1. Introduction

The observation of optical Airy Beam in 2007 [1] unveiled a category of beams with curved trajectories in free space, namely accelerating beams [2]. Such light beams have attracted wide and lasting research attention [3] for its novel properties such as self-bending, self-healing [4] and non-diffracting [5]. To flexibly design accelerating beams beyond the para-curve shape of Airy beams, a caustic method [6–10] has been proposed and developed which extends spatial beam trajectories from paraxial to non-paraxial [11], convex to non-convex [12,13], and two-dimensional (2D) to three-dimensional (3D) [14,15]. Developments in theoretical methods have promoted exploitation of accelerating beams for a broad range of applications, including energy autofocusing [16,17], plasmonic [18,19] and electron accelerating [20], micro manipulating [21] and machining [22].

Spatially structured light also found a promising role in recent research in optical communications. Space and/or mode division multiplexing (SDM/MDM) employs light beams with orthogonal wavefront structures (such as vector [23] or orbital angular momentum beams [24,25]) as a new degree of freedom to increase channel count. SDM using plane waves has also been reported with comparable channel density with other orthogonal mode sets [26].

For free-space image transmission and optic communication, compared to normal Gaussian beams, accelerating beams have unique appeals as information carriers in their higher flexibility to bypass obstacles (self-bending) and stronger robustness against environment turbulences (self-healing) [27–32]. Existing demonstrations were mainly concerned about realization of the bendable transmission of data in a single accelerating beam.

A natural approach to further increase the information capacity of bendable free-space optical links is to use multiple accelerating beams in a SDM or MDM scheme. The key to realizing such a scheme is the nontrivial task of constructing a family of accelerating beams that can share the same set of optics, which allows to convert spatially diverse Gaussian beams into the accelerating beam family (i.e., multiplexing) and, after propagation in the same free-space link following the desired bent trajectories, back into Gaussian beams (i.e., demultiplexing).

In this letter, we propose methods for simultaneously generating such a beam family from a fiber array. Since propagation of light is reciprocal, the setup for beam generation and multiplexing inversely can also be used for demultiplexing. We set up an optical system to verify our SDM scheme for Airy plane wave beams (AipwBs), and extend it to other special light beams with different bendable trajectories. Finally, we build a multiplexing communication system based on 7 AipwB channels and demonstrate bendable transmission of 56-Gbits/s OOK data in each channel, confirming the feasibility of such optical multiplexing communication scheme, which may find applications in a hybrid optoelectronic fashion.

2. Methods and experiments

2.1 Manipulations of accelerating plane-wave beams

The transverse engineering of the Airy beam began before it was experimentally implemented. As early as 2007, Airy beams were verified to be able to combine with other wave packets in a theoretical paper [33]. From a Fourier spectrum perspective, the ballistic dynamics of Airy beams were demonstrated both theoretically and experimentally [34], and optimized in control method [35] for both one- and two- dimensional configurations. In 2009, a work introduced the complete theory of 2D accelerating beams, providing some special cases of orthogonal Airy beam families with their one-to-one corresponding Fourier spectrums [36]. On this basis, accelerating beams with arbitrary on-demand transverse patterns was realized later [37]. These approaches bring an inspiration that well-constructed accelerating beams might be used as spatial modes to serve in free-space MDM optical communications.

The transverse cross-sectional fields of AipwB are given by

As $\omega ^2\varpropto k_v^2$, the component $Ai(u+\omega ^2)$ in Eq. (1) is a function of both $u$ and $v$. However, for simpler light conversion, we expect the structured light fields to have separable variables in both dimensions. Reference [37] proposed an idea in which by scaling the spatial density in the v-axis, the frequency $k_v$ is lowered so that deviation in the $u$-axis associated with $\omega ^2$ becomes sufficiently small to be ignored. AipwBs can be represented in a separable form. This approximation is used here to generate a configurable mode basis.

In subsequent steps, we adapt Airy plane wave beams as:

In this manner, beams in an AipwB family are shaped with the same Airy-function shape in the $u$-axis, and distinguished by $\omega$ or the phase gradient in the $v$-axis.

