# Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Theory

#### 2.1. Principle and Design of PG-CSSI

_{sub}, h, and β, and is usually required to reach more than 25%. Thanks to the virtual–real combination algorithm proposed in Section 2.2, PG-CSSI has no such requirement and only needs to ensure that all subaperture can cover the full aperture, which results in a larger h or β and thus a higher scanning efficiency given the same D

_{sub}. According to the full coverage requirement, h and β can be determined in combination with the help of D

_{sub}, and then θ can be obtained by Equation (1). The deflection angle θ of the light is the diffraction angle of the normal incident light on the PG, which follows the grating equation

**R**is the rotation matrix. The diffraction efficiency of the m order is

_{0}of PG used in this paper is zero, so only ±1st orders diffracted light exist. Ordinary diffraction gratings cannot achieve such performance. Because PG can ensure that no other unwanted orders exist in interferometry and can achieve any scanning angle θ by changing the period Λ, it has a huge advantage in subaperture scanning compared to ordinary diffraction gratings.

#### 2.2. Virtual–Real Combination Algorithm in PG-CSSI

## 3. Simulation and Discussions

#### 3.1. Feasibility with Different Surface Parameters

^{−8}or lower. The specific test results are shown in Figure 7 and Table 1. The distribution of the test results in each case and the difference between them and the true value are basically the same, so only the result in the case where F/# is the smallest, corresponding to R/D = 1, K = 0, is given in Figure 7. It can be seen from Table 1 that the PV and RMS values of the test results can be regarded as consistent with the true value within the allowable error range. The PV and RMS values of test error are both in the order of 10

^{−5}λ and below.

#### 3.2. Modeling Error Tolerance Analysis

- Each alignment or manufacturing error has an obvious impact only on the low-order aberration of the image plane wavefront;
- The greatest influence on the tilt aberration comes from the lateral translation or tilt error of the SUT;
- The greatest influence on the power aberration comes from the axial translation error of the LP or the SUT;
- Only the axial rotation error of the PG has an obvious impact on astigmatism and coma aberrations, and the order of magnitude is small.

_{ni}(i = 2~8) error in Equation (18) to the test results, we changed Z

_{ni}(i = 2~8) to Z

_{ni}+ ΔZ

_{i}in turn. The results are as follows.

_{i}(i = 2~4) and their point-to-point differences with the true value. Among them, ΔZ

_{2}= 10λ, ΔZ

_{3}= 10λ, ΔZ

_{4}= 1λ is the maximum possible variation that ΔZ

_{i}(i = 2~4) can take in the real experiment according to Table 3, respectively. In these cases, there is a large error in the test results of surface figure error. When ΔZ

_{2}, ΔZ

_{3}, ΔZ

_{4}are reduced to 1λ, 1λ, and 10

^{−2}λ, respectively, the PV and RMS of the test error are, respectively, 1% of the total PV and RMS of the figure error and below. It is worth noticing that power aberrations have a greater impact on test results than tilt aberrations.

_{2}, the same order of magnitudes of ΔZ

_{3}, which is also tilt aberration, introduces a larger error to the test results. Besides, in the case of ΔZ

_{2}and ΔZ

_{4}, the spatial distribution of test errors is irregular but similar for different values of ΔZ

_{i}, but a larger ΔZ

_{3}introduces an obvious high-order aberration to the test results compared with a smaller ΔZ

_{3}, as shown in Figure 9. These phenomena may be because two subapertures, subaperture No. 2 and No. 5 in Figure 5b, are located in the y direction, but no subapertures are located in the x direction. These two subapertures are more sensitive to ΔZ

_{3}, which causes y tilt, than ΔZ

_{2}, which causes x tilt, and have a greater impact on the optimization results when performing SROR.

