Background blur vs focal length, absolute aperture and relative aperture

from Shutter, by Lewis Collard

This article was written in 2012. The mathematics are the same, but references to contemporary cameras are a capsule of their time.

For normal people

Don't lose any sleep over any of this, because I haven't. I'm just intoxicated with the ability to generate data files programatically and have another program generate graphs from them. Any cheap lens on a digital SLR will blur backgrounds fine for situations where you want that (like portraits). How about a £15 Soviet 58mm f/2ski?

Serious Cat.

Does that just fine! Shoot wide-open, and if you've got a zoom, stand back and zoom in. Stop thinking about this and take some photographs instead. That's more fun than a bunch of boring graphs.

Now, with the normal people gone..

Oh look, some pretty graphs!

Everybody knows that a wider aperture will give you more background blur:

Background blur from f/0.7 to f/5.6

Here, we have a selection of hypothetical 50mm lens focused on a subject 1 metre away, with a background four metres from the camera. The numbers at the bottom are absolute aperture sizes, which you get by dividing the focal length by the f/number. The leftmost point on the line corresponds to f/5.6, the rightmost point to an f/0.7 lens.

(If you don't like pretend examples, imagine that I'm referring to the Zeiss 50mm f/0.7, Nikon 50mm f/1.4 AF-S, Zeiss Jena 50mm f/2.8, and Sigma 18-50mm f/3.5-5.6 at the tele end. I'm not, but you can pretend that.)

The y axis is the size of the defocus blur circle for the background. Larger number = more blurred backgrounds.

What we see isn't surprising: the relationship between the degree of background blur and maximum aperture is a simple, linear one. All else being equal, an f/1.4 lens shot wide-open will give you a defocus blur circle precisely twice the size of an f/2.8 lens shot wide-open, the latter being twice as much as an f/5.6 lens.

We could do this with any given focal length, subject distance, and background distance, and we would come to exactly the same conclusion: the relationship between maximum aperture and background blur is always linear. Simples!

Things get more interesting (trust me) when we throw focal length into the mix:

Background blur from 50mm to 400mm

Here we've taken exactly the same subject distance and background distance, and tried it with lenses of the same absolute aperture (50mm f/1.4, 100mm f/2.8, 200mm f/5.6, etc up to 400mm). We're looking at the absolute aperture so that we can examine the relationship between focal length alone and background blur.

So this is interesting, because a direct correlation between focal length and background blur would suggest that a 400mm lens would blur the background 8 times as much as much as a 50mm lens with the same absolute aperture, but it actually does so over twelve times as much.

Thus it is established: an increase in focal length results in background blur disproportionate to the ratios of the focal lengths.

But wait there's more!

One thing we've forgotten to factor in is the different distances from the subject necessary to get the same framing. There's a direct correlation with focal length and the distance at which you stand; to get the same subject framing that you'd get with a 50mm lens standing 1 metre away, you'll need to stand 4 metres away with a 200mm lens. There's two factors fighting each other here. First, the background blur circle (all else being equal) becomes greater the closer you are to a subject, which would give the 50mm lens the edge. But at the same time, that's also moved the background further away from the camera, which would blur it further.

So let's take a subject that you shot from 1 metre away with a 50mm f/1.4 lens, then try that with different lenses and moving around to keep the same framing. For added fun, we're going to try moving the background as well; we'll try it at 4, 8, 16, 32 and 64 metres. Once again, we're using absolute aperture sizes, not relative ones, just so we can see these effects.

Defocus blur at various focal lengths and background distances, maintaining same absolute aperture.

Did you see that? Surprise conclusion! If you maintain the same framing and keep everything else the same (i.e. the same absolute aperture), the amount of background blur decreases as you increase focal length!

Another thing worth noting is that the closer the background is to the subject, the more pronounced this effect becomes. Flip it the other way round: as the background distance approaches infinity, the blur circles become increasingly similar. At infinity, the size of the blur circles of a 50mm f/1.4 and a 400mm f/11.2 (!) are identical if you keep the same subject framing; the amount of background blur increases in direct proportion to the absolute aperture size.

Practical application of this: if you have a long, slow lens (like the f/5.6 long end of a cheap 70-200mm zoom lens), you can get almost the same amount of background blur as you would with a 50mm f/1.4 by just moving some place where the background is far away. Woohoo!

But wait there's more!

So where does the rule "longer focal lengths blur the background more" come from, if it's so obviously contradicted by the above? Simple: we've dealt with absolute apertures, while lenses are marked, sold, and spoken about in terms of their relative apertures, or f/numbers. There's no such thing as a 400mm f/11.2, that I know of.

So what happens if we take our above experiment, but use a relative aperture of f/1.4? This does:

Defocus blur at various focal lengths and background distances, with same absolute aperture.

