# Optical Design for Aberration Correction of Ultra-Wide Spectral Range Echelle Spectrometer

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## Abstract

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## 1. Introduction

- (1)
- Spherical aberration: the field of view of the echelle spectrometer is generally small, and the influence of spherical aberrations can be reduced through optical design. There are two common solutions: first, according to the Rayleigh criterion, if the system F-number is defined as ${F}^{\#}={f}^{\prime}\u2215D$ and the relationship between the system focal length ${f}^{\prime}$ and the F-number ${F}^{\#}$ satisfies ${f}^{\prime}\le 256\lambda {\left({F}^{\#}\right)}^{4}$, the spherical aberration has little influence [14]; second is the use of off-axis parabolic mirrors, which can focus collimated light without introducing spherical aberrations [15] at the expense of a worse coma for off-axis beams.
- (2)
- Coma: the most common method for eliminating coma in C–T spectrometers is the combination of collimating and focusing mirrors with the same curvature radius and opposite off-axis angles proposed by inventors Czerny and Turner. Later, the asymmetric structure was developed, which can also eliminate the coma of a specific wavelength when certain conditions are met [16]. However, these methods are limited to a single wavelength and the problem can only be partially offset in the rest of the wavelength range by the collimating and focusing mirrors with the opposite sign of the coma.
- (3)
- Astigmatism: to eliminate the astigmatism of the C–T grating spectrometer, it is necessary to add additional elements such as a lens [17], e.g., an aspherical or free-form cylindrical lens [18], and toroidal mirrors [19]. In addition, Wood and Lawler devised a method of rotating auxiliary dispersive elements [20]. However, the aforementioned methods impose high demands on optical processing and assembly, resulting in high costs and long cycles, which are not conducive to product industrialization.

## 2. Optical Layout of the Echelle Spectrometer

## 3. Theory and Design Method

#### 3.1. Collimating Mirror

#### 3.2. Detector

#### 3.3. Echelle Grating

#### 3.3.1. Principles of Design or Selection

_{0}is the blaze angle, β’ is the diffraction angle, and γ is the off-axis angle. In particular, for the central wavelength λ

_{cen},

#### 3.3.2. Matching of Echelle Gratings and Detectors

_{max}, regardless of the inclination and nonlinearity of the stripes, the largest free spectral region corresponding to the lowest order covering a single row (column) is used for calculation. According to Equation (1), the largest free spectral region should satisfy the following:

#### 3.3.3. Comparison of Dispersion Capability of Different Echelle Gratings

#### 3.4. Dispersive Prism

_{i}and L

_{i}are the Sellmeier coefficients that can be obtained by fitting the measured data using the least squares method.

_{2}, and LiF. At 20 °C and atmospheric pressure, the Sellmeier coefficients and wavelength–refractive index curves of the four materials are presented in Table 2 and Figure 3a, respectively. As shown in Figure 3a, the optical glass materials, such as BK7, are limited in the UV band. As a result, the requirements for a spectral range of 200–300 nm cannot be achieved. The refractive index changes of F-Silica, CaF

_{2}, and LiF at 200–1100 nm are 0.1014, 0.0669, and 0.0503, respectively. It is obvious that the refractive index span of fused silica material is larger in this spectral range.

_{2}, and BK7 can be estimated based on Equation (9). Figure 3b shows the three-dimensional relationships between the angular dispersion, wavelength, and prism apex angle of these four materials. It is shown that angular dispersion increases with the apex angle of the prism at a constant wavelength for each of the materials. However, as the wavelength increases, the angular dispersion of the prism declines and tends to be uniform for a constant apex angle. This result suggests that the long-wave region features higher requirements for tolerance and image recognition. Comparing the angular dispersion of the four materials, it is worth noting that BK7 shows the highest angular dispersion at a constant wavelength in the spectral range of 320–1100 nm, followed by F-Silica. In fact, F-Silica shows the highest angular dispersion for the full spectral range of 200–1100 nm. Furthermore, given its lower cost and small thermal expansion coefficient, F-Silica is the best choice for auxiliary dispersion components within the spectral range of 200–1100 nm.

#### 3.5. Focusing Lens

#### 3.5.1. Determination of Focal Length

_{coll}and f

_{foc}are the focal lengths of the collimating mirror and focusing mirror, respectively.

_{foc}can be calculated as:

#### 3.5.2. Determination of Field of View

_{cen-max}corresponding to the lowest order m

_{min}is determined by Equation (4) and can be expressed by the following equation:

#### 3.5.3. Determination of Aperture Size

## 4. Results and Discussion

#### 4.1. Verification of Spectral Resolution

#### 4.2. Evaluation of Image Quality and Spectral Resolution

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Optical path through the designed spectrometer. (

**a**) Overall optical path of the spectrometer, (

**b**) UV channel (channel 1), (

**c**) visible channel (channel 2), and (

**d**) NIR channel (channel 3). 1: slit; 2: off-axis parabolic collimating mirror; 3, 6, and 10: folding mirror; 4 and 5: beam splitter; 7, 11, and 14: echelle grating; 8, 12, and 15: dispersive prism; 9, 13, and 16: camera.

