Relationship between minimum focusing distance, maximum reproduction ratio, and focal length?

I'm having a hard time understanding these specs of these two Nikon (Z-mount) lenses (or rather, reconciling them).

One of them is the Z-mount 105mm macro: https://imaging.nikon.com/imaging/lineup/lens/z-mount/z_mc105mmf28_vr_s/. It shows a minimum focusing distance of 0.29m with a 1x maximum reproduction ratio.

The other one is the 24-120mm zoom: https://imaging.nikon.com/imaging/lineup/lens/z-mount/z_24-120mmf4s/, with maximum reproduction ratio of 0.39x and minimum focusing distance 0.35m at all zoom positions (the .com site does not say this, but the Canadian one does)

How can that be? At the same focal distance, the projected image of an object at the same distance will be the same size (the actual physical size of the projected image on the sensor, regardless of how many pixels that corresponds to). If the MFD of the 24-120mm is 0.35m (close to the MFD of the 105mm macro lens), then at a focal distance of 105 x 0.35 / 0.29 = 126.7mm, the reproduction ratio would be 1x (the 0.35/0.29 factor accounts for the larger distance to the subject, following the pinhole camera model).

So, ok, the maximum 120mm on the zoom lens means that it will not quite get to 1x, but to 120/126.7 = 0.94x instead .... Certainly not as low as 0.39x

I assume that the pinhole model becomes less and less accurate as we deal with subjects that are closer to the lens, and the discrepancy between pinhole model and the reality may be different for prime lenses optical design vs. zoom lenses optical design (which would then explain discrepancies in specs related to minimum distances). Could this account for this 1x vs. 0.39x discrepancy?

Am I missing something else? (is it an error on the Canadian website tech specs, and the minimum focusing distance gets larger on the tele side of the zoom range?)

The pinhole model isn't accurate for complex lenses, but can still be used for some aspects.

The key factor is that you don't know where exactly the "optical center" of the lens is located, and that some enlargement is achieved by some "rear magnifier". So, the rear distance need not be as long as with the pinhole model. You can imagine the rear side to get some additional virtual length, or the optical center viewed from the rear side to be at a different point, further away from the sensor than the "front-view" optical center.

But, if you ignore this difference between rear side and front side, the pinhole model is still useful for computing enlargement from distances and/or focal length.

Certainly, if the 24-120 were able to achieve a reproduction ratio better than 0.39x, Nikon would announce that, as it can be a significant selling point.

Let's calculate/estimate:

• We don't know the effective optical center of the lens, but can assume it to be somewhere inside the lens, maybe 80mm from the sensor plane.
• With a 0.35m SFD (and the sensor plane being the reference for the "focus distance"), this means a front distance of 350mm - 80mm = 270mm. This defines the "front half" of the projection geometry.
• A front distance of 270mm with reproduction ratio of 0.39x gives a rear distance of 105mm for the pinhole model. So, the rays hit the sensor in a geometry equivalent to a pinhole 105mm in front of the sensor plane.
• With a 270mm object distance and a 105mm image distance, this matches a 75mm focal length in the classical model.
• With the classical model and a 120mm focal length, we'd get an image distance of 216mm, meaning a reproduction ratio of 0.8x

However you define "focal length" for non-infinity distances (and that's not a trivial thing to define), the estimations do not match the data given.

• Either the assumption is wrong that the MFD of 0.35m applies to all zoom levels.

• Or the focal length isn't constant, meaning that by focussing close-up, you no longer have effective 120mm.

When dealing with the simplest of lenses such as a single element convex – convex or a pin-hole, subject distance and image distance are measured from their center. When dealing with multi-element camera lens, subject distance is measured from a point called the front nodal. Likewise, the image distance is measured from the location of the rear nodal to the focused image. The focal length is a measurement taken rear nodal to focused image at infinity focus (imaging a star can be artificial).

Now a complex lens consists of many elements, some air-spaced from the others, some cemented to another element. Some have positive power and some negative power. Bottom line, the user will likely never know the location of either nodal.

It gets even more complicated. Likely the position of the nodal(s) are flipflopped. As an example, in a telephoto design, the forward nodal is intentionally shifted frontward, this allows a reduced barrel length making the telephoto less awkward. Likewise, a wide-angle will likely have its rear nodal shifted so that the back-focus distance is lengthened to accommodate mechanisms like a reflex mirror etc.

All I am saying is the pin-hole and simple lens formula often fail. That being said, a 1:1 set-up usually places the lens 2X focal lengths from object and 2X focal lengths from the image. Likewise, the distance subject to focused image is 4X the focal length.