Light’s orbital angular momentum (OAM) is a conserved quantity in cylindrically symmetric media. However, it is easily destroyed by free-space turbulence or fiber bends, because anisotropic perturbations impart angular momentum. We observe the conservation of OAM even in the presence of strong bend perturbations, with fibers featuring air cores that appropriately sculpt the modal density of states. Analogous to the enhanced stability of spinning tops with increasing angular velocity, these states’ lifetimes increase with OAM magnitude. Consequently, contrary to conventional wisdom that ground states of systems are the most stable, OAM longevity in air-core fiber increases with mode order. Aided by conservation of this fundamental quantity, we demonstrate fiber propagation of 12 distinct higher order OAM modes, of which eight remain low loss and pure from near-degenerate coupling after kilometer-length propagation. The first realization of long-lived higher order OAM states, thus far posited to exist only in vacuum, is a necessary condition for achieving the promise of higher dimensional classical and quantum communications over practical distances.
© 2015 Optical Society of America
CorrectionsP. Gregg, P. Kristensen, and S. Ramachandran, "Conservation of orbital angular momentum in air-core optical fibers: erratum," Optica 4, 1115-1116 (2017)
Quantum numbers are usually assigned to conserved quantities; hence it appears natural that paraxial light traveling in isotropic, cylindrically symmetric media, such as free space or optical fibers, be characterized by its angular momentum :2] and represents its spin angular momentum (SAM), commonly known as left- or right-handed circular polarization, , such that in units of per photon. forms a countably infinite-dimensional basis, spawning widespread interest in OAM beams [3 –5]. In particular, this enables a large alphabet for hyperentangled quantum communications or high-capacity classical links. The information capacity of a classical or quantum communications link increases with the number of distinct, excitable, and readable orthogonal information channels. Degrees of freedom that conserve their eigenvalues are required, because perturbations that cause eigenstate rotation (mode coupling) are debilitating. In classical communications, computational algorithms can partially recover information for some limited perturbations, albeit with energy-intensive signal processing . For low-light-level applications such as quantum communications or interplanetary links, the information is lost. With the use of wavelength and polarization as degrees of freedom virtually exhausted, the recent past has seen an explosion of interest in a new degree of freedom—orthogonal spatial modes that are stable during propagation, of which OAM is one interesting choice [7,8].
In practice, this choice of quantum numbers is questionable. Although large ensembles of OAM modes can be generated [7 –11], they are easily destroyed by anisotropic perturbations such as atmospheric turbulence  in free space, or bends in fibers , limiting OAM transmission experiments to primarily laboratory length scales (meters) [8,14]. Practical communications distances, over fiber or free space, have been achieved only for the special case of the lowest order () states [15,16]. OAM transmission is hampered by near-degeneracies of the desired OAM state with a multitude of other modes [17,18] possessing different or radial quantum numbers. These near-degeneracies in linear momentum, or equivalently longitudinal wavevector, (in waveguides, also represented by effective index, , given by where is the free-space wavelength, and signifies propagation coordinate), phase match orthogonal modes and couple them in the presence of perturbations. Since any multimodal system would, by definition, have a high density of states, this is a fundamental problem, and exploiting the infinite-dimensional basis afforded by OAM beams requires a medium in which this modal degeneracy is addressed.
Here, we report the design of a general class of optical fibers, featuring an air core that enables conservation of OAM (Fig. 1). The air core acts as a repulsive barrier, forcing the mode field to encounter the large index step between ring and cladding [Fig. 1(b)]. This lifts polarization near-degeneracies  of OAM states with the same , though the states with OAM and SAM aligned (of the same handedness) remain degenerate with each other, but separate from those with OAM and SAM anti-aligned [Fig. 1(c); more information in Supplement 1, Section 2]. This splitting generally increases with [Fig. 1(d)]. A key feature of the air-core fiber  is the existence of modes with large but the prevention of modes with a large radial quantum number whose may be close to the desired OAM states, by appropriate sculpting of mode volume to vastly decrease the density of states (see Supplement 1, Section 4). The effect is similar to the restriction of the mode structure in microtoroid resonators, in which devices preferentially support equatorial modes .
Using the experimental apparatus  in Fig. 2(a), we excite and propagate 12 OAM states over 10 m of our air-core fiber at 1530 nm. Fiber output fields are imaged onto a camera through a circular polarization beam splitter, separating and into the right and left bins, respectively. Excitation of, for example, modes yields clean ring-like intensities that remain in the circular polarization selected by the quarter-wave plate before the fiber. As the radial envelopes of the modes are nearly identical, we interfere with a Gaussian beam [Fig. 2(a), orange path] to reveal their phase structure. For each interference pattern, the number of spiral arms indicates the mode’s , while the handedness indicates the sign of . Combined with sorting by circular polarization, we unambiguously identify OAM states. Clean spiral images (see Supplement 1, Section 5) in Fig. 2(b) indicate negligible coupling among, and hence clean transmission of, all 12 OAM states.
