Determining lens and sensor distance for narrow focus

Question

I have the following items:

• a round target object, approx. 6mm in diameter
• a lens, focal length 20mm, F/2.8
• a camera sensor, optical size 1/4" (note: I know this camera has a fixed-at-infinity lens, but I can pop it off so it has no lens at all)

I need to take a picture of the target object, through the lens, using the camera sensor. I'm trying to figure out how to position these items (distances between them) such that the object is in focus and fills the camera sensor.

This online calculator suggests that for a lens with the above specifications, and a target area of 6mm x 8mm, the object should be 4.44cm from the front of the lens. How, though, do I calculate the distance from the back of the lens to the camera sensor? I know I could use the focal length for a focus at infinity, but that would require the object to be at least 5m or so beyond the lens, which is far outside of the allowable distance for this application.

Answer 1

You can use the thin lens equation

$$\frac{1}{d} + \frac{1}{d'} = \frac{1}{f}$$

where $f$ is the focal length ($20\;\mathrm{mm}$ in your case), $d$ is the lens-object distance and $d'$ the lens-image distance. This calculation will be a bit off as your lens is not thin, but you can adjust the position by hand.

The magnification is $m = d'/d$ by the way, this calculation is easy enough that you can do it on your own (solve the system of the two equations for the given $m$ and $f$).

Said this, in my opinion: if for your application you can do it, finding the position by hand is the easiest way. Place the object in front of the lens and the camera in the back. Keep object fixed, and move camera till image is sharp. If the image of the object is too large, push the object away from the lens and refocus; if it is too small, push the object towards the lens.