Formulas for combining two lenses

Formulas have been devised to determine what the focal length of a 
combination of lenses is and where the focus of the combination 
lies.  (When I say focus here, I am talking about the focus for an 
object located at infinity.)  It is perhaps surprising that 
combining two positive lenses does not always create a stronger 
(shorter focal length) lens.  There are actually only two 
interesting formulas on the spreadsheet, LensCombo.xls.  They are:

                                         f1 * f2
the focal length of the combination = -------------
                                       f1 + f2 - d

where f1 is the focal length of one lens, f2 is the focal length 
of the other lens, and d is the distance between them.  Actually d 
is the distance between the lenses' principal points.  The + signs 
inside the lenses in the diagram are the positions of their 
primary and secondary principal points, reading from left to 
right.  So d is the distance from the secondary principal point of 
the first lens to the primary principal point of the second lens.
How do you locate a principal point?  Just measure from a focus 
back toward the lens a distance equal to the focal length of the 
lens and you're there.

                              (f1 - d) * f2
distance to the back focus = ---------------
                               f1 + f2 - d

In this formula, f1 is the focal length of the first lens the 
light hits and the distance to the back focus is the distance from 
the secondary principal point of the second lens to the back focus 
for the combination of lenses.

Since we now have the focal length and the position of the back 
focus, we can calculate the position of the secondary principal 
point of the combination.  That's what the less interesting 
formula on the spreadsheet does.

Since light can be run through a lens combination in either 
direction, I have also included on the spreadsheet the 
calculations for the condition where light is run through in the 
opposite direction from normal.  The focal length is the same 
either direction, but of course the front focus will be located in 
a different place than the back focus.

Application

It is interesting to note that the front focus of most 50 mm SLR 
lenses is just about right at the filter ring.  This is good 
because then if you screw a filter on, by the first formula, d 
will be equal to f2 and so even if the filter isn't perfectly 
flat, the combination of it and the 50 mm lens will still have a 
focal length of 50.  Same is true if you screw on a closeup lens.  
The significant effect is on the position of the back focus.  It 
won't be much different for a bad filter but it can be a lot 
different for a closeup lens, just as intended.  A closeup lens 
moves the secondary principal point of the lens away from the film 
just as if you had added an extension tube between the lens and 
the camera body.

John B  98/09/29