The Airy-shape intensity on real plane along $u$-axis can be generated through a cubic phase (a phase that is cubic power with respect to $u$) in Fourier space followed by a Fourier transform. A cylindrical lens is placed vertically to implement 1D Fourier transform in the $u$-axis and avoid unwanted influence on the $v$-component. The cubic phase is defined as $\psi _{cubic}=\exp [i(u_F/a)^3/3]$, where $u_F$ denotes the $u$-coordinate on Fourier plane, which satisfies a relationship with $k_u$ (the spatial frequency on the real plane): $k_u=2\pi u_F/\lambda f_C$. Due to the property of Fourier transform, the constant $a$ would scale the Airy distribution and further affect the ballistic feature of AipwB.

The challenge lies in how to tailor the gradient phase. For single beam generation, a specific gradient phase in the $v$-axis can be directly superimposed on the cubic phase to modulate a Gaussian light field, as shown in Fig. 1(a). Alternatively, the Fourier transform function of a lens can convert a spatial position into a gradient phase, thus can generate a group of plane waves with diverse tilted angles [26] from a fibre array facet on its focal plane. Figure 1(b) shows our method in which an AipwB family can be simultaneously generated from, or inversely, coupled into a fiber array. This enables a space-division multiplexing/demultiplexing scheme using accelerating plane-wave beams (showing self-accelerating and plane-wave characteristics respectively in two orthogoanal transverse dimensions) for multi-channel bendable free-space optical communications.

 

Fig. 1. Design of apparatuses for (a) tailoring a single AipwB, and (b) multiplexing/demultiplexing a AipwB family with a fiber array.

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Using spatial light modulators (SLMs) for phase control, an optical circuit to verify above methods for generating and demultiplexing AipwBs is designed as shown in Fig. 2(a). Two cylindrical lenses (CLs) with the same focal length $f_{C1}=f_{C2}=200$ mm are set upright to implement 1D Fourier transform in the $u$-axis, which are arranged as a 4f system together with SLM1 and SLM2. On SLM1 the mixed phase patterns $\exp [ik_v v-i(u/a)^3/3]$ is displayed, where we define $k_v=m\alpha =2m\pi /w_0$, with $w_0$ indicates the waist radius of the Gaussian beam, and $m = −3, −2, −1, 0, 1, 2, 3$ for spatial modes 1–7, as shown in Figs. 2(b)–2(h). On SLM2, an inverse cubic phase pattern $\exp [i(u/a)^3/3]$ is loaded for phase compensation in the $u$-axis [Fig. 2(i)]. SLM2 and the camera are placed at the front and rear focal planes of the lens ($f_L=400$ mm), to satisfy a Fourier transform relationship. We take $a=800$ $\mathrm{\mu} \rm {m}$ in following simulations and experiments.

 

Fig. 2. Simulation of AipwB generating and demuxing processes: (a) Layout of demonstration setup; (b)-(h) Blend phase patterns on SLM1 for pre-modulation of spatial modes 1–7; (i) Inverse cubic phase pattern on SLM2 for compensation; (j) Simulated sectional light field dynamics of 7 spatial modes in an AipwB family during propagation. (Pol: polarizer; BE: beam expander; SLM: spatial light modulator; BS: beam splitter; CL: cylindrical lens.)

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In the beam generation module, the expanded Gaussian beam is first subjected to the mixed phase modulation on SLM1 and then converted to an Airy plane wave beam through the vertically placed CL1. One AipwB with different phase gradients can be precisely generated at a time by switching the mixed phase pattern on the SLM1. In the optical path between the two CLs, an AipwB is formed, transmitted along a parabolic trajectory. As the beam arriving at the demultiplexing module, its mixed phase is recovered by CL2, with its cubic phase in the $u$-axis converted back to the planar phase by SLM2. The Gaussian-like beam carrying gradient phase is mapped to a specific position in the imaging plane, after passing through the lens.

