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After having spent some time reading about DOF and the derivation of its formula, I tried calculating the theoretical DOF of different lenses. Now to my knowledge, a camera using a lens with a focal length of 100mm, stopped down to an f-stop of 16, and the same camera on a 25mm with an f-stop of 4, should produce the same DOF. I am not trying to achieve the same framing and the object distance to the camera would remain the same, which means that both DOF and perspective should be the same. However, when plugging my values into the calculator, they are not?

  • Circle of confusion (c) = 0.029mm
  • Object distance (o) = 5000mm
DOF 1 DOF 2
f = 25 f = 100
N = 4 N = 16
DOF = 109m DOF = 2.49m

I also am aware that the variable of the circle of confusion can be very subjective, with all the aspects of visual acuity, viewing distances, and image enlargement. Nevertheless, as I am in theory using the same camera and assuming the output of the photo would be the same, I would think it is fair to assume that c is constant. In this case, I am using a c value of 0.029 (common full frame value from Wikipedia).

There obviously must be some confusion on my part, but after seeing and reading through Manuel Luebbers's tests (https://manuelluebbers.com/large-format-look-alexa-65-vs-alexa-mini/), and Steve Yedlins's article on matching lens blur on different format sizes, I can't see why the calculations shouldn't match up.

I also have checked that an object distance of 5m is not the Hyperfocal distance when using the 25mm lens, as for reasons I don't quite understand yet, that tends to mess the calculations up.

An explanation from an expert or anyone knowledgeable on this topic would be greatly appreciated.

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    \$\begingroup\$ Focal length DOES affect depth of field, but It is overly simplistic to say focal length alone affects depth of field. It is actually the magnification resulting from both focal length, AND distance, that affects depth of field. It seems you are leaving out the distance and/or magnification in your calculations. \$\endgroup\$
    – Mike Sowsun
    Commented Sep 30, 2023 at 16:08
  • \$\begingroup\$ Thank you for the response, I am not the best at formulating my questions, which may lead to confusion. I did take into account the distance which I put as 5m. Also on Manuel Luebbers test you can see that he was able to virtually achieve the same DOF between a 18mm at T2.8 and 35mm at T5.6 by keeping the distance the same. So based on that I would think it would be fair to assume that focal length doesn’t have an affect on DOF. I should probably also clarify that I’m not trying to achieve the same frame or AOV, I realise that the image on the 75mm will look magnified (cropped). \$\endgroup\$
    – vannira
    Commented Sep 30, 2023 at 17:32
  • \$\begingroup\$ After reading Steven Kerstings answer I now understand what you meant by the fact that it’s more so the magnification from the lens that leads to the difference in DOF. Thank you :) and sorry for my previous misunderstanding \$\endgroup\$
    – vannira
    Commented Sep 30, 2023 at 20:45

4 Answers 4

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Because magnification (focal length and subject distance) affects the Depth of Field more than aperture does (approx twice as much generally).

Start with 50mm @ f/8 focused at 10 ft. If you halve the aperture # (2 stops) it will result in ~ half the DOF (~3ft from ~6ft).

If instead you double the focal length (f/8/10ft) to 100mm the resulting DOF is ~ one quarter (~1.5 ft from ~6ft)... it would require 4 stops of aperture to counter this effect, not two.

Because magnification is constant for a given composition, focal length and subject distance tend to negate each other; leaving only aperture as an effective control... but that is only if you consider keeping the same composition. And similarly, cropping in post and outputting the same image (size/composition) is also a change of the final magnification and also affect the resulting DOF (as does viewing distance/visual acuity/etc).

And then there is the hyperfocal distance of a lens/aperture. Once you focus beyond that distance the DOF immediately jumps to "infinite." And the hyperfocal distance is much closer for a shorter FL lens. E.g. if you focused the 50mm f/4 at 6m instead. How close your settings and focus are to the hyperfocal distance will affect how much impact other changes make. E.g. focus a little long and the DOF remains infinite, but you lose a little from the near limit. But focus a little short and you lose a lot from the distant limit and the DOF is no longer infinite.

In your examples the physical size of the aperture openings are the same. So the depth of focus is the same.

