Charts of Depth of field vs focal length, scaling f/#

[Nodda Duma]

I know this subject has been discussed at length over the years, but I don't think the information has been presented in quite this manner before.

This week I had an opportunity at work to refresh my depth of focus calculations, and in the process created a spreadsheet which allows me to compare depth of focus for several different focal lengths.

With such a spreadsheet, I found myself wondering how the depth of field for longer focal lengths compare to 50mm f/1.4. So I plugged in some numbers to generate some charts.

Nomenclature:
object space = everything in front of the lens. Depth of field relates to object space.
Image space = everything behind the lens. Depth of focus relates to image space (see the difference?)

Key to understanding the charts:

1) For work I use exact equations, essential for doing actual lens design / optical engineering work. That said, the typical approximation is very close. What is the approximation? From wikipedia:

The depth of field can be determined by focal length, distance to subject, the acceptable circle of confusion size, and aperture.[2] The approximate depth of field can be given by:

for a given circle of confusion, C, focal length, f, F-number, N, and distance to subject, u.

Note I use "circle of confusion" and "blur circle" interchangeably. Optical engineers use one or the other depending on which lens design school they're from -- Rochester or Tuscon. I didn't go to either so I can use both if I want.

1b) Everybody always argues about what circle of confusion should be defined as. If you understand charts, you'll note that the curves are relative... the actual value used for circle of confusion will just shift everything up or down. The shape of the curves remain the same. If you don't understand charts then, well... I can't really help you. I'm not a math teacher.

1c) I approached circle of confusion in two different ways:

Set the circle of confusion of all focal lengths equal to 50mm / 1720 (Kodak's rule of thumb). This is what I call the "scanning film" approach... that is, you scan your negatives to take advantage of the greater resolution provided by larger formats than 35mm. So blur circle = 0.029 micrometers for all focal lengths.​

set the circle of confusion to the specific focal length / 1720 (Kodak's rule of thumb). This is what I call the "enlarging each negative to an 8x10". So a 35mm negative needs greater resolution to get to an 8x10 print, while an 8x10 negative can be contact printed and so the circle of confusion can be much larger than that of a 35mm. (Think about it for a bit and you'll understand what I'm getting at).​

2) The depth of fields are calculated for objects at portrait distances ... i.e. 1-10 meters. The equations then tell you how much object space will be in focus. For example, imaging an object at 2 meters with a 50mm f/1.4 lens will have ~127mm (about 5 inches) in focus... 2.5" in front and 2.5" behind the plane of focus. Btw 1" = 25.4 mm. You should memorize that conversion if you deal with lenses in America.
3) Depth of field is independent of the film format being used. As a linear system, the imaging lens doesn't care what's behind it.
4) I account for bellows factor in the curves (the impact on depth of field was surprisingly small, even as bellows factor has a large effect on working f/#).
5) These are paraxial relationships... which is fine since the type of subjects that you like to shoot with thin DoF are typically at center, but it doesn't account for design aberrations which will cause the DoF to decrease further depending on what specific lens you are using and its wide-open performance.

The first chart is a comparison of typical "wide open" f/#'s for different focal lengths that you'll find on 35mm cameras on up to 8x10 format cameras. I wanted to see how the depth of field compares for the different lenses that I use when I'm scanning in negatives. So for this information, I defined all blur circles as 50mm / 1720 = 0.029 um.

For reference, as you stop down a lens, its depth of field curve would shift upward on this chart.

The second chart is for people who print to 8x10 paper. I was answering the question "What f/# should I set my lens to in order to have equivalent depth of field to a 50mm f/1.4 lens?" So for this data, the blur circle is equal to each lens' focal length / 1720. i.e. for a 305mm lens, the blur circle is 0.177 um using Kodak's rule of thumb.

You can see the answer to my question in the legend. The values are mathematically exact (round to two decimal places). In practice you obviously set your lens to the nearest whole stop setting.

I can slice the data a bunch of different ways, but this is how I sliced it to answer the questions I had on my mind. In particular I was digging into why it seems all the old 6x9 folders out there had 105mm f/3.5 lenses but never really much faster. Well... makes perfect sense now looking at this chart.

The numbers are all in metric because lens designers work in metric, and as mentioned these charts were for real work. Which is also why you know this is good information.

The horizontal axis is distance to your object (technically from the entrance pupil) in meters. 1 meter ~ 1 yard so there you go. The vertical axis is the total range of distances (millimeters) that are in focus ... "in focus" defined as less blurring than what you care about .. i.e. your circle of confusion or blur circle. ~25mm = 1 inch. The vertical axis is a logarithmic scale.

