# How does blur radius change with distance from the point of focus at macro distances?

In a macro situation, is there a known function that maps the blur radius as a function of the physical distance from the plane of focus?

I'd expect this function to give near zero (in pixels) at zero distance (say in mm), and grow significantly outside the DOF region. Is there a known form for this function? Is it quadratic?

By blur radius I mean the approximate radius of the disc in pixels a point source at the prescribed distance shows on the sensor, assuming that we have a perfectly circular aperture. Note that I'm looking for the answer specifically in the macro situation. For the thin lens approximation we have for the diameter of the blur circle:

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Which I don't think applies in the real lens macro situation.

• I think it depends mostly on the lens elements, and they would be very different for each lens. That's why some lenses have a 'great bokeh' and others not. – Aganju Jul 8 '18 at 13:26
• @Aganju - the bokeh quality is indeed a function of the lens element configuration, but the size should be independent. – nbubis Jul 8 '18 at 13:47
• "Bokeh" is a quality of blur, and as such does not have a size. The proper term for what you're referencing is blur radius, also known as the circle of confusion. – Michael C Jul 8 '18 at 14:46
• @MichaelClark - that's correct, I am looking for the blur circle diameter which reduces to the CoC when points at the edges of the DOF are considered. The problem is that all formula I could find are for thin lenses, and I'm specifically targeting the macro case. I'll edit my question. – nbubis Jul 8 '18 at 14:55
• The math is the same (similar triangles are easier), but for macro, we don't know f or S1 or S2, and maybe not f'/stop. Macro calculations are inaccurate because we don't know extended focal length, and maybe not f/stop increase, and likely not the location of the nodal points of the lens to know working distance. At the close focus, these are large factors. Accuracy depends on knowing the numbers. More details at scantips.com/lights/fieldofview.html (page bottom). Blur diameter is computed in terms of CoC at scantips.com/lights/dof.html (but NOT for macro). – WayneF Jul 8 '18 at 15:45

I've just been researching this topic myself so here's a late answer to your question.

The equation you provide doesn't only apply to thin lens, it also applies to ideal thick lens with no abberations. It even applies to macro photography, assuming you have symmetric lens (pupil magnification = 1). Unfortunately, you probably don't.

This false assumption is completely negligible for non-macro photography, but the error becomes significant as magnification grows.

In case you have information on your lens' principal (or nodal) points and pupil magnification (or positions), you can use this formula for more accurate results: • f is focal distance
• p is pupil magnification
• u is the position of the focused point (measured from the front principal (or nodal) point of the thick lens)
• u_d is the position of the defocused point
• m is magnification at focused point: f / (u - f)
• N is f-number at infinity focus
• k is the diameter of the blur spot in the image plane

This is from page 23 of Depth of Field in Depth by Jeff Conrad, referenced by the related Wikipedia page.

As you can see, the relationship is not exactly linear, but my mathematical analysis skills are not that great so I can't really tell you more about its behaviour.

• Awesome! Is the formula for the magnification still correct though in a macro situation? – nbubis Jul 17 '18 at 21:11
• On the one lens I tried it on, calculated distance for the required magnification was accurate enough (within about a millimeter at a distance of about 75 mm), but it wasn't really extreme macro (magnification about 0.5). The trick is to measure the distance from the front principal point. I looked up its location in a data sheet from the manufacturer's web site. It's an industrial equipment lens, though, and I don't know if that kind of information is available for the usual, "artistic-use" lenses. – relatively_random Jul 17 '18 at 22:01
• Also, the confusing bit was that the front principal point was actually behind the rear one. – relatively_random Jul 17 '18 at 22:02
• I also just noticed that you want sizes in pixels. The formulas we are talking about are all in millimetres. To get pixel sizes of the blur, divide k by the size of your sensor's pixels in millimetres. – relatively_random Jul 17 '18 at 22:15