_{i}(i = 2~4) of the image plane wavefront takes the maximum value in Table 3, great errors will exist in the test results. Further, Table 3 shows that ΔZ

_{2}and ΔZ

_{3}are mainly introduced by the alignment error of the SUT, and ΔZ

_{4}is mainly introduced by LP-Δz and SUT-Δz. On the contrary, if the three aberrations ΔZ

_{i}(i = 2~4) are at the magnitude of the errors introduced by the other components, except for the SUT and LP, test results that are highly consistent with the true value can be obtained.

_{2}and ΔZ

_{3}within 1λ, SUT-Δx and SUT-Δy need to be reduced to at least 0.01 mm. If only LP-Δz and SUT-Δz are considered, in order to keep ΔZ

_{4}within 0.01λ, the two errors must be reduced to at least 5 µm. The actual situation is usually more complicated, considering that the aberrations introduced by the components will couple with each other. It is difficult to realize such a high control accuracy in actual implementation, so in order to reduce the control difficulty and not reduce the accuracy of the measurement, an error correction procedure is suggested to be introduced in the data processing process.

_{i}(i = 5~8). Table 5 shows the figure error test results under different ΔZ

_{i}(i = 5~8) and their point-to-point differences with the true value. Since the ΔZ

_{i}(i = 5~8) values are extremely small at the order of magnitude of 10

^{−4}λ, ideal test results can be obtained in each case. It can be seen from Table 3 that the variations of the 5–8 Zernike aberrations of the image plane wavefront are only caused by PG-θ

_{z}, that is, the rotation angle error of the PG during off-axis subaperture scanning. PG-θ

_{z}is relatively easy to control in real experiments, and the probability of it exceeding the maximum value is very small. Considering the coupling effect of PG-θ

_{z}and other alignment and manufacturing errors on the image plane wavefront, even after increasing ΔZ

_{i}(i = 5~8) by two orders of magnitude, the PV and RMS of the test errors are still on the order of 1% of the total PV and RMS of the figure error, respectively. In summary, it can be inferred that the PG-θ

_{z}has a small probability of significantly reducing the test results in the real experiment.

#### 3.3. Overall Consideration

- The virtual–real combination SROR method proposed in this paper can eliminate the inherent retrace error caused by the non-null structure of the system and obtain high-precision test results, which demonstrated the feasibility of the algorithm.
- The measurement accuracy is not affected by the subaperture overlap ratio, and the typical overlap ratio is about 15% to fully cover the SUT.
- Different alignment and manufacturing errors have different effects on the test results. Supposing that the alignment and manufacturing errors are not considered in the virtual interferometer modeling, it is necessary to precisely align the components, especially LP-Δz and SUT-Δz, which should be reduced to at least 5 µm. Only then can the PV and RMS of the test errors be in the order of 1% of the PV and RMS of the figure error, respectively.

## 4. Preliminary Experiment and Discussions

#### 4.1. System Adjustment

#### 4.2. Subaperture Scanning Experiment and Analysis

^{−1}λ, which were much larger than the error tolerance determined in Section 3.2. Therefore, it was difficult to obtain correct test results through the SROR algorithm.

#### 4.3. Possible Solution

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Optical path diagram of the off-axis subaperture (between the focus O of the TS and the SUT). (

**b**) Schematic diagram of the subaperture layout. O is the focus of the TS. θ is PG’s deflection angle to the chief ray. R is the radius of curvature of the SUT. h is the center distance between the off-axis subaperture and the center subaperture. β is the rotation stepping angle of the PG. D

_{sub}is the diameter of the central subaperture. D

_{full}is the diameter of the SUT.

**Figure 3.**(

**a**) Front-view and (

**b**) top-view of the microstructure of PG. The blue rod-like structures in the figure are LC molecules. d is the thickness of the LC layer. Λ is the space period of PG.

**Figure 6.**The simulated image plane wavefront of subaperture No. 6 of the (

**a**) aspheric surface with R/D = 2.5, K = 2, and (

**b**) spherical surface with R/D = 1.

**Figure 7.**(

**a**) The test result of the figure error of the SUTs with different surface parameters. (

**b**) The difference between the test results and the true value.

**Figure 8.**Variations of the Zernike coefficients describing the image plane wavefront of the subaperture No. 3 in Figure 7b caused by the SUT’s lateral translation error.

