User:Tohline/Appendix/CGH/KAH2001

Hologram Reconstruction Using a Digital Micromirror Device

In a paper titled, Hologram reconstruction using a digital micromirror device, T. Kreis, P. Aswendt, & R. Höfling (2001) — Optical Engineering, vol. 40, no. 6, 926 - 933), hereafter, KAH2001 — present some background theoretical development that was used to underpin work of the group at UT's Southwestern Medical Center at Dallas that Richard Muffoletto and I visited circa 2004.

Optical Field in the Image Plane

Labeling it as their equation (5), KAH2001 present the following Fresnel transform expression for the "optical field, <math>~B(x, y)</math>, in the image plane at a distance <math>~d</math> from the" aperture:

<math>~B(x,y)</math>	<math>~=</math>	<math>~ \frac{e^{i k d}}{i k d} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} U(\xi,\eta) \times \exp\biggl\{ \frac{i \pi}{d \lambda} \biggl[ (x - \xi)^2 + (y-\eta)^2 \biggr] \biggr\} d\xi d\eta </math>
	<math>~=</math>	<math>~ \biggl[\frac{e^{i k d}}{i k d} \biggr] I_\xi(x) \cdot I_\eta(y) \, , </math>

with,

<math>~I_\xi(x)</math>	<math>~=</math>	<math>~ \int_{-\infty}^{\infty} U(\xi) \times \exp\biggl[ \frac{i \pi}{d \lambda} (x - \xi)^2 \biggr] d\xi \, , </math>
<math>~I_\eta(y)</math>	<math>~=</math>	<math>~ \int_{-\infty}^{\infty} U(\eta) \times \exp\biggl[ \frac{i \pi}{d \lambda} (y - \eta)^2 \biggr] d\eta \, , </math>

and where <math>~U(\xi,\eta)</math> is "… the optical field immediately in front of the DMD" — i.e., the aperture. Following KAH2001, if we evaluate the square and substitute <math>~\mu = x/(d \lambda)</math>, the expression for <math>~I_\xi(x)</math> may be written as,

<math>~I_\xi(x)</math>	<math>~=</math>	<math>~ \int_{-\infty}^{\infty} U(\xi) \times \exp\biggl[ \frac{i \pi x^2}{d \lambda} \biggl(1 - \frac{2 \xi}{x} + \frac{\xi^2}{x^2} \biggr) \biggr] d\xi </math>
	<math>~=</math>	<math>~ \int_{-\infty}^{\infty} U(\xi) \times \exp\biggl[ \frac{i \pi x^2}{d \lambda} \biggr] \times \exp\biggl[- \frac{i \pi x^2}{d \lambda} \biggl(\frac{2 \xi}{x} \biggr) \biggr] \times \exp\biggl[ \frac{i \pi x^2}{d \lambda} \biggl( \frac{\xi^2}{x^2} \biggr) \biggr] d\xi </math>
	<math>~=</math>	<math>~ \exp( i \pi d \lambda \mu^2 ) \int_{-\infty}^{\infty} U(\xi) \times \exp (- i 2\pi \mu \xi ) \times \exp \biggl[\biggl( \frac{i \pi }{d \lambda}\biggr) \xi^2 \biggr] d\xi \, . </math>

Note that all three of the exponential terms in this expression can be found in equation (7) of KAH2001.

See Also

• Updated Table of Contents
• Tohline, J. E., (2008) Computing in Science & Engineering, vol. 10, no. 4, pp. 84-85 — Where is My Digital Holographic Display? [ PDF ]

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