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We present an optical fiber supporting 36 information bearing orbital angular momentum (OAM) states spanning 9 OAM orders. We introduce design techniques to maximize the number of OAM modes supported in the fiber; while avoiding LP mode excitation. We fabricate such a fiber with an air core and an annular index profile using the MCVD process. We introduce a new technique for shaping OAM beams in free-space to obtain better coupling efficiency with fiber with annular index profiles. We excite 9 orders of OAM in the fiber, using interferometry to verify the OAM state on exiting the fiber. Using polarization multiplexing and both signs for the topological charge, we confirm support of 36 states, exploiting to our knowledge the highest number of OAM modes ever transmitted in optical fiber.
© 2014 Optical Society of America
1. Introduction
Spatial multiplexing holds great promise for increasing the throughput of a single optical fiber while reducing the cost per bit [1]. The exploitation of multiple modes can be applied to either a single core or multiple cores within a single fiber. Modal-multiplexing can be implemented with linearly polarized (LP) or orbital angular momentum (OAM) modes. Both LP and OAM modes are formed from linear combinations of vector eigenmodes of the fiber. In the case of an LP mode, the component eigenmodes have distinct and disparate propagation constants. OAM modes, on the other hand, have exactly two component eigenmodes with identical propagation constants. This distinction has tremendous impact on fiber design as well as the demultiplexing reception strategy.
LP modes, being each individually composed of multiple eigenmodes, see their transmitted signals spread temporally due to the disparate propagation constants of those eigenmodes. In addition, multiple LP modes share common component eigenmodes, hence energy flows from one LP mode to another during propagation. The significant interactions between LP modes would normally preclude their utility as information channels, however, digital signal processing (DSP) can be brought to bear to undo the mixing taking place in the fiber using multiple-input, multiple-output (MIMO) techniques [2, 3]. This signal processing comes at some cost, both in complexity and power consumption, as well as signal degradation due to noise enhancement.
Research into OAM for optical communications began with free-space communications [4] as fiber was not available. Most free-space experiments covered only a very short distance (about 1 m); in general, free-space communications have only limited applications. Optical fiber supporting OAM modes was first discussed in [5]. Only 10 years later, however, was the first OAM data transmission experiment performed in a special designed fiber [6]. This fiber supported the fundamental mode and one OAM order (LP0,1 and OAM1,1, in both left- and right-circular polarization) to create a total of four channels. Reliable transmission over one kilometer without MIMO equalization was demonstrated for 50 Gbaud QPSK transmission [7]. A few other optical fibers designed to support OAM modes were reported: an air-core fiber supporting 3 OAM orders (OAM7,1, OAM8,1, OAM9,1) for a total of 12 information bearing states [8, 9], an inverse parabolic graded index fiber supporting 2 OAM orders (OAM1,1, OAM2,1), for a total of 8 states [10], and the air-core fiber we report in this paper [11] supporting 9 OAM orders and 36 states.
While free-space systems can theoretically exploit an unlimited number of OAM states, optical fiber only supports a limited number, the number of modes being determined by the fiber index profile. In this paper, we will present experimental results for a fiber with a hollow center and an highly doped annular core, and designed to support a large number of OAM modes. To the best of our knowledge, our fiber supporting 36 information bearing states has the highest reported number of OAM orders transmitted through optical fiber. We designed, fabricated and characterized our fiber at our laboratory at COPL.
To validate that our fiber indeed supports the targeted OAM modes, we generate OAM beams in free-space and couple them into the fiber. Conventional OAM beams, however, have a ring diameter that varies with order or topological charge ℓ; these diameters are incompatible with our fabricated annular fiber dimensions. We therefore developed a technique to generate perfect OAM beams whose ring dimensions could be adjusted to optimize fiber coupling.
