A common knowledge item in photography is that tele-lenses allow you better subject separation. I had been interested in how much we are talking about.

You have for sure noticed that the subject separation (aka DoF, Depth of Field) is more pronounced at close distances. To make things comparable we have to define comparable scenes and from a photographer’s point of view the common denominator is the magnification. In common portrait scenes you have a magnification of 1:10 to 1:100. This means that that 1cm (or inch) on the sensor will represent 10 to 100cm (or inches) of the subject.

The basic optical formula tells us that the magnification is the relation of the subject distance and the focus distance. The constant coupling these lengths is the focal length of the lens. Short: Double the focal length and all other lengths will have to double too. If you shot your friend so that the face fills the whole frame with a 50mm, your will have to step back four times as far to achieve the same with a 200mm lens.

Here two images shot at 70 (right) and 210mm (left). The magnification of the subject is roughly the same and both pictures had been taken at f/5.6:

This comparison makes it easy to see the difference: The diameter of the bokeh circles on the water is three times wider. In other words: The blur that causes the subject separation is proportional to the focal length - at constant magnification and aperture!

How can we understand this? What you have to understand is that at the same aperture the opening angle cone of the focused light does not depend on the focal length - this is a bit by definition, because the f-number is defines as the diameter of the aperture divided by the focal length. Blur occurs when the tip of the cone is not on the sensor plane and basic geometry tells us that the diameter will be proportional to the focus error.

Now at constant magnification, say our distance from the lens is 3m and 30cm behind we have some background. Our magnification is 1:50, so the focus distance is 3m/50=6cm.  Without much ado: The focus error to the background will be ~1mm. Now we change to a lens with the double focal length but keep the magnification, now all dimensions will multiply by two, hence distance from lens 6m, focus distance 12cm. And the focus error will be ~2mm. As the angle of the blur cone is the same for both lenses (same aperture) the blur will be twice as large. Of course you can calculate it by the equation 1/f = 1/(3m+30cm) +1/(6cm +x) and 1/2f = 1/(6m+30cm) +1/(12cm +x’), but you can as well trust my calculus skills.

Reverse case

So, if I have a 50mm f/1.4 - how much better is a 200mm lens? There is no such thing as a 200mm f/1.4 lens, but at which aperture it would achieve the same level of separation (again, at given magnification)?

We have seen that the blur circles of the 200/1.4 are four times as wide as those of the 50/1.4. We can narrow the blur cones by stopping down. Fortunately this is completely proportional. We multiply the f-number by four and hence at f/5.6 the 200mm will separate like the 50mm wide open. Other way around, a 200/2.8 will separate like a 50mm f/0.7 - there is no such lens for a SLR and if it would be rather soft or ridiculously expensive. In fact there is a 50mm/0.7 from Fujinon, but it is C-mount and supports only small sensors - the gain will be compensated by the larger DoF from the smaller sensor.

To understand this relation, remember that the blur cone is spawned by the aperture, halving the diameter of this cone  will half as well the observed blur. By definition halving the diameter of the aperture means halving the f-number.

Ready for the final result?:

At given magnification the ability of a lens to separate a subject depends only of the relation focal-length / f-number. In other words: The larger the geometrical aperture, the better the separation

Perspective

Before you go out and shoot with a 800mm f/8 lens (equivalent to a 100mm f/1.0!): Long lenses have some drawbacks

1. You need much more room
2. Lightning requires off-camera flash
3. You need fast shutter speeds, thus more light
4. On the street there will be people walking into your shot

These are practical concerns, but keep in mind as well the artistic part: The longer the lens, the more distances get compressed. Portraits will come out rather flat unless you take care of light to emphasize facial features again. For the composition the FoV (Field of View) needs to be taken into consideration: With an extreme tele it might be impossible to include, for example, a tree into the composition of a portrait - the FoV is simply too narrow.  Including context is usually a good thing, otherwise you could skip on scouting good locations.