@tvdpid
Merklinger first describes how lenses work, how circles-of-confusion arise and how those circles affect DOF. Then he presents a second way of thinking about the situation and these same factors. In his first chapters, he regards a lens as making a negative, and he shows how to compute the characteristics of that image. In the second way in later chapters, he imagines the lens as PROJECTING a negative into object space and analyses the projected image. That projected negative is taken to have the same circles-of-confusion that are associated with the lens settings in the first way. That is the focal length, f-stop and focused distance are the same. So the projection of that negative will magnify the circles-of-confusion into the projected scene, and these magnified circles-of-confusions become Merklinger's disks-of-confusion, and he proceeds to calculate and interpret them. When he describes this second way, it is not correct to interpret that he "completely leaves the theories and ideas about any circle of confusion." Metaphorically speaking, he is considering the "other side of the coin." He imagines running the light backwards, and he hopes that doing so will be a useful alternative way of understanding how things work. The circles-of-confusion and their consequences are entirely governed by physical optics, they are always present and relevant, and there is no way to circumvent them. All one can do is to strive for the best (personal) way of thinking about and understanding the situation, and he offers another perspective in his second way.
I believe that one of the greatest problems with expositions of lenses, CoC and DOF (including Merklinger's and all others that I have seen) resides in how all the explanations confuse four conceptually different circles: they use the same name, the circle-of-confusion, for all four of these different circles. Using the same name for all, this conflation makes all those explantions confusing to many people, and it leads to seemingly endless misunderstandings and arguments. The four different circles are the following. There are, first and foremost, the "circles-on-the-negative" that lenses produce from objects. These circles exist in different sizes for objects at different distances from the lens. Second, there are the "circles-on-the-print" that are produced when the "circles-on-the-negative" are magnified to make the print. Third, there is the "smallest-observable-circle-on-the-print" that one can actually resolve when one looks at the print. Fourth and finally, there is the size of the "smallest-observable-circle-on-the-negative" which is the DE-magnified size of the "smallest-observable-circle-on-the-print". The first two of these circles are related in size by the print magnification. The third and fourth circles are again related in size by the print magnification. The sizes of the first, second and fourth circles are determined by physical optics. The size of the third circle is jointly determined by the magnification of the negative, by the visual acuity of the viewer and by the viewing distance.
Strictly speaking, it is only the size of the fourth circle that should be called the circle-of-confusion and which should be involved in determining DOF. After all, if you can resolve an object's circle in a print, then you will perceive it as out-of-focus; if you can't resolve its circle, then it will appear in-focus, and it will be within the DOF for that print and under those viewing conditions. This fourth circle is dividing line between "circles-on-the-negative" that you can see in the print and those that you can't and it determines the DOF. It is also the circle that lens manufacturers like Leica and Zeiss, etc employ and for which they specify a value when they determine by calculations where to put the DOF marks on their lenses. Not every manufacturer uses the exact same value for this fourth circle, but every one uses a value which is close to one-thirtieth of a millimeter. Unfortunately, the manufacturers don't often tell us what they've done when they publish DOF tables/graphs in their lens data sheets.
When Merklinger introduces his "disk-of-confusion" he is introducing a fifth circle which I described above. I don't find Merklinger's fifth circle to be useful, but that doesn't make it wrong. It is completely consistent with optical physics, and other readers may find it revelatory.
Merklinger wrote before the advent of digital technology, and he wrote almost entirely about the situation for 35mm full-frame negatives. The principles and results he discusses apply without sigificant changes to full-frame digital sensors. However, many people now use APS-C sized digital sensors which are 2/3 the linear dimensions of full-frame, and considering those requires some adjustments. In particular, Merklinger's results assume three things about a print: 1) the print size will be 8x10inch (20x25cm); 2) the magnification will be about 8x; 3) the viewing distance will be 250mm. But an 8x magnification from an APS-C image will produce only a 5.3x6.67inch print. On this print, the standard, one-thirtieth mm, circle-of-confusion will just be observable, so the DOF will match that from a full-frame sensor's identical circle-of-confusion when both are viewed from 250mm. The APS-C sensor will require a 12x magnification to produce an 8x10 print. In that case, the one-thirtieth mm circle-of-confusion will be 1.5x bigger than expected and will be easily visible. The "smallest-observable-circle-on-the-print" will be smaller, the corresponding "smallest-observable-circle-on-the-negative" will be smaller. Since one will be seeing smaller circles in the print, the DOF of this APS-C print will consequently be less than Merklinger calculates. The lesson from this is that using an APS-C sensor requires one to work with a circle-of-confusion that is 2/3 the size (2/3 * 1/30mm = 1/45mm) that Merklinger has assumed. Merklinger's equations all still apply to the APS-C sensor, but one must change the values of one or more of the values he assumes to get valid results. That is hard to accomplish in detail, because he has suppressed the relevant variables by assuming them to take the standard values appropriate only to full-frome negatives/sensors, just as the DOF marks on lenses are made using those same assumptions. But one has two easy choices available: one can take his result liteally by considering 1) 5.3x6.67inch prints from APS-C sensors; 2) 8x10inch prints but substitute 1/45mm for the circle-of-confusion everywhere in his discussion. Interpreting the DOF marks on lenses for the APS-C world involves detail that Merklinger has suppressed, so it is a complicated subject for another day.
--- Mike
