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1 Terabit free-space data transmission employing orbital angular momentum multiplexing Jian Wang 1, 2 *, Jeng-Yuan Yang 1, Irfan M. Fazal 1, Nisar Ahmed 1, Yan Yan 1, Hao Huang 1, Yongxiong Ren 1, Yang Yue 1, Samuel Dolinar 3, Moshe Tur 4 and Alan E. Willner 1 * I. Implementation details of experimental setup Supplementary figure 1 depicts the implementation details of the experimental setup. For the experiment of multiplexing/demultiplexing of information-carrying orbital angular momentum (OAM) beams (parts A, C, D and E), at the transmitter, a 10.7X4 / 42.8X4 Gbit s -1 (4 bits per symbol) quadrature amplitude modulation (16-QAM) signal at nm (International Telecommunication Union (ITU) grid) is prepared via the vector addition of two copies of quadrature phase-shift keying (QPSK) signal using a QPSK modulator, polarization controllers (PCs), a tunable differential group delay (DGD) element, and a polarizer (Pol.). When using 10.7X4 Gbit s QAM signal, part D is not used and the QPSK modulator in part A is an I/Q modulator. When using 42.8X4 Gbit s QAM signal, part D is used and the QPSK modulator in part A consists of two parallel integrated Mach-Zehnder modulators (MZMs) driven by 42.8 Gbit s -1 pseudo-random binary sequence (PRBS) patterns. Pre-filtering in the optical domain is adopted to reduce the spectral width of the signal. By dividing the 16-QAM signal into four branches via a 1x4 optical coupler (OC), introducing relative delay with fibres, collimating the four light beams with an output beam size of ~3 mm, we obtain 16-QAM signals over four Gaussian beams, which are then converted into four OAM beams with 1 Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA, 2 Wuhan National Laboratory for Optoelectronics, College of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan , Hubei, P R China, 3 Jet Propulsion Lab, 4800 Oak Grove Drive, Pasadena, California 91109, USA, 4 School of Electrical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel. * jwang@mail.hust.edu.cn; willner@usc.edu NATURE PHOTONICS 1

2 helical phase fronts by adding different spiral phase masks through four reflective nematic liquid crystal based spatial light modulators (SLM1-4). The employed SLMs have dimensions of 7.68 x 7.68 mm, 512 x 512 pixels, a wavelength range of nm, and a fast response (<20 ms), providing phase modulation for linearly polarized light with a high efficiency of 90 95%. The half-wave plates (HWPs) in front of SLMs are used to adjust the polarization states for achieving optimal response of SLMs. Three non-polarizing beamsplitters (BS1-3) are utilized for the multiplexing of four OAM beams. Two more polarizing beamsplitters (BS4, BS5) together with two mirrors (M3, M4) are employed for polarization multiplexing (pol-mux stage). Consequently, 16-QAM signals over four polarization-multiplexed (pol-muxed) OAM beams are achieved. The 16-QAM-carrying OAM beams propagate in free space over a metre-length scale. For the demultiplexing of pol-muxed OAM beams, an HWP followed by a polarizer is used to realize polarization demultiplexing. Another SLM (SLM5) loaded with a specified spiral phase mask is used to demultiplex one of the multiplexed OAM beams back to a beam with a planar phase front. A spherical lens with a focus length of 300 mm is used to adjust the beam size in order to optimize the performance of demultiplexing of OAM beams. Note that switchable polarization demultiplexing is available by adjusting the HWP preceding the polarizer in front of SLM5. Meanwhile, reconfigurable demultiplexing of OAM beams is achievable by changing the phase mask loaded into SLM5. After demultiplexing of pol-muxed OAM beams, the back-converted beam holding a planar phase front and a bright high-intensity spot at the centre, is separated from the other OAM beams having updated charges and doughnut shapes with no intensity at the centre by spatial filtering with a pinhole. The pinhole output is then coupled into a