The propagation behaviour of 7 spatial modes in an AipwB family has been simulated and their sectional fields are illustrated in Fig. 2(j), where the brightness indicates the intensity, the pseudo color the phase, with numerals I-V corresponding to the plane positions marked in Fig. 2(a). Comparison between rows III and IV shows that the beam is clearly bent when propagating 8 cm (IV) behind the focal plane (III) of CL1. Row V shows a clear and equally spaced separation at the demultiplexed end.

In the experiment, we build the setup of Fig. 2(a) to demonstrate the proposed methods for tailoring AipwBs. At the demultiplexing end, a camera is used to capture the sorted Gaussian-like spots by mode [Fig. 3(a)]. Seven demultiplexed spots are distinctly separated with equal spacing, due to gradient of equal difference between adjacent spatial modes. We record the bending characteristics of AipwBs during propagation, and illustrate the observed trajectory of the beam with m=0 labeled as ’mode 4’ in Fig. 3(b), which matches well with the expected and simulated parabolic shapes. The bending trajectories of spatial modes in the same AipwB family are identical, and they differ only in the degree of linear tilt in the $v$-axis.

 

Fig. 3. Observation of (a) demuxed spots of an AipwB family and their superposition, bending trajectories of (b) AipwB and (c)(d) two nonconvex-trajectory accelerating plane-wave beams at spatial mode 4 compared with expectation and simulation.

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Furthermore, our methods also apply to accelerating plane wave beam families with other bending trajectories by replacing the Airy function in Eq. (2) with an amplitude distribution $E_{acce}(u)$ corresponding to another 2D self-accelerating beam. Under the paraxial approximation, only a phase modulation is required to generate such a accelerating plane-wave beam, which can be calculated from a specific trajectory expression using the Wigner distribution function [9]. By replacing the function of $u$ in mixed phase on SLM1 and the cubic phase on SLM2 with the specific phase calculated from target trajectory, self-accelerating plane wave beams are transmitted along preset paths. The spatial ’mode 4’ (with 0 phase gradient) of two nonconvex accelerating plane wave beam families is shown in Figs. 3(c)–3(d), which are set in opposite curving to verify the flexibility. Thus, the (de)multiplexing scheme proves to be a viable converter between 1D-arranged Gaussian spots from a fiber array and an accelerating plane wave family in free space.

2.2 Multi-channel bendable light communications

By mirroring the two configurations shown in Fig. 1(b) as the transmitter and receiver, we set up a 7-channel bendable free-space optical communication system as represented in Fig. 4(a). Each single mode fiber in the fiber arrays has a beam waist diameter of $\sim 46 \mathrm{\mu}$m after being collimated by a microlens, with a spacing of $127$ $\mathrm{\mu}$m. The cylindrical lenses and fiber arrays are set along $v$-axis, and the curvatures of accelerating plane-wave beams would be fixed in $u$-axis. Here AipwBs are chosen as the data carrying beams for demonstration, thus the pattern to be displayed on both phase masks is a 1D cubic phase, as illustrated in the insets.

 

Fig. 4. (a) Schematic of 7-channel bendable light communication system employing AipwBs at 1550 nm, in which 56-Gbit/s OOK signal is modulated onto each channel (Number 1 and 7 denote the fiber channel sequence); (b) Beam profiles of 7 spatial modes in $v$-$s$ plane. (CL: cylindrical lens; PM: phase mask; OOK: on-off keying; AWG: arbitrary waveform generator; EA: electrical amplifier; MZM: Mach-Zehnder modulator; EDFA: Erbium doped fiber amplifier; PC: polarization controller; VOA: variable optical attenuator; BPF: band-pass filter; PD: photodiode; OSC: oscilloscope.)

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Compared with the theoretical case that a Dirac-delta function is converted to an infinite plane wave, in experiment, the small-sized Gaussian fields from the fiber array yield truncated plane waves after being collimated by lens1. As Fig. 4(b) shows, the propagating direction of each spatial mode is determined by the emitting fiber position, resulting in a mismatch of entry angle at the receiving fiber array due to the tilted truncated plane wave being off-axis. Thus uneven coupling efficiency is a general issue in plane-wave multiplexing schemes [26] to be further investigated. Coupling efficiency should be traded off against other parameters, such as beam size, channel amount, transmission distance, etc. As lenses with a focal length $f_L=100$ mm, and cylindrical lenses with a focal length $f_C=200$ mm are used, the propagation distance of the AipwBs is $400$ mm. Variable optical attenuators (VOAs) are used in channel links to balance the received optical power (ROP). Because a lens naturally has the function of sorting plane waves, crosstalk among channels is less than $-40$ dB. Note that although the seven modes have slight angular differences in the $v$ direction, they maintain a high degree of overlap during propagation as they share the same bending path in the $u$ axis. In Fig. 4(a), the bending section indicates the near-identical propagation trajectories of the 7 AipwBs.

