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With unit focussing (also called bellows focussing, although a bellows isn’t intrinsically necessary) the entire photographic objective is moved relative to the film plane to set the focus. This also effectively increases the magnification when focussing close-up, because the image circle at the distance of the film plane gets larger as the objective is moved further away. This also requires exposure compensation (sometimes called ‘bellows extension factor’).

With element focussing (also called helicoid focussing, although likewise, a helicoid/screw thread isn’t necessarily involved), individual elements within the objective are moved to set the focus. It’s my understanding that there is typically some small effect on magnification, but that on a typical general-purpose lens (setting macro lenses and some other special cases aside), designers try to minimize this effect. No exposure compensation is required.

Does unit focussing also decrease depth of field compared to element focussing due to the higher magnification/effectively longer focal length? I.e. if I have two lenses of essentially the same design and focal length, focussing on an object the same distance away, but I focus one as a unit and one using the individual elements, will the former have a smaller depth of field than the latter?

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Directly answering the question, no. Unit focusing doesn't inherently decrease the depth of field (DoF) compared to element focusing. However, there are lots of hidden or implied possible variables. But based on the wording in your question, there is some nuance that needs to be made explicit.

[Moving the objective relative to the film plane to set the focus] also effectively increases the magnification when focussing close-up, because the image circle at the distance of the film plane gets larger as the objective is moved further away. This also requires exposure compensation (sometimes called ‘bellows extension factor’).

I wouldn't word it that way. Yes, it makes sense to talk about magnification when in the close-focus regime, as opposed to talking about subject-focus distance. But there is a 1:1 (ableit nonlinear) mapping between object focus distance do and magnification M, as defined by the thin lens formula (1/f = 1/di + 1/do) and the definition of magnification, M = hi/ho = di/do = f/(dof) = (dif)/f .

The "bellows factor" exposure compensation is required for close-up focusing, whether using an actual bellows, or internally focusing (as you called, element focusing) lens. This is because the exit pupil of the lens is significantly far enough away from the image plane that the small numerical aperture (from the image plane's point of view) is dominant enough to become material in the exposure calculation.

With element focussing ... It’s my understanding that there is typically some small effect on magnification, but that on a typical general-purpose lens (setting macro lenses and some other special cases aside), designers try to minimize this effect. No exposure compensation is required.

Setting aside that magnification and focus distance are directly, formulaically, tied together, you are hitting on one thing that is true, that you didn't state: internally-focused lenses don't necessarily have constant focal lengths throughout their focusing range. This is especially true on some zoom lenses. But even unit-focused lenses, with a fixed-focal length, do not maintain a constant angle of view throughout their focusing range. This is known as focus breathing. In fact, only specifically-controlled internally-focused lenses eliminate focus breathing by changing focal length inversely to the would-be narrowing angle of view as the lens is focused closer, keeping the actual observed angle of view constant.

Does unit focussing also decrease depth of field compared to element focussing due to the higher magnification/effectively longer focal length?

(emphasis mine) This is the part of your question that I think is backwards: the focal length of a unit-focused lens doesn't change. The focal length is a physical property of the lens system, determined by the shapes of the lens elements and their relative distances from each other. Because their shapes don't change, and the relative distances of the elements don't change, the focal length f of the unit-focused lens is absolutely invariant.

Conversely (and contradictorally), loosely speaking, you can consider the narrower angle of view of a close-focused unit focused lens an apparent increase in the lens's focal length (as compared to the lens's measured angle of view at infinity focus). In this sense, an internally/element-focused lens controlled for focus breathing will have a shorter real focal length when close focused, as opposed to focused at infinity. Thus, the DoF calculation would have to use the close-focus adjusted focal length, as opposed to constant f (in the case of a unit-focused lens).

I.e. if I have two lenses of essentially the same design and focal length, focussing on an object the same distance away, but I focus one as a unit and one using the individual elements, will the former have a smaller depth of field than the latter?

