# Adaptive Petal Reflector: In-Lab Software Configurable Optical Testing System Metrology and Modal Wavefront Reconstruction

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## Abstract

**:**

## 1. Introduction

- The developer of the reflector must have access to a metrology system capable of a rapid evaluation of the manufacturing process, and to verify, without the availability of the whole telescope, that the surface figure aberration will remain within the envelope that can be corrected subsequently by the control system.
- When the primary reflector is mounted on the telescope, part of the petals are in the shade of the secondary mirror and they appear as segments (i.e., independent of each other). However, unlike segmented mirrors which require edge sensors, all the petals have a common mechanical boundary condition at the inner ring of the reflector. This paper investigates the use of this property to implement a wavefront reconstruction based on the measurement of the x-y slopes at a set of grid points (Shack–Hartmann-type sensor).

**Figure 1.**Illustration of the system design of the active petal reflector in folded and deployed states. The reflector (red) is coated with a piezoelectric material, allowing for active control in closed loop given a metrology system. Since the secondary will obstruct the primary mirror, the recovered sensor data will be segmented, and special wavefront reconstruction is needed to recover the error accurately.

#### 1.1. In-Lab Metrology

#### 1.2. Vibration Modal Wavefront Reconstruction

## 2. SCOTS

**Zero-phase recovery**: A single pixel is lit up in the $(0,0)$-phase location of the screen and its position in the recorded image is saved (Figure 2, red dot).**Determination of phase**: Successive horizontal and vertical sinusoidal fringes with varying phase offsets are displayed. For each individual Mirror Pixel, a sine function is fit on the data from the images to determine the phase value of this pixel from $-\pi $ to $\pi $ (Figure 2, blue dot).**Phase unwrapping**: as the previous step happened on a per-pixel basis, there is no global information about the phase. The global phase is recovered by a numerical phase unwrapping technique and offset according to the previously recovered zero-phase location.

#### 2.1. Raytracing

**left**) in which a 100 µ$\mathrm{m}$ tip displacement is applied. The resulting ray-traced image can be seen in Figure 3 (

**right**). The tip displacement is generated by applying a fixed displacement in an FEM software, ensuring the displacement is representative of expected deformations on the reflector. Notable parameters are as follows: radius of curvature—2.5 $\mathrm{m}$; diameter—20 $\mathrm{cm}$; camera location $({x}_{c},{y}_{c},{z}_{c})$—(−50 $\mathrm{mm},0,0)$; screen origin $({x}_{s},{y}_{s},{z}_{s})$—(50 $\mathrm{mm},0,0)$; mirror origin $({x}_{m},{y}_{m},{z}_{m})$—(0, 0,−1400 $\mathrm{mm})$.

**left**), and the error between reconstruction and nominal aberration can be seen in Figure 4 (

**right**). The main component of the error shown is from the assumption that the reflector is flat from the point of view of the camera. This assumption leads to a coma error in the case of a full reflector, but the petal integration leads to a more complex error. With this error known, it may be possible to subtract it in the slope calculation for a specific setup, but the accuracy shown is sufficient for our purposes: characterizing different manufacturing techniques and ensuring the resulting errors are within the control stroke of the adaptive reflector.

#### 2.2. Experimental Results

## 3. Modal Wavefront Reconstruction

#### 3.1. Vibration Modes

#### 3.2. Jacobian

#### 3.3. Central Obstruction

## 4. Conclusions

- The first aspect is that of measuring the surface figure error of a spherical reflector alone, during the development and manufacturing phase. The requested accuracy is modest, because the reflector is intended to be actively controlled once in operation. The SCOTS approach has been found to be fast and satisfactory. Experimental results have been presented and their consistency with a ray-tracing virtual experiment has been assessed.
- The second part of this paper is concerned with the surface figure error reconstruction from slope measurements of a petal reflector when a central part of the mirror is obscured by the secondary mirror of the telescope, making the petals appear as completely disconnected (like segments). Using the fact that all petals have the same mechanical boundary conditions, the deformed shape is expanded in a set of orthogonal modes having the same boundary conditions (the vibration modes). The modal amplitudes are reconstructed from slope data (Shack–Hartmann) and an approximation of the surface figure error is obtained.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ESA | European Space Agency |

FEA | Finite Element Analysis |

FEM | Finite Element Modelling |

PV | Peak-to-Valley |

PVDF-TrFE | Poly(vinylidene fluoride-co-trifluoroethylene) |

RMS | Root Mean Square |

SCOTS | Software Configurable Optical Testing System |

S-H | Shack–Hartmann |

SVD | Singular Value Decomposition |

## Appendix A. SCOTS Calculation

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**Figure 2.**Illustration of the SCOTS test. The two vectors $\overrightarrow{m2c}$ and $\overrightarrow{m2s}$ together describe the surface slope. The blue dot illustrates how the recorded image relates a position of the Mirror Pixel to its corresponding screen location. The red dot shows the zero-phase location. Several images are needed to accurately relate the projected and recorded images in terms of phase.

