Let’s start by assuming we are looking for an F-Theta lens without distortion.

The imaging curve is then given by the relationship

(1)

This formula coined the name for this type of lens.

Theta is the *radian measure* of the *half* desired field angle.

**Field Angle**: 185°

**Lens Type**: Distortion-free F-Theta lens

**Image Circle Diameter**: 7.5mm

Instead of the image circle diameter, you can also use a “reference diameter” and, for example, desire a 90° field angle at 4mm distance from the center.

Then the

half field angle in radians

(1) thus becomes

A *distortion-free* F-Theta lens with a 185° field angle at an image circle of 7.5mm therefore has a focal length of approximately 2.33mm.

**Field Angle**: 185°

**Image Circle Diameter**: 7.0mm

**Lens Type**: Distortion-free F-Theta lens

So, if you are looking for a distortion-free F-Theta lens with an image diameter between 7 and 7.5mm at 185°, a focal length between 2.17mm and 2.33mm is suitable.

Then at 7.5mm you have 105% of 185 degrees in the image, and the actual 185° appear at a diameter of

105% = 7.5mm

100% =

Using this in (1) gives

Then at 7.0mm you have 105% of 185° in the image, and the actual 185° appear at a diameter of

105% = 7.0mm

100% =

Using this in (1) gives

you need a distortion-free focal length of 2.17 – 2.33mm

or a focal length of 2.1 – 2.22mm with -5% distortion.