# Optical Design of a Slitless Astronomical Spectrograph with a Composite Holographic Grism

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## Abstract

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## 1. Introduction

## 2. Optical Design

- Classical grating. This is a default option, which can be produced by most manufacturers or even found as an off-the-shelf component. It represents a grating with straight equidistant fringes having a profile perpendicular to the substrate surface. Such a grating can be recorded in a setup shown in Figure 2a, where the laser beam forms two sources 1 and the beams are collimated by two off-axis parabolic (OAP) mirrors 2, which interfere in a symmetric geometry on the substrate 3. The holographic layer thickness is equal to a standard value and the modulation depth is optimized for a chief ray using scalar diffraction theory.
- Composite grating. This version represents all the capacities of the concept shown in [11]. The recording beams in this setup Figure 2b are formed by the same sources 1 and OAP’s 2, but the angles of incidence onto the substrate 3 are not symmetric. A deformable mirror (DM) 4 is introduced in one of the interferometer’s branches to control the wavefront and introduce the aberration correction. The mirror surface shape is described by the Zernike polynomials [15] up to the 2nd order (we use only the YZ-symmetric terms):$$z=\frac{{r}^{2}/R}{1+\sqrt{1-(1+k){r}^{2}/{R}^{2}}}+\sum _{i=4}^{25}{A}_{i}{Z}_{i}(\rho ,\varphi );$$
- Simplified composite grating. Producing a recording setup as described in the previous item, with all its degrees of freedom, may be very challenging. Therefore, we consider a simplified version. In this setup, the aberrated wavefront is formed by a tilted corrector plate 4 with an ordinary axisymmetric asphere on its first surface:$$z=\frac{{r}^{2}/R}{1+\sqrt{1-(1+k){r}^{2}/{R}^{2}}}+{\alpha}_{2}{r}^{2}+{\alpha}_{4}{r}^{4},$$

## 3. Image Quality Optimization and Analysis

- a rectangular function of the given width$$rect\left({y}^{\prime}\right)=\left\{\begin{array}{c}1,-{b}_{1}^{\prime}/2\le {y}^{\prime}\le {b}_{1}^{\prime}/2,\hfill \\ 0,|{y}^{\prime}|\ge {b}_{1}^{\prime}/2.\hfill \end{array}\right.$$
- a Gaussian distribution with the given width at level of 0.1$$\sigma =\sqrt{\frac{-0.5}{ln0.1\prime}}$$$$G\left({y}^{\prime}\right)={e}^{-0.5{y}^{\prime 2}/{\sigma}^{2}}.$$

- Most of the values are feasible to accomplish in practice. It appears that the classical grism is sensitive to decenters and tilts. This can be explained by the large uncompensated aberrations, which grow rapidly when deviating from the found best-fit position. However, the requirements for the individual surfaces’ tilts can be met by defining proper specifications in the drawings of corresponding elements. An angular precision at the level of ≈1${}^{\prime}$ is definitely possible for both the prism and grating.
- The composite grism is more sensitive to the prism refractive properties, but the absolute values are high enough to reach the required precision.
- The simplified version of the composite hologram recording setup does not have wider tolerances for the auxiliary components’ parameters and alignment. The reason for this is that the quality of both surfaces, their position, and material refraction index contribute to the recording of wavefront aberrations.
- The tightest tolerance for a composite gratings’ recording setup corresponds to the corrector plate surface irregularity. However, it should still be technologically feasible.

## 4. Diffraction Efficiency Optimization and Analysis

## 5. Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**General view of the slitless spectrograph optics coupled with a Dall–Kirkham telescope: 1—primary mirror, 2—secondary mirror, 3—lens corrector, 4—focal plane position in the imaging mode, 5—composite holographic grating, split into zones with independently optimized parameters as A and B, 6—prism, 7—spectral image plane position.

**Figure 2.**Holographic grating recording setups: (

**a**)—classical grating, (

**b**)—composite grating, (

**c**)—simplified composite grating. The setup elements are: 1—laser point sources, 2—collimating mirrors, 3—hologram substrate, 4—auxiliary optics for the wavefront control; A, B, C and D—composite hologram zones.

**Figure 7.**Instrument functions of the slitless spectrograph for a 3.5${}^{\u2033}$ object in the FoV center.

**Figure 10.**Spatial distribution of the DE across the beam footprint for the simplified composite grism.

**Figure 12.**Opto-mechanical design of the slitless spectrograph unit: 1—grism, 2—grism mount, 3—connecting tube, 4—camera flange, 5—telescope mechanical interface, 6—telescope filter wheel, 7—telescope camera.