The only thing that affects the depth of focus is the lens' physical aperture diameter. Which is probably where the confusion arises in focal length and depth of field. Depth of focus is how sharp details are recorded, and how unsharp other details are, at the image plane... it's also called "focus tolerance." Everything else about depth of field is based upon how the depth of focus appears to you based upon its' magnification/visibility (focal length, cropping/crop factor, display size, viewing distance, etc, etc).

If you were to equalize the two images so that the physical display size of the details are the same relative to you, you would then find the depth of field to be the same.

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  • \$\begingroup\$ I’m not sure if I’m understanding correctly, but so does this mean that because you are using a longer lens that enlarges the image, the points that seemed in focus now appear out of focus. So in reality if I viewed the image produced by the 100mm lens at a decreased image size to where the area pictured on the longer lens is the same size as that area in the picture from the 25mm lens the dof would be the same. Hence why in Manuel Luebbers example, when he used a 18mm on t2.8 it gave him the same dof as when he used the 35mm at t5.6, because he switched to a larger format sensor \$\endgroup\$
    – vannira
    Commented Sep 30, 2023 at 17:49
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    \$\begingroup\$ @vannira, I'm not familiar with the specific examples you mention, but yes. The only thing that affects the depth of focus is the lens' physical aperture diameter. Which is probably where the confusion arises in focal length and depth of field. Depth of focus is how sharp details are recorded, and how unsharp other details are, at the image plane... it's also called "focus tolerance." Everything else about depth of field is based upon how the depth of focus appears to you based upon it's magnification (focal length, cropping/crop factor, image display size, image viewing distance, etc). \$\endgroup\$
    – Steven Kersting
    Commented Sep 30, 2023 at 18:23
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    \$\begingroup\$ @vannira, I just realized your example has the same depth of focus for both lenses, so I updated my answer to include that aspect. \$\endgroup\$
    – Steven Kersting
    Commented Sep 30, 2023 at 18:56
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    \$\begingroup\$ @vannira, focus/subject distance is entirely subjective. I.e. a lens is always in-focus at some distance; whether it is where you want it to be is irrelevant to the resulting depth of focus recorded. The physical aperture size determines the depth of focus because it determines the minimum amount of diffraction for any point that is "in focus." And that amount diffraction/blur radius directly correlates to the circle of confusion in relation to depth of field. \$\endgroup\$
    – Steven Kersting
    Commented Oct 1, 2023 at 2:20
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    \$\begingroup\$ And yes, depth of field is entirely variable; it is not a fixed characteristic of an image. What most are actually referring to is depth of focus... the sharpness of details relative to each other; they just do not realize it. \$\endgroup\$
    – Steven Kersting
    Commented Oct 1, 2023 at 2:24
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Focal length totally affects DOF. That's one of the reasons that older video camera footage and current-day cell phone video doesn't look "filmic". The sensor is tiny, so a short focal length is able to produce a big enough image to fill the sensor. Film cameras, digital cinema cameras like the RED or ARRI Alexa, and large-format cameras have larger image sizes (the size of the negative or sensor), so a longer focal length or closer shooting distance is required to fill the frame. This means that if you happen to want long DOF you have to have a smaller aperture, which means very bright lighting. That was one of the unusual things Orson Welles did in Citizen Kane.

I remember back in the early 2000's, every film student wanted their video camera to look more like film. One clever hack was a device that placed a spinning disc of ground glass in front of the camera lens. One could then use longer film lenses, and project a 35mm frame onto the ground glass. The video camera would then be focused on that image on the glass, only a few inches from its own lens! It was all kind of wonky, and looked pretty grainy and fuzzy, but it did produce those nice creamy out-of-focus backgrounds that we all craved.

