs [1], the development of single-mode fibers diversity often requires robust modes during propagation, (SMFs) quickly diverted attention away from MMFs. One which, in realistic fibers, experience perturbations such important reason was that any realistic deployment of as bends and path redirections. By isolating the effects of optical fibers includes perturbations, such as bends, twists, different perturbations an optical fiber experiences, we 3D paths as well as thermal, mechanical, and environ- study the fundamental characteristics that distinguish mental fluctuations. While modes in a perfectly straight, the propagation stability of different spatial modes. Fiber circular fiber are theoretically orthogonal, perturbations perturbations can be cast in terms of the angular typically cause coupling between them, leading to poten- momentum they impart on light. Hence, the angular tial loss of signal purity or information content. momentum content of eigenmodes (including their Subsequent advances in light guidance, including micro- polarization states) plays a crucial role in how different structuring [2], photonic bandgaps [3], or antiresonant modes are affected by fiber perturbations. We show structures [4], primarily focused on means to strip out [5–7] that, accounting for common fiber-deployment condi- HOMs to effectively realize single-mode guidance. In fact, tions, including the more subtle effect of light’spath even an SMF is two-moded, accounting for the two memory arising from geometric Pancharatnam–Berry orthogonal polarization eigenmodes. As such, bend- or phases, circularly polarized orbital angular momentum geometry-induced fiber birefringence [8] can cause modes are the most stable eigenbasis for light propaga- polarization-mode dispersion in classical communications tion in suitably designed fibers. Aided by this stability, we links [9] and loss of entanglement preservation in quantum show a controllable, wavelength-agnostic means of links [10]. tailoring light’s phase due to its geometric phase arising The advent of improved signal conditioning and from path memory effects. We expect that these findings reception technologies over the last decade has, however, will help inform the optimal modal basis to use in the refocused investigations on MMFs in which modes mix. variety of applications that envisage using higher-order Because this mixing predominantly represents unitary modes of optical fibers. transformations, multi-in multi-out digital signal process- Keywords: higher-order modes; multimode fibers; orbital ing can disentangle mode mixing in the electronic domain angular momentum; Pancharatnam–Berry phase; vector [11], resulting in scaling the capacity of telecommunica- beams; propagation stability. tions links [12]. Analysis of the speckle patterns out of MMFs enables spectrometry [13]. On the other hand, adaptively controlling the speckle pattern at the input or output enables imaging with MMFs [14, 15]. Linear mixing, when combined with nonlinear coupling, leads to effects *Corresponding author: Siddharth Ramachandran, Boston such as multimode solitons [16, 17], nondissipative beam University, Boston, MA, USA, E-mail: [email protected]. https://orcid.org/ cleanup [18] and geometric parametric instabilities [19], 0000-0001-9356-6377 Zelin Ma, Boston University, Boston, MA, USA. https://orcid.org/ among a host of multimode nonlinear effects not seen in 0000-0002-2801-4004 SMFs. Open Access. © 2020 Zelin Ma and Siddharth Ramachandran, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 210 Z. Ma and S. Ramachandran: Propagation stability in fibers and the role of path memory The aforementioned benefits of a mixed ensemble of of a cylindrically symmetric fiber mix with each other? This modes notwithstanding the ability to excite and propagate is a very complex problem for long-haul networks where specific HOMs remain especially desirable. As is evident propagation over 100s–1000s of kilometers of fiber en- from any elementary solution of eigenmodes in wave- counters a wide array of stochastically varying perturba- guides, each HOM has a characteristic phase and group tions. As a result, many realistic models for such mode velocity, group-velocity dispersion, modal area (Aeff) [20, coupling are phenomenological in nature [55, 56]. For 21]. For instance, HOMs can be tailored to have large shorter lengths, spanning a few meters to kilometers normal [22] or anomalous dispersion [23], with applications (lengths representing scales of fiber usage in most appli- in dispersion control for telecom links [24, 25] and ultra- cations, such as fiber lasers and amplifiers, data-center short pulses [26]. One of the first demonstrations of links, nonlinear devices or sensors), this is a more tractable nonlinear wave mixing in optical fibers involved inter- problem that can yield some first-principles insight. We modal phase matching between HOMs [27], the diversity of show that certain classes of spatial modes are more stable HOMs yielding enhanced degrees of freedom to achieve and propagation-tolerant than others. Somewhat counter- momentum conservation [28, 29]. Raman [30] or Briliouin intuitively, we also show that this stability depends even [31] scattering have shown to be strongly dependent on, on the bases of modes used – that is, one set of modes can and hence be tailorable by, the mode(s) in which light be more stable than modes represented in a mathemati- propagates. These concepts have received increased cally equivalent but rotated basis. Fundamentally, we attention for applications such as third-harmonic genera- show that, accounting for typical perturbations an optical tion [32], extending supercontinuum generation [33], fiber encounters, the circularly polarized OAM eigenbasis power-scalable source engineering [30, 34] or new forms of represents the most stable set of modes for light quantum sources [35]. The inherently large Aeff of HOMs transmission. has led to ultralarge Aeff, low-nonlinearity flexible fibers [36], with applications in fiber lasers [37]. More recently, fiber modes with orbital angular momentum (OAM) have 2 Mode classifications and fiber been shown to yield an additional degree of freedom with which to control the nonlinear interactions of light in fibers perturbations [38–40]. Fiber sources operating in HOMs are also inter- esting for applications where a non–Gaussian-shaped 2.1 Mode classifications emission is desired, such as in nanoscale microscopy [41, 42] and laser machining [43]. Finally, perhaps most We start with a brief description of the eigenmodes of a significantly over the last few years, there has been an cylindrically symmetric, step-index profile fiber. The 2D emerging realization that individual modes, especially cross-section allows modes to be classified by two those carrying OAM, can enable signal propagation with orthogonal polarization distributions, as well as a radial low or limited mode mixing [44], as a means of increasing order index m, and an azimuthal order index L. Equa- the capacity of classical communications networks [45–48] tion (1) shows the field distributions for the class of L =0 or for enhancing the security of quantum links [49]. All modes (HE1,m) in two polarization bases: these applications have two critical requirements: (1) the x + β HE1, m HE1, m E i 0, mz ⋅ ability to accurately control mode transformations with, for 0, m e y or − (1) HE1, m HE1, m instance, fiber gratings [50], diffractive optics [51], Pan- charatnam–Berry optical elements (PBOE) [52], spatial where light modulators [53], or metasurfaces [54]; and (2) HEx HE+ σ+ 1, m ()x , 1, m () crucially, the need for linearly, stably propagating desired y F0, m r − F0, m r σ− HE , y HE , modes in fibers. 1 m 1 m ± Here, we address the latter issue – the propagation where σ x ± iy, representing left or right handed circular fi stability of optical ber modes. Any realistic analysis of an polarization, respectively; F0,m(r) represents the radial optical fiber must necessarily consider the perturbations it profile of the field, which is typically a piecewise linear encounters over the long lengths of signal transmission in combination of Bessel functions for most profiles; β is the → facilitates. The key question is, in the presence of pertur- projection of the wave vector k in the propagation direction bations, how do otherwise theoretically orthogonal modes (usually referenced by the mode’s effective index, Z. Ma and S. Ramachandran: Propagation stability in fibers and the role of path memory 211 neff λβ/2π, where λ is the free-space wavelength of light). Figure 1 shows the intensity profiles of two repre- Paying attention to its subscripts,