ChrisLilley wrote:where do our assumptions or formulae differ?
Short answer: It's in the CoC.
Long answer: It's in the CoC. The relationship between CoC and DOF is not nearly as simple as it's usually presented. Let's look deeper...
Assume a circular aperture, a perfect lens, and ignore diffraction. Then when the lens is defocused, any single point on the subject turns into a uniformly illuminated circle of light on the sensor. Conversely, every point on the sensor is illuminated by a circle of points on the subject. As a result, the image becomes blurred.
A good way to think about the blur is in terms of the Modulation Transfer Function (MTF) of the defocused lens. As with many MTFs, the MTF of a defocused lens starts near 1.0 for low spatial frequencies (coarse detail), drops to exactly zero at some high spatial frequency (fine detail), then goes negative for even higher frequencies, and oscillates with decreasing amplitude beyond that.
Here is a picture and a graph illustrating how this works for a defocused lens.
On the left we have a sine curve of intensity, as seen through circular apertures of varying sizes. On the right, we have a graph of Modulation Transfer Function as it depends on the circle diameter. Looking on the left, you can easily see that with the largest circle, equal in width to 1.5 line pairs, there is more light than dark within the circle even though the circle is centered on a dark bar. This causes contrast reversal. With the smallest circle, contrast remains high, and with the medium circle, contrast is low though still positive. Looking more closely on the right, you can see that
- For a small circle, the MTF approaches 1.
- MTF drops to 0.5 (50%) for a circle whose diameter is about 0.71 of a line pair.
- MTF drops to 0 for a circle whose diameter is about 1.22 line pairs.
We need also to touch on sampling. The Nyquist sampling theorem says that you need at least two pixels per line pair. But in practice (and in more accurate theory), you need more like three pixels per line pair to reliably give usable contrast. For purposes of discussion, let's assume three.
Now, let's go back to the numbers that you used.
As I understand your calculation, you essentially ended up using a CoC that is three pixels wide. That's 1.0 line pair, which gives an MTF of 18% -- contrast reduction of 5.5X -- for a spatial frequency that is well resolved by the sensor. At the Nyquist limit of 2 pixels per line pair, that same CoC is 1.5 line pair, which is beyond cutoff and will exhibit contrast reversal.
So, a smaller CoC is required.
Exactly how small is sort of a judgment call. If we want to retain 90% MTF from defocusing, at a frequency of three pixels per line pair, then we need a CoC of more like 0.87 pixel. Using the D700's pixel size of 8.45 μm, this gives a CoC of 7.35 μm. If we're willing to retain only 50% MTF from defocus, then we can get along with a larger CoC, more like 18 μm.
Of course I started off by saying "ignoring diffraction". It turns out that accurately determining the combined effect of diffraction and defocus is far from easy. Some months ago I snapped a copy of an article that treats it in quite a bit of detail, but I have yet to spend enough time to really digest it. (I think that would take several days.) However, I suspect that the numbers we just computed are not too far out of line.
We are fortunate that the formula you're using is the same one that Nikon uses at
http://www.microscopyu.com/tutorials/ja ... index.html . Plugging 18 μm into that page gives a DOF of 6.7 μm; plugging in 7 gives 4.3.
So where did my "5" come from? Danged if I know -- I think I just cranked the Nikon calculator down to something around 10 μm which I've found to give results that kinda sorta match what I see in practice.
But I will happily admit that this is a very imprecise process, and no way did I run through the calculations that I just now wrote up for the sake of discussion. In fact the illustration and graph that I've posted above are freshly generated from scratch, since at the moment I'm quite far from my usual resources. I think they're correct, but I'm not sure.
I hope this helps. If anything looks wrong be sure to let me know because I hate messing up the literature.
--Rik