**Figure 9.**(

**a**,

**c**) Figure error test results under different ΔZ

_{3}and (

**b**,

**d**) the test errors. ΔZ

_{3}is the error of optimization object Z

_{3}.

**Figure 10.**Experimental setup of PG-CSSI. LP is the linear polarizer. PG is the polarization grating. QWP is the λ/4 wave plate. SUT is the surface under test.

**Figure 12.**Interferograms captured during the process of system adjustment. The interferograms have been enhanced for better visual effects. (

**a**) Stray fringes caused by scanning structure. (

**b**) Dense interferogram with aliasing artifacts captured by the detector when the scanning structure was not perpendicular to the optical axis. (

**c**) Sparse interferogram after adjusting the SUT.

**Figure 14.**Wavefront maps of the off-axis subapertures in the (

**a**) real interferometer and (

**b**) virtual interferometer.

R/D | K | F/# | $\mathsf{\Lambda}$(μm) | Overlap Ratio | Result | Error | ||
---|---|---|---|---|---|---|---|---|

PV (λ) | RMS (λ) | PV (×10^{−5} λ) | RMS (×10^{−6} λ) | |||||

— ^{a} | — | — | — | — | 2.4083 | 0.5829 | 0 | 0 |

1 | 0 | 1.8 | 1.6 | 14.48% | 2.4083 | 0.5829 | 1.4846 | 2.0356 |

2 | 3.6 | 3.125 | 15.41% | 1.4784 | 2.0358 | |||

3 | 5.3 | 4.5 | 14.44% | 1.4263 | 2.0486 | |||

4 | 7.1 | 6.25 | 16.91% | 1.6270 | 2.0037 | |||

2.5 | −2 | 5.3 | 4.5 | 14.44% | 1.4801 | 2.0396 | ||

−1.5 | 1.4802 | 2.0406 | ||||||

−1 | 1.4807 | 2.0394 | ||||||

−0.5 | 1.4805 | 2.0400 | ||||||

0.5 | 1.4729 | 2.0300 | ||||||

1 | 1.4794 | 2.0400 | ||||||

1.5 | 1.4789 | 2.0397 | ||||||

2 | 1.4857 | 2.1122 |

^{a}This line is the true value of the figure error.

Error and Its Symbol | Name of the Element | |||
---|---|---|---|---|

LP | PG | QWP | SUT | |

Axial translation error Δz | 0.5 mm | 0 | 0 | 0.5 mm |

Lateral translation error Δx, Δy | 0 | 0 | 0 | 0.1 mm |

Tilt error θ_{x}, θ_{y} | 0 | 0.0225° | 0.0225° | 0.062° |

Axial rotation error θ_{z} | 0 | ±1° | 0 | 0 |

Parallelism error δ_{x}, δ_{y} | 0.00139° | 0.00139° | 0.00139° | 0 |

**Table 3.**The order of magnitudes (λ) of the variations of the Zernike coefficients describing the off-axis subaperture image plane wavefront caused by each alignment error and manufacturing error.

Element-Error | ΔZ_{2}(Tilt) | ΔZ_{3}(Tilt) | ΔZ_{4}(Power) | ΔZ_{5}(Astigmatism) | ΔZ_{6}(Astigmatism) | ΔZ_{7}(Coma) | ΔZ_{8}(Coma) |
---|---|---|---|---|---|---|---|