**Figure 2.**Wavelength–diffraction angle–angular dispersion relationship of the echelle gratings. (

**a**) Schematic of the three channels, (

**b**) channel 3, (

**c**) channel 2, and (

**d**) channel 1. dβ/dλ is the angular dispersion, β’ is the diffraction angle, m is the order.

**Figure 3.**(

**a**) Index of refraction versus wavelength. (

**b**) Angular dispersion corresponding to different wavelengths and prism apex angles.

**Figure 5.**(

**a**) Free spectrum of 14 orders on the 640 × 512 detector of channel 3. (

**b**,

**c**) Spot diagrams of the upper limit wavelength of 1100 nm and its vicinity. (

**d**,

**e**) Spot diagrams of the lower limit wavelength of 800 nm and its vicinity.

**Figure 8.**Spot diagrams of eight wavelength points on (

**a**) channel 1, (

**b**) channel 2, and (

**c**) channel 3.

**Figure 9.**The curve of each spot size in Figure 8.

Parameter | Values |
---|---|

Detection spectrum | 200–1100 nm |

Spectral resolution | <0.1 nm (entire spectrum) |

Slit size | 25 × 25 μm |

Collimator focal length | 200 mm |

Collimator aperture | 25 mm |

Material | K_{1} | L_{1} | K_{2} | L_{2} | K_{3} | L_{3} |
---|---|---|---|---|---|---|

F-Silica | 0.6837 | 0.0046 | 0.4203 | 0.0134 | 0.5850 | 64.4933 |

CaF_{2} | 0.5676 | 0.00253 | 0.4711 | 0.01008 | 3.848 | 1200.56 |

LiF | 0.9255 | 0.00544 | 6.9675 | 1075.2 | 0 | 0 |

BK7 | 1.040 | 0.006 | 0.2318 | 0.02002 | 1.0105 | 103.56 |

Parameter | Channel 1 | Channel 2 | Channel 3 | |
---|---|---|---|---|

Detection spectrum (nm) | 200–500 | 450–850 | 800–1100 | |

Groove spacing (lines/mm) | 54.5 | 79 | 42 | |

Blazing angle (°) | 46 | 63.43 | 54.74 | |

Diffraction order | 52–130 | 26–50 | 35–48 | |

Off-axis angle (°) | 10 | 10 | 10 | |

Focusing lens | Focal Length (mm) | $103\le {f}_{foc1}\le 170$ | $91\le {f}_{foc2}\le 170$ | $116\le {f}_{foc3}\le 120$ |

Field of view (°) | 2.3 | 8.9 | 4.6 | |

Aperture size (mm) | 34.8 | 66.1 | 43.0 |

Channel | m | λ (nm) | R | $\overline{\mathit{R}}$ | Δλ (nm) |
---|---|---|---|---|---|

Channel 1 | 130 | 200 | 180,299 | 16,319 | 0.012 |

74 | 350 | 102,632 | 16,256 | 0.022 | |

52 | 500 | 72,120 | 16,319 | 0.031 | |

Channel 2 | 50 | 450 | 156,110 | 31,791 | 0.014 |

34 | 650 | 106,155 | 31,226 | 0.021 | |

26 | 850 | 81,177 | 31,226 | 0.027 | |

Channel 3 | 48 | 800 | 61,734 | 22,350 | 0.036 |

41 | 950 | 52,731 | 22,670 | 0.042 | |

35 | 1100 | 45,014 | 22,408 | 0.049 |

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**MDPI and ACS Style**

Wang, Y.; Qu, Y.; Zhao, H.; Fan, X. Optical Design for Aberration Correction of Ultra-Wide Spectral Range Echelle Spectrometer. *Photonics* **2022**, *9*, 841.
https://doi.org/10.3390/photonics9110841

**AMA Style**

Wang Y, Qu Y, Zhao H, Fan X. Optical Design for Aberration Correction of Ultra-Wide Spectral Range Echelle Spectrometer. *Photonics*. 2022; 9(11):841.
https://doi.org/10.3390/photonics9110841

**Chicago/Turabian Style**

Wang, Yuming, Youshan Qu, Hui Zhao, and Xuewu Fan. 2022. "Optical Design for Aberration Correction of Ultra-Wide Spectral Range Echelle Spectrometer" *Photonics* 9, no. 11: 841.
https://doi.org/10.3390/photonics9110841