Coupling of surface roughness to the performance of computer-generated holograms

Ping Zhou; Jim Burge

doi:10.1364/AO.46.006572

Coupling of surface roughness to the performance of computer-generated holograms

Ping Zhou and Jim Burge

Applied Optics
Vol. 46,
Issue 26,
pp. 6572-6576
(2007)
•https://doi.org/10.1364/AO.46.006572

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Ping Zhou and Jim Burge, "Coupling of surface roughness to the performance of computer-generated holograms," Appl. Opt. 46, 6572-6576 (2007)

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Related Topics
Table of Contents Category
- Diffraction and Gratings
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Binary optics (050.1380)
- Gratings (050.2770)

About this Article
History
- Original Manuscript: April 23, 2007
- Manuscript Accepted: July 3, 2007
- Published: September 5, 2007

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Abstract

Computer-generated holograms (CGHs), such as those used in optical testing, are created by etching patterns into an optical substrate. Imperfections in the etching can cause small scale surface roughness that varies across the pattern. The variation in this roughness affects the phase and amplitude of the wavefronts in the various diffraction orders. A simplified model is developed and validated that treats the scattering loss from the roughness as an amplitude effect. We demonstrate the use of this model for engineering analysis and provide a graphical method for understanding the application. Furthermore, we investigate the magnitude of this effect for the application of optical testing and show that the effect on measurement accuracy is limited to $1 n m$ for typical CGHs.

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	Zero Order (m = 0)	Nonzero Order (m ± 1, ±2, …)
η	${A_{0}}^{2} {(1 - D)}^{2} + {A_{1}}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos (ϕ)$	$[{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} \cos (ϕ)] D^{2} {\sin c}^{2} (m D)$
tan(Ψ)	$\frac{A_{1} D \sin (ϕ)}{A_{0} (1 - D) + A_{1} D \cos (ϕ)}$	$\frac{A_{1} \sin (ϕ) \sin c (m D)}{[- A_{0} + A_{1} \cos (ϕ)] \sin c (m D)}$

	Zero Order (m = 0)	Nonzero Order (m ± 1, ±2, …)
$\frac{\partial η}{\partial A_{1}}$	$2 A_{1} D^{2} + 2 A_{0} D (1 - D) \cos (ϕ)$	$(2 A_{1} - 2 A_{0} \cos (ϕ)) D^{2} \sin c^{2} (m D)$
$\frac{\partial Ψ}{\partial A_{1}}$	$\frac{A_{0} D (1 - D) \sin (ϕ)}{{{A_{0}}^{2} (1 - D)}^{2} + {A_{1}}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos (ϕ)}$	$\frac{- A_{0} \sin (ϕ)}{{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} \cos (ϕ)}$

	Sensitivities	P-V Error	rms Error
Diffraction efficiency
Zero order	0.206	0.0064	0.0023
First order	0.322	0.0100	0.0035
Wavefront phase
Zero order	0.156λ	0.0048λ	0.0017λ
First order	−0.041λ	−0.0013λ	−0.0005λ

	Zero Order (m = 0)	Nonzero Order (m ± 1, ±2, …)
η	${A_{0}}^{2} {(1 - D)}^{2} + {A_{1}}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos (ϕ)$	$[{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} \cos (ϕ)] D^{2} {\sin c}^{2} (m D)$
tan(Ψ)	$\frac{A_{1} D \sin (ϕ)}{A_{0} (1 - D) + A_{1} D \cos (ϕ)}$	$\frac{A_{1} \sin (ϕ) \sin c (m D)}{[- A_{0} + A_{1} \cos (ϕ)] \sin c (m D)}$

	Zero Order (m = 0)	Nonzero Order (m ± 1, ±2, …)
$\frac{\partial η}{\partial A_{1}}$	$2 A_{1} D^{2} + 2 A_{0} D (1 - D) \cos (ϕ)$	$(2 A_{1} - 2 A_{0} \cos (ϕ)) D^{2} \sin c^{2} (m D)$
$\frac{\partial Ψ}{\partial A_{1}}$	$\frac{A_{0} D (1 - D) \sin (ϕ)}{{{A_{0}}^{2} (1 - D)}^{2} + {A_{1}}^{2} D^{2} + 2 A_{0} A_{1} D (1 - D) \cos (ϕ)}$	$\frac{- A_{0} \sin (ϕ)}{{A_{0}}^{2} + {A_{1}}^{2} - 2 A_{0} A_{1} \cos (ϕ)}$

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Figures (8)

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Equations (7)

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