In [12], Ostovsky, et al. introduced the perfect vortex beam: one having ring diameter independent of topological charge. The perfect OAM (or vortex) beam demonstrated by Ostovsky, et al., however, has periodic variation along the azimuthal direction. Recently, an axicon was also used to generate perfect vortex beams [13]. We demonstrate a new technique to form a perfect OAM beam with controllable ring-width and diameter. Our technique uses a new phase mask formed by combining axicon, and spiral phase functions. Multiple orders of perfect OAM beams are generated, multiplexed in free-space and coupled into an air-core fiber designed to support OAM mode propagation.
In section 2, we give a brief overview of the orbital angular momentum of light, and we introduce our mode notation. In section 3, we present the design and the fabrication process of our special air-core fiber. We show effective index, group index, and dispersion parameter simulated from the measured profile following fabrication. In section 4, we present a novel method to generate a perfect OAM beam that matches the fiber profile, and how to couple it into the fiber. We also present experimental results validating that our fiber supports 36 information bearing states.
2. OAM modes
An orbital angular momentum (OAM) beam of light has a helical phase front that can be expressed, in polar coordinates, as Φ(r, ϕ) = exp(jℓϕ), where ℓ is an integer value known as topological charge. Due to this rotating phase, the light has orbital angular momentum [14]. This is different from spin angular momentum (SAM), which is related to the rotation of the polarization vectors. In fact, both OAM and SAM can coexist in the same beam of light.
In this paper, OAM modes are identified as OAMℓ,m, where ℓ is the topological charge of the OAM mode, and m is the number of concentric rings in the intensity profile of the mode. OAM modes in optical fibers have an electrical field defined by:
The number of supported states includes all degeneracies in polarization and in direction of rotation of the phase front of the fields, i.e., sign of the topological charge. The two OAM0,m modes are in fact the fundamental HE1,m modes with left- or right- circular polarization. The two OAM1,m modes always have the same directions of circular polarization and rotation, hence the OAM1,m mode has two information bearing states. Higher order OAM modes yield four independent information bearing modes (or channels or states), as they can have polarization and rotation either in the same or in opposite directions (see Fig. 1).HE|ℓ|+1,m modes are the basis of OAMℓ,m modes, with a circular polarization in the same direction as the field rotation, while EH|ℓ|−1,m modes are the basis of OAMℓ,m modes with a circular polarization in the opposite direction than the field rotation. This can be summarized with the following equations:
where the sign in the superscript denotes the direction of the circular polarization, and the sign of ℓ denotes the direction of the wave-front rotation.3. Fiber design and fabrication
3.1. Design strategy
To support OAM modes, the fiber must exhibit 1) good effective index separation between vector modes (the true eigenmodes of the fiber), to minimize modal coupling and degeneracy into LP modes, and 2) a fiber profile that matches the donut shaped OAM fields.
To obtain good mode separation, we must have a high contrast in refractive indices of the fiber materials; in this way we violate the weakly guiding approximation under which LP modes are formulated. In particular, there must be great separation between the effective indices of HEℓ+1,m and EHℓ−1,m modes, otherwise those modes would couple into LPℓ,m modes, and we would lose the OAM states during propagation. Design of polarization-maintaining fibers suggests that an effective index separation on the order of 10−4 between the modes in a group will preclude LP mode formation [6, 7]. This is an order of magnitude greater separation than typically found in conventional fibers. In [9], they transmitted OAM modes over 2 m of fiber with an effective index separation around 0.6 × 10−4.
The refractive index contrast is limited by material constraints and by the fabrication process. To achieve maximum contrast, a hollow air-core was suggested [8], since air has an index of approximately 1, while doped Silica has an index that does not fall below 1.45 (at 633 nm).
The fiber designed in [9] supports a large number of vector modes, but only a few of them are separated enough (in terms of effective index) to be used for OAM modes. Hence only 3 OAM orders, or 12 information bearing states, can be supported. In contrast, we designed a fiber supporting fewer vector modes in total, but where all those vector modes can be used as a basis for OAM modes. All modes must have an effective index that resides between the refractive index of the cladding and the maximum refractive index of the fiber. A fiber having fewer vector modes allows them to be more separated (in terms of effective index), as the space of possible effective indices is exploited by all supported modes. This is achieved by carefully adjusting the width of the doped region in the fiber to limit the total number of supported vector modes. We included an external layer of material having a refractive index lower than the cladding, to increase the contrast of the refractive indices, and to limit the number of higher order modes, i.e. limit m.
