Conservation of orbital angular momentum inair-core optical fibersP. GREGG,1 P. KRISTENSEN,2 AND S. RAMACHANDRAN1,*1Boston University, 8 St. Mary’s St, Boston, Massachusetts 02215, USA2OFS-Fitel, Priorparken 680, Brøndby 2605, Denmark*Corresponding author: sidr@bu.edu

Received 17 November 2014; revised 19 February 2015; accepted 20 February 2015 (Doc. ID 226933); published 19 March 2015

Light’s orbital angular momentum (OAM) is a conservedquantity in cylindrically symmetric media. However, it iseasily destroyed by free-space turbulence or fiber bends, be-cause anisotropic perturbations impart angular momentum.We observe the conservation of OAM even in the presenceof strong bend perturbations, with fibers featuring air coresthat appropriately sculpt the modal density of states.Analogous to the enhanced stability of spinning tops with in-creasing angular velocity, these states’ lifetimes increase withOAM magnitude. Consequently, contrary to conventionalwisdom 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, wedemonstrate fiber propagation of 12 distinct higher orderOAM modes, of which eight remain low loss and >98% purefrom near-degenerate coupling after kilometer-length propa-gation. The first realization of long-lived higher order OAMstates, thus far posited to exist only in vacuum, is a necessarycondition for achieving the promise of higher dimensionalclassical and quantum communications over practicaldistances. © 2015 Optical Society of America

OCIS codes: (050.4865) Optical vortices; (060.2280) Fiber design and

fabrication; (060.4230) Multiplexing; (260.6042) Singular optics.

http://dx.doi.org/10.1364/OPTICA.2.000267

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 [1]:

J � L� S: (1)

L represents light’s orbital angular momentum (OAM) [2] and Srepresents its spin angular momentum (SAM), commonly knownas left- or right-handed circular polarization, σ̂�, such thatS � �1 in units of ℏ per photon. L forms a countably infinite-dimensional basis, spawning widespread interest in OAM beams[3–5]. In particular, this enables a large alphabet for hyperen-tangled quantum communications or high-capacity classicallinks. The information capacity of a classical or quantum

communications link increases with the number of distinct, excit-able, and readable orthogonal information channels. Degrees offreedom that conserve their eigenvalues are required, becauseperturbations that cause eigenstate rotation (mode coupling)are debilitating. In classical communications, computational algo-rithms can partially recover information for some limited pertur-bations, albeit with energy-intensive signal processing [6]. Forlow-light-level applications such as quantum communications orinterplanetary links, the information is lost. With the use of wave-length and polarization as degrees of freedom virtually exhausted,the recent past has seen an explosion of interest in a new degree offreedom—orthogonal spatial modes that are stable during propa-gation, 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 perturbationssuch as atmospheric turbulence [12] in free space, or bends infibers [13], limiting OAM transmission experiments to primarilylaboratory length scales (meters) [8,14]. Practical communica-tions distances, over fiber or free space, have been achieved onlyfor the special case of the lowest order (jLj � 1) states [15,16].OAM transmission is hampered by near-degeneracies of thedesired OAM state with a multitude of other modes [17,18] pos-sessing different jLj or radial quantum numbers. These near-degeneracies in linear momentum, or equivalently longitudinalwavevector, kz (in waveguides, also represented by effective index,nef f , given by k � 2π ·

nef fλ where λ is the free-space wavelength,

and z signifies propagation coordinate), phase match orthogonalmodes and couple them in the presence of perturbations. Sinceany multimodal system would, by definition, have a high densityof states, this is a fundamental problem, and exploiting the infin-ite-dimensional basis afforded by OAM beams requires a mediumin 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 fieldto encounter the large index step between ring and cladding[Fig. 1(b)]. This lifts polarization near-degeneracies [19] ofOAM states with the same jLj, though the states with OAMand SAM aligned (of the same handedness) remain degeneratewith each other, but separate from those with OAM and SAM

Letter Vol. 2, No. 3 / March 2015 / Optica 267

2334-2536/15/030267-04$15/0$15.00 © 2015 Optical Society of America

anti-aligned [Fig. 1(c); more information in Supplement 1,Section 2]. This nef f splitting generally increases with jLj[Fig. 1(d)]. A key feature of the air-core fiber [20] is the existenceof modes with large jLj but the prevention of modes with a largeradial quantum number whose nef f may be close to the desiredOAM states, by appropriate sculpting of mode volume to vastlydecrease the density of states (see Supplement 1, Section 4). Theeffect is similar to the restriction of the mode structure in microt-oroid resonators, in which devices preferentially support equato-rial modes [21].

