# Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Electromagnetic Force of Micro VCA

_{m}is the magnet’s radius, h

_{m}is the magnet’s height, d

_{c-in}is the coil’s inner diameter, d

_{c-out}is the coil’s outer diameter, h

_{c}is the coil’s height, and h

_{g}is the height of the air gap.

**H**of the current-carrying coil at the position vector

**r**is

_{0}is vacuum magnetic permeability, Id

**l**is the current element, and

**e**is the unit vector along a position vector

_{r}**r**. The permanent magnet in the VCA is in the magnetic field generated by the current-carrying coil, so the electromagnetic force can be expressed as

_{r}is the remanence of the permanent magnet.

#### 2.2. Power Dissipation and Efficiency of Micro VCA

^{1/2}, F is the electromagnetic force, and P is the power dissipation of the micro VCA that is equal to I

^{2}R. From Equations (2) and (3), ε could be written as

#### 2.3. Response Time of VCDM

_{c}related to the required control bandwidth of an AO system is [19]

_{r}is the uncorrected power, L is the path length, ${C}_{n}^{2}(z)$ is the refractive-index structure constant, and v(z) is the wind speed. For a single turbulence layer with constant wind speed v, the Greenwood frequency can be approximated as

_{0}of 5 cm, the Greenwood frequency is approximately 86 Hz. The operating bandwidth of the DM is limited by several factors including the response time of the VCA and the first mechanical resonance frequency of the mirror. With 100 Hz operating bandwidth as a goal, the response time of the actuator should be less than 5 ms and the first resonance frequency of the mirror is aimed at 2000 Hz.

#### 2.4. Fitting Error of VCDM

_{fit}arises from the limited number of actuators of the DM and thus the limited number of spatial frequencies that it can correct. The variance of the fitting error can be approximated by [20]:

_{0}is the Fried parameter, generally ranging from 5 cm to 20 cm.

## 3. Design and Optimization

#### 3.1. Model of Compact VCDM

#### 3.2. Optimization of Micro VCA

^{1/2}.

## 4. Discussion

#### 4.1. Aberration of the Thin Mirror Due to Thermal Effect

^{2}K) [22]. According to the mechanical model of the VCDM, as shown in Figure 2, the internal heat generation of the voice coil is introduced by the ohmic loss calculated by the Maxwell module. After that, the thin mirror temperature obtained by the Steady-State Thermal module is transferred to the Static Structural module to calculate the thermal deformation of the thin mirror.

^{2}, the temperature rise of the VCDM is about 1 Celsius degree. Figure 7b shows the thermal deformation of the thin mirror due to temperature-rising non-uniform distribution. It indicates that the peak-to-valley (PV) of the thin mirror deformation increases along with the current. When the current density is 6 A/mm

^{2}corresponding to 0.06 A, the PV is 12.36 nm, which is about 0.022λ for λ = 550 nm. The root mean square (RMS) value of the mirror deformation is around 0, and it gradually deviates from 0 as the current increases. Therefore, the maximum control current of a single VCA should not be larger than 0.06 A for good thermal stability. At this time, the temperature and the deformation of the thin mirror are shown in inserted figures of Figure 7a,b.

#### 4.2. Response Time

#### 4.3. Wavefront Fitting Precision

_{0}is 2.5 mm, and α is Gaussian index.

#### 4.4. Application in Optical Communication

## 5. Conclusions

^{1/2}. The maximum current of a micro VCA is 0.06 A, which makes the VCDM has excellent thermal stability. The temperature difference of the thin mirror is less than 0.4 degrees Celsius at the maximum current of 0.06 A, and the thermal deformation of the thin mirror is only 12.36 nm. The first resonance frequency of the 69-element compact VCDM is 2045 Hz which is more than three times that of the DM69-25. High wavefront fitting precision with a relatively low coupling coefficient of about 25% and large phase stroke is also demonstrated in the paper. The fitting results of Zernike aberrations show that the wavefront fitting precision of the compact VCDM has increased by 13% compared with a traditional VCDM. The VLC scenario was established, and the results proved that the compact VCDM can improve the coupling efficiency. The above results indicate that the compact VCDM can satisfy the requirements of optical communication systems. The design and optimization method are also valuable for the design of other kinds of DMs. Our design decreases the development cost and obtains a compact VCDM with high electro-optical performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**3D drawing of VCDM. Front shell 1, thin mirror 2, O-ring 3, springs 4, struts 5, spring fixing plate 6, permanent magnets 7, voice coils 8, substrate 9, back shell 10.

**Figure 3.**The electromagnetic force of VCA is a function of current for different magnetization directions of the permanent magnet.