3.2. Design software
To design the fiber, our software calculates the supported vector modes for a given step-index fiber profile, using the transfer-matrix method [15]. We impose constraints (such as the maximum and minimum refractive indices, and the minimum number of supported modes), then sweep several parameters of the fiber profile (the refractive index and the radius of each layer of the fiber), maximizing the minimum effective index separation of the supported modes. We start from the simplest design (a single core fiber), then we progressively add layers, observing the performance at each configuration. The final design (dotted line in Fig. 2(a)) was chosen due to the good balance between the number of supported modes and the separation of the effective indices, with a good tolerance to imperfections introduced into the index profile during the fabrication process. As the doped region is relatively thin, only modes with m = 1 (that is modes having a single ring in their intensity profile) are supported. This also simplifies the multiplexing and the demultiplexing of the OAM modes, and prevents crossing between effective indices of modes belonging to different families, allowing good mode separation across all C-band wavelengths. Finally, the fiber profile has an annular shape, which is suitable since the intensity profile of OAM beams is also ring-shaped.
Fig. 2 (a) Designed (dotted), measured (red), and simulated (green) fiber profile, at 657.6 nm. Index in the center part is 1, although measurement shows the index of the refractive index liquid used when measuring. (b) Optical microscope photo of actual fiber.
3.3. Fiber profile and fabrication
The optical fiber was fabricated at our laboratory at COPL, equipped with modified chemical vapor deposition (MCVD) for preform fabrication and two drawing towers for fiber pulling. Figure 2(b) shows a photo of one end of the fiber. The cladding diameter is around 125 μm, as in conventional fiber. The fiber preform was fabricated via MCVD in two steps. We formed the first layer by incorporating adequate concentrations of SiO2, P2O5 and F. This was followed by a deposit ring layer of SiO2 and GeO2 to produce a step index profile. Finally, the result was partially collapsed to produce the glass preform. The hollow core diameter is controlled during the fiber drawing process to achieve the target value. The refractive index profile was measured directly on the fabricated fiber using a refracted near-field analyzer (Exfo NR-9200HR), and is shown in Fig. 2(a). Fabricating a silica fiber with a very high refractive index contrast is difficult, because of the large stress that appears at the core cladding interface during fiber fabrication. Figure 2(a) shows that the fabricated fiber has a lower refractive index within the doped ring than targeted. We extracted an estimate of the fabricated index profile, green curve in Fig. 2(a), and used this estimate in new simulations of the number of modes supported.
Using our software based on the transfer-matrix method, we calculated expected effective index, group index and dispersion parameters for all guided modes, based on the measured fiber profile. Those results are presented in Fig. 3. From Fig. 3(a), we see that we can expect the fiber to support modes up to the HE8,1 – EH6,1 group, which correspond to OAM7,1 modes. We do not consider the HE9,1 – EH8,1 mode group, because it is too close from cut-off and only supported at lower wavelengths. This figure also suggests a very good separation of the effective indices of the modes and, for example, at 1550 nm, the closest modes are TE0,1 and HE1,1, with a 1.1 × 10−4 separation. According to simulation, the fiber was expected to support up to 28 different OAM modes over the whole C-band, since all mode groups support 4 OAM modes (two polarizations and two rotation directions), with the exception of the HE2,1 groups that supports 2 OAM modes, and HE1,1 that is not strictly speaking OAM, but supports two polarizations. TE0,1 and TM0,1 modes cannot be used for OAM. This predicts twice as many modes, with a 3 to 4 times better separation, than what is reported in [9]. The simulation only gives an approximation of the real fiber characteristics: as we will see, experimental results show that the fiber really support up to 36 different information bearing states. This discrepancy can be due to the fact we simplified the fiber profile using small steps instead of simulating real gradients of indices.