Using the experimental apparatus [22] in Fig. 2(a), we exciteand propagate 12 OAM states over 10 m of our air-core fiber at1530 nm. Fiber output fields are imaged onto a camera through acircular polarization beam splitter, separating σ̂� and σ̂− intothe right and left bins, respectively. Excitation of, for example,jLj � 7 modes yields clean ring-like intensities that remain inthe circular polarization selected by the quarter-wave plate beforethe fiber. As the radial envelopes of the modes are nearly identical,we interfere with a Gaussian beam [Fig. 2(a), orange path] to re-veal their phase structure. For each interference pattern, the num-ber of spiral arms indicates the mode’s jLj, while the handednessindicates the sign of L. Combined with sorting by circularpolarization, we unambiguously identify OAM states. Cleanspiral images (see Supplement 1, Section 5) in Fig. 2(b) indicatenegligible coupling among, and hence clean transmission of, all12 OAM states.

This result is counterintuitive—while the air-core design liftsdegeneracies among a host of OAM states, the modes still appearin degenerate pairs (spin-orbit aligned or anti-aligned). The co-efficient of power coupling between modes j and k is [17]

hPj;ki � k2Φ�kj − kk��Z Z

rdrdφΔn2�r;φ�ψ�j ψ k

�2

: (2)

ψ j and kz;j are the normalized electric field and longitudinal wave-vector of the jth mode, respectively, Δn2 is the index perturba-tion, and Φ�kj − kk� incorporates the perturbation’s longitudinal

behavior and is typically maximized for kz;j � kz;k (seeSupplement 1, Section 3). Thus, pairs of degenerate modesshould be susceptible to coupling within their two-mode subspacevia anisotropic perturbations such as the bends that existed on the10 m long fiber. In fact, for lower order, jLj � 1, OAM states,such coupling is possible [23] and controllably exploited [24]using a series of fiber loops, in analogy to a conventional

Fig. 1. (a) Free-space OAM states are coupled into air-core fiber and conserved despite bends and random fiber shape deformations. Each state possessestotal angular momentum, J , which comprises orbital, L, and spin, S, parts, which may be positive or negative (right or left handed). Thus, there are fourOAM states for every jLj, each of which can carry information. (b) Microscope image (top) and measured refractive index profile (bottom) for air-corefiber supporting 12 OAM states. The radius of the air core is 3 μm, and the outer radius of the ring region is 8.25 μm. (c) Example of nef f splitting amongOAM states. In conventional fibers, states of the same jLj are near-degenerate and freely couple. Via the air-core design, this near–degeneracy is brokensuch that states with spin and OAM aligned separate from states with spin and OAM anti–aligned. (d) Effective index splitting in a typical air-core fiber:∼10−4 is considered sufficient for OAM propagation [13] achieved in this design for jLj � 5, 6, and 7.

Fig. 2. (a) Setup: light from an external cavity laser (ECL; 1530 nm)or a picosecond pulsed laser (1550 nm) is converted to free-space OAMmodes via a spatial light modulator (SLM) followed by a quarter-waveplate, and launched into the air-core fiber. The fiber’s output is imagedthrough a circular polarization beam splitter (CPBS), separating σ̂− fromσ̂�. For interference measurements, the reference is tapped from a 50/50splitter at the input (orange path). For stability measurements, the air-core fiber passes through a polcon. For time-of-flight measurements (bluepath), a fast detector and oscilloscope (Rx and Osc) are used. Dashedlines indicate free space, solid lines indicate fiber. Fiber output imagesare for jLj � 7. (b) OAM states after 10 m of the air-core fiber, interferedwith an expanded Gaussian reference. Text around images indicateslaunch conditions. 12 states for jLj � 5, 6, 7 in all SAM/OAM combi-nations are shown. See Supplement 1, Section 6 for additional experi-mental details.

Letter Vol. 2, No. 3 / March 2015 / Optica 268

polarization controller (polcon) in single-mode fiber (SMF). Thepolcon may be understood as transfer of OAM from the bendperturbation to the field itself [25]. Any z-independent aniso-tropic perturbation may be expanded as

Δn2�r;φ� �X∞p�−∞

ap�r�eipφ; (3)

where ap�r� is the Fourier coefficient of the perturbation corre-sponding to angular momentum p · ℏ per photon. Coupling froma mode with L1 to one with L2 depends on the inner productbetween the initial field, ψ1 � F 1�r�eiL1φ, the perturbation,and the second field, ψ2 � F 2�r�eiL2φ. Evaluating the angularpart of this integral,

hψ1jΔn2�r;φ�jψ2i

�X∞p�−∞

hF 1�r�jap�r�jF 2�r�iheiL1φjeipφjeiL2φi; (4)

yields the selection rule:p − �L1 − L2� � 0: (5)

