# Spatial Mode Division Multiplexing of Free-Space Optical Communications Using a Pair of Multiplane Light Converters and a Micromirror Array for Turbulence Emulation

^{1}

^{2}

^{*}

## Abstract

**:**

_{n}

^{2}= 10

^{−13}m

^{−2/3}) are emulated using a micromirror array producing a time sequence of aberrating frames. The modal crosstalk between transmitter and receiver modes induced by the turbulence is presented by measuring the intensity in receiver channels for the same turbulence. Six receiver modes are used for optical communication channels with a rate of 137 Gbits/s displaying the benefits of single input multiple output (SIMO) operation for overcoming the deleterious effects of turbulence.

## 1. Introduction

^{2}) from weak (C

_{n}

^{2}= 1 × 10

^{−16}m

^{2/3}) to strong (C

_{n}

^{2}= 1 × 10

^{−13}m

^{2/3}), to be created so that the induced crosstalk between modes can be observed.

## 2. Materials and Methods

#### 2.1. Theory of Modal Overlap

_{i}is the Hermite polynomial, m and n are the mode indices in the x and y directions and ${\omega}_{0}$ is the beam waist size. The overlap between the transmitted and received mode is then:

_{n}

^{2}(and aperture) [15]. We can, thus, estimate the effects of a controlled level of atmospheric turbulence from such a set of Zernike modes by summing the modal phase effects as follows:

#### 2.2. Phase 1—Intensity Measurements of Induced Crosstalk

_{n}

^{2}) along a defined path length. The emulated turbulence was for an equivalent path length of 1 km and an equivalent capture aperture of 10 cm. The distortion is represented as a set of orthogonal Zernike mode functions whose amplitudes are selected from a random distribution and whose widths are statistically related to the values of C

_{n}

^{2}[19]. Smooth transitions between frames are created by interpolating between values [20]. Thus, a smoothly varying sequence of frames can represent changing wavefront distortions of a chosen turbulence strength, and this sequence can be repeated to provide a consistent comparison of the effects across all possible channels (modes) at a frame rate that is completely controllable; up to 10 kHz, if required. This approach of using a DMD to emulate turbulence in a controllable, reproducible way is presented in more detail in ref [8]. More sophisticated turbulence emulations operate with multiple planes to better represent turbulence effects spread along the entire path [21,22]. The loss associated with the DMD makes this infeasible, but it is sufficient to explore the effects arising from large-scale intensity variations. This arrangement forms the kernel of experiments aimed at observing the turbulence-induced modal channel mixing across all channels, measuring just the intensity, and across a selected subset of channels across a communications link.

_{3}). The result is that the light is distributed into a first-order diffraction focus, a zero-order undiffracted beam and a negative first-order defocussed beam. This results in less than 10% efficiency of the collected power, but only the first-order focus is effectively received by the MPLC. Unity magnification is found by adjusting the length of the input path to the DMD, adjusting the position of a pair of fold mirrors on a translation stage.

#### 2.3. Phase 2—Coherent Data Measurements of Induced Crosstalk

#### 2.4. SIMO Processing

## 3. Results

#### 3.1. Phase 1

_{n}

^{2}= 1 × 10

^{−13}m

^{2/3}.

_{m},

_{n}are received as HG

_{n,m}; thus, only where m = n is the receive channel, the same as the transmit channel (in this case channels 10, 11 and 12).

_{00}) matches to receiver ch11(HG

_{00}) and transmitter ch6(HG

_{02}) matches to receiver ch16(HG

_{20}). We can observe many frames where the principal channel is strongly attenuated and the received power increases in other neighbouring channels, thus demonstrating the modal redistribution effect of the turbulence. Thus, capturing multiple modes reduces the loss inflicted upon a single mode system by turbulence [28].

_{00}is the best coupled principal channel, and that coupling efficiency reduces as mode order increases. On some occasions, where the HG

_{00}channel is strongly attenuated, other principal channels show a larger intensity—for example, frame 248. Different modes, therefore, respond differently to the same turbulence.

_{00}to HG

_{22}

_{,}with a clear reduction for all the principal channels and increases in neighbouring channels. The right-hand matrix is the result of calculated values of the difference in modal overlap given the expected parameters for ZMs with a C

_{n}

^{2}= 1 × 10

^{−13}m

^{2/3}. The calculation is simplistic; it does not include the effects of mode-related divergence changes, but it is enough to give an understanding of expectations and show similarities with the measured data. With the predominant ZMs being tip and tilt [19,29], we can see, in terms of measurement and calculation, how these beam-steering modes will, for example, direct power from HG

_{00}to HG

_{10}and HG

_{01}. Intuitively, we can understand this as translating the central node of HG

_{00}to an offset node such as HG

_{10}.

#### 3.2. Phase 2

_{00}principal channel.

_{00}(ch11), HG

_{01}(ch9), HG

_{10}(ch13), HG

_{11}(ch12), HG

_{02}(ch6) and HG

_{20}(ch16)—this set allows for consistency and reciprocity for the mode index swapping that occurs between transmitter and receiver. Figure 6 shows matrix plots for the subset of channels taken for each of the chosen frames, with the plots of Figure 6b showing the intensity in the receive channel (horizontal axis) for different transmitter channels (vertical axis). The plots of Figure 6c display the phase 2 SNR of the coherent signals in the same manner, with SNR converted from the calculated Error Vector Magnitude to give linear comparisons with the plots of Figure 6b. Frames 110 and 249 represent positions of little turbulence within the sequence and we can see—particularly for ch11—that this is the strongest channel seen in both the intensity and coherent data plots. Frames of strong turbulence effect (48 and 229) reduce the SNR and see the input power distributed around other channels.

