The
diopter number of a lens
is the inverse of the focal length of the lens, measured in
meters. Thus, a lens with a focal length of 1 meter is a 1
diopter lens. A lens with a focal length of 1 inch, which is
about 1/39 meter, is a 39 diopter lens.
A negative lens is treated identically. Thus, a lens with a focal
length of 1/2 meter is a 2 diopter lens.
The reason this is a useful way to measure the focal length of a lens
is that, when two or more lenses are stacked, the focal length in
diopters of the resulting "combined lens" can be found by simply adding
up the focal lengths (in diopters) of all the lenses in the
stack. On the remainder of this page, we'll demonstrate this fact
by finding the focal length of such a stack of lenses. Along the
way we'll show that a stack of ideal lenses functions as a single ideal
lens.
A Sandwich of Two Ideal Lenses
As we found
here,
when a ray which is coplanar with the lens axis passes through an ideal
lens, it's deflected according to the formula
1)
Ideal lenses have zero thickness (they are certainly not "real"
lenses!). Consequently if we have two stacked ideal lenses,
they're both in the same plane, a ray which passes through them passes
through both at exactly the same distance from the lens axis, and
consequently we can easily see how the "sandwich" must behave: We
just apply formula (1) once for each lens. If the first lens has
focal length
f_{0}
and the second has focal length
f_{1}
then we'll obtain the formula
2)
Thus the "sandwich" deflects light according to the same formula which
describes an ideal lens, and so the combination must also function as
an ideal lens. Furthermore, if we write
f_{3}
for the focal length of the "sandwich", then, comparing (2) with (1),
we must have
3)
Note that this formula applies regardless of whether the lenses are
positive or negative.
By applying the same formula each time we add another lens to the
stack, it's easy to see that a stack of
n ideal lenses for which the focal
lengths are
f_{0}, f_{1}, f_{2}, ... f_{n} must be
4)
Lens Combinations Measured in Diopters
If a lens has focal length
f_{k}
expressed in meters, then let
D_{k}
be its focal length expressed in diopters. Let's translate (3)
into diopters:
5)
or, multiplying through top and bottom by D
_{0}D
_{1},
6)

which is what we hoped to show.
Page
created on 01/14/2009