Projection and focus sharpness

[bercov@xxxxxxxxxx (John Bercovitz)]

> Date: Mon, 25 Dec 1995 15:25:13 -0800
> From: bercov@xxxxxxxxxx (John Bercovitz)
 
Projection and focus sharpness
 
My brother-in-law and I were discussing what projection lens might
be best if you didn't want a warped slide to be too far out of focus.
I suggested that a long lens might be best because then the warpage of
the slide would be a smaller percentage of the focal length of the
projection lens.  He suggested that a shorter lens might be better
because this could be a situation analogous to using a short lens on
a camera for increased depth of focus.  Well, he was packing for a
trip so I did the math.  There were pages of derivation using Gaussian
and Newtonian forms of the lens equation which I'll spare you.  Here's
the end result.
 
             M/N
C = ------------------------
     1/F + (M + 1)/(d*M)
 
C = out-of-focus diameter on the screen of a point in the transparency
d = distance transparency surface lies back of plane of best focus
M = magnification from transparency to screen
F = focal length of projection lens
N = f/number of projection lens
 
Here's a real-world example from a Leitz projector projecting a
2x2 slide (1.4" wide) onto an 84" wide screen.  I'm guessing that
when a slide "pops"
it moves back 0.25 mm in the center.
 
F = 90 mm
N = 2.5
M = 84"/1.4" = 60
d = 0.25 mm
 
          24
C = --------------
     1/F + 4.0666
 
Solving,
C = 5.88 mm
 
So for a given screen size, if F is in the normal range of 90 to 150 mm,
and the f/number is held constant, there is no real change in the diameter,
C, with focal length.
 
John B


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