Conservation of orbital angular momentum in air-core optical fibers

P. Gregg; P. Kristensen; S. Ramachandran

doi: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 .

L

represents light’s orbital angular momentum (OAM) [2] and

S

represents its spin angular momentum (SAM), commonly known as left- or right-handed circular polarization,

{\hat{σ}}^{\pm}

, such that

S = \pm 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 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 [6]. 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 [12] in free space, or bends in fibers [13], 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 ( $| L | = 1$ ) 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 $| L |$ or radial quantum numbers. These near-degeneracies in linear momentum, or equivalently longitudinal wavevector, $k_{z}$ (in waveguides, also represented by effective index, $n_{eff}$ , given by $k = 2 π \cdot \frac{n_{eff}}{λ}$ where $λ$ is the free-space wavelength, and $z$ 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 [19] of OAM states with the same $| L |$ , 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 $n_{eff}$ splitting generally increases with $| L |$ [Fig. 1(d)]. A key feature of the air-core fiber [20] is the existence of modes with large $| L |$ but the prevention of modes with a large radial quantum number whose $n_{eff}$ 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 [21].

Fig. 1. (a) Free-space OAM states are coupled into air-core fiber and conserved despite bends and random fiber shape deformations. Each state possesses total angular momentum, $J$ , which comprises orbital, $L$ , and spin, $S$ , parts, which may be positive or negative (right or left handed). Thus, there are four OAM states for every $| L |$ , each of which can carry information. (b) Microscope image (top) and measured refractive index profile (bottom) for air-core fiber 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 $n_{eff}$ splitting among OAM states. In conventional fibers, states of the same $| L |$ are near-degenerate and freely couple. Via the air-core design, this near–degeneracy is broken such 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: $\sim 10^{- 4}$ is considered sufficient for OAM propagation [13] achieved in this design for $| L | = 5$ , 6, and 7.

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Using the experimental apparatus [22] 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 ${\hat{σ}}^{+}$ and ${\hat{σ}}^{-}$ into the right and left bins, respectively. Excitation of, for example, $| L | = 7$ 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 $| L |$ , while the handedness indicates the sign of $L$ . 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.

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 OAM modes via a spatial light modulator (SLM) followed by a quarter-wave plate, and launched into the air-core fiber. The fiber’s output is imaged through a circular polarization beam splitter (CPBS), separating ${\hat{σ}}^{-}$ from ${\hat{σ}}^{+}$ . For interference measurements, the reference is tapped from a 50/50 splitter at the input (orange path). For stability measurements, the air-core fiber passes through a polcon. For time-of-flight measurements (blue path), a fast detector and oscilloscope (Rx and Osc) are used. Dashed lines indicate free space, solid lines indicate fiber. Fiber output images are for $| L | = 7$ . (b) OAM states after 10 m of the air-core fiber, interfered with an expanded Gaussian reference. Text around images indicates launch conditions. 12 states for $| L | = 5$ , 6, 7 in all SAM/OAM combinations are shown. See Supplement 1, Section 6 for additional experimental details.

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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 $j$ and $k$ is [17]

〈 P_{j, k} 〉 = k^{2} Φ (k_{j} - k_{k}) {(\iint r d r d φ Δ n^{2} (r, φ) ψ_{j}^{*} ψ_{k})}^{2} .

ψ_{j}

and

k_{z, j}

are the normalized electric field and longitudinal wavevector of the

j

th mode, respectively,

Δ n^{2}

is the index perturbation, and

Φ (k_{j} - k_{k})

incorporates the perturbation’s longitudinal behavior and is typically maximized for

k_{z, j} = k_{z, k}

(see 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,

| L | = 1

, OAM states, such coupling is possible [23] and controllably exploited [24] 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 [25]. Any

z

-independent anisotropic perturbation may be expanded as

Δ n^{2} (r, φ) = \sum_{p = - \infty}^{\infty} a_{p} (r) e^{i p φ},

where

a_{p} (r)

is the Fourier coefficient of the perturbation corresponding to angular momentum

p \cdot ℏ

per photon. Coupling from a mode with

L_{1}

to one with

L_{2}

depends on the inner product between the initial field,

ψ_{1} = F_{1} (r) e^{i L_{1} φ}

, the perturbation, and the second field,

ψ_{2} = F_{2} (r) e^{i L_{2} φ}

. Evaluating the angular part of this integral,

〈 ψ_{1} | Δ n^{2} (r, φ) | ψ_{2} 〉 = \sum_{p = - \infty}^{\infty} 〈 F_{1} (r) | a_{p} (r) | F_{2} (r) 〉 〈 e^{i L_{1} φ} | e^{i p φ} | e^{i L_{2} φ} 〉,

yields the selection rule:

p - (L_{1} - L_{2}) = 0 .