## 4. Discussion

_{01}and HG

_{10}have a higher intensity than the HG00 mode, despite this one being much more efficiently coupled. The implication is that the same turbulence has had less effect on the coupling of HG

_{10}than on HG

_{00}. Higher-order aberration modes correlate less well through turbulence, resulting in a mode-dependent isoplanatic angle [30], so it is reasonable to assume that higher-order transmission modes will also respond differently to the same turbulent atmosphere.

_{00}Gaussian beam is laterally translated by tip/tilt and, thus, increases the overlap with HG

_{01}and HG

_{10}modes. A strong coupling of power out of HG

_{00}(ch11) and into HG

_{01}(ch9) and HG

_{10}(ch13) can be observed in the data. This tells us that power redistribution is not random and that judicious choice of receive channels is beneficial. Indeed, utilizing many modes would require more hardware and could be a limiting factor. Thus, the use of a subset of receiver channels, as was shown in Phase 2, is a sensible compromise. This enables the use of an adaptive SIMO approach where the choice of receiver channels could be actively controlled depending on the observed turbulence conditions.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) A schematic diagram of the equipment used to observe the power transfer between spatial channels selected and received by 2 MPLCs, with a DMD micromirror array creating turbulence. (

**b**) A diagram showing the relationships between MPLC device channel number and the HG mode it represents. (

**c**) A diagram showing the inverting image effect of the DMD in relating output and input channels.

**Figure 3.**The same turbulence frame sequence for 20 of the input channels (modes) for all receive modes with the 1545 nm sequence. Intensity scales are linear and the same for each plot.

**Figure 4.**Diagram (

**a**) (

**top left**) shows the received sequence–channel matrices, as shown in Figure 3, for HG

_{00}and HG

_{01}, showing how the profile for a frame sequence and the total intensity for each frame is produced for the following plots. Plot (

**b**) (

**top right**) shows the total intensity across all channels for each frame with the given transmit modes. Plot (

**c**) (

**bottom left**) shows the intensity variation received in the principal channel. Plot (

**d**) (

**bottom right**) shows the average fractional change in intensity across the whole sequence (max intensity at best transmission point minus the mean intensity across channel in linear arbitrary units) for transmit and receive channel combinations.

**Figure 5.**The left-hand matrix is the normalized mean change in the intensity of the received channel (horizontal axis) measured for the transmission mode given in the vertical axis. Values are the fractional change relative to the highest channel value. Negative values (blue) represent a loss of power, and the positive (red) a gain. The right-hand matrix is the calculated fractional change in channel overlap between transmission and receiver spatial modes calculated for C

_{n}

^{2}= 1 × 10

^{−13}m

^{2/3}.

**Figure 6.**Modal crosstalk observed in specific frames of the turbulence sequence. (

**a**) Shows the sequence of intensity variations (measured voltage) for strong turbulence at 1545 nm. (

**b**) Matrices showing the received intensity in each receiver channel horizontally, with the transmission channel vertically. (

**c**) The same but showing the SNR (linear ratio not dB) when the system is used with the coherent comms setup. (

**d**) Plots of the SNR (linear) for the SIMO-processed data compared with the largest single channel for each transmission channel.

Index (m or n) | Hermite Polynomial | Index (j) | Zernike Mode | Aberration Name |
---|---|---|---|---|

0 | $1$ | 0 | $1$ | Piston |

1 | $2x$ | 1 | $x$ | Tip |

2 | $4{x}^{2}-1$ | 2 | $y$ | Tilt |

3 | $8{x}^{3}-12x$ | 3 | $-1+2({x}^{2}+{y}^{2})$ | Focus |

4 | $16{x}^{4}-48{x}^{2}+12$ | 4 | $2xy$ | V Astigmatism |

5 | $32{x}^{5}-160{x}^{3}+120x$ | 5 | $-{x}^{2}+{y}^{2}$ | H Astigmatism |

6 | $-2x+3{x}^{3}+3{xy}^{2}$ | V coma | ||

7 | $-2y+3{y}^{3}+3{x}^{2}y$ | H coma | ||

8 | $-{x}^{3}+3x{y}^{2}$ | V trefoil | ||

9 | ${y}^{3}-3{x}^{2}y$ | H trefoil | ||

10 | ${1-6x}^{2}6{y}^{2}+6{\left(\right(x}^{4}+{y}^{4})+2({x}^{2}{y}^{2}\left)\right)$ | Spherical aberration |

IQ imbalance compensation |

Matched filter |

CD compensation for FDLs |

Timing recovery |

Frame synchronization |

Frequency estimation |

3-dimensional LMS equalization |

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**MDPI and ACS Style**

Benton, D.; Li, Y.; Billaud, A.; Ellis, A.
Spatial Mode Division Multiplexing of Free-Space Optical Communications Using a Pair of Multiplane Light Converters and a Micromirror Array for Turbulence Emulation. *Photonics* **2024**, *11*, 241.
https://doi.org/10.3390/photonics11030241

**AMA Style**

Benton D, Li Y, Billaud A, Ellis A.
Spatial Mode Division Multiplexing of Free-Space Optical Communications Using a Pair of Multiplane Light Converters and a Micromirror Array for Turbulence Emulation. *Photonics*. 2024; 11(3):241.
https://doi.org/10.3390/photonics11030241

**Chicago/Turabian Style**

Benton, David, Yiming Li, Antonin Billaud, and Andrew Ellis.
2024. "Spatial Mode Division Multiplexing of Free-Space Optical Communications Using a Pair of Multiplane Light Converters and a Micromirror Array for Turbulence Emulation" *Photonics* 11, no. 3: 241.
https://doi.org/10.3390/photonics11030241