Does background blur change with focal length when subject framing stays the same?

Question

If I photograph the same subject so it fills the frame equally with two lenses — for example a 50mm f/1.8 and an 85mm f/1.8 — I have to stand farther back with the longer lens. In that situation, depth-of-field calculators show very similar DOF for the subject. But what about the background, especially if it is far behind the subject or effectively at infinity? Will the longer focal length produce more background blur even though the subject framing and aperture are the same?

depth-of-field background-blur composition bokeh focal-length

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Photography Stack Exchange contributor

7y ago

user75526 · Accepted Answer

The formulas don't account for factors that cause real lenses to deviate from the ideal. Formulas are from Wikipedia.

Depth of Field – DOF 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 Blur – The 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 (x_d). Here is a formula for amount of background blur:

b = f m_s x_d / (N (s + x_d))

b = blur
f = focal length
N = aperture F-number
m_s = subject magnification (what's this?)
x_d = 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 m_s is "subject magnification", it is focal length (f) / subject distance (s). The blur formula can be rewritten: b = f² x_d / (N s (s + x_d))

As the subject-background distance increases, x_d/(s + x_d) 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.

Does background blur change with focal length when subject framing stays the same?

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