Zone | Composite | Simplified Composite | ||
---|---|---|---|---|

${z}_{def}^{\prime}$, mm | $\delta {z}_{PTV}^{\prime}$, $\mathsf{\mu}$m | ${z}_{def}^{\prime}$, mm | $\delta {z}_{PTV}^{\prime}$, $\mathsf{\mu}$m | |

A | −2168.9 | 13.7 | −3844.8 | 2.67 |

B | −3575.9 | 24.5 | −3744.4 | 2.77 |

C | −1074.3 | 22.4 | - | - |

D | −2984.8 | 20.9 | - | - |

Metric | Classical | Composite | Simplified Composite | ||||||
---|---|---|---|---|---|---|---|---|---|

Wavelength, nm | 450 | 700 | 950 | 450 | 700 | 950 | 450 | 700 | 950 |

FWHM Rect., $\mathsf{\mu}$m | 73.4 | 62.8 | 208.8 | 61.3 | 62 | 60.8 | 66.1 | 63.7 | 122.4 |

$\delta \lambda $ Rect. | 1.28 | 1.10 | 3.64 | 1.06 | 0.07 | 1.05 | 1.15 | 1.11 | 2.13 |

R Rect., nm | 352 | 639 | 260 | 425 | 654 | 905 | 392 | 632 | 447 |

FWHM Gauss., $\mathsf{\mu}$m | 61.8 | 41.4 | 172.8 | 35.9 | 34.8 | 39.6 | 49.6 | 37.8 | 118.8 |

$\delta \lambda $ Gauss. | 1.08 | 0.72 | 3.02 | 0.62 | 0.60 | 0.68 | 0.86 | 0.66 | 2.06 |

R Gauss., nm | 417 | 967 | 315 | 726 | 1164 | 1389 | 523 | 1067 | 460 |

$\delta {y}_{best}^{\prime}$, $\mathsf{\mu}$m | 226 | 35 | 223 | 147 | 58 | 95 | 138 | 40 | 128 |

$\delta {x}_{best}^{\prime}$, $\mathsf{\mu}$m | 309 | 425 | 441 | 74 | 5 | 15 | 61 | 7 | 22 |

$\delta {y}_{worst}^{\prime}$, $\mathsf{\mu}$m | 257 | 69 | 259 | 156 | 65 | 114 | 177 | 77 | 170 |

$\delta {x}_{worst}^{\prime}$, $\mathsf{\mu}$m | 340 | 457 | 490 | 85 | 9 | 41 | 95 | 27 | 44 |

Parameter | Classical | Comp. | Simpl. Comp. |
---|---|---|---|

${R}_{sub},fr.$ | 8 | 8 | 8 |

${R}_{grat},fr.$ | 7 | 8 | 8 |

${R}_{prism},fr.$ | 8 | 8 | 8 |

${t}_{air}$, mm | 0.68 | 1 | 1 |

${t}_{grat}$, mm | 0.22 | 0.5 | 0.5 |

${t}_{prism}$, mm | 0.20 | 0.5 | 0.5 |

$d{y}_{grism}$, mm | 0.06 | 0.2 | 0.2 |

${\alpha}_{x,grism}{,}^{\circ}$ | 0.6 | 1 | 1 |

${\alpha}_{y,grism}{,}^{\circ}$ | 1 | 1 | 1 |

${\alpha}_{x,grat}{,}^{\circ}$ | 0.013 | 0.28 | 0.27 |

${\alpha}_{y,grat}{,}^{\circ}$ | 0.26 | 0.5 | 0.5 |

${\alpha}_{x,prism}{,}^{\circ}$ | 0.023 | 0.1 | 0.1 |

${\alpha}_{y,prism}{,}^{\circ}$ | 0.94 | 1 | 1 |

$\Delta {N}_{grat},fr.$ | 0.52 | 1 | 1 |

$\Delta {N}_{prism},fr.$ | 0.52 | 0.9 | 0.9 |

${n}_{prism}\times {10}^{-3}$ | 2.22 | 1.82 | 1.76 |

${\nu}_{prism}$ | 10 | 6.1 | 5.9 |

${z}_{corr}$, mm | - | 2 | 2 |

${\alpha}_{x,corr}{,}^{\circ}$ | - | 1 | 0.3 |

${t}_{corr}$, mm | - | - | 0.5 |

$\Delta {N}_{corr},fr.$ | - | 5.7 | 1.6 |

${n}_{corr}\times {10}^{-3}$ | - | - | 6.8 |

${i}_{rec}{,}^{\circ}$ | 0.203 | 0.145 | 0.141 |

${f}_{rec}$, mm | - | 0.79 | 0.77 |

Zone | Classical | Composite | Simpl. Composite | |||
---|---|---|---|---|---|---|