So for example at 3m object distance, if the curve shows a 300mm depth of field, then everything at a distance of 3 + / - 0.150 meters is in focus.

 

[Nodda Duma]

A couple of interesting tidbits that I didn't expect but make sense once I thought about it:

1) the closer the subject is to you, the narrower the DoF. So if you are tight in on a subjects face and thus closer to the subject, then your DoF will be thinner than if you are, say, 10 meters away.
2) You can see why longer focal length lenses such as those for larger formats do not need to be as fast as your 35mm format lens. Or put another way, you can see why very fast long focal length lenses would be almost impossible to use effectively, because the DoF quickly becomes razor thin.
3) Extrapolating into the digital world, you can also see why smaller format sensors require even faster optics to provide the same DoF that we've gotten used to with full frame and larger. Or put another way (and from a design perspective), you can see why faster lenses are both practical and possible for the smaller image circles of digital sensors.



Of course, this curve extends out to infinity focus. The equations used to generate this information can also be used to derive the hyperfocal distance which gives you the greatest DoF for a given focal length and f/#.

 

[wyofilm]

Great graphs. They make it easy to see how the dial on the Voigtlander Bessa II was constructed, for example. The dial indicates acceptable focus for various f stops at various distances. Spacing at shorter distances is greater and the f/stop curves have tails on the shorter distance side.

Voigtlander isn't unique here, but I like the elegance of their presentation.

https://cameratique.com/ebay/opticxchange/30032016pic7.jpg

 

[wiltw]

To simplify things, one has only to keep in mind that when you have SAME SUBJECT SIZE at focal plane = same DOF, regardless of what FL was mounted at the time...only the f/stop used needs to be the same for same DOF.
50mm f/4 at 10' = same DOF as 100mm f/4 at 20' = same DOF as 200mm f/4 at 40' because the subject size is the same in the frame for any of those combinations.

 

[reddesert]

The second chart is for people who print to 8x10 paper. I was answering the question "What f/# should I set my lens to in order to have equivalent depth of field to a 50mm f/1.4 lens?" So for this data, the blur circle is equal to each lens' focal length / 1720. i.e. for a 305mm lens, the blur circle is 0.177 um using Kodak's rule of thumb.

You can see the answer to my question in the legend. The values are mathematically exact (round to two decimal places). In practice you obviously set your lens to the nearest whole stop setting.

View attachment 230268

I can slice the data a bunch of different ways, but this is how I sliced it to answer the questions I had on my mind. In particular I was digging into why it seems all the old 6x9 folders out there had 105mm f/3.5 lenses but never really much faster. Well... makes perfect sense now looking at this chart.

The numbers are all in metric because lens designers work in metric, and as mentioned these charts were for real work. Which is also why you know this is good information.

The horizontal axis is distance to your object (technically from the entrance pupil) in meters. 1 meter ~ 1 yard so there you go. The vertical axis is the total range of distances (millimeters) that are in focus ... "in focus" defined as less blurring than what you care about .. i.e. your circle of confusion or blur circle. ~25mm = 1 inch. The vertical axis is a logarithmic scale.

So for example at 3m object distance, if the curve shows a 300mm depth of field, then everything at a distance of 3 + / - 0.150 meters is in focus.

You don't say the key thing about this chart (IMO) - probably you knew it when constructing the chart, but I think it would be useful for others to understand. These curves overlap because the lenses at these f-numbers all have equivalent physical aperture diameters to the 50/1.4 lens. A 50/1.4 lens has an aperture of 35.7 mm, so do a 85/2.38, 105/2.94, and so on.

In the object space, there is a plane of perfect focus. A point source behind the plane of perfect focus emits rays of light in a cone that enter the physical aperture of the lens. That cone intersects the object plane of perfect focus, and the cross-section is a circle. One can think of the lens as imaging that circle at the plane of perfect focus onto the sensor/film as a circle, the blurred image of the original point. Seen that way, it's clear that the amount of blur depends strictly on the aperture diameter (and on the subject's distance, of course).

I think of it this way:
- distance to subject determines perspective (ratio of near/far sizes, elongation or flattening of facial features, etc)
- given the distance, focal length determines magnification (image size) on the sensor. You then have to think about sensor size and desired enlargement.
- given the distance, aperture diameter determines DOF and amount of blurring of the background. Obviously not all aperture diameters are available for all sensor/film formats.

