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Let's assume I have two lenses, 50mm f/1.8 and 85mm f/1.8. I want to fill the frame with the photographed object. Now, if the object is 50cm wide, I need to take the photograph at a distance of 1.112 m or 1.888 m, respectively, on a crop sensor Canon (1.6x crop factor) camera.

The depth of field is 3 cm, exactly the same with 50mm f/1.8 and 85mm f/1.8 lens.

Does the amount of blur at the background (that can be assumed to be at infinity) vary with focal length?

I know the depth of field doesn't vary with focal length, being dependent only on the aperture number. Or at least doesn't vary much: at 10 m distance, it's 2.74 m (50 mm lens) an at 17 m distance, it's 2.71 m (85 mm lens), with 10 m and 17 m giving equivalent framing on these lenses. So some very small variation can be seen with focal length.

(Side note: I originally used the term "quality of bokeh", but apparently it meant something else so I edited it away -- what I meant is amount, not quality.)

  • How far is the background from the focus distance? ("Infinity" can change with different focal lengths.) Does it stay the same, or is it proportional to the change in subject distance when different focal lengths are used to get the same framing? Your question assumes some variables that aren't spelled out. – Michael C Apr 14 at 8:20
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The formulas don't account for factors that cause real lenses to deviate from the ideal. Formulas are from Wikipedia.

  • Depth of FieldDOF stays the same because distance to subject (u) is in the numerator and focal length (f) is in the denominator. They are both squared, so changes that are proportional to each other cancel out. Here is the standard formula for DOF:

    DOF = 2 u2 N C / f2

    N = aperture F-number
    C = circle of confusion
    u = distance to subject
    f = focal length

    All of the advice people give to minimize DOF are in the formula – use larger apertures, use longer focal lengths, and get closer to the subject.

  • Background BlurThe amount of blur does change with focal length even though the subject is kept the same size in the frame. Although focal length (f) is in the numerator and distance to subject (s) is in the denominator, the changes don't cancel out because they are modified differently by distance between subject and background (xd). Here is a formula for amount of background blur:

    b = f ms xd / (N (s + xd))

    b = blur
    f = focal length
    N = aperture F-number
    ms = subject magnification (what's this?)
    xd = distance between subject and background
    s = subject distance

    WayneF explains magnification ratio:

    Magnification is just ratio of distance behind lens (focal length) / distance in front of lens... See [Circle of confusion] for a simpler formula, including for infinity.

    Since ms is "subject magnification", it is focal length (f) / subject distance (s). The blur formula can be rewritten: b = f2 xd / (N s (s + xd))

    As the subject-background distance increases, xd/(s + xd) approaches 1. The formula simplifies to: b = f2 / N s

    If the changes in focal length (f) and subject distance (s) are proportional, to maintain subject size within the frame, background blur is proportional to f/N. If we consider a superzoom 18-200/3.5-6.3, we can see that background blur at 18/3.5 (5.14) is less than at 200/6.3 (31.75). For my 18-55/2.8-4 kit lens, the amount of background blur at 18/2.8 (6.43) is about half that at 55/4 (13.75).

    Maximum background blur on variable-aperture zooms is usually at max focal length rather than max aperture (with minimum focal length) because zoom ratios are usually greater than 2, while the max-aperture ratio is usually less than 2.

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    Magnification is just ratio of distance behind lens (focal length) / distance in front of lens (background distance in this case). See en.wikipedia.org/wiki/… for a simpler formula, including for infinity. And 99999 feet works pretty well for infinity (about 19 miles). – WayneF Apr 14 at 20:58
  • They do say m = f1/fS1. I always think subject should logically be at the focus point S1 and background at S2, but it says an out of focus subject at S2 when focused at S1. There are two result formulas there. First one computes out of focus blur at S2 when focused at S1. See the diagram there with it, the capital letter C. To me, that's background. Then second formula is in regard to infinity, where if infinity, their assumption seems kinda S2/S2 = 1, but infinity has no valid actual single value answer. I'd instead call it 99999 in the first one. :) – WayneF Apr 14 at 22:42
  • @WayneF - I don't know this topic well enough to reconcile the formulas on the two different wikipedia pages... I'm going to let it stew in my mind a bit... – xiota Apr 14 at 22:53
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Yes, but it depends a lot on the amount of background separation. A less technical explanation is that when you back up in order to create the same subject composition you negate the increased magnification at that distance, but not at the distance of the background (BG) itself.

