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A little bit about diffraction

Doug Kerr

Well-known member
[My original post has been greatly edited to correct an inappropriate premise I used to derive the "diffraction limit" relationship.]

Diffraction is an optical phenomenon in which, even with a perfect lens, a point on an object (that is in perfect focus) is imaged not as a point but as a blur figure. This blur figure, if of "consequential" extent, causes degradation of the image.

If the aperture is circular, this blur figure is also circular. In theory it has the form of an Airy disk (often called Airy circle). It has the greatest luminance at its center. The luminance then declines until it reaches zero at a certain radius. Then it increases, to a much lower value, and then decreases again, this process in theory continuing to an infinite radius.

It turns out that this "diameter" of the Airy disk is a function both of the relative aperture (the formal name for the lens parameter we characterize with the f-number) and the wavelength of the light involved. Of course for an actual situation, the light from a point comprises a range of wavelengths (which is in fact part of the whole story here). But let's consider, to keep this manageable, light of a wavelength of 500 nm (a light wavelength in the "green" portion of the spectrum).

Then, if the object is at a very great distance and the camera is focused at that distance, the "diameter" of the Airy disk will be given by:

A = 1.22 N​
where A is the "diameter" of the Airy disk (in µm) and N is the f-number of the aperture.

As we can see from that formula, the larger is the f-number, the greater will be the diameter of the Airy disk. And we can certainly realize that the larger the diameter of the Airy disk, the more the resulting diffraction will contribute to a "blurring" of our image. which we can think of as a decline in resolution compared to what we would otherwise have.

So it is said that if we try to use an aperture smaller than a certain value, the effect of diffraction will be "bad enough to be 'unacceptable' ".

But what should be our criterion for the effect of diffraction being considered "unacceptable"?

Simplistically, we say that we consider an "unacceptable" effect of diffraction to be when the diffraction substantially reduces the resolution of the system below its otherwise-attainable resolution.

But how do we define resolution? It seems that the custom is to consider the "resolution" of the system to be the spatial frequency at which the MTF is 50% of what it is for low spatial frequencies (a definition commonly used in other contexts).

So the "diffraction limit" f-number is that leading to an Airy disk of such size that it would just limit the resolution (as defined just above) to the resolution value we would have in the absence of diffraction.

[Continued]
 
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Doug Kerr

Well-known member
[This post has also been amended as discussed in the original post of this thread.]

[Part 2]

Tables showing how this works have been calculated by various organizations, Here is one useful one (I believe done by Luminous Landscape):


Suppose we have a camera (such as my Panasonic ZS100) with a pixel pitch of 2.4 µm. The "geometric" resolution that would lead to would be 208 line pairs/mm. But such factors as the Kell effect mean that the actual resolution we might attain might only be about 175 lp/mm.

Again, using the "MTF of 50% of the low frequency value" definition of resolution (the second column from the right), we can see that this "potential resolution" would be sustained in the face of diffraction for an f-number not over f/4.

If we then consider a camera with a pixel pitch of 4.8 µm, the "geometric" resolution that would lead to would be 104 line pairs/mm. But such factors as the Kell effect mean that the actual resolution we might attain might only be about 85 lp/mm.

Again, using the "MTF of 50% of the low frequency value" definition of resolution, we can see that this "potential resolution" would be sustained in the face of diffraction for an f-number not over f/8.

In fact, we see that the function in this column is linear with f-number. If we plug in a typical relationship between sensel pitch and resolution, we can generalize the relationship this way:

N = 1.66p​

where N is the "diffraction limit" f-number and p is the pixel pitch in µm.

Best regards,

Doug
 
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Asher Kelman

OPF Owner/Editor-in-Chief
Doug,

As I promised Bart’s writing here and a further discussion and extension here.

You may have already thought about this aspects, but these are here nonetheless just for reference!

Asher
 

Asher Kelman

OPF Owner/Editor-in-Chief
Next, we need Bart Van Der Wolfe's wonderful insight into sense pitch Raleigh thoughts and wave theory...and his table!

Asher
 
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