## Figuring Field of View Mathematically

For Linescan and Non-Standard cameras or Non-Standard lenses including extension tubes

The basic lens formula is

**f**= focal length of lens (for example – 50 mm lens)**d**= distance from detector plane to the principle point of the lens_{1}**d**= distance from the principle point of the lens to the object_{2}

**Note**: if d_{2}= infinity then d_{1}= f. So the principle point is**f**away from the array when the lens is focused at infinity.**Note**: lenses focus by moving the optics further from the detector (increasing d_{1}. If you know the minimum object distance (d_{2}) for a lens, you can figure out where d_{1}is when the lens is focused close. But it usually varies just a few millimeters from the focal length f.

**What is a principle point?**

Most lenses act geometrically like a pinhole lens placed at the principle point. we will use this model. If you are designing the next Hubble telescope, go find a good textbook on optics. Otherwise this will get you started just fine. The principle point is usually inside the lens glass, but in a telephoto lens, this point is in front of the lens!

**So how do I figure my field of view?**

Now since this lens acts like a pinhole, the field of view (FOV) is found by :

**A**= distance on the detector (array size)

The good news is you don’t have to find d_{1} or the principle point. Combining formulas

or

Watch your units! This formula lets you figure out field of view and working distance for a given camera.

Notice that d_{2} is roughly the “working distance” from the lens to the object. Strictly speaking the working distance is from the front of the lens to the object.

**For large working distances :**

so the formula reduces to

or

where theta is the angular field of view. This formula lets you figure out the angle of view of the camera.

**Let’s take an example : **What is your working distance to get an 10″ field of view with a 50mm lens on a linescan camera with array length 0.6″?

This is the approximate working distance.

**Things to watch out for:**Make sure your lens will image a large enough area. Small C-Mount lenses for 1/2″ CCD are not large enough for a 0.6″ array. Use a 2/3″ or 1″ format lens.

- If d
_{2}is small, then your lens may not focus that close. You may want to figure out d_{1}and then figure out what extension tubes you will need.

**Let’s take another example : ** Suppose you want to use a 75mm lens to get 1:1 magnification. What extension tube is needed?

Here the assumption that working distance = d_{2} may not be close enough.

Since FOV = A, then

So your distance from the array to your object is d_{1} + d_{2} = 4f = 300mm

Since d_{1} needs to be pushed out to 150mm, and your 75mm lens only moves d_{1} from 75 to 80mm (figure this out based on working distance range) then you will need extension tubes that are 70-75 mm long!

You can figure your working distance based on the physical measurements of the lens. Remember the principle point is usually inside the lens! This is usually easier to setup and measure, but it can be calculated.

**Good Luck!**