t, $\mathsf{\mu}$m | $\delta n$ | t, $\mathsf{\mu}$m | $\delta n$ | t, $\mathsf{\mu}$m | $\delta n$ | |

A | 20.0 | 0.012 | 18.5 | 0.022 | 20.0 | 0.019 |

B | - | - | 17.5 | 0.023 | 20.0 | 0.022 |

C | - | - | 22.9 | 0.018 | - | - |

D | - | - | 20.0 | 0.028 | - | - |

Field | Classical | Composite | Simpl. Composite | |||
---|---|---|---|---|---|---|

max | av | max | av | max | av | |

$-17.{8}^{\prime},-3.{7}^{\prime}$ | 0.690 | 0.303 | 0.843 | 0.383 | 0.846 | 0.394 |

${0}^{\prime},{0}^{\prime}$ | 0.690 | 0.303 | 0.881 | 0.386 | 0.853 | 0.396 |

$17.{8}^{\prime},3.{7}^{\prime}$ | 0.690 | 0.303 | 0.878 | 0.387 | 0.845 | 0.386 |

Parameter | Classical | Comp. | Simpl. Comp. |
---|---|---|---|

${i}_{rec}$, deg | 0.082 | 0.046 | 0.067 |

t, $\mathsf{\mu}$m | 1.6 | 0.9 | 1.3 |

$\delta n\times {10}^{-4}$ | 6.3 | 3.5 | 5.1 |

Instrument | Comp. Grism | FOCAS | EFOCS | MAO | AFA | Galex-NUV |
---|---|---|---|---|---|---|

Wavelengths | 450–950 nm | 365–900 nm | 350–1000 nm | 400–1000 nm | 350–700 nm | 177–283 nm |

Spectral resolving power | R425–905 | R250–2000 | R250–7140 | R100 | R103–147 | R90 |

Throughput | 18–51% | 57–82% | 18–65% | 25–75% | 25–75% | 10–82% |

FoV | 35.6${}^{\prime}$ × 7.2${}^{\prime}$ | 6${}^{\prime}$ × 6${}^{\prime}$ | 3.5${}^{\prime}$ × 5.7${}^{\prime}$ | 10.8${}^{\prime}$ × 10.8${}^{\prime}$ | 14${}^{\prime}$ × 14${}^{\prime}$ | ⌀1.24${}^{\circ}$ |

Number of elements | 1 | 15 | 7 | 1 | 1 | 2 |

Hosting telescope | CDK500 | Subaru | ESO | Zeiss-600 | DFM | Galex |

0.5 m ground | 8.2 m ground | 3.6 m ground | 0.6 m ground | 0.4 m ground | 0.5 m space |

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## Share and Cite

**MDPI and ACS Style**

Muslimov, E.; Akhmetov, D.; Kharitonov, D.; Ibatullin, E.; Pavlycheva, N.; Sasyuk, V.; Golovkin, S. Optical Design of a Slitless Astronomical Spectrograph with a Composite Holographic Grism. *Photonics* **2023**, *10*, 385.
https://doi.org/10.3390/photonics10040385

**AMA Style**

Muslimov E, Akhmetov D, Kharitonov D, Ibatullin E, Pavlycheva N, Sasyuk V, Golovkin S. Optical Design of a Slitless Astronomical Spectrograph with a Composite Holographic Grism. *Photonics*. 2023; 10(4):385.
https://doi.org/10.3390/photonics10040385

**Chicago/Turabian Style**

Muslimov, Eduard, Damir Akhmetov, Danila Kharitonov, Erik Ibatullin, Nadezhda Pavlycheva, Vyacheslav Sasyuk, and Sergey Golovkin. 2023. "Optical Design of a Slitless Astronomical Spectrograph with a Composite Holographic Grism" *Photonics* 10, no. 4: 385.
https://doi.org/10.3390/photonics10040385