single-mode fibre, mixed with a local oscillator (LO), and sent to a coherent detection set for post digital signal processing (DSP), constellation analyses, error vector magnitude (EVM) and bit-error rate (BER) measurements. To analyse the characteristics of the multiplexing/demultiplexing of OAM beams, a camera is used to view the intensity profiles of different light beams. When observing intensity profiles and interferograms of OAM beams for multiplexing, SLM5 functions as a mirror. In 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION order to confirm the charge of OAM beams for multiplexing, one needs to take into consideration the charge sign flip due to reflection and the number of reflections after multiplexing and before the camera. For the multiplexing/demultiplexing of OAM beams encoded with 42.8X4 Gbit s QAM signals, an electroabsorption modulator (EAM) driven by a synchronized 10.7 GHz clock, is used to enable 42.8X4-to-10.7X4 Gbit s -1 demultiplexing before the coherent detection. We assess the loss budget for the free-space link (thick blue line in Suppl. Fig. 1) by measuring optical powers at each part of the process. In our proof-of-concept experiments, the total equivalent loss budget for the free-space link is observed to be less than 20 db for the branch of each information-carrying OAM beam, including <0.5 db loss per SLM, <3.5 db loss per non-polarizing beamsplitter, <3.5 db loss for polarization multiplexing using two polarizing beamsplitters, <0.3 db loss per HWP, <0.4 db loss per mirror, <0.5 db loss per lens, <0.4 db loss of the Pol. before SLM5, and <5 db loss passing through the pinhole for spatial filtering. For the demultiplexing of information-carrying OAM beams, the cross talk is measured to be less than -20 db. A 10.7X4 / 42.8X4 Gbit s QAM 10.7 Gbit s -1 QPSK or 42.8 Gbit s -1 Laser Fast axis PC QPSK modulator B E Off-line DSP ADC ADC ADC ADC EDFA BPF 50 Gbit s -1 TDL DQPSK modulator Coherent detection Hybrid LO Slow axis Tunable DGD Pol. 100 Gbit s -1 DQPSK F Rx AM Direct detection 50-GHz DLI 20ps 16-QAM DQPSK 1x4 OC Att Multiplexing/demultiplexing of information-carrying OAM beams C D Fibre SLM1 HWP Col. HWP EAM SLM2 BS1 BS3 Col. BS2 Pol-mux M3 M4 BS4 BS5 HWP SLM4 Mirror (M1) Col. (SLM6) HWP SLM3 Lens (300 mm) Data exchange between OAM beams 42.8-to-10.7G demux 10.7-GHz clock Col. PM HWP Pinhole 1% tap Pol. Camera SLM5 Lens (400 mm) M2 Lens Col. (200 mm) Supplementary Figure 1 Implementation details of the experimental setup. A,C,D,E, multiplexing/demultiplexing of information-carrying OAM beams; B,C,F, data exchange between OAM beams. (D)QPSK, (differential) quadrature phase-shift keying; 16-QAM, quadrature amplitude modulation; PC, polarization controller; EDFA, erbium-doped fibre amplifier; BPF, band-pass filter; DGD, differential group delay; Pol., polarizer; TDL, tunable delay line; AM, amplitude modulator; OC, optical coupler; Col., collimator; HWP, half-wave plate; SLM1-6, spatial light modulator; BS1-3, non-polarizing beamsplitter; BS4, BS5, polarizing beamsplitter; M1-M4, mirror; PM, power metre; EAM, electroabsorption modulator; Att, attenuator; DLI, delay-line interferometre; Rx, receiver; LO, local oscillator; ADC, analog-to-digital converter; DSP, digital signal processing. NATURE PHOTONICS 3

4 As shown in Suppl. Fig. 1, besides the experiment of multiplexing/demultiplexing of information-carrying OAM beams involving parts A, C, D and E, the setup can also be configured for the experiment of data exchange between OAM beams (parts B, C and F), in which parts B and F are connected (red dotted lines) to the setup as the transmitter and receiver instead of parts A and E. In part B, 100 Gbit s -1 differential quadrature phase-shift keying (DQPSK) signal is generated by feeding a continuous wave (CW) at nm into a 100 Gbit s -1 DQPSK transmitter, consisting of two parallel integrated MZMs driven by 50 Gbit s -1 PRBS patterns. An additional amplitude modulator (AM), acting as a pulse carver, is used to produce