Fig. 3 Simulated (a) effective index, (b) group index, (c) dispersion parameter of supported modes, over C-band, calculated from the measured fiber index profile.
In Fig. 3(b), we plotted group indices, calculated from the first derivative of the effective indices. This figure suggests that the maximum group delay between the modes should be around 27 ps/m. We see that OAM modes with higher topological charge usually propagate slower than those with lower topological charge. Furthermore, in OAM modes with the same topological charge, those with polarization in the same direction than the rotation propagate slower than those with polarization in opposite direction. Finally, dispersion parameter, calculated from the second derivative of the effective index, is plotted in Fig. 3(c). This figure suggests that both the dispersion and the slope of dispersion are very reasonable.
In Fig. 4, we see simulated intensity and phase for OAM7,1 mode. In fact, because of the strong confinement of the mode, the intensity profile of all the different modes is almost the same. Dashed lines represent the boundaries between the different regions of the fiber.
4. Exciting OAM modes in annular fiber
4.1. Perfect OAM beam generation
The OAM modes in fiber can be excited by free-space OAM beams [7]. Free-space coupling is optimized when the beam profile and fiber annular core dimensions are matched; light that does not come through the annular core is lost. Our fiber is characterized by a very thin annular section (2 μm width) where the modes are confined. Conventional free-space OAM beams, however, have a ring diameter that varies with order of topological charge ℓ (the number of twists in a wave-front per unit wavelength of vortex beam), per [16]:
where w is the Gaussian beam radius at focus. This makes coupling of multiple OAM beams into a fiber with a fixed annular index profile problematic. Perfect OAM beams can be used to assure all OAM beams have the same size, and that size can be optimized for efficient coupling into a given fiber profile.Our goal is to generate an OAM beam with a donut-shaped intensity whose two dimensions (ring diameter rd and ring width rw, as shown in Fig. 4) can be controlled independently. The generation of a perfect OAM beam is divided into two stages; the first controls the external beam width rw, while the second converts the Gaussian beam to an OAM beam with a tunable ring diameter rd.
The first stage is made with two lens, focal lengths f1 and f2, to build a beam expander. The distance between the two lens must be L = f1 + f2. Adjusting f1 and f2, we can control the diameter w of the incident Gaussian beam, which will control the rw parameter.
The second stage is made of an SLM and a Fourier transform lens (L1). The SLM implements a phase mask (PM) that combines axicon and spiral phase functions:
where a is the axicon parameter. The axicon function forms a ring beam having the diameter rd varying with the axicon parameter a. Finally, the spiral phase function gives the vortex nature with topological charge ℓ to this ring beam. When combining the beam expander with the axicon/spiral phase function, we adjust w and a to change the resultant ring geometry. As the beam radius w decreases, the ring-width increases in both directions, inward as well as outward. The ring diameter of perfect vortex decreases with axicon parameter a for fixed Gaussian beam width.To verify the generation of the perfect OAM beams, we split a collimated Gaussian beam generated from a semiconductor laser into two parts. The first part goes through the beam expander, is reflected by a SLM, and is Fourier transformed by a lens of 50 cm focal length. The Fourier transformed field forms a perfect OAM beam at the focus point. We captured the intensity of this beam, as well as the interference of the beam with the second part of the incident beam, to get an interference pattern. The top row of Fig. 5 shows two examples of perfect OAM beams generated, with their topological charges as indicated and their corresponding interference patterns are given in the bottom row. As the topological charge increases, the diameter of the beam increases slightly due to diffraction. This can be corrected by changing the axicon parameter. The rings exhibit small ellipticity, due either to misalignment of the lens or angled incidence on the SLM. The generated beam has a vortex nature, which is confirmed by the interference pattern. Due to the intensity null at the center of the ring beam, and large ring diameter, it is difficult to observe spiral patterns, but the ℓ fringes, are still clearly visible.