**Figure 4.**Force and efficiency as a function of VCA’s structural parameters. (

**a**) Permanent magnet radius. (

**b**) Permanent magnet height. (

**c**) Coil inner diameter. (

**d**) Coil outer diameter. (

**e**) Coil height. (

**f**) Air gap.

**Figure 5.**Electromagnetic force and efficiency as a function of current. The red circle is before optimization, and the black triangle is after optimization. (

**a**) Force. (

**b**) efficiency.

**Figure 7.**Temperature and thermal deformation of VCDM’s thin mirror as a function of current. (

**a**) The temperature of the thin mirror is a function of currents. The inset shows the temperature of the thin mirror when the current is 0.06 A. (

**b**) The thermal deformation of the mirror surface as a function of currents. The inset shows the deformation of the thin mirror when the current is 0.06 A.

**Figure 13.**Wavefront Fitting and error of some typical Zernike modes. (

**a**) The 4th Zernike mode Z4. (

**b**) The 7th Zernike mode Z7. (

**c**) The 10th Zernike mode Z10. (

**d**) The 13th Zernike mode Z13.

**Figure 15.**Fitting errors of the first 14 Zernike terms. (

**a**) Comparison among four VCDMs listed in Table 4. (

**b**) Comparison between P3 and DM69-25.

Serial Number | Component |
---|---|

1 | Front shell |

2 | Thin mirror |

3 | O-ring |

4 | Springs |

5 | Struts |

6 | Spring fixing plate |

7 | Permanent magnets |

8 | Voice coils |

9 | Substrate |

10 | Back shell |

Parameters | Unit | Values | Step |
---|---|---|---|

Magnet radius | [mm] | 0.1 ≤ r_{m} ≤ 1.1 | 0.1 |

Magnet height | [mm] | 0.05 ≤ h_{m} ≤ 1 | 0.05 |

Coil inner diameter | [mm] | 0.2 ≤ d_{c-in} ≤ 1 | 0.2 |

Coil outer diameter | [mm] | 0.4 ≤ d_{c-out} ≤ 2.2 | 0.2 |

Coil height | [mm] | 0.1 ≤ h_{c} ≤ 1 | 0.1 |

Air gap | [μm] | 50 ≤ h_{g} ≤ 100 | 10 |

Thermal Conductivity | Coefficient of Thermal Expansion | Density | Young’s Modulus | Poisson’s Ratio | |
---|---|---|---|---|---|

Material | [W/m/°C] | [/°C] | [kg/m^{3}] | [Pa] | [/] |

CP1 Polyimide | 0.25 | 5.1 × 10^{−5} | 1540 | 2.1 × 10^{9} | 0.34 |

316 Stainless Steel | 13.44 | 1.478 × 10^{−5} | 7954 | 1.95 × 10^{11} | 0.25 |

FR-4 Epoxy | 0.294 | 1.688 × 10^{−5} | 1900 | 2.64 × 10^{10} | 0.1543 |

NdFe35 | 7.7 | 3.2 × 10^{−6} | 7450 | 1.6 × 10^{8} | 0.24 |

Copper | 112.1 | 1.999 × 10^{−5} | 8267 | 9.995 × 10^{10} | 0.345 |

Aluminum Alloy | 114 | 2.3 × 10^{−5} | 2770 | 7.1 × 10^{10} | 0.33 |

Pattern | Thickness | Stiffness | First Resonance | k | ω | α |
---|---|---|---|---|---|---|

P1 | 35 μm | 55 N/m | 2019.9 Hz | 9.958 | 19.78% | 2.191 |

P2 | 40 μm | 61 N/m | 2010.8 Hz | 8.214 | 23.22% | 2.111 |

P3 | 45 μm | 70 N/m | 2045.2 Hz | 6.694 | 25.81% | 2.049 |

P4 | 50 μm | 80 N/m | 2085.8 Hz | 5.525 | 28.03% | 1.995 |

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**MDPI and ACS Style**

Jiang, L.; Hu, L.; Hu, Q.; Xu, X.; Wu, J.; Yu, L.; Huang, Y.
Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication. *Photonics* **2023**, *10*, 322.
https://doi.org/10.3390/photonics10030322

**AMA Style**

Jiang L, Hu L, Hu Q, Xu X, Wu J, Yu L, Huang Y.
Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication. *Photonics*. 2023; 10(3):322.
https://doi.org/10.3390/photonics10030322

**Chicago/Turabian Style**

Jiang, Lv, Lifa Hu, Qili Hu, Xingyu Xu, Jingjing Wu, Lin Yu, and Yang Huang.
2023. "Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication" *Photonics* 10, no. 3: 322.
https://doi.org/10.3390/photonics10030322