Bends and shape deformations additionally induce birefringence,which couples spins, as does a conventional polcon for SMF [26].Allowing for spin coupling, transitioning between higher orderdegenerate states (F 1�r� � F 2�r�) requires a perturbationelement of order p � 2jLj, which becomes increasingly negligiblefor large jLj [Fig. 3(a)]. To experimentally interrogate this curiouseffect, we build a polcon [27], but with the air-core fiber [purplecircles in Fig. 2(a)] with bend radius ∼2.8 cm. We define a deg-radation factor, α, as the ratio of the maximum power in σ̂� tothat in σ̂− when σ̂� is launched, or vice versa. For high jLj states,such as L � −7 σ̂� [Fig. 3(b)], degradation factors are typically

<10%. As expected, for the L � 0 mode in SMF [Fig. 3(c)],α ≅ 1, indicating complete coupling between the two degenerateSAM states. Due to the rapid decrease of japj as L increases, theobserved degradation factor decreases, with ratios as low as−12 dB (∼6%) for higher jLj states relative to SMF [Fig. 3(d)].Thus, we find that, for high jLj states in air-core fibers, OAM istruly a conserved quantity even in the presence of anisotropic per-turbations, since transitions among degenerate states are forbid-den, based on conservation of OAM [Fig. 3(e)]. This behaviorparallels forbidden transitions between electron spin states withan externally applied electric field. Here, anisotropic bends as-sume the role of electric field perturbations, leaving the initialstate unchanged.

Over long enough interaction lengths, light may encounterother perturbation symmetries due to twists and imperfectionsin the draw process. We experimentally study long-length propa-gation by transmitting OAM states with a picosecond pulsed(70 GHz bandwidth) laser at 1550 nm [Fig. 2(a)] and measuringtime-of-flight traces. At the output of fiber length z, modes j andk are temporally separated by Δtj;k � Δnj;kg z∕c. As all of theOAM modes in this fiber have similar group-velocity dispersions,relative mode purity, α, conventionally referred to as multipathinterference (MPI) [28], is given by

α � 10 log

�Ppeak 1 − P̄noise

Ppeak 2 − P̄noise

�; (6)

where P̄noise is the average noise power and Ppeak k is the peakpower of the kth mode. Combined time-of-flight measurementsfor modes in the jLj � 5, 6 families are shown in Figs. 4(a) andS5(a) (see Supplement 1 for Fig. S5), with close-ups of individual

Fig. 3. (a) Theoretical prediction of degenerate-state coupling for different OAM orders due to a 2.8 cm radius-of-curvature fiber bend. Coefficientsrapidly decrease with increasing OAM content, p. (b) Illustration of power binning measurement for L � 7, σ̂�. As the polcon paddles [Fig. 2(a)] aretuned, negligible coupling from σ̂� to σ̂− is observed, indicating degenerate-state stability. Legend “pol 1” indicates launched polarization; “pol 2”indicates parasitic polarization. (c) Polcon measurement for SMF, indicating complete degenerate state mode coupling. (d) Experimentally measuredaverage values of degradation factor α for each jLj, plotted against a shifted 1∕jLj trend line (dashed line). Degradation drops with increasing OAM. Thisconcept was tested experimentally on states for which spin-orbit aligned to spin-orbit anti-aligned coupling is suppressed. (e) Schematic indicating theperturbation OAM content necessary to couple degenerate fiber states with opposite values of L.

Letter Vol. 2, No. 3 / March 2015 / Optica 269

traces in Figs. 4(b)–4(e) and Figs. S5(b)–S5(e). We find thatMPIs of −18 dB or greater (>98% purity) can be achieved forany jLj � 5, 6 mode relative to the background, the jLj � 7modes being too lossy for 1 km transmission at 1550 nm. Weobtain similar results from 1530 to 1565 nm in wavelength, thusconfirming that the OAM states are wavelength agnostic. Loss forthe jLj � 5 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 betweenspin-orbit aligned and anti-aligned states, as the degenerate stateshave identical group delays. When OAM states are propagatedover kilometer lengths, we observe σ̂� to σ̂− transitions at the fiberoutput. This potentially indicates twist perturbations, known toaffect OAM stability [29]. Extending the quantum-mechanicalanalogy, twists would assume the role of magnetic perturbations,which couple electronic spin states. However, this coupling con-stitutes a unitary transformation within the two-mode subspaceand may be disentangled with devices such as q–plates [30], thusstill yielding a medium in which all eight of the states may beinformation carriers.

Conservation of light’s OAM in air-core fibers enableskilometer-length-scale propagation of a large ensemble of spatialeigenstates, in analogy to the perturbation resistance ofspinning tops and electron spin states. Therefore, this new pho-tonic degree of freedom, having attracted much recent attentionon account of its potentially infinite-dimensional basis, remainsa conserved quantity over lengths practical for optical communi-cations in appropriately designed fiber. Hence, we expect suchfibers and their OAM states to play a crucial role in the generalproblem of increasing the information content per photon.