We're going to have to be a bit more precise. Assuming that a particular internally-focused lens's focal length is kept constant throughout its focal range, then if we focus that lens at infinity, then mount it on a bellows focusing rack onto a camera, and comparing to the same exact lens on a camera without a bellows (i.e., relying on the lens's internal focusing), then at a given focus distance, no: the unit (bellows) focused lens will have an apparent longer focal length, and therefore deeper DoF, than the internally (element) focused lens, with the constant (i.e., marginally wider) angle of view (and correspondingly marginally shorter focal length).

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As you know, the f-number expresses the speed of the lens (image brightness). We calculate this value by dividing the focal length of the lens by its working diameter. Thus, a lens. 50mm focal length with aperture set to f/8 had a working diameter of 6.25mm.

The dimension we call focal length is only valid if the object being imaged is at infinity. If the object is closer than infinity, the distance lens to image increases with the focus which is a repositioning. For most picture tanking setups, this projection distance increase only minimally effects image brightness (it is moot). However, when doing close-up work, the projection distance greatly increases. In other words, the lens must be of the macro design, or we must dismount the lens and remount using a bellows or extension tubes.

As an example - When imaging at unity (life-size often called 1:1) the lens must be repositioned 2x focal lengths forward of its infinity focus position. If the lens is set to f/8, this increased lens to image distance invalidates the engraved f-number. We must now compensate 4x (2 f-stops). This is one example of the bellows factor impact.

Bottom line – It makes no difference if the lens is repositioned for close focus by changing the spacing of lens elements or moving the whole lens as a unit. It is the increased lens to image distance that must be compensated for. We calculate this by the formula M+1^2 (M + 1 squared when M = magnification). We work this problem to find out a revised shutter speed or f-number or combination of both.

Now a maco lens has a trick up its sleeve. The front element group acts as a magnifier that causes the apparent diameter of the iris (aperture) to appear to change as we focus. In other words, as we focus the distance front group to the iris changes causing the iris to appear larger or smaller. If the outside world sees an enlarged iris, its revised diameter allows more light to enter. In this way the macro applies the bellows factor compensation.

No significant difference in depth-of-field either way.

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Depth of field is determined by two things:

  • Aperture (specifically, f-number)
  • Total magnification

The factors that determine total magnification are:

  • subject distance
  • lens focal length (that is, focal length defined as the distance behind the optical rear nodal point of the lens at which light arriving from the focus distance comes into sharpest focus, rather than defined as the distance behind the rear nodal point of the lens at which collimated light will come into sharpest focus)
  • sensor size (because it determines the enlargement ratio for any particular display size)
  • enlargement ratio
  • display size
  • viewing distance
  • viewer's visual acuity

If focusing, either by unit/bellows focusing or by element spacing focusing, changes either total magnification or actual f-number (actual focal length divided by actual entrance pupil size) then the Depth-of-Field will also change. Most of the time with many lens designs, both of those factors will change as the lens is focused. This is true for both unit/bellows focusing designs as well as for element focusing designs.

Keep in mind, though, that DoF is not some absolute point at which everything within it is equally sharp and everything outside of it is equally blurry. It's just the point at which we define any blur less than a specific amount of blur is acceptably sharp and any blur more than the same specific amount of blur is not acceptably sharp.

A word about what depth-of-field is and is not:

In a way, depth-of-field is an illusion. There is only one plane of focus. Everything in front of or behind the point of focus is out of focus to one degree or another. What we call DoF is the area where things look, to our eyes, like they are in focus. This is based on the ability of the human eye to resolve certain minute differences at a particular distance. If the slightly out-of-focus blur is smaller than our eye's capability to resolve the detail then it appears to be in focus. When you magnify a portion of an image by making it larger or moving closer to it you allow your eye to see details that before were too close together to be seen by your eyes as separate pieces of the image.

Since things are gradually blurrier the further they are from the point of focus, as you gradually magnify the image the perceived depth of field gets narrower as the near and far points where your eyes can resolve fine details moves closer to the focus plane.

Since depth-of-field is dependent upon viewing size and distance as well as the visual acuity of the viewer even if we don't change the lens' focus at all we can change DoF, in either direction, by changing the display size or viewing distance.

So will element focusing increase or decrease DoF compared to unit/bellows focusing?

It depends on whether element focusing increases or decreases total magnification and/or increases or decreases entrance pupil size. It could go either way depending on a particular lens design.

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