the 50% return-to-zero DQPSK (RZ-DQPSK). In part C, the SLM2 and SLM3 branches together with the pol-mux stage and the polarizer before SLM5 are removed. The mirror (M1) is replaced by another SLM6. When measuring intensity profiles and interferograms of OAM beams before exchange, SLM5 and SLM6 in Suppl. Fig. 1 function as mirrors. When observing intensity profiles and interferograms of OAM beams after exchange, SLM5 is loaded with a specified spiral phase mask to enable the data exchange between OAM beams and SLM6 functions as a mirror. In order to confirm the charge of two OAM beams, one needs to take into account the charge sign flip due to reflection (SLM, mirror) and the number of reflections that OAM beams undergo before they are captured by the camera, i.e., reflections between BS3 and camera for OAM beams before exchange while between SLM5 and camera for OAM beams after exchange. Hence, one more reflection from SLM5 is introduced for the OAM beams before exchange compared to those after exchange. In part F, 100 Gbit s -1 DQPSK is demodulated through a 50 GHz delay-line interferometre (DLI) with a relative delay of 20 ps between the two arms. The in-phase (Ch. I) and quadrature (Ch. Q) components of the DQPSK signals are obtained by adjusting the bias voltage ( ± π /4 biased) of the DLI. The DLI outputs are then sent to the receiver (Rx) for direct detection, followed by the analyses of the demodulated temporal waveforms, balanced eye diagrams, and BER performance. 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION II. Supplementary results of 16-QAM signals over four OAM beams Supplementary figure 2 shows the experimental and theoretical results of the multiplexing/demultiplexing of four OAM beams with different l ( l : topological charge of OAM beam), e.g., OAM -8, OAM +10, OAM +12, OAM -14. The pol-mux stage and the polarizer before SLM5 are not used in part C of Suppl. Fig. 1. Four computer-generated spiral phase masks with charges of +8, +10, +12 and +14, as shown in Suppl. Fig. 2a1-a4, are loaded into SLM1-4, respectively. After reflecting off the four SLMs, four Gaussian beams are transformed by adding azimuthal phase terms of exp( il θ ) ( l =+8, +10, +12, +14) and converted into four OAM beams (OAM -8, OAM -10, OAM -12, OAM -14 ). One can see that the charge sign of the OAM beam reflected off the SLM is opposite to that of the spiral phase mask loaded into the SLM. This is because the reflection (i.e., reflective-type SLM) flips the charge sign of an OAM beam. Similarly, one more reflection in the SLM2 and SLM3 branches while two more reflections in the SLM4 branch are introduced by non-polarizing beamsplitters BS1-BS3 during the multiplexing of four OAM beams. As a result, the multiplexed four OAM beams after the BS3 are OAM -8, OAM +10, OAM +12 and OAM -14, respectively. For multiplexing/demultiplexing of information-carrying (i.e., 10.7X4 Gbit s QAM) OAM beams, supplementary figure 2b depicts measured BER curves for the demultiplexing of OAM +10 and OAM +12 beams without and with crosstalk. Less than 1.2 db optical signal-to-noise ratio (OSNR) penalty at a BER of 2X10-3 (enhanced forward error correction 1 (EFEC) threshold) is measured without crosstalk. Less than 2.2 db OSNR penalty at a BER of 2X10-3 is observed with crosstalk. We also theoretically analyse the multiplexing/demultiplexing of OAM beams using an angular-spectrum propagation method, with the same parameters as those adopted in the experiment. Shown in Suppl. Fig. 2c1-c5 are 3-D surface plots of the intensity profiles of four OAM beams (OAM -8, OAM +10, OAM +12, OAM -14 ) and their superposition. Supplementary figure 2d1-d3 depicts 3-D surface plots of the intensity profiles for the demultiplexing of an OAM beam (e.g., OAM -8 ). Shown in Suppl. NATURE PHOTONICS 5

