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Ting Lei

1

†, Meng Zhang

2

†, Yuru Li

3

†, Ping Jia

2

, Gordon Ning Liu

4

, Xiaogeng Xu

4

,

Zhaohui Li

3

*, Changjun Min

1

, Jiao Lin

5

, Changyuan Yu

6

, Hanben Niu

1

, Xiaocong Yuan

1

*

I.

The Dammann grating has the ability to divide an incident beam into

*N*

parts with approximately equal power

1, 2

. Here, with the diffraction order

*N*

as an even number, we consider the target function as

*n*

*N*

2

*N*

2

*A n*

exp

*T*

(1) where

*T*

is the period of the grating,

*n*

is the diffraction order from -

*N*

/2 to

*N*

/2, and

*A n*

2

1/

*N*

is the power of the

*n*

th order normalized with reference to the total power.

Because the function Ψ(

*x*

) is not a pure phase structure, we must find a binary pure phase

(0–

) grating

(

*x*

) with exp(i

(

*x*

)) approaching Ψ(

*x*

). We can optimize the parameters of the positions of the changing points for the 0 to

phase shift

*a m*

and the

to 0 phase shift

*b m*

, where

*m*

indicates the number of changing points as shown in Fig. S1.

Fig. S1

**.**

Design principle of the Dammann gratings.

Therefore, the conventional Dammann grating could be expressed as

*M m*

1

π

*m*

*m*

(2) where

*u*

(

*x*

) is a step function. The optimization process adjusts the values of

*a m*

and

*b m*

to make exp(i

(

*x*

)) close to Ψ(

*x*

). We use an annealing algorithm to find the optimized

*a m*

and

*b m*

values and obtain the desired

(

*x*

).

Based on the Dammann grating, we replace the term

2π

*x*

with

*T*

2π

*x*

*l*

*T*

, in which

*l*

is the interval of the topological charges

3

. Therefore, we obtain the DOVG as

*N*

2

*n*

*N*

2

1

*N*

exp

*in*

2π

*x*

*T*

*l*

(3)

When the incident light beam passes through the DOVG that is described by the above functions, it is diffracted into

*N*

orders with equal distributed powers and a topological charge of

*n*

×

*l*

in the

*n*

th order direction. Fig. S2 shows a conventional Dammann grating with 10

diffraction orders and the corresponding DOVG encoded with OAM states =±3, ±9, ±15, ±21,

±27.

Fig. S2. DOVG converted from a conventional Dammann grating.

In our system, only two DOVGs and a single aperture are required to realize OAM division multiplexing (OAM-DM) and the insertion loss of the multiplexing grating is ~10×lg(

*N*

). It is noted in the literature that OAM multiplexing was generally achieved using beam splitters or complicated composite CGH (Computer Generated Hologram) in most cases

3–5

. With the beam splitter method,

*N*

SLMs (Spatial Light Modulator) or spiral phase plates are required to generate

*N*

OAM states. For example, to multiplex ten OAM channels, ten SLMs or spiral phase plates and 11 beam splitters would be needed. We estimate the insertion loss to be more than 14 dB, including losses of ~3.5 dB per beam splitter and ~ 0.5 dB per SLM or spiral phase plate. In our

system, a DOVG and an aperture can replace all the optical elements listed above, and the insertion loss is ~10 dB. With the composite CGH method, the OAM channels are generated by a single composite CGH, which means that the

*N*

OAM channels cannot be modulated independently. In our system, the

*N*

channels of the Gaussian beams can be modulated independently and subsequently multiplexed into the different OAM channels by the DOVG. We therefore believe that this method has vast application prospects in the FSO communications field.

Table S1 summarizes the parameters of the 1×10 DOVG. The ten OAM channels have topological charges of ±3, ±9, ±16, ±21, and ±27.

Table S1. Parameters of our designed 1×10 DOVG .

-3

-2

-5

-4

-1

+1

+2

+3

+4

+5

Total

+21

+27

-3

+3

+9

+15

-27

-21

-15

-9

Modeled

Diffraction Topological diffraction order charge efficiency

(%)

10.41

7.33

9.29

11.18

7.98

7.98

11.18

9.29

7.33

10.41

92.38

Measured diffraction diffraction power

Modeled angle (°)

(dBm)

-15.92

-15.36

-16.200

-12.533

-16.03

-15.23

-15.47

-15.65

-8.917

-5.336

-1.776

1.776

-15.70

-15.66

-15.41

-15.37

5.336

8.917

12.533

16.200

Fig. S3. Scanning profile of the DOVG. The scale bar indicates the length of 100

m .

We designed a phase mask with dimensions of 5.12 mm × 5.12 mm, and 1024 pixel× 1024 pixel with a grating period size of 50

m. The mask pattern was then transferred to an AR-N4340 photoresist by an ultraviolet (UV) lithography technique. We spin coated the AR-N4340 photoresist on a quartz substrate with a photoresist thickness of 1.285

m (the

phase difference was calculated as

*H*

2

*n*

1

1285.2 nm ; the refractive index of AR-N4340 given in the instruction manual is 1.603 at 1550 nm wavelength). After a soft bake, the photoresist was exposed using an MJB4 Mask Aligner (Karl Suss) with a UV wavelength of 365 nm and a dose of 20 mW cm

-2

. After post-exposure baking (PEB) and development, we obtained the DOVG as shown in Fig. S3.

