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Aberrations, defocus, diffraction, and the MTF

Doug Kerr

Well-known member
In this report, I take a rather different approach in showing how certain factor affecting resolution interact.

Readers may first wish to read this series of essays:

http://www.openphotographyforums.com/forums/showthread.php?p=145367#post145367
In this report, I will take the liberty of adapting just a couple of figures from Modern Optical Engineering, Second Edition, by Warren J. Smith. This is done under the doctrine of fair use.

Aberrations

By aberrations we mean, simplistically, the ways in which a real lens does not provide what we consider "theoretically ideal" image performance.

Some classical aberrations are spherical aberration and coma.

One general manifestation of most aberrations is that if we have a point on an object, its image will not be a point.

Misfocus

Misfocus can be considered a special type of aberration. It is not due to any imperfection on the behavior of the lens itself, but rather an error in how it is (at the moment) deployed in the camera

The basic manifestation of misfocus is that if we have a point on an object, its image will not be a point.

Diffraction

Diffraction is not generally considered to be an aberration, as it is not due to any "imperfect" behavior of the lens.

Nevertheless, the basic manifestation of misfocus is that if we have a point on an object, its image will not be a point.

The modulation transfer function

The modulation transfer function of an optical system tells us how the "contrast" of a region on an object (that is, the change in luminance through which detail is portrayed) is "conserved" into the final output of the system (perhaps into a digital image), as a function of many system parameters, one of which is the spatial frequency of the modulation of interest (which we can colloquially think of as being the quantitative "fineness" of the detail.

In our work here, we will look the modulation transfer function as plots of the modulation transfer ratio (often also called modulation transfer function) against spatial frequency.

The resolution of our camera

We speak of the resolution of a camera as being the spatial frequency of a test target that is just "nicely" resolved.

We typically find that to be about 0.375/p, where p is the sensel/pixel pitch (we assume that we take the camera's highest resolution output).

Diffraction and the diffraction cutoff frequency

We will consider the classical diffraction model, which begins with the assumption that the exit pupil of the lens is a virtual disk of uniform luminance. The result is that, for a point source on the object, the image will be what is called an Airy figure, which has a certain distribution of luminance as we move out from the axis. Its overall diameter is infinite!

The inevitable result of any situation in which the image of a point on the object is not a point is that the modulation transfer function will decline with increasing spatial frequency; this is therefore true of a system with diffraction (and of course any system has diffraction).

In figure 1, we see the plot of MTF vs. spatial frequency in the face of classical diffraction.

Fig_11-15-02.gif

Figure 1. MTF with classical diffraction.

We see that the MTF drops to zero (meaning that detail is no longer portrayed at all) at the spatial frequency noted at the lower right of the figure.
The lower-case lambda is the wavelength of the light being considered. "(f#)" represents the f-number of the lens.​
That frequency is considered the "diffraction cutoff frequency" of the system.

Suppose that our system aperture is f/8. Then the cutoff frequency (1.0 on the lower scale of spatial frequency) is 272.3 cy/mm.

We would of course not consider operation at anywhere near the diffraction cutoff frquency would be appropriate.

If our pixel pitch is 10 µm (0.010 mm), then the frequency we think of as representing the resolution of the sensor is 37.5 cy/mm. That is 0.137 times the diffraction cutoff frequency.

The effect of misfocus

Figure 2 shows effect on the MTF of a system with misfocus.

Fig_11-17-01.gif

Figure 2. MTF with misfocus.

The curves ignore the effect of diffraction (but the basis of the scale of spatial frequency is still the diffraction cutoff frequency, in order to allow the various charts to to be compared).

The six curves A-F represent different degrees of misfocus, as follows:

A: No misfocus (thus no decline in MTF).

B: Misfocus to produce a blur figure diameter of 0.0088 mm (in our example, 0.88 times the pixel pitch)

C: Misfocus to produce a blur figure diameter of 0.0176 mm (1.76 pixel pitch)

D: Misfocus to produce a blur figure diameter of 0.0352 mm

E: Misfocus to produce a blur figure diameter of 0.0704 mm

Consider misfocus curve B (blur circle of 0.88 times the pixel pitch). Consider the frequency we estimate as the basic resolution of the camera, 37.5 cy/mm. At that frequency, the MTF is degraded by misfocus by about 20%. We might think that is not significant.

Next consider misfocus curve C (blur circle of 1.76 times the pixel pitch). Again, consider the frequency we estimate as the basic resolution of the camera, 37.5 cy/mm. At that frequency, the MTF is degraded by misfocus by about 40%. We might think that is significant.

All this suggests that the choice of a COCDL of 1.0 pixel pitch as the premise for "outlook B" depth of focus planning is reasonable.

Best regards,

Doug
 
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