6 Fig. 2d1 is the demultiplexed beam from the OAM -8 beam as other OAM beams are blocked (w/o crosstalk). Shown in Suppl. Fig. 2d2 is the crosstalk of the demultiplexing as the OAM -8 beam is blocked while other OAM beams are on. Shown in Suppl. Fig. 2d3 corresponds to the demultiplexing result when all OAM beams are on (w/ crosstalk). To better understand the orbital angular momentum, as shown in Suppl. Fig. 2e1-e4, we further simulate and depict the phase fronts of four OAM beams located in front of the camera. Those OAM beams are corresponding to the generated four OAM beams (OAM -8, OAM +10, OAM +12 and OAM -14 ) for multiplexing. One can clearly see helical phase fronts with the number of intertwined helices and the handedness depending on the magnitude and the sign of l, respectively. Remarkably, for the demultiplexing of OAM beams, in addition to the back-conversion from an OAM beam of interest to a beam with a bright high-intensity spot at the centre, other superposed OAM beams are not converted into such a beam with central high intensity but transformed into new OAM beams with updated charge values of l and doughnut shapes. Hence, the back-converted beam with a bright high-intensity spot at the centre can be filtered from the other OAM beams with doughnut shapes through a pinhole together with the spatial filtering of the received single-mode fibre. Taking the demultiplexing of OAM -8 beam as an example, after reflecting off the SLM5 loaded with a spiral phase mask with a charge of +8, OAM -8 beam is back-converted to a beam with a bright high intensity spot at the centre. Meanwhile, OAM +10, OAM +12 and OAM -14 beams are transformed into OAM -18, OAM -20 and OAM +6 beams, respectively, considering the additive reflection of the reflective-type SLM5. These three OAM beams (OAM -18, OAM -20 and OAM +6 ) with updated charge values of l are verified by the simulated helical phase fronts as shown in Suppl. Fig. 2f1-f3, respectively. We also simulate the interference between these three OAM beams and a Gaussian beam reference as shown in Suppl. Fig. 2g1-g3. The obtained interferograms further indicate that the three updated OAM beams are OAM -18, OAM -20 and OAM +6, respectively. The bending of the interferograms is caused by the radius-dependent phase contained in the OAM beam and Gaussian beam reference. 6 NATURE PHOTONICS

7 a1 a2 SUPPLEMENTARY INFORMATION b a3 a4 c1 c2 c3 c4 c5 x y x y x y x y x y d1 d2 d3 x y x y x y e1 e2 e3 e4 z z z z f1 f2 f3 z z z g1 g2 g3 Supplementary Figure 2 Experimental (b) and theoretical results (c1-c5,d1-d3,e1-e4,f1-f3,g1-g3) of the multiplexing/demultiplexing of four OAM beams (OAM -8, OAM +10, OAM +12, OAM -14 ). a1-a4, Four computer-generated spiral phase masks with charges of +8, +10, +12 and +14 loaded into SLM1-4. b, BER curves for the demultiplexing of 10.7X4 Gbit s QAM carrying OAM beams (OAM +10, OAM +12 ) without and with crosstalk. c1-c5, 3-D surface plots of the intensity profiles of four OAM beams (OAM -8 (c1), OAM +10 (c2), OAM +12 (c3), OAM -14 (c4)) and their superposition (c5). d1-d3, 3-D surface plots of the intensity profiles for the demultiplexing of OAM -8 beam. d1, OAM -8 beam on while other OAM beams off (w/o crosstalk). d2, OAM -8 beam off while other OAM beams on. d3, all OAM beams on (w/ crosstalk). e1-e4, Phase fronts of four OAM beams located in front of the camera, corresponding to four OAM beams (OAM -8, OAM +10, OAM +12 and OAM -14 ) for multiplexing. f1-f3, Phase fronts of three updated OAM beams (SLM2-4 branches) after the demultiplexing of OAM -8 beam. g1-g3, Interferograms from the interference between OAM beams and a Gaussian beam reference corresponding to f1-f3, respectively. NATURE PHOTONICS 7

