All-in-Fiber Dynamically Reconfigurable Orbital Angular Momentum Mode Sorting

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■ INTRODUCTION
The orbital angular momentum (OAM) modes of light 1,2 have captured the interest of the scientific community due to their many possible applications, such as optical manipulation, 3,4 information transfer, 5 data multiplexing, 6 quantum key distribution, 7 and high-dimensional quantum information processing. 8OAM modes are commonly represented using a Laguerre−Gaussian (LG) basis set, which has a ring-like amplitude profile and an azimuthal phase dependence e ( ) .LG modes are characterized by azimuthal and p radial integer numbers. 1 In contrast, the modes that only carry OAM have only an azimuthal phase structure and are described only by the index .The fact that ± OAM modes are mutually orthogonal and can generate an infinite state space makes them a suitable platform for applications requiring d spatial modes (or d dimensions), such as classical and quantum communications.For many applications, it is crucial to develop OAM-supporting devices that are compatible with optical fibers, can be integrated into other systems, and are scalable.Direct compatibility with optical fibers was not an issue in the past, but since the recent advent of space-division multiplexing (SDM) optical fibers that can support the propagation of OAM modes, 9 the use of bulk optical elements can now be seen as a major disadvantage.
Although some efforts have been made to generate OAM modes using fiber systems, 10−16 the efficient detection of OAM modes using all-in-fiber platforms remains an ongoing challenge.An ideal OAM mode sorter is a device capable of deterministically demultiplexing different incoming electromagnetic fields that carry OAM.Its internal operation depends on the spatial information carried by the incoming mode. 17,18igure 1 schematically shows the behavior of a general OAM mode sorter, where several OAM modes with different azimuthal indexes can be sorted at the output of the system.
OAM mode sorters have been implemented using several techniques, such as with the use of spatial light modulators, 17,19−21 q-plates, 22,23 geometrical transformations, 24 log-polar coordinate transformations, 25,26 spiral transformations, 27,28 multiplane converters, 29−31 and Porro prisms. 32−35 All of these systems are based on performing operations on the OAM modes using bulk optics, which makes their integration into current telecommunication networks difficult.Furthermore, when the number of spatial modes to be distinguished is scaled up, this type of scheme becomes complex.A fiber-based technique recently presented is a sorter using optical fiber delays, 36 taking advantage of the fact that the different modes propagating in a fiber will separate due to modal dispersion.This method has a limited bandwidth due to the long delays needed.Finally, a static multiplexer/ demultiplexer based on a photonic lantern has been presented; 37 however, as it has no reconfigurable capabilities, its applications in quantum information processing or in dynamic optical networks are impractical.
In this paper, we present a novel all-in-fiber platform with ultrafast reconfigurability for interferometric sorting of arbitrary incoming ± OAM modes.Our approach is based on the fact that any ± OAM mode can be decomposed into a combination of an odd and an even linearly polarized (LP) mode, with a relative phase difference of ±π/2. 38,39The key components of our platform consist of an optical fiber supporting multiple transverse spatial propagation modes and a photonic lantern (PL). 40,41We can immediately obtain complete information about the parameter ± of the incoming OAM mode by detecting which output of our system the light beam exits through.Our scheme can be scaled up by employing components that support more spatial modes.In addition, we can run our system from passive operation, where the modes are sorted to active, where we show that the information on the OAM mode can be arbitrarily redirected to any other route.To experimentally demonstrate the feasibility of the scheme, we successfully implemented an all-in-fiber reconfigurable mode sorter for the OAM +1 and OAM −1 modes.As we used off-the-shelf components, our system can be easily integrated into fiber-optical telecommunication networks.Our sorter can operate at high speeds, being able to perform routing operations with a switching time as low as 7 ns, limited by the driving electronics.To the best of our knowledge, this represents a record speed response for OAM mode sorters, making them excellent candidates for active (or passive) demultiplexing systems in current optical networks and for applications in quantum information processing.

