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.” See below for an example:

The light from the lens comes to a focus and then expands again, as it should. The diameter of the illumination cone (called the spot size) is very small at the focus but it is larger at either side. The following figure, called a “through focus spot diagram”, shows its variation on either side of focus.

The central spot is at the focus of the lens and the spots 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. The F# of the optical system will determine how fast this happens. Any photographers reading this may be familiar with this term, but let me explain it for the others.

### f/#

f/# (pronounced “EFF number”) is the focal length of a lens divided by the diameter of the beam of light that enters it. If you’d like to learn more about that concept, head to this page. For a simple lens, such as the one pictured above, the diameter of the beam is the diameter of the lens. More complicated lenses, like those on digital cameras, often have an entering beam diameter that differs from the lens diameter. The lens above has a focal length that’s twice the diameter of the entering beam, so the f/# is 2. This is written as f/2 (pronounced EFF 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 represents data from the image’s center, red from the edge, and green from in between. To get these curves, the user specifies a certain spatial frequency that’s commonly half to three fourths of the maximum. Then the user 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 plot shows how fast the MTF falls off as the image plane moves from its optimal location.

### Interpreting the Data

Let’s start by observing the blue line. On it we can see that the MTF peaks at around 0.06 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. For more information on aberrations, click here. Assuming that an MTF of 0.2 is acceptable, the image plane can be moved about 0.4 units to the left of best focus or 0.16 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, known as “field curvature”, is caused by one of two aberrations: Petzval curvature or astigmatism. See our Aberrations page to learn about these. Also, the sagittal and tangential curves do not peak at the same focal position. 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. This means your optical engineer would tell you that the Petzval curvature is more of a problem than the astigmatism. If you can’t get the entire image in focus at the same time, this is likely the problem. If you focus on the edge of the image, the center looks fuzzy (and vice versa) as seen here.

### Measuring Petzval Field Curvature

We can measure Petzval field curvature several different ways, but we prefer to rely on the through focus MTF plot. It generally gives the most accurate representation of the real field curvature. See the two plots below as an example. One is a plot of the through focus MTF and the other is a simple field curvature plot of the same lens. Both of these plots are for monochromatic light at 550nm.

According to the Petzval field curvature plot, the total field curvature from the center of the image (0 degrees) to the edge (9.4 degrees) is about 0.23 mm. We can measure the Petzval curvature on the corresponding through focus MTF chart by measuring the distance from the peak of the 0 degree line to the peak of the 9.4 degree line. These peaks as seen on the plot are about 0.15mm apart. The field curvature plot exaggerates the real field curvature by 150%! This is because Petzval field curvature plots rely on approximations.

The Petzval curvature plot doesn’t always differ significantly from the through focus MTF plot, but it’s always a good idea to specify the maximum allowable focal shift in MTF when designing a custom lens. This is because just specifying the Petzval curvature approximation may differ significantly from the real field curvature.

### Uses for Through-Focus MTF

Through-focus MTF can be used to determine the depth of focus for a lens. In the above plot, all three fields have an MTF above 0.2 from -0.14 units to +0.07 units. This means that the total depth of focus is 0.21 units. If the units are mm, this would be very easy for a mechanical designer to accommodate, but if the units are microns, it would be exceedingly difficult. That’s why it’s important for your optical engineer to communicate this information to you as you design your optical system.