Fig. 5 Intensity profile of free-space perfect OAM beams of different orders (top row), and corresponding interference spiral-fringe patterns confirming OAM (bottom row).
4.2. Experimental validation of OAM in fiber
Figure 6 shows the experimental setup for coupling of a perfect OAM beam with different topological charges into annular-core fiber. The beam from a semiconductor laser with a single mode fiber patch cord is collimated by a collimating lens. The beam is split in two parts. A single perfect OAM mode is generated per the method described in the previous section. As the SLM requires linearly polarized light, a quarter wave plate converts the beam to right or left circularly polarized light, before it is coupled into the fiber. At the output of the fiber, a quarter wave plate and an polarizing beam splitter are used to bring the beam back to linear polarization. Finally, the beam is interfered with the second part of the incident beam, to get the characteristic spiral patterns that allows the identification of OAM beams.
Fig. 6 Experimental setup for the generation and the transmission of OAM through optical fiber. CL: fiber colimator; BS: non-polarizing beam-splitter; BE: beam expander; M: mirror; SLM: spatial light modulator; PM: phase mask; L: lens; QWP: quarter wave plate; FL: focusing lens; ACF: air-core fiber; PBS: polarizing beam splitter; CCD: ccd camera.
In Fig. 7 and 8, we present experimental results after propagation of OAM modes in 85 cm of our air-core fiber. We excited and observed nine different OAM orders (one to nine), as well as the fundamental mode. The presence of OAM in the generated modes is confirmed through interferometry. The number of spirals is equal to the order of the mode, as seen in the bottom row of Fig. 5. We also confirmed excitation of both positive and negative order OAM modes, in both right- and left- circular polarization, confirming that a total of 36 information bearing modes, with topological charges from ℓ = 0 to ℓ = 9, are supported by this ACF fiber.
4.3. Discussion
Experiments demonstrated that the number of supported modes in the air-core fiber is greater than what was expected from simulation. This could be due to 1) imprecisions in the measurement of the fiber profile, or 2) the slight differences between the real fiber profile and the simplified profile used for the simulation. We suspect these higher modes are less stable and slowly leaking. Despite the short length of the fiber tested, we confirm this hypothesis experimentally; we consistently observed heightened sensitivity to fiber perturbations for modes OAM8,1 and OAM9,1. Clearly the number of supported modes is very sensitive to the fiber profile, and only a slight variation in the profile can lead to changes in the characteristics of the fiber.
Modes were observed to be stable over time (hours). However, they are sensitive to fiber bends, especially for lower order modes (OAM0,1 through OAM3,1). More simulations and measurements are needed on this fiber to better characterize performance and achieve a deeper understanding of OAM mode transmission. For instance, effective indices could be measured using the reflectogram of a Bragg grating [17]. Bending and twist losses could be investigated more systematically experimentally, and compared to numerical simulations of these impairments. Our experiments were limited to a short length of fiber as we were not able to efficiently transmit the light on longer spans of fiber. We observed high loss in of our fiber, that we estimate to a few dB per meter, and we are still investigating the origin of this loss. Losses must be reduced to be able to measure channel crosstalk using the fiber in an optical communications setup.
5. Conclusion
We proposed a fiber profile able to guide a large number of OAM modes. Such fiber could have applications in short-reach telecommunications, or in any field where the transmission of different OAM modes through optical fiber could be useful. We proposed a novel way of shaping OAM beams, to form perfect vortex beams that can match annular fiber profiles. This kind of beam shaping is necessary because conventional free-space OAM beams usually do not match the profile of annular-core fibers. Using perfect OAM beam generation, we now have a tool that gives the freedom of being able to couple OAM beams with any desired annular fiber profile. Finally, we successfully demonstrated the transmission of OAM modes through a special designed fiber. To the best of our knowledge, this is the highest reported number of OAM modes transmitted through an optical fiber.
Acknowledgments
The authors would like to thank Dr. Bora Ung for many fruitful discussions on OAM for spatial multiplexing, particularly fiber design and simulation of fiber performance.
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