■ ALL-IN-FIBER OAM MODE SORTER
As shown in Figure 2a, we present the experimental setup of our proposal for sorting the OAM states.The inputs to our scheme are any state between + OAM and OAM , where can only take integer values.The mode sorter consists of a fewmode fiber 42 connected to a photonic lantern. 43The physical principle behind our experiment relies on the fact that any ± OAM mode within a few-mode fiber can be decomposed into the following LP modal superposition: 14,39,44,45 = ± The photonic lantern maps the LP modes that follow the above decomposition from a specific ± OAM mode coming from the few-mode fiber with a one-to-one correspondence into a pair of single-mode fiber outputs only supporting the fundamental LP 01 mode (Gaussian).The lantern also preserves the phase relation between the LP mode pair.The single-mode fibers are then recombined interferometrically with a multiport beam splitter (MBS). 46Phase modulators (PMs) are placed in each arm of this high-dimensional interferometer to adjust the internal relative phases.It is therefore possible to identify uniquely the state's topological charge ± by detection at the output of the interferometer.This method is completely infiber, and this makes it perfect for integration to OAM sources and fiber systems. 9,13,47,48Observe that if the appropriate phases are set in all of the phase modulators, 46 it is possible to route the mode information to any output.Furthermore, this reconfigurable setup allows one to implement any unitary transformation onto a high-dimensional OAM-encoded quantum state, which is highly sought after in quantum information.In this work we perform an experimental demonstration of the scheme to sort the OAM +1 and OAM −1 modes (Figure 2b), based on 3-mode few-mode fibers and photonic lanterns.

■ EXPERIMENTAL SETUP
We now describe in detail the experimental setup for the 2D version of our scheme (Figure 3).We generate the OAM ±1 modes using the well-known technique based on a computergenerated hologram on a spatial light modulator (SLM) mimicking the desired optical element capable of converting a Gaussian laser beam into a helical mode. 49Although this is done in free space for convenience, our mode-sorting system is completely in-fiber, and it would work equally well if the OAMs were generated with an in-fiber method. 14A continuous wave (CW) laser operating at 1546.92 nm is spatially filtered through a single-mode fiber (SMF) and collimated with a 10x objective lens to reach a waist of ω p ≈ 1250 μm.A polarizing beam splitter (PBS) is used to set the laser's polarization state to horizontal for the optimal operation of the SLM.A manual polarization controller placed before the objective is used to optimize the optical power following the objective.Note that ω p is sufficient to illuminate all of the -forked holograms, which consist of a helical phase profile superposed with a linear phase ramp to spatially isolate the encoded field from the Gaussian mode, resulting in a diffraction grating that produces the ± OAM mode in the first diffraction order. 50We use a Holoeye Pluto-Telco-013 phase-only SLM with a resolution of 1920 × 1080 pixels and an 8 μm pixel pitch to implement the holograms.Each pixel acts as a programmable phase shifter between 0 and 2π, which is controlled by the grayscale value of the corresponding pixels of the hologram.
The SLM reflects the beam onto a 4f system composed of two lenses, L 1 and L 2 , each with a focal length of 150 mm.The Fourier transform of the reflected field is formed in the focal plane of L 1 , where there is a pinhole.This spatial filter aims to pick out the diffracted first order, corresponding to the Fourier spectrum of the encoded field.The second lens (L 2 ) performs an inverse Fourier transform, resulting in the OAM ±1 modes.A beam splitter (BS) is placed at the end of the 4f system in order to split the beam and allow us to image the generated mode during the experiment.The transmitted mode is imaged by an InGaAs CCD camera placed in the image plane of L 2 (Figure 3 and inset).Both the amplitude and phase profiles of the OAM modes are imaged on the camera.For the phase profiles, we create a reference mode by splitting the laser's beam with a 50:50 fiber coupler before the input 10x objective (not shown in the figure for the sake of simplicity).This reference beam is connected to the fiber launcher named "control input" in Figure 3, which consists of another 10x objective, where it is superposed with the OAM beam on the beam splitter.Therefore, the phase profiles are imaged as the interference pattern between these two beams and shown in the inset of Figure 3.This control laser input is otherwise used as a launch device to a second diode laser as part of an active phase stabilization system for the mode sorter interferometer (further explained below).
On the reflected output of the BS, there is a 20x objective used to couple the OAM ±1 mode into a FMF.The FMF used in this experiment is a commercially available graded index telecom fiber (OFS 80730) that can support three modes: the fundamental Gaussian LP 01 and the higher-order LP 11a and LP 11b modes.The linearly polarized modes are particular modes that propagate in a weakly guided medium.Therefore, when the 20x objective couples the OAM ±1 mode into the FMF, it is decomposed into LP modes propagating in the FMF.A manual fiber polarization controller, where the FMF is wound, is used to correctly adjust the modal demultiplexing in the photonic lantern.This fiber carrying the incoming OAM ±1 modes is then connected to a commercial PL (Phoenix Photonics 3PLS-GI-15) consisting of one FMF at its input and three single-mode fibers at its output, corresponding to each of the supported LP modes.The internal structure of the PL is made of an adiabatic region that provides a low transition from the input FMF to the output of the three SMFs in such a way that each LP mode is mapped to the appropriate output port. 40,41This mapping is produced by a matching process between the effective indices of the tapered region and the incoming spatial modes. 39,51,52Since the incoming OAM mode only decomposes into the LP 11a and LP 11b modes, we will use only the two outputs of the PL corresponding to these modes, leaving out the optical port that decomposes the LP 01 mode.After the OAM ±1 modes are decomposed, the following mapping is done: LP 11a → E 1 ; LP 11ab → E 2 , where E 1 and E 2 represent both electrical fields propagating through the two different paths after the PL.
A lithium niobate pigtailed telecom phase modulator (PM) (Thorlabs LN65S-FC) is placed on the upper arm.The PM is driven by an electrical signal from a function generator, so any arbitrary relative phase ϕ can be added to the upper arm between 0 and 2π.A variable optical attenuator (not shown for simplicity) is located in the lower arm to equalize the optical power from both arms of the interferometer before recombining on the 50:50 fiber coupler (FC).We also have manual polarization controllers before the FC to ensure path indistinguishability. D +1 and D −1 are amplified p−i−n photodiodes to measure the optical intensity at the two outputs, with the results recorded using a digital oscilloscope.
Due to the natural instability of the interferometric configuration, we inject a laser at a different wavelength (1546.12nm) through the free port of the beam splitter before the InGaAs CCD camera (Figure 3).This laser beam copropagates with the OAM modes through the sorter and is used as a feedback signal for a phase stabilization system.This reference beam is split from the OAM information before D +1 with a dense wavelength division multiplexer (DWDM) and is detected by a third amplified p−i−n photodiode D r , whose electrical output is read using a microcontroller running a perturb-and-observe algorithm 53 to stabilize the environmentally induced phase drift in the mode sorter interferometer.The phase controller employs a second phase modulator (PM c ) placed in the opposing arm to compensate for the phase drift.The control system operates continuously but is briefly switched off when there is a need to operate the sorter in the routing mode.The optical loss from the PL input to the detectors is 8.3 dB, which is mainly given by the PM and the internal losses of the PL (∼5 dB).These losses can be greatly reduced by new lantern designs that have much lower losses (up to 0.2 dB), 54 showing a path to having a mode sorter with very low losses.