 

[Nodda Duma]

To simplify things, one has only to keep in mind that when you have SAME SUBJECT SIZE at focal plane = same DOF, regardless of what FL was mounted at the time...only the f/stop used needs to be the same for same DOF.
50mm f/4 at 10' = same DOF as 100mm f/4 at 20' = same DOF as 200mm f/4 at 40' because the subject size is the same in the frame for any of those combinations.

Only correct if you assume the same blur circle for all your focal lengths.... i.e. you are shooting strictly within the same format. I point out that distinction because I was exploring DoF for different formats, so the wrinkle is that the blur circle may not or usually is not assumed the same when you move from one format to the next (especially if you print in the darkroom).

DoF is a function not just of object distance and f/# but also your chosen blur circle.

In the chart below your relationship doesn't hold because the blur circle isn't the same for each focal length ... i.e. the photographer is switching between formats when he/she selects another focal length, with a (likely unconscious) assumption that the size of the acceptable spot blur is changing.

Your relationship holds here because the blur circle is assumed to be the same i.e. the photographer is shooting within the same format.

You don't say the key thing about this chart (IMO) - probably you knew it when constructing the chart, but I think it would be useful for others to understand. These curves overlap because the lenses at these f-numbers all have equivalent physical aperture diameters to the 50/1.4 lens. A 50/1.4 lens has an aperture of 35.7 mm, so do a 85/2.38, 105/2.94, and so on.

Yeah... sort of. Strictly speaking it’s the working aperture diameter (or effective aperture) after bellows factor and pupil magnification have been factored in, but a decent approximation esp. at longer object distances is the physical diameter. In that chart you’re referring to, you are backing out the effective aperture (which can be thought of as changing with distance due to bellows factor and pupil magnification). Obviously it starts getting pretty esoteric pretty quick.

I didn't point it out because I've never seen the aperture of a lens marked in actual aperture diameters... only f/#. So relating to f/# is more useful from an "in the field" perspective (as well as a design perspective).

 

[foc]

To simplify things, one has only to keep in mind that when you have SAME SUBJECT SIZE at focal plane = same DOF, regardless of what FL was mounted at the time...only the f/stop used needs to be the same for same DOF.
50mm f/4 at 10' = same DOF as 100mm f/4 at 20' = same DOF as 200mm f/4 at 40' because the subject size is the same in the frame for any of those combinations.

Like this?

 

[wiltw]

Like this?

Yup, same DOF (i.e., what is IN focus) regardless of FL (and of course, assuming a single format size!).
What changes with FL is the degree to which the Out of Focus Background is blurred...longer FL = more blurred OOF Background

 

[ic-racer]

Why leave out magnification? In large format and close up photography its effects on DOF are visible in prints.

 

[Nodda Duma]

Who says I left out magnification?

“Exact” means “did not leave any shit out”

 

[Sirius Glass]

To simplify things, one has only to keep in mind that when you have SAME SUBJECT SIZE at focal plane = same DOF, regardless of what FL was mounted at the time...only the f/stop used needs to be the same for same DOF.
50mm f/4 at 10' = same DOF as 100mm f/4 at 20' = same DOF as 200mm f/4 at 40' because the subject size is the same in the frame for any of those combinations.

I discovered this decades ago photographing a fountain while wanting to get a greater depth of field while preserving the subject size and f/stop. I kept changing lenses on the camera moving closer and further. It bothered me for years, so when I was working at Kodak I asked my boss. He opened one of my optic books and copied the depth of field equation. Then he took another equation and solve it for a variable. He substituted the second equation in the first and the focal length dropped out of the equation. I would appreciate it someone would reproduce those equations and the substitution in this thread.

 

[reddesert]

Yeah... sort of. Strictly speaking it’s the working aperture diameter (or effective aperture) after bellows factor and pupil magnification have been factored in, but a decent approximation esp. at longer object distances is the physical diameter. In that chart you’re referring to, you are backing out the effective aperture (which can be thought of as changing with distance due to bellows factor and pupil magnification). Obviously it starts getting pretty esoteric pretty quick.

I didn't point it out because I've never seen the aperture of a lens marked in actual aperture diameters... only f/#. So relating to f/# is more useful from an "in the field" perspective (as well as a design perspective).

Yes, when I said "physical aperture diameter" I should have said "entrance pupil diameter," because one could reasonably think I meant the actual mechanical diameter of the iris inside the lens. But from a photographer's point of view the entrance pupil diameter is what he or she sees; most photogs think of the entrance pupil as the aperture. Optical scientists are different.