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    \$\begingroup\$ Interesting technique, thank you for the comment. \$\endgroup\$
    – vannira
    Commented Oct 1, 2023 at 11:17
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    \$\begingroup\$ Why does it have to be a "spinning" disc of ground glass? \$\endgroup\$
    – Nayuki
    Commented Oct 2, 2023 at 6:32
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    \$\begingroup\$ @Nayuki it has to spin, or the diffusion pattern of the glass will be noticeable as an artifact. This is similar to the difference between repeating a single frame or using 100 consecutive frames. There's always noise in an image, and the single frame, or non-spinning diffuser, will "freeze" noise in a visible way. \$\endgroup\$
    – Carl Witthoft
    Commented Oct 2, 2023 at 23:12
  • \$\begingroup\$ @CarlWitthoft Right. And conversely, spinning it simulates film grain. \$\endgroup\$
    – Robert M.
    Commented Oct 5, 2023 at 19:03
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In short: there are two reasons your expectations don't hold true:

  1. In order for crop 'equivalence' to hold true, everything must scale with crop factor, including subject distance and subject size. And truth be told, even light frequency (wavelength) should scale (but let's ignore that).
  2. F-number is a ratio of two distances (focal length to entrance pupil diameter); as such, it is scale-invariant to crop factor, which is 1-dimensional distance scale factor.
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  • \$\begingroup\$ I’m very interested in your answer, however I can’t seem to really grasp what those two points mean? What do you mean by crop factor since in theory I am using the same sensor size. \$\endgroup\$
    – vannira
    Commented Sep 30, 2023 at 21:02
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    \$\begingroup\$ Well, first of all, your original statement/question, "Since we know that focal length does not affect DOF" makes absolutely no sense. But in terms of comparative focal lengths between crop factors, it's implied (at least to me) that there's an argument that DoF is not dependent upon (crop-factor equivalent) focal length. But instead, perhaps you should post your DoF equations that purportedly are independent of focal length. Hint: you won't be able to. \$\endgroup\$
    – scottbb ♦
    Commented Sep 30, 2023 at 23:05
  • \$\begingroup\$ Thank you. Yes I’ve read many articles stating that focal length isn’t a factor affecting DOF or that it is, but in their reasoning they are actually talking about about the aperture size (f-stop). So when I then saw the example of a frame produced by the Alexa 65 with a 35mm lens being matched by the Alexa mini on a 18mm, I assumed that it was true. Nevertheless I now understand why that isn’t the case. \$\endgroup\$
    – vannira
    Commented Oct 1, 2023 at 8:10
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Circle of confusion is an absolute value. Circle of confusion can be drawn on the image itself. If you were to scale the image from 25mm lenses to match the image from 100mm you will find out that circle of confusion does not match any more.

So, if you are going to scale the image 4x for comparison, then the circle of confusion should be 4 times smaller for 25mm.

Measuring DOF through defining circle of confusion is correct. It is impossible to have same DOF for two cases you suggested because DOF should reflect perceived sharpness of the image.

In more words: let's say that you use one camera to take pictures and photos to one person (viewer) in some specific setup (monitor/printing/distance, same processing, no cropping). In that scenario the viewer won't be able to tell spot light source from small enough circle light source. It's the viewer who is confusing everything.

And maximum size of that light source can be measured in milimetres of displayed image. That's the actual circle of confusion which is important albeit not the one which is usually discussed. You can then convert it to camera sensor millimetres if you know sensor dimensions and that would be the commonly known circle of confusion.

There can be some average value for this metric which suits most use cases but you can never get a universal value. Even processing the image differently might change viewer's perception.

So, if you want to compare two images this way, one taken with 100mm F16 and other taken with 25mm F4 and then cropped to same content, you will find out the circle of confusion for the viewer is almost the same (provided that quality did not deteriorate too far because of cropping), and if you then want to know what this circle of confusion on the sensor is, it would be different, because different area of the sensor is used to present same image.

In your comparison, the absolute DoF is different but sharpness of object relatively to each other is same to the extent the quality of optics and resolution permits it.

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  • \$\begingroup\$ I am a little confused. So as I have just found out, it is correct that when I scale the image of the 25mm to the size of the 100mm, the two images would now have the same DOF. Is this also what you are saying ? (Apologies that I don’t understand) \$\endgroup\$
    – vannira
    Commented Sep 30, 2023 at 21:17
  • \$\begingroup\$ @vannira more or less, yes. I tried explaining it, added text to answer. \$\endgroup\$
    – Euri Pinhollow
    Commented Oct 1, 2023 at 10:47
  • \$\begingroup\$ thank you, I think I understand what I couldn't seem to understand from the first text \$\endgroup\$
    – vannira
    Commented Oct 1, 2023 at 18:11

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