How to determine the needed focal length for a Fisheye lens?

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) r = f \cdot \theta

This formula coined the name for this type of lens.

Theta (\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 \text{image circle radius} = \frac{\text{image circle diameter}}{2} = \frac{7.5\text{mm}}{2} = 3.75\text{mm}

\text{Half field angle} = \frac{185\ensuremath{^{\circ}}}{2} = 92.5\ensuremath{^{\circ}}

\theta = half field angle in radians = \frac{92.5\ensuremath{^{\circ}}}{180\ensuremath{^{\circ}}} \cdot \pi = 1.61..

(1) thus becomes

3.75 = f \cdot 1.61

\Leftrightarrow f = \frac{3.75mm}{1.61} = 2.33mm

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

3.5 = f \cdot 1.61

\Leftrightarrow f = \frac{3.5mm}{1.61} = 2.17mm

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.

If, for example, -5% distortion can be accepted, you can roughly calculate with 100% – (-5)% = 105% = 7.5mm,
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% = \frac{100 \cdot 7.5}{105} = 7.14mm

Using this in (1) gives

\frac{7.14}{2} = 3.57 = f \cdot 1.61
\Leftrightarrow f = \frac{3.57}{1.61} = 2.22mm

If, for example, -5% distortion can be accepted at a 7mm image circle, you can roughly calculate with 100% – (-5)% = 105% = 7.0mm,
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% = \frac{100 \cdot 7.0}{105} = 6.67mm

Using this in (1) gives

\frac{6.67}{2} = 3.34 = f \cdot 1.61
\Leftrightarrow f = \frac{3.34}{1.61} = 2.1mm

So, if you want an image circle diameter between 7mm and 7.5mm at 185° field of view,
you need a distortion-free focal length of 2.17 – 2.33mm
or a focal length of 2.1 – 2.22mm with -5% distortion.