**Figure 3.**(

**left**) A 100−micron tip displacement aberration simulated with FEA. (

**right**) The resulting ray-traced image from the simulation. Note that, on the right side, the fringes do not line up exactly due to the tip displacement.

**Figure 4.**(

**left**) Reconstructed aberration from simulated images of 100−micron tip displacement. The geometry is slightly eroded to avoid edge problems. This also reduces the apparent PV slightly. (

**right**) Error between the nominal aberration of Figure 3 (

**left**) and the measured aberration of Figure 4 (

**left**). The main discrepancy arises from the assumption of zero curvature on the primary mirror.

**Figure 5.**Illustration of the different operating modes of the software. (

**a**) The raytracing can be used to test the software implementation by comparing to a known aberration. (

**b**) the ray-tracing can be used to verify the hardware implementation by reconstructing a virtual image from the measured surface.

**Figure 6.**Demonstrators of the previous project [3] with individual electrode control.

**Figure 7.**Image overlay of the real image (green) and ray-traced simulation of the measured geometry (red). The two images closely match, showing that the geometry reported by the SCOTS implementation corresponds closely to the real world.

**Figure 8.**Petal reflector, Shack–Hartmann sensor array, and secondary mirror obscuration. The petals appear disconnected, like on a segmented mirror.

**Figure 9.**Example of vibration modes of the petal reflector, which will be used in the reconstruction of the aberration.

**Figure 10.**Evolution of the relative RMS residual error as a function of the number of modes in the expansion. Comparison between the mode shapes of the petal spherical shell and the corresponding petal plate. The surface figure error is a tip displacement of 100 µm in the corner of one petal.

**Figure 11.**Reconstructed surface (

**left**) and residual error (

**right**) of a 100 micron tip displacement.

**Figure 12.**Petal reflector with 30% obstruction. Reconstructed surface (

**left**) and residual error (

**right**) of a 100 micron tip displacement.

**Table 1.**Error between the nominal aberration and the measured aberration including geometric tolerances.

Geometric Change (Camera) | RMS Error |
---|---|

Nominal | 0.15 µ$\mathrm{m}$ |

$2\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ X | 0.18 µ$\mathrm{m}$ |

$2\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ Y | 0.17 µ$\mathrm{m}$ |

$2\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ Z | 0.21 µ$\mathrm{m}$ |

2° X | 0.65 µ$\mathrm{m}$ |

2° Y | 1.18 µ$\mathrm{m}$ |

**Table 2.**Relative RMS error after modal reconstruction based on orthogonality condition (8) and slopes measurements, for 0% and 30% central obstruction with 50 and 100 modes in the reconstruction.

Modes | Equation (8) 0% | Slopes 0% | Slopes 30% |
---|---|---|---|

50 | 1.52 × ${10}^{-3}$ | 1.84 × ${10}^{-3}$ | 3.47 × ${10}^{-3}$ |

100 | 6.47 × ${10}^{-4}$ | 6.47 × ${10}^{-4}$ | 6.48 × ${10}^{-4}$ |

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**MDPI and ACS Style**

Nielsen, C.J.G.; Preumont, A.
Adaptive Petal Reflector: In-Lab Software Configurable Optical Testing System Metrology and Modal Wavefront Reconstruction. *Sensors* **2023**, *23*, 7316.
https://doi.org/10.3390/s23177316

**AMA Style**

Nielsen CJG, Preumont A.
Adaptive Petal Reflector: In-Lab Software Configurable Optical Testing System Metrology and Modal Wavefront Reconstruction. *Sensors*. 2023; 23(17):7316.
https://doi.org/10.3390/s23177316

**Chicago/Turabian Style**

Nielsen, Carl Johan G., and André Preumont.
2023. "Adaptive Petal Reflector: In-Lab Software Configurable Optical Testing System Metrology and Modal Wavefront Reconstruction" *Sensors* 23, no. 17: 7316.
https://doi.org/10.3390/s23177316