Doug Kerr
Well-known member
Often, especially in connection with discussions of the phenomenon of diffraction and its effect on the "resolution" of a photographic system, we find references to the "Dawes criterion" (or limit) and (more frequently) the "Rayleigh criterion" (or limit).
I thought I would talk about what these are and what is their connection with our usual interests (the answer to the latter will turn out to be, "not much").
Resolution
We often speak of the resolution of a photographic system. The term comes from the verb "to resolve", which in this context means "to distinguish; to recognize as distinct."
We generally use it to mean "the ability of the system to capture fine detail" (and the actual meaning of the term does not well fit that), and of course there is no single definition of exactly how we would quantify that (the source of a lot of grief in this area of work).
The basis of the term is in fact most clearly applicable to a matter of historical concern in astronomical optics (a field that of course existed long before there was photography). An important part of that field deals with objects that are point sources (namely distant stars): objects whose angular size from the telescope location is infinitesimal.
There is thus no concept of being able to capture the detail on the face of a star (as we would be concerned with if, for example, observing the moon or another planet in the Solar System).
Rather, the concern was with the ability to recognize as separate stars two stars that were very close together in angular terms. This was spoken of as the ability to "resolve" the two stars. Telescope makers strove to improve the ability of their instruments to resolve stars separated by smaller and smaller angles (so that more stars could be discovered and cataloged, and perhaps named after the astronomers' girlfriends).
Diffraction
Early astronomers did not have photographic plates on which to record the images, but instead had to observe them visually. In order for stars with a low photometric intensity to be seen, we needed a telescope with a large "light-gathering capability", which meant a large entrance pupil, which meant a large objective lens or primary mirror.
Of course, achieving this was costly, in time and money. But a further benefit was soon noticed, and the reason for it demonstrated mathematically. This was the impact of the phenomenon of diffraction.
If the exit pupil of the telescope was not of infinite diameter, then even if the instrument were optically perfect, the image of a point source object would not be a point of light but rather a finite figure. The distribution of illuminance across this figure is the familiar Airy figure.
This "spreading" of the image illuminance would of course interfere with our ability to visually distinguish two stars with small angular separation.
The larger the diameter of the objective lens or primary mirror, that smaller is the Airy figure (measured out to any feature, such as the first minimum - of course in theory the overall diameter of the figure was always infinite). Thus a larger instrument could allow us to not only see "dimmer" stars but in fact to recognize as distinct stars separated by smaller angular distances. As is so often the case, "inches count."
Astronomers (or their financial backers) were keen to know how large a diameter instrument would be needed to resolve stars at some certain small angular separation.
But (and this should have a familiar ring to it), how "distinct" did the components of the image from the two stars have to be before we would say, "ah, yes, this instrument resolves two stars at <some stated angular separation>."
For example, if we saw the first of these three images:
Courtesy of Wikimedia Commons
we could without any doubt say, "that's two stars, all right - I can see 'em both." If we saw the second, we would probably say, "that must be two stars, all right." If we saw the third, we might say, "That might be two stars."
So which of these fulfills the concept of "distinguishing" the two stars, rather than just "there are probably two stars there, but I can't distinguish them?"
Dawes' criterion
The British astronomer William Rutter Dawes, interested in this matter, conducted extensive tests, using various pairs of nearby stars with similar intensities (between the pair), but with different angular separations, observed through instruments of different diameters by numerous observers. In each trail, the observer was asked whether he saw two nearby stars or only one star.
From the results of these tests, Dawes concluded that typically the two stars could be "resolved" if their angular separation was at least 5.61×10^-7/D, where D is the aperture diameter in meters.
Now, if we consider a wavelength of 555 nm (considered "typical for visible light"), we find that the "radius" of the Airy figure from diffraction (the distance to the first "minimum", or dark band) is 5.61×10^-7/D.
Thus, Dawes' result is that typically the two stars can be considered resolved if the spacing of the centers of their images as ay least 0.82 times the radius of the Airy figure resulting from the expected diffraction for the instrument of interest.
This shows such an image:
Courtesy of Wikimedia Commons
Rayleigh's criterion
The British physicist Lord Rayleigh (John William Strutt, 3rd Baron Rayleigh) considered Dawes' criterion to be too optimistic, and felt that in predicting the resolving capability of telescopes we should look to a more foolproof visible distinction - that is, use as our criterion for the separation of the two stars' images a larger fraction on the radius of the Airy disk than Dawes had adopted.
Various thoughts lead him to a certain "neighborhood" for the value, and, being a practical fellow, in order to make all the math more "handy", he adopted as the minimum separation of two stars that would be said to be resolved by an instrument with certain diffraction precisely the radius of the Airy disk.
This separation is in fact that shown in the middle panel of the first illustration above.
What does this mean to us?
How does all this fit in with our quest to find a way to describe "how much diffraction can we have before it significantly degrades our image from what it would be with negligible diffraction?"
Not really at all. Our concern (usually) is not with whether the viewer of our image can tell whether a little figure on the image comes from a point object or two nearby point objects.
So if we read in some article that "this guideline [some suggested rule of thumb for planning a shot in terms of the impact of diffraction] is consistent with the Rayleigh limit", that has no real meaning. With a stretch, we can usually concoct a relationship, generally from the fact that the "rule of thumb" may suggest a sensel pitch with the radius of the Airy figure. Then, we could fancifully visualize the images of two stars centered on adjacent sensels.
Other connections can be made with the interpretation of the effect of diffraction in terms of an MTF. But they are all forced and without real meaning.
