Measuring Focal Length

Prepared 2005-11-15 by Bill Claff

### Introduction

Extension (_{}) (as in bellows or extension tubes), magnification (_{}), and focal length (_{}) are interrelated as follows:

_{}

Where _{}

And the values _{} and _{} are the magnification
and extension of the lens at the current focus setting of the lens.

Note that lens extension due to focusing can be internal and the value of _{} is never positive.

Typically, when the current focus setting is infinity, _{}, and _{}. So _{}, and _{}.

### Application

At infinity focus a minimum of one magnification measurement
at a non‑zero extension is required to determine focal length.

At focus settings other than infinity a minimum of two
magnification measurements at different non‑zero extensions are required
to determine focal length and extension of the lens.

In practice, even with precise measurements, multiple data
points are required to obtain satisfactory results.

Because extension is known and magnification is measured we
take our data points and do a linear least squares fit to:

_{}

Where _{} and _{} are the slope and
intercept from out least squares fit.

Then _{} and _{}

In practice, slight positive values for _{} at infinity focus may
arise due to imprecise data and because due to mechanical tolerances and lens
design the lens may in fact focus slightly beyond infinity.

This analysis can be performed on a lens that is reverse
mounted and for many lenses this data is easier to collect.

For a reverse mounted lens keep in mind that the value of _{} is relative to where
the lens attaches to the camera body, adapter ring, or extension tube.

### Field Of View

Naturally, once we have a more precise measurement of focal
length we also have a more precise measurement of field of view.

We can calculate focal length and field of view for any focus
setting regardless of whether an extension ring is present.

Application to panoramic corrections for a lens set to a hyper‑focal
distance rather than infinity would be a good example.

### A Final Note

In cases where I have collected both normal mounted and
reverse mounted data the computed reverse mounted focal lengths have been
slightly less than the computed normal focal lengths.

These differences are less than 1mm. I have no explanation for this phenomenon.