To test the communication performance, in each channel a 56-Gbit/s OOK signal is generated by an arbitrary waveform generator (AWG), electrically amplified, and applied to a Mach-Zehnder modulator (MZM) to modulate optical carrier at a wavelength of 1550 nm. Then the optical signal is transmitted into free space after optical amplification by an Erbium-doped fiber amplifier (EDFA). At the receiving end, after a VOA and a pre-EDFA, the optical signal is detected and converted into electrical signal by a photodiode (PD), then sampled and stored by a real-time oscilloscope (OSC). Finally, the bit error rate (BER) of the communication link is estimated according to the data transmission performance. For simplification, the links of transmitter and receiver in each channel are represented by green and yellow squares respectively in the Fig. 4(a), while the specific connections are shown in the dotted boxes. Figure 5(a) illustrates the curves of BER versus ROP of 7 AipwB channels, showing a linear relationship after taking the logarithm of both axes. The curve trend is almost identical to the case of the Gaussian beam channels, which is commonly characterized in free space optical communications. Seven channels show a small variation of $\sim$ 1 dB in term of the ROP (of around $-30$ dBm) required to reach the hard-decision forward error correction (HD-FEC) threshold [26], a BER of $3.8\times 10^{-3}$.

 

Fig. 5. (a) BER of seven AipwB channels against ROP; (b) BER of the system against location of obstacle with Airy and Gaussian plane-wave beams.

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Next, an opaque obstacle is placed at the midpoint of the free-space optical path, and the performances of AipwBs and Gaussian plane waves are compared as the obstacle is moved into the beams. As bending characteristics on the u-axis are identical for different channels, their performances in under-occlusion communications are similar. Channel 4 is used as the representative for test with the results illustrated in Fig. 5(b), where a positive location indicates that the edge of the obstacle crossed the optical axis. The communication performances of both beam types degrade as obstruction increases, while AipwBs show both a much slower rate and a postponed position in deterioration. It is worth mentioning that, for the negative location Airy beam is still better than Gaussian beam because a small part of the Gaussian light field is blocked when the obstacle gets close to the central axis, while the main valve of AipwB is not. At the hard-decision FEC threshold, Gaussian plane waves can tolerate an obstruction of up to $w_0/2$, while AipwBs can maintain good performance with obstruction of more than $w_0$ as they bend around the obstacle as shown in the inset of Fig. 4(a), rising above the threshold at $\sim {1.1w_0}$. Therefore, at the premise of maintaining the BER, an increase ability of 0.6 beam waist to evade obstruction of AipwBs over Gaussian plane waves is demonstrated. As the bending path of the beams can be readily adjusted by replacing the cubic phase on SLMs with a purposely calculated phase [9], the scheme has the flexibility of accommodating carrying degrees of obstruction.

3. Conclusions

In summary, we have proposed and verified efficient methods for generating and sorting multiple co-propagating accelerating plane-wave beams. We demonstrated a bendable free-space SDM optical communication system with seven AipwB channels. Each channel sustains a BER below the HD-FEC threshold of $3.8\times 10^{-3}$ while transmitting a 56-Gbit/s OOK signal with an increased ability to evade obstruction for more than half beam waist size (from $\sim 0.5 w_0$ to $\sim 1.1 w_0$) over the Gaussian beams. The bendable SDM scheme may pave the way to optical communications with higher space utilization and larger data capacity in obstructed free space optical links, which might find further applications in short-distance interconnections in data centers or photonic chipsets on a hybrid optoelectronic integrated circuit board.