Dawes and Rayleigh made important contributions. Let's not demean them by struggling to apply them where they do not fit.
Time for breakfast.
Best regards,
Doug
I thought I would talk about what these are and what is their connection with our usual interests (the answer to the latter will turn out to be, "not much").
Resolution
We often speak of the resolution of a photographic system. The term comes from the verb "to resolve", which in this context means "to distinguish; to recognize as distinct."
We generally use it to mean "the ability of the system to capture fine detail" (and the actual meaning of the term does not well fit that), and of course there is no single definition of exactly how we would quantify that (the source of a lot of grief in this area of work).
The basis of the term is in fact most clearly applicable to a matter of historical concern in astronomical optics (a field that of course existed long before there was photography). An important part of that field deals with objects that are point sources (namely distant stars): objects whose angular size from the telescope location is infinitesimal.
There is thus no concept of being able to capture the detail on the face of a star (as we would be concerned with if, for example, observing the moon or another planet in the Solar System).
Rather, the concern was with the ability to recognize as separate stars two stars that were very close together in angular terms. This was spoken of as the ability to "resolve" the two stars. Telescope makers strove to improve the ability of their instruments to resolve stars separated by smaller and smaller angles (so that more stars could be discovered and cataloged, and perhaps named after the astronomers' girlfriends).
Diffraction
Early astronomers did not have photographic plates on which to record the images, but instead had to observe them visually. In order for stars with a low photometric intensity to be seen, we needed a telescope with a large "light-gathering capability", which meant a large entrance pupil, which meant a large objective lens or primary mirror.
Of course, achieving this was costly, in time and money. But a further benefit was soon noticed, and the reason for it demonstrated mathematically. This was the impact of the phenomenon of diffraction.
If the exit pupil of the telescope was not of infinite diameter, then even if the instrument were optically perfect, the image of a point source object would not be a point of light but rather a finite figure. The distribution of illuminance across this figure is the familiar Airy figure.
This "spreading" of the image illuminance would of course interfere with our ability to visually distinguish two stars with small angular separation.
The larger the diameter of the objective lens or primary mirror, that smaller is the Airy figure (measured out to any feature, such as the first minimum - of course in theory the overall diameter of the figure was always infinite). Thus a larger instrument could allow us to not only see "dimmer" stars but in fact to recognize as distinct stars separated by smaller angular distances. As is so often the case, "inches count."
Astronomers (or their financial backers) were keen to know how large a diameter instrument would be needed to resolve stars at some certain small angular separation.
But (and this should have a familiar ring to it), how "distinct" did the components of the image from the two stars have to be before we would say, "ah, yes, this instrument resolves two stars at <some stated angular separation>."
For example, if we saw the first of these three images:
Courtesy of Wikimedia Commons
we could without any doubt say, "that's two stars, all right - I can see 'em both." If we saw the second, we would probably say, "that must be two stars, all right." If we saw the third, we might say, "That might be two stars."
So which of these fulfills the concept of "distinguishing" the two stars, rather than just "there are probably two stars there, but I can't distinguish them?"
Dawes' criterion
The British astronomer William Rutter Dawes, interested in this matter, conducted extensive tests, using various pairs of nearby stars with similar intensities (between the pair), but with different angular separations, observed through instruments of different diameters by numerous observers. In each trail, the observer was asked whether he saw two nearby stars or only one star.
From the results of these tests, Dawes concluded that typically the two stars could be "resolved" if their angular separation was at least 5.61×10^-7/D, where D is the aperture diameter in meters.
Now, if we consider a wavelength of 555 nm (considered "typical for visible light"), we find that the "radius" of the Airy figure from diffraction (the distance to the first "minimum", or dark band) is 5.61×10^-7/D.
Thus, Dawes' result is that typically the two stars can be considered resolved if the spacing of the centers of their images as ay least 0.82 times the radius of the Airy figure resulting from the expected diffraction for the instrument of interest.
This shows such an image:
Courtesy of Wikimedia Commons
Rayleigh's criterion
The British physicist Lord Rayleigh (John William Strutt, 3rd Baron Rayleigh) considered Dawes' criterion to be too optimistic, and felt that in predicting the resolving capability of telescopes we should look to a more foolproof visible distinction - that is, use as our criterion for the separation of the two stars' images a larger fraction on the radius of the Airy disk than Dawes had adopted.
Various thoughts lead him to a certain "neighborhood" for the value, and, being a practical fellow, in order to make all the math more "handy", he adopted as the minimum separation of two stars that would be said to be resolved by an instrument with certain diffraction precisely the radius of the Airy disk.
This separation is in fact that shown in the middle panel of the first illustration above.
What does this mean to us?
How does all this fit in with our quest to find a way to describe "how much diffraction can we have before it significantly degrades our image from what it would be with negligible diffraction?"
Not really at all. Our concern (usually) is not with whether the viewer of our image can tell whether a little figure on the image comes from a point object or two nearby point objects.
So if we read in some article that "this guideline [some suggested rule of thumb for planning a shot in terms of the impact of diffraction] is consistent with the Rayleigh limit", that has no real meaning. With a stretch, we can usually concoct a relationship, generally from the fact that the "rule of thumb" may suggest a sensel pitch with the radius of the Airy figure. Then, we could fancifully visualize the images of two stars centered on adjacent sensels.
Other connections can be made with the interpretation of the effect of diffraction in terms of an MTF. But they are all forced and without real meaning.
Dawes and Rayleigh made important contributions. Let's not demean them by struggling to apply them where they do not fit.
Time for breakfast.
Best regards,
Doug