I'll start with the conclusions, then justify them with numbers computed by a spreadsheet you can play with yourself.

Assume we're comparing camera and lens setups such that:

- All setups have the same field of view for the in-focus subject.
- All setups use apertures chosen to give the same nominal DOF.

- Sensor size has nothing to do with the appearance of background.
- Focal length makes no difference in the diameter of blur circles as measured in the scene being photographed.
- Shorter lenses do give backgrounds that
*look*sharper. This is because the background is magnified less, leading to smaller blurs in the image even though blurs are the same size in the scene. - Effective f-number is directly proportional to sensor size. This implies that the tradeoff between DOF and diffraction blur does not depend on sensor size. This is because the Airy disk scales in proportion to effective f-number and thus in proportion to sensor size, so diffraction blur remains constant in the final image as viewed.

My strategy this time is to compute the sizes of blur circles based on very simple "first principles" analysis using nothing more than similar triangles and the thin lens formula 1/f = 1/o + 1/i.

The computation is packaged as a spreadsheet that makes it easy to compare the results of up to three different setups that all capture the same field of view with the same nominal DOF.

Input to the computation are just a few numbers:

- Field of view at the in-focus subject (same for all setups)
- Distance from subject to some background point (same for all setups)
- Sensor size (may vary between setups)
- Distance from lens to subject (may vary between setups)
- Nominal f-number for the first setup.

- Determine required magnification, given the sensor size and field of view.
- Determine distance from lens to sensor to yield that magnification.
- Determine lens focal length needed to focus at those distances.
- Determine effective f-number and ratio between effective f-number and sensor size.
- Determine "nominal DOF" using a standard formula.
- Determine lens-to-subject distance at rear limit of that nominal DOF.
- Determine blur diameters for a point at that rear limit.
- Determine lens-to-background distance.
- Determine blur diameters for a point on the background.

Here are some spreadsheet results that illustrate the conclusions.

First, let's vary the sensor size from 4 to 100 mm, while keeping the lens at the same position. Notice that the lens focal length, nominal f-number, and effective f-number must change to maintain constant field width and constant DOF. But the ratio of effective f-number to sensor size ends up being the same in every case (row 14). Notice also that the blur circles are constant both in the image and in the scene (rows 20:21 and 27:28).

Second, let's keep the sensor size constant, while changing the lens-to-subject distance. Notice in this case that there are changes in the sizes of blur circles as measured in the image, that is, as fraction of field width (rows 21 and 28). These changes are tiny for points that are near the focus plane (row 21), but large for points in the far background (row 28). On the other hand, as measured in the scene, the blur circles are the same size in each setup (rows 20 and 27). Hence my comment that shorter lenses give backgrounds that

*appear*sharper because they give less magnification for backgrounds.

In other words, it's just a matter of perspective. Shorter lenses give a wider angle of view, which changes the subject-to-lens distance in such a way that the background looks smaller. Making the background look smaller doesn't let you see more detail in any given part of the background, but it does give you more background to see detail in. It's worth noting that the results I'm showing here are exactly the same as those shown by this Luminous Landscape tutorial for telephoto versus wideangle lenses. Backgrounds shot with the wideangle look sharper, but that's only because the background is rendered with lower magnification. The level of detail visible in the background is unchanged.

You can download a copy of the spreadsheet HERE and play with it yourself. If you don't like my choices for distances and lens lengths -- and you probably won't, because they're pretty extreme -- then plug in your own and see what happens.

For completeness, here are the formulas:

I hope this helps.

--Rik

P.S. If you find a mistake, please let me know. I've cross-checked this thing as well as I can by myself, but that's still only one set of eyes.