# Numerical aperture of a lens

Using a laser setup, I was asked to determine the aperture of a given lens and then use some geometrical arguments and compare the theoretical value from the manufacturer and the experimental value. However, when I did this, the values differed by orders of magnitude. The way I got the aperture was to let the laser light pass through the lens onto a white screen and then keep the screen at a fixed distance from the lens and measure the radius of the light spot obtained. Then I used geometry to determine the angle and thus its sine. Could diffraction be the reason of such a large error?

• What do you mean by "lens attenuation" ? Can you please post a drawing of your setup? Include the collimation (or divergence) of your source laser beam, and how you expect to go from radius of a focussed or unfocussed spot to numerical aperture. – Carl Witthoft Mar 18 '14 at 11:57
• Is "attenuation" a typo? Did you mean to type "aperture" instead? – DumpsterDoofus Mar 18 '14 at 15:19
• @DumpsterDoofus haha yes I am so sorry for that:) Thank you so much for pointing it out. It has been changed now :) – Artemisia Mar 19 '14 at 17:08

It sounds as though you might not have "filled" the lens properly. An infinite conjugate microscope objective, for example, has a specified numerical aperture when it is driven by for a collimated beam of a specified beamwidth and apodisation (i.e. whether Gaussian, uniform or so forth). See my drawing below:

whence you can understand the fundamental relationship

$$NA = \frac{\frac{W}{2}}{\sqrt{\frac{W^2}{4}+f^2}}\approx \frac{W}{2\,f};\;NA\ll1$$

so that you can see that your measured numerical aperture is going to be proportional to the input beamwidth. So the manufacturer might be foreseeing an input beamwidth of $W$, whereas you might only be giving it $W_0$: the outcome is obvious from the drawing.

Some modern objectives have very wide beamwidths indeed: this makes for very long working distances / focal lengths whilst achieving high NA. Depending on your laser, your beamwidth may only be less than a millimetre (I'd say this is more than likely), whereas I think Zeiss and Olympus use about a 6mm beamwidth and Nikkon have even given up on the standard "Royal Screw" (Royal Microscope Society RMS thread) used for over 100 years as a standard to hold objectives on microscope turrets with and chosen instead to use much bigger bores and I think their beamwidth may be approaching 12mm.

Otherwsie, the simple measuring of numerical aperture as you have is in general not meaningful, even if you do fill the lens until you can't get any more light in, i.e. vignetting limits your input beamwidth. Manufacturers do NOT mean that the sine of the angle of the output light cone is the NA when the input fills up all the available space. What they mean is their objectives will achieve the rated aberration performance when they are filled such that the output lightcone is the quoted NA. Practically, you can often fill a lens a great deal more that its rated beamwidth, so if you measure NA by filling till you see vignetting, you may get an overly high value. Yes the lens can output this high NA, but it won't be meaningful for imaging because the aberration performance will be crap in this case.

You also need to be very careful of NA measurements with lasers because they output Gaussian beams whereas manufacturers generally talk about their specifications with a uniformly apodised beam. If you use a Gaussian beam, work out numerical apertures using the $1/e^2$ beamwidth, i.e. the diameter of the beam where its intensity has fallen to $1/e^2$ of its peak, on-axis intensity.

• @Artemisia No worries. – WetSavannaAnimal Mar 20 '14 at 1:45