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.

As a point of comparison, in our accompanying discussion of 1D parallel apertures (specifically, the subsection titled, Case 1), we have presented the following expression for the y-coordinate variation of the optical field immediately in front of the aperture: <math>~A(y_1)</math> <math>~\approx</math> <math>~ e^{i 2\pi L/\lambda }\biggl[ \frac{w}{2\beta_1} \biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot e^{-i \Theta } d\Theta \, , </math> where, <math>~\frac{1}{\beta_1}</math> <math>~\equiv</math> <math>~\frac{\lambda L}{\pi y_1w} \, ,</math>       <math>~L</math> <math>~\equiv</math> <math>~ Z \biggl[1 + \frac{y_1^2}{Z^2} \biggr]^{1 / 2} \, , </math>       and,       <math>~\Theta</math> <math>~\equiv</math> <math>~\biggl(\frac{2\pi y_1 Y}{\lambda L} \biggr) \, .</math> In other words, making the substitution, <math>~(2\pi/\lambda) \rightarrow k</math>, and recognizing that, <math>~d \leftrightarrow Z</math>, our expression becomes, <math>~I(y) \equiv \biggl[i k d e^{-i k d} \biggr] A(y_1)</math> <math>~\approx</math> <math>~ \biggl[i k d e^{-i k d} \biggr] e^{i kL }\biggl[ \frac{L}{k y_1} \biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i \frac{2\pi y_1 Y}{\lambda L} \biggr] \biggl[ \frac{k y_1 }{L} \biggr] dY </math>   <math>~=</math> <math>~ (i k Z) e^{i k (L-Z)} \int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY </math>   <math>~\approx</math> <math>~ (i k Z) \exp\biggl[i k \biggl( L-Z \biggr)\biggr] \int a_0(\Theta) e^{i\phi(\Theta)} \cdot \exp\biggl[-i 2\pi Y \biggl(\frac{y_1 }{\lambda L}\biggr) \biggr] dY </math>

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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