Why is depth of field not always symmetrical about the object, yet depth of focus is practically symmetrical?

I have found an explanation on this, however I cannot seem to picture what the explanation is saying. From a Toshiba technical article:

Remember that the previous subsection mentioned that the depth of focus is the conjugate of the depth of field. A change in the object-side depth of focus caused by a displacement of the image plane by half the depth of focus is equal to half the depth of field. A slight displacement of an image plane causes the object-side focal point to shift by as much as a change in the image-side depth of focus divided by the longitudinal magnification of the lens. Because the depth of focus is symmetrical around the image plane, the depth of field is also symmetrical around the image plane.

However, if the optical magnification of a lens differs greatly at both ends of the depth of focus, the front and rear depths of field are not equal as described in 3 and 4 above. Equations for the depth of field are presented in the following subsections. In the case of machine vision and other applications requiring relatively close focusing, we recommend using an equation with an optical magnification term. Otherwise, we recommend using an equation with a subject distance term.

• They are BOTH asymmetrical. Commented Oct 3, 2023 at 0:43
• Rereading your question, I see you said "practically" symmetrical. So I guess you understood both had asymmetry. I somehow missed that. The stronger asymmetry of one to the other has more to do with distance from the lens than which depth it is. Consider that for a symmetrical lens, the two are identical at 1:1 magnification. At higher magnification, it is the depth of field that is more symmetrical. Commented Oct 3, 2023 at 3:48

It's because depth of field is based on human perception.

At short focal distances DOF is symmetrical, and it is shallow. It is shallow because there is a higher degree of magnification...i.e. details are presented larger to the lens to start with. And it is symmetrical because those details w/in the DOF are all very close to being at the same distance from the lens.

At median focal distances the DOF increases and divides ~ 1/3 - 2/3; and at focal distances beyond the hyperfocal distance the DOF becomes "infinite," with unlimited depth beyond the point of focus.

The reason the DOF extends more greatly behind the point of focus is because those details are at a greater distance, and therefore their magnification is less. And it becomes "infinite" because the details beyond the point of focus are so far away, and the magnification so low, that you cannot tell the difference between a detail that is in focus, and one that is not.

It's similar to displaying an image that is a little soft on your computer and then moving away from that image... as you increase the distance (reducing magnification/relative size) the details that were soft begin to appear sharp to you, while the points that were already sharp remain sharp... the DOF increases simply because you have changed your perception of it (ability to see the softness).

Another way of understanding it is that the depth of FOCUS is always extremely shallow and basically symmetrical (ignoring optical aberrations). I.e. the plane of actual maximum sharpness at the image plane (and resulting image) is extremely shallow. Short distances and higher magnification allow you to see that; longer distances and lower magnification do not.

• "At median focal distances the DOF increases and divides ~ 1/3 - 2/3" that's only at $H/3$ when the front/rear focal distances are 1:2. I mention this because it's often mis-stated as a rough 2:1 rule. But really, the rule is front:rear = $(m-1)/2m$ : $(m+1)/2m$ for $H/m$.
– scottbb
Commented Oct 2, 2023 at 23:16

Depth-of-field calculations are based on a selected circle-of-confusion size. Such a treatment goes something like this --- Using a full frame 35mm format camera format 24mm height by 36mm length, I desire to make an 8X10 inch image. This image will be viewed from a standard reading distance of 500mm (20 inches).

From experience, I know that a circle size of ¼ mm (0.25 mm) on the displayed image will be acceptable I also know that to make this enlarged image, I must apply 8X magnification to the camera’s tiny projected image. Therefore, I set the maximum circle size at the camera’s image plane at 0.25 / 8 = 0.03 mm. I will set the aperture at f/16, the mounted lens has a focal length of 50 mm.

Now a tack sharp image of a point on the object is in fact a bright spot of light surrounded by a series concentric circles or rings, bright, then dark, repeating, and getting dimmer. This is called the Airy disk (best focused smudge) named for its discoverer George Airy (Astronomer Royal). If focusing is in error in any direction, the smudge becomes enlarged.

Now the light from a point on the object will be directed by lens, is takes the shape of a cone. Sharp focus is only achieved if the apex of this cone just kisses the surface of the imaging chip (or film). If this cone of light is intercepted anywhere other than at the apex, it will become a spreads out circle of light.

Imagine two sharpened pencils set pointy end to pointy end. A slice at any point other than the apex returns a circle. The further from the apex the slice, the larger the circle realized.

Should the size of the circle slice approach the maximum permissible size stipulated its name changes from Airy disk to circle-of-confusion. The math to find this distance limit (focus error) is f number times max circle diameter. Thus, the math is f/16 times 0.03 = 16 X 0.03 = 0.48 mm.

Now this distance error of 0.48 mm can happen both forward or behind the adjoining apexes, thus the possible error is twice this distance or 0.96 mm.

0.96 mm is the span of depth of focus for this lash-up. The distance fore and aft of exact focus is symmetrical because we are considering just one object point.

Whereas depth-of-field is considering a span of object points at diverse distances. In other words, depth-of-field may be non-symmetrical because the subject distances are dissimilar.