How NOT to get fooled by the results of aerial image resolution testing of lenses

The resolution of a lens can be tested in a number of different ways. One way some people have chosen is to measure the resolution in the aerial image formed by the lens. To do this a resolution chart is used as the target and the image formed in the film plane is examined by a microscope. This does away with the need for film, but the results of aerial image resolution testing may be highly misleading if you don't realize all the factors involved.

There is a formula for diffraction limited (perfect) lenses which tells you that the maximum resolution of a lens is given by (x/f-stop), where x is somewhere in the 1600-1800 range (depending on exactly what assumptions are made regarding the wavelength of light, what exactly constitutes "resolved" etc.). If a lens tests out to show the maximum theoretical resolution at any given aperture (let's say it shows 400 lp/mm at f4) then some people have concluded the lens must be close to diffraction limited. After all, if a lens resolves line patterns at the theoretical diffraction limite, then it must be diffraction limited, right? WRONG!. Nothing of the sort is true. A lens can be severely aberrated and still resolve line patterns close to the diffraction limit in the aerial image. This is not intuitive, but nevertheless it's true.

The figure below shows the MTF of an f2.8 lens with various amounts of wavefront error. MTF is a measure of the contrast of a lens at any particular spatial frequency (lp/mm reading). It is related to resolution, since resolution is defined as the spatial frequency at which MTF drops below some given value. It's not the same as overall "image contrast", which is basically related to the MTF at low spatial frequencies (say 1-10 lp/mm). Spatial frequency should, strictly speaking, be given in cycles/mm for a sine wave pattern, but for practical purposes it's OK to approximate it with lp/mm and a square wave pattern (i.e. a bar pattern such as is found on most resolution test charts). Wavefront error is simply a convenient way of defining the amount of aberation present - a measure of how close to "diffraction limited" or "perfect" the performance of the lens is. A perfect lens has 0 wavefront error. A wavefront error of 0.25 wavelengths is good. It's about the error of a pretty decent astronomical telescope. A really good telescope may have only 0.1 wavelenghts of wavefront error. At 0.5 wavelenghts of wavefront error image degradation becomes quite noticable. At 0.75 and 1 wavelength of wavefront error, image degradation is very noticable. Studies of a number of 50mm lenses have shown that wide open, fast lenses can easily have 1 wavelength of wavefront error.

Now look closely at the figure, especially at the region at which the MTF falls close to zero. Where the MTF drops below some very small value is the point at which the resolution of the lens is reached. How close to zero the MTF must drop depends on a number of things such as the brightness of the image and the nature of the target pattern, but 1% (0.01) isn't an unreasonable value for a bright image and a high contrast bar pattern. As you can see, even with very large aberrations (large wavefront errors), the MTF at frequencies close to the resolution limit hardly changes! Large wavefront errors don't greatly affect the "ultimate" resolution of the lens, though they make it FAR from "diffraction limited". An expansion of the high frequency region of the plot is shown below.

As you stop down, wavefront error tends to decrease, though as you can see from the "x/f-stop" formula, diffraction limited resolution also decreases. What this means is that up to some point, stopping down will improve lens performance. Past that point, diffraction (x/f-stop) will start to decrease image quality (lower resolution). The plot below shows the results for an f2.8 lens with 0.6 wavelengths of wavefront error (perhaps a typical fast 50mm lens) and an f5.6 lens with no wavefront error. As you can see, although the ultimate resolution of the lens has dropped (in fact by a factor of 2), the performance of the lens in the region which matters (film can only record data in the region from 0-150 lp/mm) has greatly improved.

The final conclusion I would like you to come away with is as follows:

You cannot make any judgements about the imaging ability of a lens (how close to "diffraction limited" it is) by measuring the maximum resolution in the aerial image

Measuring the resolution in the aerial image may be interesting, and indeed, if you know what you are looking for you can make some judgements about some aspects of the quality of the lens. However you can't measure imaging quality or the level of aberration in a lens from the MTF cutoff frequency (the "resolution").