The amount of background blur is dependent only on the aperture opening, as some of the answers here already mentioned. From Wikipedia:
b = f*m_s/N * x_d/(s ± x_d) = f*m_s/N / (s/x_d ± 1)
For background bokeh as opposed to foreground bokeh, the ± sign should be + sign. If the background is at infinity, subject distance
s is much smaller than subject-to-background distance
x_d so the first term vanishes. So,
b = f*m_s/N
m_s is subject magnification, i.e.
x_sensor/x_s (assuming the subject fills the frame, more on that later). So,
b = f*x_sensor/x_s / N
b/x_sensor = f/N / x_s
The figure you're interested in is the ratio of blur disc size
b to sensor size
x_sensor. It is dependent on the subject size
x_s you can't affect and
f/N which you CAN affect.
To achieve as large background blur as possible, you're looking to maximize
f/N. This means:
- Prefer telephoto lenses that give larger
- Prefer wide-aperture lenses that give smaller
Sometimes, these two may be at odds with each other. For example, do you choose 135mm f/2, 200mm f/2.8 or 400mm f/5.6? All of them have about 70mm aperture opening.
Now if the background isn't at infinity preference would be on slightly shorter wide-aperture lenses. For example, if the background is at 30 meters from subject, a telephoto requiring you to take the picture from 10 meters divides the infinity blur by (10/30+1) = 1.3 whereas a shorter lens allowing you to take the picture from 3.5 meters divides the infinity blur by (3.5/30+1) = 1.1167. So, while 400mm f/5.6 is slightly better than 135mm f/2 if the background is at infinity, for background at 30 meters from subject, 135mm f/2 wins. Indoors, I say even 85mm f/1.2 might be better.
Also do keep in mind the depth of field. You may want the entire subject to be in focus. Depth of field is
DoF = 2*u^2*N*c/f^2
u is the distance to subject,
N is the aperture number,
c is the circle of confusion (a certain fraction of sensor size), and
f is the focal length.
For a particular sensor, you have
u/f constant if you use equal framing. Thus, to maximize depth of field, you are looking to increase
N from the smallest possible value (but don't increase it so much that background blur vanishes!). Focal length does not matter.
You can play around with the parameters here: https://dofsimulator.net/en/
For example, full frame 200mm f/2.8 american shot of man 2 (1.80 m) has 6.98% background blur and 6.7 cm depth of field in front of the subject and 6.9 cm behind the subject.
In contrast, APS-C 135mm f/2.0 american shot of man 2 (1.80 m) has 6.51% background blur (slightly less due to 135mm/2 being smaller than 200mm/2.8) but now the depth of field is 7.8 cm in front of the subject and 8.0 cm behind. If you had f/1.8 available, the background blur would be 7.24% but depth of field would still be 7.1 cm in front of the subject and 7.2 cm behind the subject.
What if you take 200mm f/2.8 american shot of man 2 (1.80 m) then on APS-C? You need greater distance, but now the depth of field is 11cm in front of the subject and 11.2cm behind. However, 200mm f/2.8 is equivalent to 320mm f/4.48 on full frame so you are gaining the extra depth of field due to longer effective focal length.
So, APS-C can give more depth of field and larger background blur at the same time, if you have the same lens. But the same lens is effectively a longer lens on APS-C, so you could achieve approximately the same by using a longer lens on full frame.
So, all in all:
- Select full frame vs APS-C based on which has the lens selections you want. Generally, smaller sensors have smaller aperture opening lenses, so something like micro four thirds might not be entirely ideal. I think most who want lots of indoors background blur choose full frame because fast short primes are available only for full frame (example: 50mm f/1.2 is available but APS-C equivalent 31.25mm f/0.84 or anything close to it isn't). For long telephotos primes the situation is better on APS-C, and outdoors telephotos are more convenient, so for outdoors APS-C vs full frame does not matter as much.
- Use a telephoto prime lens, especially long if the background-to-subject distance is large (although 70-200/2.8 zoom at 200/2.8 or 100-400/4.5-5.6 zoom at 400/5.6 might replace a prime at cost of larger price and heavier weight). Indoors, the lens should not be as long as outdoors.
- Select the aperture to have the desired depth of field. Ideally with long telephotos, you can achieve both the desired depth of field and the desired background blur at the same time.
It should also be noted that longer lenses result in compressed background. You may or may not want this. Also, the subject should fill the frame for strong bokeh. So, at the selected focal length, you should be as close to the subject as possible. Some of the assumptions I made in the calculations are no longer valid if the subject does not fill the frame.