How TeleConverters Affect Magnification
Prepared 2007-02-28 (169/12917) by Bill Claff

TeleConverters (TCs) operate by the same lens combination principles as close-up lenses.

The basic formulas are:
P = P₁ + P₂ - d_m * P₁ * P₂
Where
P, P₁, and P₁ are lens powers in diopters
d_m is the distance between lens nodes of P₁ and P₂ in meters

And
H₁₁H₁ = d_m * P₂ / P
H₂₂H₂ = -d_m * P₁ / P
For the distance the resulting lens nodes shift as a result of a non-zero d_m

For close-up lenses we often disregard d_m which simplies calculations considerably.
(See How Close-up Lenses Affect Magnification)
However, the design of a TC is more complex since in addition to changing focal length the rear node must also move to maintain infinity focus.
Because of the placement and movement of the rear node, the term d_m in the power equations cannot be disregarded.
So without complete lens and TC information, only the effect on focal length at infinity focus and the effect on magnification at any focus position can be easily calculated.

In a suprising twist, the effect of the TC multiplier decreases as the magnification of the primary lens increases and can go below 1 resulting in a decrease of focal length as you focus closer.
For example, the 105mm f/2.8G ED-IF AF-S VR Micro-Nikkor has infinity and closest focus focal lengths of 105.2mm and 75.1mm.
With the addition of a Kenko 1.4xTC the infinity focal length increases a factor of 1.4 from 105.2mm to 148.8mm but the closest focus focal length drops from 75.1mm to 52.9mm.
All of the above values are actual measurements with the exception of the 148.8mm value which has been computed.

Although focal length may not be as expected at closest focus, magnification is increased as expected; and a TC can be a suprisingly good way to get more magnification for close-ups.