How Close-up Lenses Affect Magnification
Prepared 2007-02-02 (169/12614) by Bill Claff

We already learned that
m = v / f - 1

Extension tubes increase m by increasing v.
Close-up lenses increase m by reducing f.

The mathematics is governed by the formula
P = P₁ + P₂ - d_m * P₁ * P₂
Where
P, P₁, and P₂ are lens powers in diopters
d_m is the distance between lens nodes of P₁ and P₂ in meters

Diopters are the reciprocal of focal length in meters (we use a capital D to indicate units of diopters).
Since d_m is seldom known we will assume d_m = 0 so
P = P₁ + P₂

Note that the powers of lenses are additive.
This is why when you take two close-up lenses, for example a 1D lens and a 2D lens, and put them together you can simply add the powers, to get 3D in this case.

Using P = 1000 / f, P₁ = 1000 / f₁, and P₂ = 1000 / f₂ we can restate the combination formula as

f = ( f₁ * f₂ ) / ( f₁ + f₂ )

This form is usually easier to use than the power formula.

So, what happens when we put a Canon 500D, a 500mm 2D close-up lens on a 300mm f/4D ED-IF AF-S Nikkor?
(BTW, the D in 500D does not stand for diopter but rather for duplet)

At infinity focus, ( 300mm * 500mm ) / ( 300mm + 500mm ) = 187.5mm
At closest focus, ( 242.9mm * 500mm ) / ( 242.9mm + 500mm ) = 163.5mm

Since m = v / f - 1 that's
At infinity focus, 300mm / 187.5mm - 1 = .60x
At closest focus, 308.5mm / 163.5mm - 1 = .89x

I won't derive the formula but for infinity focus it can be shown that
m = f₁ / f₂

So note that 300mm / 500mm = .60x, which matches the infinity result above.

Finally, substituting back into
S = ( 1 / m + 1 + 1 + m ) * f

At infinity focus, S = ( 1 / .60 + 1 + 1 + .60 ) * 187.5mm = 800mm
At closest focus, S = ( 1 / .89 + 1 + 1 + .89 ) * 163.5mm = 656mm

Note that in terms of magnification mounting a reversed lens is just a way of attaching a high powered close-up lens.
For example, a 50mm reversed lens acts like a 20D (1000mm/50mm) close-up lens.

I won't cover it here but it's straightforward to determine the effect of extension tubes on a lens that has close-up lens(es) attached.