Fabrication error analysis and experimental demonstration for computer-generated holograms

Ping Zhou; James H. Burge

doi:10.1364/AO.46.000657

Applied Optics
Vol. 46,
Issue 5,
pp. 657-663
(2007)
•https://doi.org/10.1364/AO.46.000657

Fabrication error analysis and experimental demonstration for computer-generated holograms

Ping Zhou and James H. Burge

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Ping Zhou and James H. Burge, "Fabrication error analysis and experimental demonstration for computer-generated holograms," Appl. Opt. 46, 657-663 (2007)

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Abstract

Aspheric optical surfaces are often tested using computer-generated holograms (CGHs). For precise measurement, the wavefront errors caused by the CGH must be known and characterized. A parametric model relating the wavefront errors to the CGH fabrication errors is introduced. Methods are discussed for measuring the fabrication errors in the CGH substrate, duty cycle, etching depth, and effect of surface roughness. An example analysis of the wavefront errors from fabrication nonuniformities for a phase CGH is given. The calibration of these effects for a CGH null corrector is demonstrated to cause measurement error less than 1 nm.

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	Zero Order (m = 0)	Nonzero Order (m ± 1, ±2, …)
Diffracted Wavefront
η, diffraction efficiency	${A_{0}}^{2} {(1 - D)}^{2} + {A_{1}}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) cos (ϕ)$	$[{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} cos (ϕ)] D^{2} sin c^{2} (m D)$
$tan (Ψ)$ Ψ, wavefront phase	$\frac{A_{1} D sin (ϕ)}{A_{0} (1 - D) + A_{1} D cos (ϕ)}$	$\frac{A_{1} sin (ϕ) sinc (m D)}{[- A_{0} + A_{1} cos (ϕ)] sinc (m D)}$
Sensitivity Functions
$\frac{\partial η}{\partial D}$	$- 2 {A_{0}}^{2} (1 - D) + 2 {A_{1}}^{2} D + 2 A_{0} A_{1} (1 - 2 D) cos ϕ$	$2 [{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} cos (ϕ)] D sinc (2 m D)$
$\frac{\partial η}{\partial ϕ}$	$- 2 A_{0} A_{1} D (1 - D) sin ϕ$	$2 A_{0} A_{1} sin ϕ D^{2} sin c^{2} (m D)$
$\frac{\partial Ψ}{\partial D}$	$\frac{A_{0} A_{1} sin ϕ}{{A_{1}}^{2} D^{2} + {A_{0}}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) cos ϕ}$	${\begin{matrix} \infty, \\ 0, \end{matrix} \begin{array}{l} for sinc (m D) = 0 \\ otherwise \end{array}$
$\frac{\partial Ψ}{\partial ϕ}$	$\frac{{A_{1}}^{2} D^{2} + A_{0} A_{1} D (1 - D) cos ϕ}{{A_{1}}^{2} D^{2} + {A_{0}}^{2} {(1 - D)}^{2} + 2 A_{0} A_{1} D (1 - D) cos ϕ}$	$\frac{{A_{1}}^{2} - A_{0} A_{1} cos ϕ}{{A_{1}}^{2} + {A_{0}}^{2} - 2 A_{0} A_{1} cos ϕ}$
$\frac{\partial Ψ}{\partial (A_{0} / A_{1})}$	$\frac{- D (1 - D) sin ϕ}{{(A_{0} / A_{1})}^{2} {(1 - D)}^{2} + D^{2} + 2 D (1 - D) (A_{0} / A_{1}) cos ϕ}$	$\frac{sin ϕ}{{(A_{0} / A_{1})}^{2} + 1 - 2 (A_{0} / A_{1}) cos ϕ}$

	D	t (waves)
Point 1	0.4897	0.3274
Point 2	0.4906	0.3315
Point 3	0.4897	0.3317
Point 4	0.4904	0.3326
Point 5	0.4881	0.3275
Average	0.4897	0.33014
rms variation	0.0009	0.0022
rms variation in percentage	0.18%	0.67%

	Unetched Area		Etched Area		Ratio of amplitudes $\| A_{0} / A_{1} \|$
	rms roughness (nm)	I _scat	rms roughness (nm)	I _scat	Ratio of amplitudes $\| A_{0} / A_{1} \|$
Point 1	0.950	0.00002	2.188	0.00012	1.000048
Point 2	1.090	0.00003	1.540	0.00006	1.000015
Point 3	1.110	0.00003	1.840	0.00008	1.000027
Point 4	2.000	0.00010	2.663	0.00017	1.000038
Point 5	1.738	0.00007	2.169	0.00012	1.000021
			Average		1.000030
			rms variation		0.000033
			rms variation in percentage		0.000033

Source of Errors	Description	Measured Variation	Sensitivities	rms Wavefront Errors at First Order (nm)
CGH substrate	Transmitted wavefront error	13.6nm	1:1	13.6
Etching depth	Effect on diffracted wavefront	0.67%	$\frac{δ Ψ_{m = 1}}{δ ϕ}$	0.700
Duty cycle	Effect on diffracted wavefront	0.18%	0	0
Amplitude ratio	Effect on diffracted wavefront	0.0033%	$\frac{δ Ψ_{m = 1}}{δ (A_{0} / A_{1})}$	0.001
Root-sum-square error				13.6

Source of Errors	Measured Variation	Sensitivities	rms Wavefront Errors (nm)
Etching depth error	0.67%	$\frac{\partial Ψ_{m = 1}}{\partial ϕ} - \frac{\partial Ψ_{m = 0}}{\partial ϕ}$	0.056
Duty-cycle error	0.18%	$- \frac{\partial Ψ_{m = 0}}{\partial D}$	0.300
Amplitudes error	0.0033%	$\frac{\partial Ψ_{m = 1}}{\partial (A_{0} / A_{1})} - \frac{\partial Ψ_{m = 0}}{\partial (A_{0} / A_{1})}$	0.004
Root-sum-square error			0.3

	rms Error: Zero Order	rms Error: First Order	Measured Difference
	(nm)	(nm)	(nm)
Configuration 1	1.677	1.648	0.714
Configuration 2	7.596	7.801	0.779
Configuration 3	16.82	17.43	1.291

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Applied Optics