This result is counterintuitive—while the air-core design lifts degeneracies among a host of OAM states, the modes still appear in degenerate pairs (spin-orbit aligned or anti-aligned). The coefficient of power coupling between modes and is Supplement 1, Section 3). Thus, pairs of degenerate modes should be susceptible to coupling within their two-mode subspace via anisotropic perturbations such as the bends that existed on the 10 m long fiber. In fact, for lower order, , OAM states, such coupling is possible  and controllably exploited  using a series of fiber loops, in analogy to a conventional polarization controller (polcon) in single-mode fiber (SMF). The polcon may be understood as transfer of OAM from the bend perturbation to the field itself . Any -independent anisotropic perturbation may be expanded as 26]. Allowing for spin coupling, transitioning between higher order degenerate states () requires a perturbation element of order , which becomes increasingly negligible for large [Fig. 3(a)]. To experimentally interrogate this curious effect, we build a polcon , but with the air-core fiber [purple circles in Fig. 2(a)] with bend radius . We define a degradation factor, , as the ratio of the maximum power in to that in when is launched, or vice versa. For high states, such as [Fig. 3(b)], degradation factors are typically . As expected, for the mode in SMF [Fig. 3(c)], , indicating complete coupling between the two degenerate SAM states. Due to the rapid decrease of as increases, the observed degradation factor decreases, with ratios as low as () for higher states relative to SMF [Fig. 3(d)]. Thus, we find that, for high states in air-core fibers, OAM is truly a conserved quantity even in the presence of anisotropic perturbations, since transitions among degenerate states are forbidden, based on conservation of OAM [Fig. 3(e)]. This behavior parallels forbidden transitions between electron spin states with an externally applied electric field. Here, anisotropic bends assume the role of electric field perturbations, leaving the initial state unchanged.
Over long enough interaction lengths, light may encounter other perturbation symmetries due to twists and imperfections in the draw process. We experimentally study long-length propagation by transmitting OAM states with a picosecond pulsed (70 GHz bandwidth) laser at 1550 nm [Fig. 2(a)] and measuring time-of-flight traces. At the output of fiber length , modes and are temporally separated by . As all of the OAM modes in this fiber have similar group-velocity dispersions, relative mode purity, , conventionally referred to as multipath interference (MPI) , is given by4(a) and S5(a) (see Supplement 1 for Fig. S5), with close-ups of individual traces in Figs. 4(b)–4(e) and Figs. S5(b)–S5(e). We find that MPIs of or greater ( purity) can be achieved for any , 6 mode relative to the background, the modes being too lossy for 1 km transmission at 1550 nm. We obtain similar results from 1530 to 1565 nm in wavelength, thus confirming that the OAM states are wavelength agnostic. Loss for the and 6 mode groups, measured via conventional fiber-cutback, is 1.9 and 2.2 dB/km, respectively (see Supplement 1, Section 1). Note that this measures only cross coupling between spin-orbit aligned and anti-aligned states, as the degenerate states have identical group delays. When OAM states are propagated over kilometer lengths, we observe to transitions at the fiber output. This potentially indicates twist perturbations, known to affect OAM stability . Extending the quantum-mechanical analogy, twists would assume the role of magnetic perturbations, which couple electronic spin states. However, this coupling constitutes a unitary transformation within the two-mode subspace and may be disentangled with devices such as –plates , thus still yielding a medium in which all eight of the states may be information carriers.
Conservation of light’s OAM in air-core fibers enables kilometer-length-scale propagation of a large ensemble of spatial eigenstates, in analogy to the perturbation resistance of spinning tops and electron spin states. Therefore, this new photonic degree of freedom, having attracted much recent attention on account of its potentially infinite-dimensional basis, remains a conserved quantity over lengths practical for optical communications in appropriately designed fiber. Hence, we expect such fibers and their OAM states to play a crucial role in the general problem of increasing the information content per photon.
Defense Advanced Research Projects Agency (DARPA) (W911NF-12-1-0323, W911NF-13-1-0103); National Science Foundation (NSF) (DGE-1247312, ECCS-1310493).
The authors would like to acknowledge J. Ø. Olsen for help with fiber fabrication; N. Bozinovic, S. Golowich, and P. Steinvurzel for insightful discussions; and M. V. Pedersen for help with the numerical waveguide simulation tool.
See Supplement 1 for supporting content.
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