Objective Lens

In a binocular, the first thing a light ray hits is the objective lens.  As observers watching the object through the objective lens, they would notice that image is inverted and reversed.  So therefore it is necessary to revert the image back to its original orientation using prism systems or else all binoculars will have upright, reversed images.  In order to know where to put the prisms at a certain distance, one must know the focal length f, the image distance i, and the distance from the object to the lens s. Which leads to this relationship:

To trace the ray and find the focal length, it can be found by using a linear approximation, which can be understood further in here: http://www.math.ubc.ca/~cass/courses/m309-03a/text/systems/systems.pdf  In order to calculate the focal length, it can be found using a transfer matrix.  In Figure 1,  the transfer matrix is T = T5*T4*T3*T2*T1, so

since the height at the focal length is 0 so the upper right entry is equal to 0, so f is,

so all that is needed is to know s, and i can be found.

 
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