8 III. Supplementary results of 16-QAM signals over four pol-muxed OAM beams a1 a2 c1 c2 a3 a4 b c3 c4 d1 d2 d3 d4 d5 d6 d7 d8 d9 Supplementary Figure 3 Experimental results of the multiplexing/demultiplexing of four pol-muxed 10.7X4 Gbit s QAM-carrying OAM beams (OAM +4, OAM +8, OAM -8, OAM +16 ). a1-a4, Four computer-generated spiral phase masks with charges of -4, +8, -8 and -16 loaded into SLM4, SLM3, SLM2 and SLM1. b,c1-c4, Measured spectra (b) and BER curves (c1-c4) for demultiplexing of four pol-muxed OAM beams. c1-c9, Constellations of 16-QAM for demultiplexing of four pol-muxed OAM beams. c1, Back to back. d2-d5, x-pol. d6-d9, y-pol. d2,d6, OAM +4 beam. d3,d7, OAM +8 beam. d4,d8, OAM -8 beam. d5,d9, OAM +16 beam. x-pol., x-polarization; y-pol., y-polarization. Supplementary figure 3 shows the experimental results of the multiplexing/demultiplexing of four pol-muxed 10.7X4 Gbit s QAM-carrying OAM beams, i.e., OAM +4, OAM +8, OAM -8, OAM +16. As shown in Suppl. Fig. 3a1-a4, four computer-generated spiral phase masks with charges of -4, +8, -8 and -16 are loaded into SLM4, SLM3, SLM2 and SLM1, respectively. Taking into account the 8 NATURE PHOTONICS

9 SUPPLEMENTARY INFORMATION reflections of reflective-type SLMs, mirrors, and beamsplitters, four pol-muxed OAM beams of OAM +4, OAM +8, OAM -8, OAM +16 are achieved for multiplexing. Supplementary figure 3b shows measured spectra of back-converted beams from different polarizations of pol-muxed OAM beams. A power suppression of ~30 db is achieved at a frequency offset of 12.5 GHz from the centre. A spectral efficiency is measured to be 25.6 bit s -1 Hz -1 for a total capacity of (10.7X4X4X2) Gbit s -1, i.e., 10.7X4 Gbit s QAM signals over four pol-muxed OAM beams (12.5-GHz grid), including the 7% forward error correction (FEC) overhead. Shown in Supp. Fig. 3c1-c4 are measured BER curves for demultiplexing of OAM beams (OAM +4, OAM +8, OAM -8, OAM +16 ) along x- and y-polarizations without (x- or y-polarization of only one pol-muxed OAM beam is on) and with (all four pol-muxed OAM beams are on) crosstalk. An OSNR penalty less than 1.4 db at a BER of 2X10-3 is observed without crosstalk. A total OSNR penalty of less than 3.5 db at a BER of 2X10-3 is measured with crosstalk. Supplementary figure 3c1-c9 depicts observed constellations of 16-QAM for demultiplexing of OAM beams along x- and y-polarizations with measured error vector magnitude (EVM (%rms)). IV. Supplementary results of 16-QAM signals over eight pol-muxed OAM beams in two groups of concentric rings In another experiment, we explore the scalability of the multiplexing/demultiplexing of OAM beams in the spatial domain. Supplementary figure 4 shows the concept, experimental blocks, and results of the multiplexing/demultiplexing of eight pol-muxed 20X4 Gbit s QAM-carrying OAM beams (OAM ±10, OAM ±12, OAM ±14 and OAM ±16 ) in two groups of concentric rings. As depicted in Suppl. Fig. 4a, two groups of OAM beams 1, 2,, N with the same charges but different beam sizes, i.e., one group (outer rings) is expanded compared to the other (inner rings), can be spatially multiplexed together as concentric rings with small inter-ring crosstalk. These two groups of OAM beams 1, 2,, N can carry independent two sets of Data 1, 2,, N and Data N+1, N+2,, 2N, which adds an additional NATURE PHOTONICS 9

10 freedom to increase the capacity. For demultiplexing, spatial filtering can be used to separate the inner and outer rings. In order to demonstrate the multiplexing and demultiplexing of two groups of concentric rings of eight pol-muxed OAM beams (OAM ±10, OAM ±12, OAM ±14, OAM ±16 ), two additional experimental blocks, as shown in Suppl. Fig. 4b,c, are added in Suppl. Fig. 1 before and after the pol-mux stage, respectively. Supplementary figure 4b shows the block used to prepare eight OAM beams. The superposed four OAM beams generated from four SLMs are split into two copies, relatively delayed, and recombined again by two non-polarizing beamsplitters (BS). The beams in the upper path experience five reflections, three of which are from mirrors (M) and two from beamsplitters. As a result, the OAM beams in the upper path flip their charge sign compared to the lower path due to odd times of reflections. Thus, eight OAM beams (OAM ±10, OAM ±12, OAM ±14, OAM ±16 ) can be emulated. Supplementary figure 4c shows the block used to prepare two groups of concentric rings for spatial multiplexing. The incoming eight pol-muxed OAM beams are divided into two branches, relatively