# Through Focus MTF

Now that you understand MTF, the next important concept is how refocusing a lens affects it. This is called the through focus MTF.  It is important that you get this information from your optical engineer, because without it you could end up with an optical system that does not meet your needs.

# Through Focus Spot Diagram

The first basic principle of through-focus MTF is the geometrical limitation. Any lens that focuses light produces an “illumination cone”. An example is shown below.

The light from the lens comes to a focus and then expands again, as it should. The diameter of the illumination cone at the focus is very small, but it is larger at either side. The diameter of the illumination cone is also known as the spot size. Its variation on either side of focus is shown in the following figure, which is known as a “through focus spot diagram”.

The central spot is at the focus of the lens and the ones to the left and right correspond to the illumination cone before (to the left) and after (to the right) focus. As you can see, the spot size increases dramatically as you move away from focus. This forms an upper limit on the through focus MTF.  How fast this happens depends on the F/# of the optical system. Those readers who are familiar with photography may well be familiar with this term, but let me explain it for the others.

F/# (pronounced “F number”) is the focal length of a lens divided by the diameter of the beam of light that enters it. For a simple lens, such as the one pictured above, the diameter of the beam is the diameter of the lens. More complicated lenses, such as one would find on a digital camera, often have an entering beam diameter that differs significantly from the diameter of the lens. The lens pictured above has a focal length that is twice the diameter of the entering beam, so the F/# is 2, which would be written F/2 (pronounced F two).

The spot size at a given distance from focus can never be smaller than the diameter calculated from the F/#. Take, for example, the F/2 lens shown above. The spot size 1 mm away from the focus can never be smaller than 1/2 mm. If the lens were F/2.5, the spot could be no smaller than 1/2.5 or 0.4 mm. Applying what we learned about MTF, the MTF at 2 cy/mm would be essentially zero for an F/2 lens 1 mm from focus. An F/2.5 lens would have a nonzero 2 cy/mm MTF at 1 mm from focus, but it would drop to zero at 1.25 mm from focus.

# Through Focus MTF for a Real Lens

Now that you understand the theoretical limit of through focus MTF, it’s time to get practical. A picture of the through-focus MTF of a real lens is displayed below.

As before, the blue curve is data for the center of the image, while red is at the edge and green is in between. To get these curves, the user specifies a certain spatial frequency (commonly half to three fourths of the maximum) and has the lens design software plot the variation of the MTF at this spatial frequency versus focus. The zero point is typically the “best focus” for the lens. The through focus MTF plot shows how fast the MTF falls off as the image plane is moved from its optimal location.

Starting with blue, we can see that the MTF peaks roughly 0.05 units to the right of the “best focus” and drops off on either side. The dropoff is not symmetric, which is common for systems with spherical aberration (see the discussion of aberrations). Assuming that an MTF of 0.2 is acceptable, the image plane can be moved about 0.1 units to the left of best focus or 0.18 units to the right before the resolution at the center of the image is no longer acceptable.

Unfortunately, the best focus for the light at the edge of the image (red) and even part way out (green) does not lie at the same place as it does for the center of the image. This condition is known as “field curvature”, and is caused by one of two aberrations: Petzval curvature or astigmatism (see the page on aberrations for a description of these aberrations). In addition, the sagittal and tangential curves do not peak at the same place. This is typical of astigmatism. However, the difference between the sagittal and tangential curves is much less than the difference between the center and edge of the image, so your optical engineer would tell you that the Petzval curvature is more of a problem than the astigmatism. The way this would show up in an image is that you can’t get the entire image in focus at the same time. If you focus on the edge of the image, the center looks fuzzy and vice versa.

Through-focus MTF can be used to determine the depth of focus for a lens. In the above example, all three fields have an MTF above 0.2 from -0.10 units to +0.02 units. This means that the total depth of focus is 0.12 units. If the units are mm, this would be very easy for a mechanical designer to accomodate, but if the units are microns, it would be exceedingly difficult. That is why it is important for your optical engineer to communicate this information to you as you design your optical system.