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Holography Transmission Equations Part I

, 16:02, 31 October 2015
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Introduction
These two papers, plus a follow up in the succeeding 1985 ISDH proceedings by Benton, form the basis for computation of almost all the geometrical problems facing the display holographer. With these papers, the geometry of reference and object angles for pseudo-color holograms (multi-colored replay made with a single color laser) of the transmission and reflection configurations can be plotted. (I have also used them to successfully record HOE’s for a variety of applications.)
Alas and alack, these papers as written are only available in the out of print and hard to find Symposia proceedings. But the equations are available in the sites below, hopefully into perpetuity, as they are the cornerstones of the mathematical foundations of holography, although diagonally opposite.('''Update:''' Courtesy of the author, Stephen McGrew: [http://nli-ltd.com/publications/graphical_method.php ''Graphical Methods''])
Benton’s approach is a trigonometric one, and McGrew’s is a geometric one; they are equivalent, giving the same result, except with Benton you plug and chug through equations, then map out the results. McGrew is more Euclidean; the diagram is made with straightedge and compass.
==Spatial Frequency==
:: $\displaystyle f = \frac{sin(\theta_1) - sin(\theta_2)}{\lambda}$
Frequency, how often something happens, the frequencies most often coming to mind are the 20 to 20,000 cycles per second (abbreviated as Hertz) of good human hearing (not mine, for sure!) or the radio frequencies of 550 kiloHertz to 1800 kHZ of the AM band, or the GigaHertz processing speed of computer chips.
Related to this equation is finding the distance between fringes; instead of fringes per mm, how big is a fringe cycle? (= the distance from the center of one bright fringe to the next.) It is simply the inverse of the spatial frequency equation:
:: $\displaystyle d = \frac{\lambda}{sin(\theta_1) - sin(\theta_2)}$
(And don’t forget theta 2 is a negative angle, otherwise you will have an error!) (Or to make life easier, forget about the sign convention, and just add the sines of the two angles as positives together! But watch your step when getting too far off the track!)
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