delayed, and recombined together through two non-polarizing beam splitters. The beams in the two branches have the same set of charges and the lower branch is expanded using a 1:2 telescope. Consequently, two groups of concentric rings are achieved with the inner and outer rings corresponding to the upper and lower branches, respectively. Spatial demultiplexing of two groups of concentric rings is illustrated in Suppl. Fig. 4d. Two kinds of spatial filters are employed to demultiplex the outer and inner rings, respectively. For demultiplexing from two groups of concentric rings of eight pol-muxed 20X4 Gbit s QAM-carrying OAM beams (i.e., 32 channels in total), supplementary figure 4e1,e2 shows measured BER curves of the back-converted beams from y-polarized OAM -16 beam in inner rings and x-polarized OAM +10 beam in outer rings. The BER curves of back to back and without crosstalk (i.e., all other channels are blocked) are also plotted for references. The obtained results show that the OSNR penalty caused by the crosstalk between inner and outer rings is less than 0.5 db at a BER of 2X NATURE PHOTONICS

11 SUPPLEMENTARY INFORMATION a Data 1, 2,, N OAM beams 1, 2,, N Data 1, 2,, N OAM beams 1, 2,, N b M M M BS BS Spatial multiplexing Data N+1,, 2N Spatial demultiplexing Data N+1,, 2N c BS BS OAM beams 1, 2,, N ( beam expanded ) OAM beams 1, 2,, N M 1:2 Expansion M d e1 e2 Supplementary Figure 4 Concept, experimental blocks, and results of the multiplexing/demultiplexing of eight pol-muxed 20X4 Gbit s QAM-carrying OAM beams (OAM ±10, OAM ±12, OAM ±14, OAM ±16 ) in two groups of concentric rings. a, Concept of spatial multiplexing/demultiplexing of two groups of concentric rings. b, Experimental block for the emulation of eight OAM beams. c, Experimental block for the emulation of two groups of concentric rings. d, Principle of spatial demultiplexing of two groups of concentric rings. e1,e2, Measured BER curves for y-polarized OAM -16 beam in inner rings (e1) and x-polarized OAM +10 beam in outer rings (e2) after demultiplexing. V. Propagation of OAM beams (angular-spectrum propagation method) Under the Fresnel approximation to the scalar diffraction theory, the near-field optical wave propagation from an input-plane ( x 1, y 1 ) to an output-plane ( x 2, y 2 ) can be described by the Fresnel diffraction integral 2, exp( ikδz) in k = iλδz 2Δz out U ( x, y ) U ( x, y ) exp{ i [( x x ) ( y y ) ]} dx dy (1) in out where U ( x1, y 1) and U ( x2, y 2) are the input-plane field and output-plane field, k is the wavenumber, λ is the optical wavelength, and Δ z is the propagation distance from the input-plane to the output-plane. NATURE PHOTONICS 11

12 Note that equation (1) is recognized to be a convolution, expressible in the form out in in 2 2 = = U ( x, y ) U ( x, y ) h( x x, y y ) dx dy U ( x, y ) h( x, y ) (2) where exp( ikδz) k 2 2 hx ( 1, y1) = exp[ i ( x1 + y1 )] iλδz 2Δz is the Fresnel diffraction impulse response. The convolution in equation (2) can be further expressed as the angular-spectrum form of the Fresnel diffraction integral, out -1 in 2 2 = 1 1 x1 y1 U ( x, y ) F { F [ U ( x, y )] H( f, f )} (3) where F and -1 F denote the spatial Fourier transform operation and its inverse operating only on the transverse coordinates. The transfer function valid for Fresnel diffraction is expressed as H( f, f ) = F [ h( x, y )] = exp( ikδz) exp[ iπλ Δ z( f + f )] (4) 2 2 x1 y1 1 1 x1 y1 The angular-spectrum propagation method computes the output-plane field from the knowledge of the input-plane field. The multiplexing/demultiplexing of OAM beams and the exchange between OAM beams can be theoretically analysed by consecutively using the angular-spectrum propagation method through the system as shown in the experimental setup. 1. Chang, F., Onohara, K. & Mizuochi, T. Forward error correction for 100 G transport networks. IEEE Commun. Mag. 48, S48 S55 (2010). 2. Goodman, J. W. Introduction to Fourier Optics (McGraw-Hill, New York, 2005). 12 NATURE PHOTONICS