We notice that as the background distance increases, there's an increasingly direct correlation between focal length and blur circle size. This is the same as earlier: as the background distance approaches infinity, the size of the blur circle grows in an increasingly linear way with absolute aperture size. In other words, as the background gets further away (independently of the subject), "longer lens means more background blur" becomes more true. (And once again, with a background at infinity, that's exactly what happens.)

But wait, what's happening with the background at 4 metres? It's starting to level out. Let's extend our focal length range to 3200mm (at f/1.4!) and see what happens:

Defocus blur at various focal lengths and background distances, with same absolute aperture.

Notice how the red line goes nearly flat, and the blue and green lines are well on their way there? Beyond a certain point, with background distances that you may well encounter in the real world (and with lenses you certainly won't), increasing focal length has very little effect on background blur.

(That's even though we're pretending that there's such a thing as a 3200mm f/1.4: there isn't. But you can feed in any other relative aperture value for the same effect.)

Defocus blur at various focal lengths and background distances, with same absolute aperture.

Background blur on compact cameras

So why is it so hard to get background blur on point and shoot cameras?

Here's the thing. Compact cameras invariably have much smaller sensors, which means to get the same angle of view with any given lens, you need a shorter focal length. But with shorter focal lengths come smaller blur circles at any given relative aperture. At the same time, a blur circle of any given absolute size will "go further"; it'll cover a larger proportion of the sensor. Shouldn't the two balance each other out?

We don't need to speculate about this. In fact, we can throw in another pretty graph!

Relative blur circles of various film and digital sensor formats.

Once again, we've taken a subject distance of 1m on a 50mm lens on a 35mm camera (but at f/2.8 this time), with a background 4 metres away. Once again, we're shooting the same scene with lenses of different focal lengths, and maintaining the same framing in all of them by moving backwards as the focal length increases. What we've done differently this time is tried the same thing with different, hypothetical cameras as well: one DX digital SLR, and two compact cameras with frames about half and quarter the width of a full 35mm frame. We've also taken a medium-format camera for kicks.

The correct response is crikey. Even at a massive 400mm equivalent focal length on the quarter-frame camera, the background blur is barely what it would be with your dad's 28mm f/2.8 wide-angle lens on his Olympus OM-1!

This gets worse, too. We're being generous and assuming that it's f/2.8 throughout the range, when it isn't, just like cheap SLR zooms. In reality, even if you did have a super-zoom that went out to 400mm equivalent on your compact, it's likely to give you f/5.6 at the long end if you're lucky, which means that even if you figure out how to hold a small camera steady, you'll still be getting probably less than a third (do the maths yourself kids) of the background blur than you do with a 50mm f/1.8 on a digital SLR.

Why is this happening? We can only explain this because of some effect of using a shorter focal length that isn't anything that comes from using a smaller absolute aperture size. (The relative aperture of f/2.8 would translate into a smaller absolute aperture because of the focal length, which would cast a smaller blur circle, but the sensor is smaller to precisely the degree that the aperture is smaller, therefore it'll cast a blur circle that is proportionally the same.

So this effect can only be explained by some effect that comes with an increase in focal length that is disproportionate to the increase in focal length. Hey, remember this from earlier?

Background blur from 50mm to 400mm

Yup, we established just that in our second graph. This was with an absolute aperture size, though, so what we really want to do is this: Take a fixed subject, a fixed background, then examine the effects on blur circle size as we increase or decrease the focal length, but this time use a relative aperture. Here's how things look from 5mm to 50mm at f/2.8, with a subject distance of 1 metre and a background distance of 5 metres:

Why compacts don't do background blur!

I've got your dramatic and disproportionate effect of focal length on the defocus blur circle right here! Let's take a real-world example here: The Four Thirds camera system has a sensor of 17.3mm, which is so close to 18mm (and half of a 36mm frame) that we'll call it that. The blur circle of a 25mm f/2.8 lens is less than a quarter of that of a 50mm f/2.8, but the sensor is only half the size. Therefore, with this awesome "digital from the ground up" technology, you'll only be able to blur the background half as much as you will an equivalent lens on a full-frame 35mm camera.

(I can't help but wonder if even Olympus believe that "digital from the ground up" stuff, since even they admit that they've named the system after the standard method of measuring vacuum tubes from analog video cameras, not after the actual sensor size, and using old-school inches at that. Teehee.)

That's with a relatively big sensor (for compacts) like you get in the Four Thirds system. If we take a Canon S95, which has a crop factor of (so I've seen it stated) about 4.66, which means that to get the same framing as a 50mm lens on a full-frame camera, you'll be using a real focal length of about 11mm. That'll get you a defocus blur circle in this scenario of about 0.035 millimeters. That's about twenty-one times smaller than a 50mm lens on a 35mm camera; proportionally, this works out to 4.6 times less background blur. Or to put it another way, shooting the S95 at 50mm equivalent wide-open (f/2) would be like shooting a 50mm lens on a 35mm camera at f/9. Not that blurred at all.

