Numerical calculation of the mosaic error between mosaic gratings

Guojun Yang; Guojun Yang; Xiangdong Qi; Xiaotao Mi; Shanwen Zhang; Hongzhu Yu; Haili Yu; Xiaotian Li; Shuo Yang

doi:10.1364/AO.389274

Applied Optics
Vol. 59,
Issue 13,
pp. 4048-4054
(2020)
•https://doi.org/10.1364/AO.389274

Numerical calculation of the mosaic error between mosaic gratings

Guojun Yang, Xiangdong Qi, Xiaotao Mi, Shanwen Zhang, Hongzhu Yu, Haili Yu, Xiaotian Li, and Shuo Yang

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Guojun Yang, Xiangdong Qi, Xiaotao Mi, Shanwen Zhang, Hongzhu Yu, Haili Yu, Xiaotian Li, and Shuo Yang, "Numerical calculation of the mosaic error between mosaic gratings," Appl. Opt. 59, 4048-4054 (2020)

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Abstract

The theoretical calculation model for a mosaic error was established based on the plane equation for a grating surface and the relationship equation for a mosaic grating surface. A mosaic grating was obtained based on this model. In the experiment, the mosaic error was calculated based on the diffraction wavefronts of two groups of mosaic gratings that were obtained simultaneously with a Zygo interferometer. The difference between the wavefront of the mosaic grating and the average wavefront of the mosaic grating element was $0 . 031 \lambda$. The maximum far-field intensity of the mosaic grating was 90% of that without an error. This model provides a theoretical basis for the numerical mosaic between gratings. In addition, the mosaic error can be calculated with this model, and the quality of the mosaic grating can be evaluated.

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Components	Parameters	Performance Index
Zygo interferometer	Wavelength	632.8 nm
	Beam diameter	100 mm
	Type	GPI XP/D
Mirror	Diameter	100 mm
	Reflectivity	$> 90 %$
	PV	$0.05 λ (λ = 632.8 n m)$
Prism	Material	Quartz
	Pass light area	$50 m m \times 50 m m$
	Wedge angle	$14^{\circ}$
Mosaic grating element	Area	$35 m m \times 35 m m$
	Groove density	79 line/mm
	Blaze angle	$64^{\circ}$
Mosaic device	Rotation accuracy	0.3 µrad
Mosaic device	Translation accuracy	1 nm

Mosaic Errors	$Δ θ_{x}$	$Δ θ_{y}$	$Δ θ_{z}$	$Δ x$	$Δ z$
Step (nm)	30	30	30	1	1
Rotation radius (mm)	100	70	65	—	—
Trip	12.7 mm	12.7 mm	12.7 mm	250 µm	250 µm
Step angle (µrad)	0.300	0.429	0.462	—	—
Step number	$Δ θ_{x} / 0.3$	$Δ θ_{y} / 0.429$	$Δ θ_{z} / 0.462$	$Δ x / 1$	$Δ z / 1$

First Incident Angle		Second Incident Angle
Wavefront1	Wavefront2	Wavefront1	Wavefront2
$a_{11} = - 4.9411 \times 10^{- 5}$	$a_{12} = - 1.4919 \times 10^{- 5}$	$a_{21} = - 8.7551 \times 10^{- 5}$	$a_{22} = - 4.4425 \times 10^{- 5}$
$b_{11} = - 8.0342 \times 10^{- 6}$	$b_{12} = - 6.4541 \times 10^{- 6}$	$b_{21} = 3.4163 \times 10^{- 7}$	$b_{22} = 3.0188 \times 10^{- 6}$
$c_{11} = 6.7847 \times 10^{- 8}$	$c_{12} = 2.4079 \times 10^{- 8}$	$c_{21} = 1.0109 \times 10^{- 7}$	$c_{22} = 4.1856 \times 10^{- 8}$

First Incident Angle		Second Incident Angle
Wavefront1	Wavefront2	Wavefront1	Wavefront2
$a_{11} = 1.6708 \times 10^{- 5}$	$a_{12} = 1.6756 \times 10^{- - 5}$	$a_{21} = 1.6386 \times 10^{- 5}$	$a_{22} = 1.6250 \times 10^{- 5}$
$b_{11} = - 2.7606 \times 10^{- 7}$	$b_{12} = - 1.1491 \times 10^{- 6}$	$b_{21} = - 1.6521 \times 10^{- 6}$	$b_{22} = - 2.5050 \times 10^{- 6}$
$c_{11} = - 1.6194 \times 10^{- 8}$	$c_{12} = - 1.5222 \times 10^{- 8}$	$c_{21} = - 1.7349 \times 10^{- 8}$	$c_{22} = - 1.5539 \times 10^{- 8}$

Components	Parameters	Performance Index
Zygo interferometer	Wavelength	632.8 nm
	Beam diameter	100 mm
	Type	GPI XP/D
Mirror	Diameter	100 mm
	Reflectivity	$> 90 %$
	PV	$0.05 λ (λ = 632.8 n m)$
Prism	Material	Quartz
	Pass light area	$50 m m \times 50 m m$
	Wedge angle	$14^{\circ}$
Mosaic grating element	Area	$35 m m \times 35 m m$
	Groove density	79 line/mm
	Blaze angle	$64^{\circ}$
Mosaic device	Rotation accuracy	0.3 µrad
Mosaic device	Translation accuracy	1 nm

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Applied Optics