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Diopters, and the Focal Length of Stacked Lenses

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)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_QoDklD.png

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 f0 and the second has focal length f1 then we'll obtain the formula

2)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_AWCp5y.png

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 f3 for the focal length of the "sandwich", then, comparing (2) with (1), we must have

3)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_ceejwU.png

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 f0, f1, f2, ...  fn must be

4)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_Uiqfrp.png

Lens Combinations Measured in Diopters

If a lens has focal length fk expressed in meters, then let Dk be its focal length expressed in diopters.  Let's translate (3) into diopters:

5)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_Xe9Djf.png

or, multiplying through top and bottom by D0D1,

6)    file:///media/disk/home/slawrence/website/physics_insights/physics/formulas/f2/eqe_temp_image_X5oyB2.png

which is what we hoped to show.

Page created on 01/14/2009