Defense Advanced Research Projects Agency (DARPA)(W911NF-12-1-0323, W911NF-13-1-0103); National ScienceFoundation (NSF) (DGE-1247312, ECCS-1310493).

The authors would like to acknowledge J. Ø. Olsen for help withfiber fabrication; N. Bozinovic, S. Golowich, and P. Steinvurzelfor insightful discussions; and M. V. Pedersen for help with thenumerical waveguide simulation tool.

See Supplement 1 for supporting content.

REFERENCES

1. S. J. Van Enk and G. Nienhuis, Europhys. Lett. 25, 497 (1994).2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman,

Phys. Rev. A 45, 8185 (1992).3. D. L. Andrews, Structured Light and Its Applications (Academic, 2008).4. L. Yan, E. Auksorius, N. Bozinovic, G. J. Tearney, and S. Ramachandran,

in CLEO, OSA Technical Digest (online) (Optical Society of America,2013), paper CTu3N.2.

5. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, Science341, 537 (2013).

6. R.-J. Essiambre, G. Kramer, and P. Winzer, J. Lightwave Technol. 28,662 (2010).

7. J. Wang, J.-Y. Yang, I. M. Fazal, M. Ahmed, Y. Yan, H. Huang, Y. Ren,Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, Nat. Photonics 6, 488(2012).

8. A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev.Lett. 91, 227902 (2003).

9. Y. Tanaka, M. Okida, K. Miyamoto, and T. Omatsu, Opt. Express 17,14362 (2009).

10. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss,and P. St. J. Russell, Science 337, 446 (2012).

11. Y. Awaji, N. Wada, and Y. Toda, in CLEO, OSA Technical Digest (online)(Optical Society of America, 2012), paper JTu2K.3

12. B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M.Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, Opt. Lett.37, 3735 (2012).

13. S. Ramachandran and P. Kristensen, Nanophotonics 2, 455 (2013).14. C. Brunet, P. Vaity, Y. Messaddeq, S. LaRochelle, and L. A. Rusch, Opt.

Express 22, 26117 (2014).15. G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F.

Sciarrino, and P. Villoresi, Phys. Rev. Lett. 113, 060503 (2014).16. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E.

Willner, and S. Ramachandran, Science 340, 1545 (2013).17. A. Bjarklev, J. Lightwave Technol. 4, 341 (1986).18. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and

Hall, 1983).19. S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525

(2009).20. K. Oh, S. Choi, and J. W. Lee, J. Lightwave Technol. 23, 524 (2005).21. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, Nature

421, 925 (2003).22. P. Gregg, P. Kristensen, S. Golowich, J. Olsen, P. Steinvurzel, and S.

Ramachandran, in CLEO, OSA Technical Digest (online) (OpticalSociety of America, 2013), paper CTu2K.2.

23. C. N. Alexeyev, M. S. Soskin, and A. V. Volyar, Semiconductor Phys.Quantum Electron. Optoelectron. 3, 501 (2000).

24. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, Opt.Lett. 37, 2451 (2012).

25. P. Z. Dashti, F. Alhasen, and H. P. Lee, Phys. Rev. Lett. 96, 043604(2006).

26. J. N. Blake, H. E. Engan, H. J. Shaw, and B. Y. Kim, Opt. Lett. 12, 281(1987).

27. P. Gregg, P. Kristensen, and S. Ramachandran, inCLEO, OSA TechnicalDigest (online) (Optical Society of America, 2014), paper SM2N.2.

28. S. Ramachandran, J. W. Nicholson, S. Ghalmi, and M. F. Yan, IEEEPhoton. Technol. Lett. 15, 1171 (2003).

29. C. N. Alexeyev, J. Opt. 14, 085702 (2012).30. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E.

Nagali, and F. Sciarrino, J. Opt. 13, 064001 (2011).

Fig. 4. (a) Time-of-flight measurements, using setup of Fig. 2(a), forfour different OAM states. Traces vertically offset for visual clarity, inorder of increasing group delay: L � �5σ̂−, L � �5σ̂�, L � �6σ̂−,and L � �6σ̂�. Inset: fiber output image after 1 km propagation.(b) Close-up of time-domain trace for spin-orbit anti-aligned L � �5σ̂−

mode (peak around 519.5 ns is spurious from the detector’s electricalimpulse response). (c) Close-up of time-domain trace for spin-orbitaligned L � �5σ̂�. The time difference between the two L � �5 peaks,0.75 ns, corresponds well to the theoretical value of 0.7 ns. (d) and(e) show close-ups of L � �6σ̂− and L � �6σ̂� traces, indicating evenbetter parasitic mode suppression. The peaks from (d) and (e) wouldoverlap in conventional step-index fibers due to mixing. In each case,the excited mode is approximately 18–20 dB pure relative to the back-ground. See Supplement 1, Section 7 for additional details.

Letter Vol. 2, No. 3 / March 2015 / Optica 270