Bokeh basics and intuition from lens equations

SUMMARY:
The amount of background blur is...
... linearly proportional to focal length.
... inversely proportional to F-number.
... inversely proportional to the focal distance.
... directly proportional to lens diameter.
... depends on the square root of lens cost.

If you disagree with any of these statements, I politely suggest that you read or browse the following before commenting.

Obtaining perfect bokeh is a skill that can only learned through experimentation and practice. The two parameters which are most often discussed about bokeh are its quality (shape) and quantity (amount). I hope to shed some light on the latter and how it is fundamentally controlled by the photographer. Also, what methods can be used to gain and advantage and what is gained by simply throwing money at the problem. What follows is insight from examining the physics governing nature.

While some people have seen the basic lens equations (wikipedia is a good reference), many more are familiar with derived statements or rules-of-thumb, some true, some untrue, that circulate. Pick your favorite and investigate it further to ensure you're not perpetuating lies! While I make no claims of being and expert, I am an engineer; naturally I am familiar with the practice of using approximations and making simplifications.

I realize that my analysis is not completely rigorous, particularly in cases of close focus (macro photography), but I am only trying to gain intuition from the mathematics. If you have some deep technical understanding I welcome comments on errors or inconsistencies in my assumptions.

QUALITY:

The quality of bokeh is primarily influenced by the lens itself. The Out-of-focus (OOF) point-spread function defines the shape and hence it's quality. This shape is easily examined by taking a picture with highlights in the OOF regions. Qualifiers for describing the quality range from harsh or nervous to the highly-desired creamy and buttery smooth. The math behind this is quite technical but it is largely determined by the aperture shape and the lens design. With those comments in mind I now turn my attention to the second parameter of bokeh.

QUANTITY:

A quantitative description for background blur relates the level of defocus applied to points being bokeh'd (TM:). Every photographer can tell you that this depends on factors such as the focal length [f], focus/subject distance [u], and F number [N]. These variables also directly affect the depth-of-field (DoF). The DoF is a description of the region having an amount of defocus which is below an arbitrary threshold set by the circle-of-confusion (CoC) [c].

The following analysis examines how these 3 variables, over which the photographer has control, dictate the amount of background blur. The best examples of bokeh often have a background that is much more distant than the focal plane. This is the conditions I will use to establish a measure for the amount of blur.

From these 4 parameters (f,u,N,c) another useful variable can be derived. This is the magnification [m]. It can be used to specify a consistent subject framing for different focal length and subject distance combinations (adjustment is necessary if different crop factors are used).
m = f/(u-f)

This can be replaced with f/u when u> > f (u much larger than f). This approximation has a diminishing error as the ratio u/f increases. For example, at a subject distance 10x the focal length, there is a 10% error and at 100x, 1%.

The method I propose for characterizing the quantity of bokeh is to examine the size of the blur disk of points at infinity, given a specific subject distance. The equation for calculating the hyper-focal distance can be modified to measure this blur. The Hyper-focal distance is that which if focused upon gives the desired amount of blur (circle of confusion) to objects at infinity. The equation is shown below.
u_hyp = (f^2) (Nc) + f

By reorganizing this equation as shown below, it now calculates the circle of confusion at infinity (blur diameter) necessary to obtain a desired hyper-focal distance.
c_inf = (f^2) (N(u-f))

Taking the ratio of this value to the standard CoC used for DoF calculations a blur factor [b] is defined.
b = c_inf/c
b = (f^2) (Nc(u-f)) [equation 1]

By itself, this blur factor relationship reveals the inverse proportionality of bokeh quantity to F-number. Additional insight is gained by making substitutions which yield the next two results.
b = (fm) (Nc) [equation 2]
b = (f^2) (uNc) [equation 3]

Equation 2 removes the subject distance but now includes the magnification. Now, another statement can be made. For a constant F-number, if the subject distance is changed to maintain a constant magnification then the blur amount is linearly proportional to the focal length. Obviously the perspective will change because of the varying subject distance, but the subject's framing will remain constant. Interestingly these conditions will yield an approximately constant Depth of Field. I think this result surprised me the most. Think about it - using a constant F-number, if you move further away, but increase the focal length the depth of field will stay constant but objects at infinity will blur more!.

Equation 3 shows two facts. First, blur is inversely proportional to subject distance. Second, if subject distance is held constant, background blur increases as the square of focal length.

(continued in next post)