Fig. S4. Details of the experimental setup with the QPSK/16-QAM modulators and the coherent detection systems.

Fig. S4 depicts the implementation details of the modulator and detector systems. The upper optical path indicates a 100 Gbit s

-1

quadrature phase-shift keying (QPSK) signal (including two polarization states). The lower optical path indicates a 200 Gbit s

-1

quadrature amplitude modulation (16-QAM) signal (including two polarization states). We use QPSK as an example here to show the experimental setup details of our FSO communication system. A single wavelength laser source is aligned with a specific 50 GHz ITU-T grid channel in the 1529.16 nm to 1560.61 nm range. The beam, containing both

*x*

- and

*y*

-polarization components, is modulated into a 100 Gbit s

-1

QPSK signal by a QPSK modulator. As shown in the constellation of the

QPSK signal, QPSK is a multivariate digital frequency modulation signal, with four possible discrete phase states in the carrier phase, and it carries two binary symbols. The QPSK signal in

the N slot can be expressed as

*S*

QPSK

*A*

*f t*

c

*n n*

1

*T*

s

*nT*

s

(4) where

*A*

is the amplitude of the signal, which can be considered to be constant,

*n*

is the modulated phase with four possible values,

*f*

c

is the carrier wave frequency, and

*T*

s is the time interval of the quaternary signal. In our experiment, the possible values of

*n*

are π/4, 3π/4, 5π/4 and 7π/4, corresponding to the binary symbols 00, 01, 10 and 11, respectively. The signal beam is magnified by an erbium-doped fiber amplifier (EDFA) and passes through a narrow band filter.

The beams with different wavelengths are multiplexed using an arrayed waveguide grating

(AWG, not shown in the figure 4). After modulation and the WDM system, the signal beams are coupled into free space by an optical coupler for OAM multiplexing. After passing through the transmission link in free space, the beam is then coupled back into a fiber for coherent detection.

Unlike direct detection, coherent detection mixes the signal beam with a local oscillator beam with a power that is much larger than that of the signal to improve the OSNR of the detected signal. Coherent detection technology also makes full use of powerful digital signal processing

(DSP) technology to deal with polarization- and mode-multiplexed signals. The reconstructed signal can subsequently be compensated using a digital signal. The characteristics of the transmitted signal (polarization mode, amplitude and phase) can be restored, thus largely eliminating the transmission impairments produced in the transmission link. The structure of the

coherent detector is shown in the figure 4. The signal beam and the local oscillator beam are mixed and initially interfere in the 90° hybrid. The output beam from the hybrid is then transformed into an analog electrical signal by the photodiodes. The electrical signal is then sampled and converted into digital signals by the high speed ADC (analog-to-digital converter).

The data recovery of the digital signal is completed in the DSP.

Fig. S5. Experimental results for OAM-based FSO communication using 16-QAM signals.

**(a)**

Measured

BERs at OAM state 15 with

*x*

-polarization.

**(b) **

Measured BERs at OAM state 15 with y-polarization.

Constellations of the QPSK signals with

**(c)**

*x*

- and

**(d)**

*y*

-polarizations: back to back

**(c1, d1)**

; without OAM

crosstalk at a BER of 1.2×10

−3

**(c2, d2)**

; with OAM crosstalk at a BER of 1.2×10

−3

(

**c3, d3)**

; without OAM crosstalk at a BER of 3×10

−3

**(c4, d4)**

; with OAM crosstalk at a BER of 3×10

−3

**(c5, d5)**

.

We also achieved an information capacity of sub-Pbit s

-1

order using 1600 channels of 100

Gbit s

-1

16-QAM signals. In this experiment, we used ten OAM channels, each with two polarizations and 80 WDM channels. Fig. S5a, S5b show the measured BERs of

*x*

- and

*y*

-polarized data carried by the OAM beams ( = +15,

= 1544.53 nm) as a function of the

OSNR with/without the OAM crosstalk. The dashed line indicates the soft decision FEC

(SD-FEC) threshold of 2.4×10

−2

BER. Without the OAM crosstalk, the OSNR penalties are less than 1 dB. With the OAM crosstalk, the OSNR penalties are less than 1.5 dB. We can also attribute the additional penalty due to OAM crosstalk to the superfluous OAM states that are caused by inaccurate grating fabrication. Fig. S5c, S5d depict the observed 16-QAM constellations for demultiplexing of the OAM state = +15. Fig. S5c shows the constellations of the

*x*

-polarized signals and Fig. S5d shows those of the

*y*

-polarized signals. Fig S5c1, S5d1 are the back to back measurement results. Fig. S5c2, S5d2 show the constellations without the OAM crosstalk at a BER of 1.2×10

−3

. Fig. S5c3, S5d3 show the constellations with the OAM crosstalk at a BER of 1.2×10

−3

. Fig. S5c4, S5d4 show the constellations without the OAM crosstalk at a

BER of 3×10

−3

. Fig. S5c5, S5d5 show the constellations with the OAM crosstalk at a BER of

3×10

−3

. The area of the dot matrix increases with increasing BER. There are no obvious data overlaps in the constellations, because the BER is still less than the SD-FEC threshold and can effectively be corrected in each case. In the FSO system, we achieved a total data capacity of 160

Tbit s

-1

(100 Gbit s

-1

16-QAM × 80 wavelengths × 10 OAM states ×2 polarizations).

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