■ RESULTS AND DISCUSSION
The first goal is to demonstrate successful sorting between the OAM ±1 modes with our scheme.Given that the mode sorter Mach−Zehnder interferometer (MZI) is actively stabilized by the controller, the power detected in D +1 and D −1 will depend on the additional relative phase ϕ applied by the PM and also of the topological charge = ±1 of the incoming OAM beam.It is well known that in an MZI, the optical intensity registered in D +1 (D −1 ) is proportional to From eq 1, we notice that the incoming OAM ±1 beam carries a relative phase between both LP modes of ± π/2.By adjusting the controller to maintain a relative phase between both arms of π/2, if the incoming beam is OAM +1 (OAM −1 ), we obtain maximum optical intensity detected at D +1 (D −1 ).We then continuously switch between the OAM +1 and the OAM −1 modes every second by changing corresponding forked diffraction gratings in the SLM. Figure 4 shows the electrical signal corresponding to the optical intensity measured at D +1 , when fixing ϕ = π/2 is fixed in the interferometer, with periodic switching of the OAM modes.From this result, we show the deterministic identification of the direction of rotation of the OAM mode and thus its topological charge.
To demonstrate the stability of the sorting process over longer time periods, we repeat this experiment continuously for 1 h.We calculate the optical visibility using the consecutive optical intensities between the maxima and minima at the output of D +1 as the input OAM mode is continuously switched.For any two consecutive maxima and minima, we calculate the visibility V as V = (I + − I − )/(I + + I − ), where I + (I − ) is the detected optical intensity when the OAM +1 (OAM −1 ) mode is sent.We plot the visibilities calculated for every mode transition as a function of time, as shown in Figure 5, while also displaying in the inset, the histogram of all of the measured visibilities giving an average of 92.59 ± 1.87%.
Our system can also operate as an active router for OAM modes by applying fast relative phase changes through the PM.For this demonstration, we fix the OAM +1 in the source while electrically driving a train of ten 100 μs-long pulses to the PM.During the time the pulse train is sent, the phase stabilization algorithm is paused.The amplitude and offset of these driving pulses were previously calibrated in order to have a relative phase applied between the paths equivalent to ϕ = π/2 and ϕ = −π/2, respectively.We observe the output of the system by recording the optical intensity at D +1 .By applying the sequential pulses to the PM, the OAM +1 mode can be routed to either D +1 or D −1 , depending on the value of the driving relative phase, with these results shown in Figure 6a, where we can clearly see the sequential routing of the OAM +1 mode between the two outputs.The system works equally well if we now observe the D −1 output, as shown in Figure 6b, with complememtary routing operation, as expected.The average visibility for both cases is 92.75 ± 1.27%, with the main limitation to the optical visibility originating from the modal cross-talk in the photonic lantern.
The ultrafast reconfigurability is a major advantage of our configuration, stemming from the use of telecom electrooptical fiber pig-tailed phase modulators.For this demonstration, we replace the p−i−n photodiode, which has a limited bandwidth of 10 MHz, with an InGaAs single-photon counting module (IdQuantique id210) having a timing resolution of 200 ps.The detector runs with 10% detection efficiency and a 2.5 ns wide gate window.We then modulate the internal PM in the mode sorter with a series of 20 ns wide transitions between π/2 and − π/2 and synchronize the detector's gate window with these transitions.We then vary the electrical delay to the trigger signal fed to the InGaAs detector in steps of 1 ns, allowing us to scan the optical phase pulse which switches between the OAM modes, and the result is shown in Figure 6c, where we show a rise/fall time of 7 ns, which is limited by our homemade electrical driver amplifying the electrical signal feeding the phase modulator.This value can be further improved with appropriate electronics as the response bandwidth of the phase modulator is within the 10 GHz range.For this measurement, the OAM laser beam is attenuated such that only a few photons per detection window are present and, thus, the detector is not saturated.

