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.

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One final pointer: 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. Tolerances are discussed on another page.

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. It is important to remember that spatial frequency increases as the period gets smaller.

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

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Tangential: related to resolution for periodic patterns with the period tangent to a circle centered on the optical axis.

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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.

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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.

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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:

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:

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Based on these definitions, we can state that MTF is a plot of contrast versus spatial frequency. 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.

MTF stands for Modulation Transfer Function. It is a function that calculates the sharpness or fuzziness of an image. It is important to understand several terms before we can fully describe MTF.

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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 Saggital 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.

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So how does that relate to spatial frequency? Actually, dpi is spacial 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.

Saggital: related to resolution for periodic patterns with the period parallel to a line through the center of the image.

Spatial Frequency is defined as one divided by the period of a repeated structure. A picture would probably be helpful here.