depth of field - Does amount of background blur change with focal length given equal framing?

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.)

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.