Diffractive optics calibrator: measurement of etching variations for binary computer-generated holograms

Wenrui Cai; Ping Zhou; Chunyu Zhao; James H. Burge

doi:10.1364/AO.53.002477

Applied Optics
Vol. 53,
Issue 11,
pp. 2477-2486
(2014)
•https://doi.org/10.1364/AO.53.002477

Diffractive optics calibrator: measurement of etching variations for binary computer-generated holograms

Wenrui Cai, Ping Zhou, Chunyu Zhao, and James H. Burge

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Wenrui Cai, Ping Zhou, Chunyu Zhao, and James H. Burge, "Diffractive optics calibrator: measurement of etching variations for binary computer-generated holograms," Appl. Opt. 53, 2477-2486 (2014)

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Abstract

We present a new device, the diffractive optics calibrator (DOC), for measuring etching variations of computer-generated holograms (CGHs). The intensity distribution of the far-field diffraction pattern is captured and fitted to a parametric model to obtain local etching parameters such as the duty cycle, etching depth, and grating period. The sensitivity of each etching parameter is analyzed, and design choices are provided. For the wavefront created by the CGH, the DOC is capable of measuring variations in these parameters that cause 1 nm peak-to-valley phase errors. System performance is verified by measurements from a phase shift Fizeau interferometer. This device will be used primarily for quality control of the CGHs. The measurement results can be used to evaluate the fabrication performance and guide future design. DOC is also capable of generating an induced phase error map for calibration. Such calibration is essential for measuring free-form aspheric surfaces with 1 nm root-mean-square accuracy.

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	Zero Order ( $m = 0$ )	Nonzero Order ( $m = \pm 1, \pm 2, \dots$ )
Diffracted wavefront
$η$ diffraction efficiency	$A_{0}^{2} {(1 - D)}^{2} + A_{1}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ$	$(A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D^{2} {sinc}^{2} (m D) {sinc}^{2} (m P)$
$\tan Ψ$ ( $Ψ$ , wavefront phase)	$\frac{A_{1} D \sin ϕ}{A_{0} (1 - D) + A_{1} D \cos ϕ}$	$\frac{A_{1} \sin ϕ}{- A_{0} + A_{1} \cos ϕ}$
Sensitivity functions
$\partial η / \partial D$	$- 2 A_{0}^{2} (1 - D) + 2 A_{1}^{2} D + 2 A_{0} A_{1} (1 - 2 D) \cos ϕ$	$2 (A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D sinc (2 m D) {sinc}^{2} (m P)$
$\partial η / \partial ϕ$	$- 2 A_{0} A_{1} (1 - D) \sin ϕ$	$2 A_{0} A_{1} \sin ϕ D^{2} {sinc}^{2} (m D) {sinc}^{2} (m P)$
$\partial η / \partial P$	0	$(A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D^{2} {sinc}^{2} (m D) \cdot \frac{2}{P} [sinc (2 m P) - {sinc}^{2} (m P)]$
$\partial Ψ / \partial D$	$\frac{A_{0} A_{1} \sin ϕ}{A_{1}^{2} D^{2} + A_{0}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ}$	0
$\partial Ψ / \partial ϕ$	$\frac{A_{1}^{2} D^{2} + A_{0} A_{1} D (1 - D) \cos ϕ}{A_{1}^{2} D^{2} + A_{0}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ}$	$\frac{A_{1}^{2} - A_{0} A_{1} \cos ϕ}{A_{1}^{2} + A_{0}^{2} - 2 A_{0} A_{1} \cos ϕ}$
$\partial Ψ / \partial P$	0	0

Duty Cycle Range	41% to 53% (1% change/grating)
Etching depth	0.5 μm
Grating period	15.0 μm
Grating size	$3 mm \times 20 mm$
Overall pattern size	$39 mm \times 20 mm$
Number of sets	3

	Average	P-V	RMS
Duty cycle (%)	48.00	3.60	0.40
Etching depth (nm)	511.8	19.8	3.1
Fringe spacing (μm)	8.2	15.8	2.6

	Zero Order ( $m = 0$ )	Nonzero Order ( $m = \pm 1, \pm 2, \dots$ )
Diffracted wavefront
$η$ diffraction efficiency	$A_{0}^{2} {(1 - D)}^{2} + A_{1}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ$	$(A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D^{2} {sinc}^{2} (m D) {sinc}^{2} (m P)$
$\tan Ψ$ ( $Ψ$ , wavefront phase)	$\frac{A_{1} D \sin ϕ}{A_{0} (1 - D) + A_{1} D \cos ϕ}$	$\frac{A_{1} \sin ϕ}{- A_{0} + A_{1} \cos ϕ}$
Sensitivity functions
$\partial η / \partial D$	$- 2 A_{0}^{2} (1 - D) + 2 A_{1}^{2} D + 2 A_{0} A_{1} (1 - 2 D) \cos ϕ$	$2 (A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D sinc (2 m D) {sinc}^{2} (m P)$
$\partial η / \partial ϕ$	$- 2 A_{0} A_{1} (1 - D) \sin ϕ$	$2 A_{0} A_{1} \sin ϕ D^{2} {sinc}^{2} (m D) {sinc}^{2} (m P)$
$\partial η / \partial P$	0	$(A_{0}^{2} + A_{1}^{2} - 2 A_{0} A_{1} \cos ϕ) D^{2} {sinc}^{2} (m D) \cdot \frac{2}{P} [sinc (2 m P) - {sinc}^{2} (m P)]$
$\partial Ψ / \partial D$	$\frac{A_{0} A_{1} \sin ϕ}{A_{1}^{2} D^{2} + A_{0}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ}$	0
$\partial Ψ / \partial ϕ$	$\frac{A_{1}^{2} D^{2} + A_{0} A_{1} D (1 - D) \cos ϕ}{A_{1}^{2} D^{2} + A_{0}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) \cos ϕ}$	$\frac{A_{1}^{2} - A_{0} A_{1} \cos ϕ}{A_{1}^{2} + A_{0}^{2} - 2 A_{0} A_{1} \cos ϕ}$
$\partial Ψ / \partial P$	0	0

Duty Cycle Range	41% to 53% (1% change/grating)
Etching depth	0.5 μm
Grating period	15.0 μm
Grating size	$3 mm \times 20 mm$
Overall pattern size	$39 mm \times 20 mm$
Number of sets	3

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Applied Optics