LP-Δz | — * | — | 1 | — | — | — | — |

LP-δ_{x} | — | 10^{−2} | — | — | — | — | — |

LP-δ_{y} | 10^{−2} | — | — | — | — | — | — |

PG-θ_{x} | 10^{−1} | 10^{−1} | — | — | — | — | — |

PG-θ_{y} | 10^{−1} | 10^{−1} | — | — | — | — | — |

PG-θ_{z} | 10^{−4} | 10^{−4} | — | 10^{−4} | 10^{−4} | 10^{−4} | 10^{−4} |

PG-δ_{x} | 10^{−2} | 10^{−2} | — | — | — | — | — |

PG-δ_{y} | 10^{−2} | 10^{−2} | — | — | — | — | — |

QWP-θ_{x} | 10^{−2} | 10^{−2} | — | — | — | — | — |

QWP-θ_{y} | 10^{−2} | 10^{−2} | — | — | — | — | — |

QWP-δ_{x} | 10^{−3} | 10^{−3} | — | — | — | — | — |

QWP-δ_{y} | 10^{−3} | 10^{−3} | — | — | — | — | — |

SUT-Δz | 1 | 1 | 1 | — | — | — | — |

SUT-Δx | 10 | — | — | — | — | — | — |

SUT-Δy | — | 10 | — | — | — | — | — |

SUT-θ_{x} | — | 10 | — | — | — | — | — |

SUT-θ_{y} | 10 | — | — | — | — | — | — |

ΔZ_{i} (λ) | Test Result | Test Error | |||
---|---|---|---|---|---|

PV (λ) | RMS (λ) | PV (λ) | RMS (λ) | ||

0 (True value) | 2.4083 | 0.5829 | 0 | 0 | |

ΔZ_{2} | 10 | 2.5115 | 0.5888 | 0.2360 | 0.0476 |

1 | 2.4183 | 0.5840 | 0.0239 | 0.0047 | |

ΔZ_{3} | 10 | 2.6009 | 0.5077 | 0.7036 | 0.1265 |

1 | 2.3953 | 0.5812 | 0.0432 | 0.0070 | |

ΔZ_{4} | 1 | 3.5673 | 0.7145 | 1.2783 | 0.1820 |

10^{−2} | 2.4190 | 0.5841 | 0.0119 | 0.0018 |

ΔZ_{i} (λ) | Test Result | Test Error | |||
---|---|---|---|---|---|

PV (λ) | RMS (λ) | PV (λ) | RMS (λ) | ||

0 (True value) | 2.4083 | 0.5829 | 0 | 0 | |

10^{−4} | ΔZ_{5} | 2.4082 | 0.5829 | 6.4749 × 10^{−4} | 7.4275 × 10^{−5} |

ΔZ_{6} | 2.4085 | 0.5829 | 5.0569 × 10^{−4} | 6.7166 × 10^{−5} | |

ΔZ_{7} | 2.4083 | 0.5829 | 9.9981 × 10^{−4} | 1.7070 × 10^{−5} | |

ΔZ_{8} | 2.4082 | 0.5829 | 1.1147 × 10^{−4} | 1.9517 × 10^{−5} | |

10^{−2} | ΔZ_{5} | 2.3980 | 0.5837 | 0.0649 | 0.0074 |

ΔZ_{6} | 2.4275 | 0.5826 | 0.0495 | 0.0066 | |

ΔZ_{7} | 2.4101 | 0.5831 | 0.0097 | 0.0017 | |

ΔZ_{8} | 2.4051 | 0.5830 | 0.0106 | 0.0019 |

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**MDPI and ACS Style**

Hu, Y.; Wang, Z.; Hao, Q. Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer. *Sensors* **2022**, *22*, 9129.
https://doi.org/10.3390/s22239129

**AMA Style**

Hu Y, Wang Z, Hao Q. Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer. *Sensors*. 2022; 22(23):9129.
https://doi.org/10.3390/s22239129

**Chicago/Turabian Style**

Hu, Yao, Zhen Wang, and Qun Hao. 2022. "Circular Subaperture Stitching Interferometry Based on Polarization Grating and Virtual–Real Combination Interferometer" *Sensors* 22, no. 23: 9129.
https://doi.org/10.3390/s22239129