If you really must get background blur on your compact, the best thing you can do is to find somewhere with the background further behind your subject. Zooming in helps a bit, but not much. Better yet, be a better artist than I am by finding some way of contextualising your subject within his or her surroundings such that you don't need to blur the background.

Hey, I'm not bashing point-and-shoots; I love them because the same optical phenomena that make it near-impossible to get blurred backgrounds in sensible conditions is the same thing that gives them enormous depth of field (background blur and depth-of-field are related, but not identical concepts). It's also what keeps medium-and-large-format film shooters on tripods all the time, since more often than not we're down at f/16 to ensure enough depth of field. Honestly, I'd probably be better off with a compact camera instead of a digital SLR for most of what I shoot, for that reason. The main reason I carry big, heavy cameras is because they're harder to lose, no joke.

In the real world

Just knowing the focal length and maximum aperture will tell you relatively little about how nice the background blur will look. One thing frequently overlooked is that we perceive all things in a photograph relative to the other things in a photograph. In this case, humans will, to some degree, perceive the amount of background defocus relative to how sharp the subject is. Nearly all lenses are less sharp shot wide-open, thus we'll perceive the defocused parts of a photograph as being sharper as the subject area becomes softer.

Then there's the issue of "bokeh", which has been done to the death by so many people that I won't go into it very far, other than to mention that it's about the quality of the background blur. Ultra-fast lenses (in the 50mm domain, around f/1.4 and faster) often have pretty ugly background blur when shot wide-open; therefore under certain circumstances and with certain lenses, the background becomes less distracting as you stop down, despite the blur circle being smaller.

It happens that two random lenses that I could be bothered to find illustrated this quite well. Here's a boring test shot of a very small teddy bear, taken with a Nikon 50mm f/1.8D wide-open on a Nikon D2H:

Tiny teddy test shot 2!

And here's almost the same framing (I accidentally pulled some sky into the above shot, which is bad) with a 55-200mm VR, at 200mm and f/5.6:

Tiny teddy!!

Calculations show that with a background 6 metres behind the subject and keeping the same framing as a 50mm lens with a subject distance of 1 metre, the difference between the size of the defocus blur circles on the two lenses should be negligible (1.25mm for the 50mm f/1.8 vs 1.13mm for 200mm and f/5.6). Yet the backgrounds look completely different. It's up to you which you prefer. For me, the 55-200mm VR looks slightly more pleasant, especially on the repeating patterns, though the 50mm defocuses the twigs (actually tomato plants) in the background somewhat better. If they were the only two lenses I owned, I'd still grab the 50mm for a portrait, since on DX digital the crop factor gets me just far away enough that the perspective is pleasing while not being so far away that communication with the subject becomes difficult without shouting. Real-world factors win every time.

If any of this bothers you, make your own experiments if you already own a lens, or find real-world photographs (not boring test shots) taken by a skilled photographer if you do not. As for me, I know that I would go nuts worrying about all the factors involved, so I don't hesitate to shoot any lens, even the bad Ukrainian lenses that I got with my Kiev 88, wide-open if I really need to.

Reader Chris Raymond wrote in to point out another real world factor when it comes to comparing smaller formats with larger ones: despite the longer focal lengths required for the larger formats, lenses for large formats tend to be slower. For example, the standard fixed lens for a medium format camera is usually a 80mm f/2.8, whereas for a 35mm camera it's a 50mm f/1.7, f/1.8 or f/2. I've never even heard of an f/1.8 lens for larger formats! The trick there is that in very plausible shooting scenarios, the wider relative aperture of a 50mm f/1.8 lens makes up for the shorter focal length compared to the 80mm f/2.8. I've run the calculations and so did Chris; I don't feel like making another graph so you'll just have to trust me.

Yup, the real world is complicated!

Summary (in which I tell you what you knew already)

Generalities are nice. Once the volume of information increases beyond a certain point, we need generalities to stop our heads from exploding. Things like "increasing your focal length gives you more background blur" simplifies a much more complicated situation such that we can make intelligent decisions within a limited scope.

So what did we learn today, children?


I didn't enter a single value into a spreadsheet to generate these graphs, because life is short. I wrote a Python script to generate data files, which was then fed into a gnuplot script to generate these tables. Data at random points generated by the Python script were fed into VWDOF to verify that they were correct. (All errors made on this page are entirely mine, not VWDOF's author.)

In the spirit of openness, you can have the source code, including all generated data files in a tar.gz file here; this is so you don't have to take my word for it. The source code and files generated therefrom are in the public domain to be used without any restrictions, so feel free to hack this to generate graphs for your own shooting scenarios. (A simple "sh" will generate the files on Unix systems; you'll need Python, gnuplot and rsvg installed.)

Credits: Thanks Elias Benkhodja for catching a technical inaccuracy (and, indirectly, a graph that confused the narrative rather than clarified it) in an earlier version of this article.