We are often asked, “What MTF is good enough?” The answer is, of course, “it depends…” But a good rule of thumb is that when the MTF drops below 0.2 (20%), it is probably unacceptable. Beyond that, lenses are rarely as good as the ideal model in the computer. How much difference depends on the manufacturer and the tightness of tolerances. So, there is a short answer, but perhaps breaking down what MTF is will provide more depth on this popular subject.
As you may already know, MTF stands for Modulation Transfer Function. It is a function that calculates the sharpness or fuzziness of an image.
Spatial Frequency is defined as one divided by the period of a repeated structure. A picture would probably be helpful here.
In the square wave pictured above, the spatial frequency would be 1.0 divided by the distance shown. For example, if one period is 10mm, the spatial frequency would be 0.1 cycles/mm, sometimes written 0.1cy/mm. For square waves or pixels, a cycle is a “line pair” (one bright line next to a dark one on a display), so line pairs/mm or lp/mm means roughly the same thing. It is important to remember that spatial frequency increases as the period gets smaller.
Another way to get a picture of spatial frequency is to think in terms of dots per inch (dpi). Computers went through a revolution when Hewlett Packard introduced its first laser printer that produced 300 dpi. A few years later 600 dpi became the norm and now printers put out so many dpi that we don’t care anymore. The same thing is happening in cell phones. My first cell phone had a 1.75″ display with 320 x 240 resolution. That works out to 230 dpi. Now you can get a cell phone with over 400 dpi, so the picture is much sharper.
So how does that relate to spatial frequency? Actually, dpi is spatial frequency! Let’s change the units to make this clear. To make one cycle, we need two pixels so one can be on and the other off. This means that 400 dpi = 200 cycles per inch. Since optics is done in metric, we still need to convert to millimeters. There are roughly 25 mm in 1 inch, so we just divide cycles per inch by 25. Thus 400 dpi = 200 cy/in = 8 cy/mm.
Modulation: Assume the above square wave plot is the amount of light in an image, and the peaks represent bright portions, and the valleys represent dark portions. Let’s call the amount of light in the peaks “max” and the amount in the valleys “min”. Then the modulation, or contrast, is given by:
\( modulation = \frac{max-min}{max+min} \)
From this equation, you can see that if min = 0, modulation will be 1.0, and if max = min, the contrast will be 0.0. When the modulation is near one, the contrast is very high, and the image looks sharp. Conversely, when the modulation is near zero, it is so blurred that it is difficult or impossible to see details.
Based on these definitions, we can state that MTF is a plot of modulation versus spatial frequency.
At some point or another we have all been introduced to the eye chart, officially the Snellen chart. When our vision is tested, almost everyone can read the E at the top of the chart and people with 20/20 vision can read the eighth line down. But as you go down the chart, each line gets progressively more difficult to read.
Going down the chart, to smaller and smaller letters, corresponds to “spatial frequency” increasing. Letters getting more difficult to read corresponds to the “contrast” or “modulation” decreasing. So, a typical MTF chart shows a line that starts at 1 and slopes down to the right, which means that modulation or contrast is decreasing as spatial frequency increases.
Experience tells us that smaller details are harder to see. In terms of MTF, this is expressed by saying that the modulation decreases as spatial frequency increases. Somewhat surprisingly, this is true even for lenses that are perfect. Such lenses are called “diffraction limited” because a physical phenomenon called diffraction keeps them from resolving infinitesimal detail. A plot of MTF for a perfect lens is shown below.
Note that this curve is different for every wavelength and F/# (or NA). However, the only thing that changes is the scale of the abcissa (x-axis). The cutoff frequency is given by:
\(\nu_c = \frac{1}{\lambda (f/\#)}\)
If a lens has any aberrations, either by design or because of manufacturing variations, the MTF will be less than diffraction limited. For aberrations that amount to less than roughly a quarter of a wavelength of light, the MTF drops by the same amount regardless of the type of aberration.
The plot below shows how the MTF of an optical system drops as a function of the amount of aberration. Although it is for an F/8 lens at a wavelength of 0.55 µm, you can use the above formula to scale the spatial frequency axis to match your lens. The plot doesn’t change, just the spatial frequency scale.
Obscuration affects the diffraction limit of optical systems. It rarely applies to lenses, but is quite common in reflecting and catadioptric systems. The obscuration percent refers to the diameter of the component that blocks light compared to the diameter of the beam of light.
The chart below shows that for 10% obscuration, the MTF scarcely differs from the 0% case, but by the time obscuration reaches 50%, the MTF drops dramatically. Off-axis systems avoid this problem, but they are far more difficult and expensive to manufacture than rotationally-symmetric ones.
Some types of lenses, such as telescopes and microscopes, commonly reach the diffraction limit. Others, for example camera lenses, rarely approach the diffraction limit. A camera lens might have an MTF plot like the one below:
Note that there are now five curves on the plot rather than just one and also that the maximum spatial frequency is only 100 cy/mm rather than 840. The blue curve (labeled TS 0.00 deg) is for the center of the image. Before discussing the green and red curves, we need two more definitions.
Tangential: related to resolution for periodic patterns with the period tangent to a circle centered on the optical axis.
Sagittal: related to resolution for periodic patterns with the period parallel to a line through the center of the image.
As you may have surmised by now, periodic patterns away from the center of the image have an MTF that depends on their orientation. If you orient a periodic pattern of parallel lines so one of the lines passes through the center of the image, the MTF of that pattern will be determined by the Sagittal MTF curve, which is the one labeled S.
MTF also varies with the distance from the axis of symmetry (optical axis) of the system. That’s why there are separate curves for 0.00 deg (the optical axis or center of the image), 14 deg and 20 deg. This particular lens was analyzed by tracing rays through the lens that made a 14 degree angle with the optical axis and another set at 20 degrees. In this lens, things don’t get much worse between 14 and 20 degrees, but they are different. Other lenses will behave differently, depending on the aberrations of the optical system. But that’s another topic.
Another way to look at the MTF of a lens is in terms of information transfer. At very low spatial frequencies, nearly all of the information gets transferred by the lens from the object being viewed to the camera pixels – the MTF is 1 (100% transfer). At high spatial frequencies (very fine detail), the MTF drops to 0 (0% transfer), which means that no information is being transferred to the pixels by the lens.
In addition to the amount of information transferred by the lens, the receiver of that information must be considered. This concept is the rarely discussed Aerial Image Modulation (AIM) curve. For pixels, the AIM curve is given by a sinc (sin x / x) curve. Again, at very low spatial frequencies, no information is lost. At the Nyquist frequency (500 divided by the pixel width in microns), 36% [1 – sin(π/2) / (π/2)] of the information is lost, and at twice the Nyquist frequency, all information is lost. Beyond that, there is aliasing, but let’s ignore that for now.
We can define MTF, draw apt analogies, and explore its deeper implications and applications. The fundamental question our customers want to know concerns the overall quality of their lenses or design. While MTF performance has proven to be an important metric for high-resolution imaging, there is always more to consider.
It may be sufficient to specify the MTF at the Nyquist frequency of your image sensor, but that begs a question about intermediate spatial frequencies. In the end, having an experienced team of engineers behind you can give your next lens a competitive advantage.
To learn more about how we MTF test our lenses, click here. As always, we’re interested in finding creative ways to solve new and challenging problems. If we can help you with your next project, let us know!