I.e. if your subject is at 10ft and your BG is at 50ft. When you double the FL and subject distance (100% increase) you have only added 10ft to the BG distance (20% increase)... nowhere near doubled so you maintain ≈ 80% of the increased magnification at the BG location.

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Yes, background blur increases with focal length. This is pretty much the whole point of this page, with example charts and DOF calculators that look at background too, found at my website: Understanding Depth of Field, with Depth of Field Calculator (DOF with a plus)

Standing back with a longer lens can then still stop down a bit giving better DOF at the subject where it matters greatly, but yet with more blur at the background. 50 mm f/1.8 seems about the worst DSLR choice, for multiple reasons.

People get troubled when I link my site here, but it's way too huge to copy here. The point is to confer information, and that's where the info is.

The blur effect at infinity is large, but maybe not as huge as might be expected, not nearly infinite. But at 40 or 100 feet, it seems a very large effect, as compared to shorter lens at f/1.8.

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    I wish you would link your site with the titles of the pages, not just the naked URLs. – xiota Apr 14 at 21:06
  • No one gets troubled as long as you reveal your association with the link. – Michael C Apr 16 at 6:30
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The answer is: yes, the amount of background blur varies. A depth of field calculator gives only the depth of field, namely the depth of the distances where the picture is acceptably in focus (what "acceptably" means of course can vary depending on how large the image is printed, so depth of field isn't an absolute truth).

However, how much exactly the background is out of focus differs from one focal length to another.

There is a video on YouTube (jump to 8 minutes 0 seconds) that demonstrates the effect:

  • 24mm focal length, f/10, 2.4 feet away gives a depth of field of 2.07 feet on a full frame camera,
  • whereas 100mm focal length, f/11 and 10 feet away gives a depth of field of 2.02 feet, practically the same

Despite the same depth of field, the background is far more blurred with the 100mm focal length.

If you want blurred background, prefer longer focal lengths and fast apertures instead of blindly trusting a depth of field calculator.

You can derive the following equations if the background is at infinity:

b   = f^2 / (x_d * N) 
    = 2 * x_d * C / DoF

DoF = 2 * x_d^2 * N * C / f^2 
    = 2 * x_d * C / b

where b is background blur, DoF is depth of field, f is focal length, x_d is subject distance, N is aperture F-number, and C is circle of confusion (usually considered to be 0.019 mm for Canon crop sensors and 0.030 mm for full frame sensors). You can see that long subject distance (which means long focal length given equal framing) allows both deep DoF and high background blur at the same time.

Since a photo is worth more than a thousand words on this site, here are two examples with approximately same background blur, yet very different depths of field.

Here's an image with x_d = 8400 mm, f = 250 mm, N = 5.6:

Image of a seagull

We can calculate blur disc size at the sensor is 1.33 mm (about 5% of sensor diagonal). We can also calculate DoF is 240 mm. Thus, the whole fence is in focus, and so is the whole seagull.

Here's an image with x_d = 1000 mm, f = 50 mm, N = 1.8:

Image of a cactus

We can calculate blur disc size at the sensor is 1.39 mm (about 5% of sensor diagonal again). We can also calculate DoF is 27 mm. Thus, parts of the cactus are out of focus. Note that strictly speaking, in this second example the background is not as far away than in the first example, so the real background blur isn't the same as the blur for a background at infinity.

  • You are addressing the amount of blur. That's not what bokeh is. Bokeh is the quality of blur. That is, it is the characteristics of the blur, not how much blur there is. – Michael C Apr 14 at 8:10
  • Oh, I'll edit the question: I meant the amount of blur. – juhist Apr 14 at 8:11

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