■ CONCLUSIONS
Appropriate tools for manipulating the OAM states are highly sought in many areas of photonics, opening up new possibilities based on this degree of freedom of light.Sorting of the OAM modes has been one particular area of considerable intense research in the last years, using many different techniques.Common to most of these previous methods is the use of bulk optical elements, to a greater or lesser degree.The downside of these schemes is the difficulty of integration with optical fiber systems due to the additional losses and complexity needed to couple the light to and from a fiber.This is especially acute in cases where long-distance transmission is desired, such as in optical and quantum communications.In the few cases where optical-fiber-based techniques were used, no reconfigurability or high-speed responses were possible, severely limiting those to applications in which dynamic operation is essential, such as in quantum information processing.Here, we solve this issue by combining photonic lanterns with an interferometric analysis of the components of an OAM mode.Our sorter is designed to receive an OAM mode coming from a few-mode fiber and it decomposes it into its linearly polarized mode components with a photonic lantern.Then, by recombining these modes interferometrically onto a beam splitter, we are able to sort the incoming modes.Our scheme is generalizable to higher dimensionalities by employing few-mode fibers, photonic lanterns, and multiport beam splitters that support more transverse spatial modes.For instance, MBSs with 7 ports 46 or, alternatively, arrays of 2-port beam splitters on a photonic-Figure 5. Long-term performance of the OAM mode sorter.We plotted the visibilities during a period of 1 h calculated from the constant switching between the OAM +1 and OAM −1 modes from the source through the SLM.The inset shows a histogram based on these results, showing that the average visibility is 92.59 ± 1.87%.Figure 6.OAM mode routing with the source transmitting the OAM +1 state.(a) Optical intensity at the D +1 output when the internal PM in the mode sorter changes ϕ from π/2 to − π/2 ten times consecutively in a period of 1.1 ms, generated from a field programmable gate array (FPGA) electronic module.During the switching time, the phase stabilization algorithm is suspended and resumed after the last switching pulse is applied.In this way, it is possible to route the OAM +1 mode to any detector arbitrarily.(b) Same measurement, but observed in detector D −1 , where the complementary nature of the interferometer can be observed, as expected.The routing speeds in the measurements shown in panels (a) and (b) are limited by our p−i−n photodetectors.(c) Demonstration of the speed response of our mode sorter by changing the p−i−n photodiode at D +1 for an InGaAs single-photon detector, which has a timing resolution of 200 ps.For this measurement, we attenuated the output power of the source to a few photons per detector gate window of 2.5 ns.We measured a rise/fall time of approximately 7 ns, limited by our employed driving electronics.integrated circuit functioning as a general d-port beam splitter 55 as well as lanterns with 15, 56 19, 57 and 61 58 ports can all be used, showing the potential of our scheme.Furthermore, very recent results have demonstrated photonic lanterns with very low loss (<0.2 dB), 54 further showing the feasibility of our proposal for practical applications.
Finally, due to the interferometric nature of our configuration, it can also be used in an active/reconfigurable mode, where the input mode can be routed to a different output on demand, or a unitary transformation can be applied by the phase modulator, allowing the measurement in different mutually unbiased bases in quantum information. 46In this active mode, we have demonstrated, to the best of our knowledge, the fastest response in an OAM router, with the potential to reach subnanosecond response times due to the use of fast telecom phase modulators.In addition, we demonstrated the stability of our mode sorter, which can be further improved in future by making the interferometer more compact (using integrated photonic circuits for example 59 ), implementing modal correction before the photonic lantern 60 and motorized polarization control inside the interferometer.For practical applications, robustness to degradation of state fidelity during transmission is an issue.Two common sources of noise, namely, modal distortion from propagation and added noise from the decrease in the signal intensity as well as other background noise, would lead to a lowering in the visibility pattern.Further studies are needed to quantify these effects and evaluate the sorting performance of the degraded OAM states.At last, if one wishes to recover the OAM spatial information at each output, it is possible to split the beam into two paths and interferometrically reconstruct the OAM state with a photonic lantern. 14Our results show how the use of space-division multiplexing technology (i.e., few-mode fibers and photonic lanterns) can be used to implement a fast and reconfigurable OAM mode sorter that is fully compatible with optical fiber networks, further paving the way for this popular degree of freedom to be used as a reliable transport medium for information in classical and quantum networks.

