I find that the smaller the pixel-size, the greater the spatial sampling-rate - therefore the more resolved is the blur for a given aperture-diameter.
There is a phrase often used in lens reviews that diffraction "sets in" above some stated f-number, completely ignoring the effect of pixel-size on diffraction-visibility in the captured image.
Your formula for the disc of confusion is not quite correct. For it to work correctly, D must be the distance from the OOF object to the aperture. Here is a diagram that (hopefully) will make things clearer:
P is a point on the surface of the moon. The point P appears as a blur that extends from Q to R in the plane of focus (the mountain range 4km away).
E is the distance from the OOF point P to the plane of focus, while D is the distance from the OOF point P to the aperture (indicated by the line from T to U).
S is the distance from Q to R, which is the size of the blur produced by the OOF point P in the plane of focus, i.e. the disc of confusion.
The plane of focus (subject plane) is parallel to the plane of the aperture, hence the triangles PQR and PTU are similar.
Hence S = E/D*d.
For your example, d = 45mm, D = 400,000km and E = 399,996km, so S = 45mm. This is the disc of confusion produced in the plane of focus (4km away from the camera). 45mm will not be a visible blur at 4km distance, so the moon will look sharp if the camera is focused on mountains 4km away.
If the plane of focus is at the surface of the moon and the mountains 4km away are now OOF, then D (the distance from the OOF point to the aperture) is now 4km.
E is the distance from the OOF point to the plane of focus, so E is now 399,996km. Hence the disk of confusion is now 399,996/4*45mm, which is 4.5km (near enough). Now 4.5km is too small a distance to be distinguished on the surface of the moon (with an ordinary camera and 1000mm f/22 lens), so the mountains at 4km distance will also appear sharp.
This is Merklinger's diagram that his formula and my post is based upon:
Your D is a different distance, so the same formula gives a different result. I'll stick with Merklinger, thank you ...
Merklinger's diagram is essentially the same as mine, but you have misunderstood what it represents. His "point of exact focus" is the position of the moon, the point which you aim to focus on, i.e. the point of exact focus to get the sharpest possible shot of the moon. What his diagram illustrates is the size of the blur if you make an error in focussing, where E is the focus error.
So if you focus at the distance D-E instead of focussing at the intended distance D, then the object at distance D is blurred by the amount S, as measured in the actual plane of focus (which is at a distance D-E from the camera). The same blur size would apply if you focussed at a distance D+E instead of focussing at distance D.
Of course, the best way to resolve any disagreement on the science is to do some experiments. I have done my own experiments on measuring blur sizes in practical situations and my experiments were in agreement with my interpretation. However, I think you will be much more convinced by doing some experiments of your own.
The simplest experiment is just to photograph a ruler (with the ruler in the plane of focus). An object in the background will be blurred and you can use the scale on the ruler to measure the size of the blur. If you measure the distances D (from camera to background object) and E (from ruler to background object), then you can check if the formula gives the correct blur size. Measuring the size of the blur is easiest if the background object is a small point of light. It is blurred into a circular disc, whose diameter is easily measured.
Nope, it is you who are misunderstanding his diagram.
Please note that I did say "What if you shoot the moon at f/22 and 1000mm lens but focus on a tree-line for effect say 4 km away, how will the moon look?"
And when I say where the lens is focused, kindly don't change that to somewhere else!
The disk of confusion S at the moon would be approx 4,500 km, according to Merklinger's diagram, would it not?
Ted, this sort of discussion could go on for ever. Your explanation and my explanation give different results. One of us is wrong (and hopefully the other is right).
For God's sake, just do the experiment and see who is right and who is wrong: is the moon blurred (as you claim) or sharp (as I claim) if you focus on something 4km away when using a 1000mm f/22 lens?
(I cannot do it as I don't have a 1000mm lens)
I can not do it either.
So instead, a 50mm macro lens at f/2.8 = 17.9 mm aperture diameter. 19 mm diameter flashlight at 3 m distance.
Focused manually on flashlight:
Focused at 0.188 m(1:1):
Images 2268x1512px.
Merklinger's E/D*d = ((3-0.188)/0.188)*17.9 = S = a disk of confusion of 268 mm at 3 m.
You too could shoot an object at some distance, focused on it then focused much closer and post it here with your conclusions ...
Sorry, Ted, my mistake! I now think your statement of Merklinger's formula was perfectly correct and I was mistaken in suggesting that you had misinterpreted it.
I think my misunderstanding arose from two sources. Firstly, I find it much simpler to think in terms of a disc of confusion in the plane of focus. I have never studied Merklinger's papers in detail as I find his thinking unnecessarily convoluted (that's just my opinion). He seems to prefer to think of a disc of confusion in the plane of the out-of-focus object. I was automatically thinking about the disc of confusion being in the plane of focus (and that is what his diagram appears to show).
Secondly, I was thrown by a mistake in the arithmetic in calculating the disk of confusion at the moon, which gave a result that I did not think was realistic.
Anyway, I hope we can bury the hatchet!
Here is an image I took seven years ago.
The ruler intersects a circular blur disc produced by a single LED light about 14 metres away. Using the ruler, it is easy to see that the diameter of the blur disc is 16mm approximately (in the plane of focus). The entrance pupil of the lens is 25/1.4 = 18mm approx. when focussed at infinity. However, the lens was focussed at around its closest focus and the entrance pupil may have been reduced slightly by the internal focussing mechanism.
OK, Tom, and thanks for the interesting test image.
Dept of field is not a exact thing in everyday photography. In my experience the resolution of the sensor/film (not the size) is a major fact in pixelpeeping. There is a huge difference in DoP between my Nikkormat with 35mm on TriX, and my Sony A7R2 with 35mm.
Here's a simple answer to your first question, Alan.
The Hyperfocal Distance Sh is that at which the "far" distance Sf in the DOF becomes infinity.
This occurs when and only when the focal distance S times the angle of confusion e** becomes equal to the aperture diameter d.
Mathematically, Sh = d/e. However, the Hyperfocal Distance is constant for any given aperture diameter and CoC.
therefore it is not possible to determine the focusing distance from Sh alone.
** the angle of confusion is that subtended by the CoC at the internal focusing distance f' i.e. CoC/f' radians. Apart from close focusing, the said distance is about the same as that marked on the lens.
