# Easy method to measure the NA of a lens system?

I have a lens system made up of some components of the Leica Z16 APO zoom system. This includes a built-in iris diaphragm to adjust the Numerical Aperture (NA) and therefore the depth-of-field (DOF).

The iris is marked with some repeatable settings labelled 1-5. I can't find any documentation to tell me what the actual size of the iris is at these settings but I need to work out the NA at each one. Is there a simple way of doing this?

## 1 Answer

The answer linked by @tmwilson26 is certainly helpful, but it may not be what you are looking for. There are a couple of different ways to approach this problem, and it depends a little bit on the equipment you have available. Note that if you have a zoom lens, leaving the iris at a fixed value will not, in general, result in a constant NA - instead the NA will depend on the focal length as well as the iris aperture.

A simple setup would focus a point source of light onto a ground surface (a focusing screen) that is mounted on an optical rail (so you can move it along the optical axis). Once you have found the focal point, you move the surface a known distance away, and measure the size of the focal spot (I would recommend using an eyepiece graticule to make the measurement straightforward). Repeat this for a few different distances, and determine the slope of the straight line through the plot of spot radius vs distance to focal point. This slope represents the tangent of the half-angle of the iris subtended at the focal point.

Now the numerical aperture is usually the $\sin$ of the angle - so if you care about the difference (which is important as NA gets larger) you need to do a bit of math:

$$NA = \sin\left(\tan^{-1}s\right)$$

where

$$s = \frac{r}{D}$$

The slope of your plot.

Now you can repeat this measurement for different settings of the focal length of the zoom lens, and for different values of the iris setting. Note that by defocusing to various points and measuring the slope you avoid the problem of not knowing the focal length of the lens.

EDIT
Since your lens is connected to a CMOS sensor, the above approach will not work for you. I recommend instead that you put a point source in front of the lens at a controlled (variable) distance, and measure once again the spot size.

Now we can use the formula for the distance to the focal point. If the source is a distance $s$ from the lens, and the focal length is $f$, then we expect the point to come into focus at a distance $d$ where

$$\frac{1}{f} = \frac{1}{s} + \frac{1}{d}\tag1$$

Which can be rewritten as

$$f = \frac{s\cdot d}{s+d}$$

or alternatively

$$d = \frac{s\cdot f}{s - f}\tag2$$

Assuming you know the focal length, and that you focused the lens on infinity (so you set $d' = f$), then your imaging plane would be away from the in-focus plane by a distance $d - f$. In the small angle approximation, the point would become a circle with a radius $r = NA\cdot (d-f)$.

We can rewrite (2):

\begin{align}d - f &= \frac{s\cdot f}{s - f} - f \\ &= \frac{s\cdot f}{s - f} - \frac{f(s-f)}{s-f}\\ &=\frac{f^2}{s-f}\end{align}

So if you plot the spot radius as a function of $\frac{f^2}{s-f}$ for different values of $s$ you should get a straight line whose slope is the tangent of the numerical aperture.

This does require you to know the focal length of the lens, and the lens to be focused on infinity. Measuring the distance to the optical center of the lens can be tricky - a zoom lens is a composite lens, and they don't always have an obvious "center". You can use the distance to the focal plane instead - that should be $s+f$ .

I am not absolutely sure that these formulas work for compound lenses... perhaps somebody can either confirm this or point out the correction that is needed.

• Thanks very much, the linked question is interesting however my set-up is slightly different. I have a lens system attached to a CMOS camera, it is difficult to detach the lens from the camera so I'm not sure how to implement your suggestion. Could I move a point source along the optical axis on the object side and measure the spot size on the CMOS image for a fixed focal length? – Ciara Oct 26 '15 at 17:22
• @Ciara I have updated my answer to account for the constraint you added. – Floris Oct 26 '15 at 18:27