Figure 1 .
Figure 1.General ideal OAM mode sorter.The purpose of an OAM mode sorter is to sort incoming OAM modes to a predetermined output, which depends on both the topological charge of the incoming beam and any operations carried out by the sorter.

Figure 2 .
Figure 2. Proposed OAM mode sorter.(a) In this general high-dimensional case, there is a source-emitting OAM mode with different azimuthal indices.Each OAM mode is mapped onto different output paths from the photonic lantern (PL), which are then superposed by a multiport beam splitter (MBS).Each output path is connected to a detector to determine the topological charge of each OAM mode.(b) Experiment realized on the two-dimensional case allowing the sorting of the OAM +1 and OAM −1 modes, with detectors placed at the two outputs.D: detector; FMF: fewmode fiber; PL: photonic lantern; and PM: phase modulator.

Figure 3 .
Figure3.Experimental setup.The setup for demonstrating the sorting of the OAM ±1 modes can be divided into two main parts: the OAM ±1 mode source and the OAM ±1 mode sorter.The OAM source can switch between the OAM +1 and the OAM −1 modes by changing the forked diffraction grating on the spatial light modulator (SLM).The mode sorter is an all-fiber interferometer based on space-division multiplexing fiber technology.The computer-generated holograms used to create the OAM ±1 modes are a superposition of a helical phase and a diffraction grating.The inset shows the generated spatial intensity and phase profiles (interference patterns) of the OAM modes.The phase profiles are obtained from the interference pattern between the corresponding OAM mode and a Gaussian beam to distinguish the topological charge = ±1.These spatial patterns are detected by an InGaAs CCD camera.A control laser at a different wavelength is injected onto the FMF through the beam splitter (BS) before the CCD camera and is used as a feedback signal for a microcontroller (MC) running a control algorithm to stabilize the phase drift noise inside the mode sorter interferometer.A phase modulator (PM c ) placed in one of the arms is used as the actuator for this control system.The other phase modulator (PM) is used to dynamically reconfigure the mode sorter.D: photodiode; FC: fiber coupler; HWP: half-wave plate; L: lens; PC: personal computer; and WDM: wavelength demultiplexer.Please see the text for more details.

Figure 4 .
Figure 4. OAM mode sorting.Here, we show how our setup can sort the OAM modes when the source is constantly changing between the OAM +1 and the OAM −1 modes with a period of 1 s.A relative phase of π/2 is being applied inside the interferometer by the PM, such that the OAM +1 mode is sent to D +1 and the OAM −1 mode is sent to D −1 .The intensity curve shown above as a function of time is measured at D +1 .The OAM mode labels indicate when each particular OAM mode is being sent.