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 order degenerate states (

F_{1} (r) = F_{2} (r)

) requires a perturbation element of order

p = 2 | L |

, which becomes increasingly negligible for large

| L |

[Fig. 3(a)]. To experimentally interrogate this curious effect, we build a polcon [27], but with the air-core fiber [purple circles in Fig. 2(a)] with bend radius

\sim 2.8 cm

. We define a degradation factor,

α

, as the ratio of the maximum power in

{\hat{σ}}^{+}

to that in

{\hat{σ}}^{-}

when

{\hat{σ}}^{+}

is launched, or vice versa. For high

| L |

states, such as

L = - 7

{\hat{σ}}^{+}

[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 degenerate SAM states. Due to the rapid decrease of

| a_{p} |

as

L

increases, the observed degradation factor decreases, with ratios as low as

- 12 dB

(

\sim 6 %

) for higher

| L |

states relative to SMF [Fig. 3(d)]. Thus, we find that, for high

| L |

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.

Fig. 3. (a) Theoretical prediction of degenerate-state coupling for different OAM orders due to a 2.8 cm radius-of-curvature fiber bend. Coefficients rapidly decrease with increasing OAM content, $p$ . (b) Illustration of power binning measurement for $L = 7$ , ${\hat{σ}}^{+}$ . As the polcon paddles [Fig. 2(a)] are tuned, negligible coupling from ${\hat{σ}}^{+}$ to ${\hat{σ}}^{-}$ 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 measured average values of degradation factor $α$ for each $| L |$ , plotted against a shifted $1 / | L |$ trend line (dashed line). Degradation drops with increasing OAM. This concept was tested experimentally on states for which spin-orbit aligned to spin-orbit anti-aligned coupling is suppressed. (e) Schematic indicating the perturbation OAM content necessary to couple degenerate fiber states with opposite values of $L$ .

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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 $z$ , modes $j$ and $k$ are temporally separated by $Δ t^{j, k} = Δ n_{g}^{j, k} z / c$ . As all of the OAM modes in this fiber have similar group-velocity dispersions, relative mode purity, $α$ , conventionally referred to as multipath interference (MPI) [28], is given by

α = 10 \log (\frac{P_{peak 1} - {\bar{P}}_{noise}}{P_{peak 2} - {\bar{P}}_{noise}}),

where

{\bar{P}}_{noise}

is the average noise power and

P_{peak k}

is the peak power of the

k

th mode. Combined time-of-flight measurements for modes in the

| L | = 5

, 6 families are shown in Figs. 4(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

- 18 dB

or greater (

> 98 %

purity) can be achieved for any

| L | = 5

, 6 mode relative to the background, the

| L | = 7

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

| L | = 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 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

{\hat{σ}}^{+}

to

{\hat{σ}}^{-}

transitions at the fiber output. This potentially indicates twist perturbations, known to affect OAM stability [29]. 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

q

–plates [30], thus still yielding a medium in which all eight of the states may be information carriers.

Fig. 4. (a) Time-of-flight measurements, using setup of Fig. 2(a), for four different OAM states. Traces vertically offset for visual clarity, in order of increasing group delay: $L = + 5 {\hat{σ}}^{-}$ , $L = + 5 {\hat{σ}}^{+}$ , $L = + 6 {\hat{σ}}^{-}$ , and $L = + 6 {\hat{σ}}^{+}$ . Inset: fiber output image after 1 km propagation. (b) Close-up of time-domain trace for spin-orbit anti-aligned $L = + 5 {\hat{σ}}^{-}$ mode (peak around 519.5 ns is spurious from the detector’s electrical impulse response). (c) Close-up of time-domain trace for spin-orbit aligned $L = + 5 {\hat{σ}}^{+}$ . 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 $L = + 6 {\hat{σ}}^{-}$ and $L = + 6 {\hat{σ}}^{+}$ traces, indicating even better parasitic mode suppression. The peaks from (d) and (e) would overlap in conventional step-index fibers due to mixing. In each case, the excited mode is approximately 18–20 dB pure relative to the background. See Supplement 1, Section 7 for additional details.

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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.

FUNDING INFORMATION

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

ACKNOWLEDGMENT

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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Conservation of orbital angular momentum in air-core optical fibers

Abstract

Corrections

FUNDING INFORMATION

ACKNOWLEDGMENT

REFERENCES

Supplementary Material (1)

Cited By

Figures (4)

Equations (6)

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