I think aperture (pupil) diameter is useful just to understand what is going on. A 50/2 lens on 35mm has just about the same DOF as a 100/4 lens on 6x9 and a 200/8 lens on 4x5. If you lined up all those cameras and looked at them from the subject's point of view, they'd be looking into the same diameter light-hole.

The bellows factor is an effect, but mostly a minor perturbation (apart from macro distances), until one starts to try to work at, say, portrait distances with larger formats. Over on LFPF, people will say they want to do head-and-shoulders portraits on 8x10 with a long focus lens, and most have not thought clearly about magnification. A head and shoulders could be say 20" wide, so 1:20 on 35mm, but 1:2.5 on 8x10, so the bellows factor and extension required starts to bite hard.

 

[Nodda Duma]

Optical folks usually just say aperture, too.. only splitting hairs about entrance pupil vs exit pupil vs aperture when necessary or when in a tough detailed review with customers trying to show they know more than the designers.

Bellows factor has an effect on working f/# and thus the exposure as we all know, and you see how close distances to the object also affects depth of field. The effect on exposure is outside the scope of a DOF scaling discussion so I only briefly touched on it. To address all the aspects of imaging would require volumes of books.

As far as including bellows factor (magnification) in the exact calculation of DoF vs leaving bellows factor out of the approximation shown above, the approximation error is negligible.

I discovered this decades ago photographing a fountain while wanting to get a greater depth of field while preserving the subject size and f/stop. I kept changing lenses on the camera moving closer and further. It bothered me for years, so when I was working at Kodak I asked my boss. He opened one of my optic books and copied the depth of field equation. Then he took another equation and solve it for a variable. He substituted the second equation in the first and the focal length dropped out of the equation. I would appreciate it someone would reproduce those equations and the substitution in this thread.

Focal length drops out with your assumptions, yep. You can see that in that last chart above.

I specifically didn’t include the equations in this thread because trying to type equations into forums is as frustrating as (insert here the most frustrating thing you can think of). You couldn’t pay me enough. If I have some free time I can replicate what your kodak guy derived on a sheet of paper and post a pic of the paper, but honestly as a sharp fellow you could probably derive it yourself.

 

[Sirius Glass]

Optical folks usually just say aperture, too.. only splitting hairs about entrance pupil vs exit pupil vs aperture when necessary or when in a tough detailed review with customers trying to show they know more than the designers.

Bellows factor has an effect on working f/# and thus the exposure as we all know, and you see how close distances to the object also affects depth of field. The effect on exposure is outside the scope of a DOF scaling discussion so I only briefly touched on it. To address all the aspects of imaging would require volumes of books.

As far as including bellows factor (magnification) in the exact calculation of DoF vs leaving bellows factor out of the approximation shown above, the approximation error is negligible.



Focal length drops out with your assumptions, yep. You can see that in that last chart above.

I specifically didn’t include the equations in this thread because trying to type equations into forums is as frustrating as (insert here the most frustrating thing you can think of). You couldn’t pay me enough. If I have some free time I can replicate what your kodak guy derived on a sheet of paper and post a pic of the paper, but honestly as a sharp fellow you could probably derive it yourself.

I have tried for years and cannot come up with the right substitution and I know that the same size subject requires an assumption.

 

[reddesert]

I have tried for years and cannot come up with the right substitution and I know that the same size subject requires an assumption.

This is the depth-of-field equation screen captured from https://en.wikipedia.org/wiki/Depth_of_field
u is subject distance, N is f-number, C is circle of confusion, f is focal length.

Let's assume the subject is pretty far away, like >10 focal lengths. Then the magnification is small, bellows factor is negligible, and the magnification is simply m = f / u. Meaning if your subject is say 1 meter tall, the height on sensor is 1 meter * f/u.

Substitute m into the DOF equation and we get DOF = 2 * N * C / m^2.
f and u are subsumed into the magnification.

If you change focal length and then move your feet so as to keep magnification (image size) constant, then the DOF remains constant.

 

[Nodda Duma]

That’s the derivation.

What I’m not sure about (but think it’s true) is if focal length completely falls out or is just negligible at short distances, where magnification and pupil magnification have to be considered.

 

[Sirius Glass]

View attachment 230333

This is the depth-of-field equation screen captured from https://en.wikipedia.org/wiki/Depth_of_field
u is subject distance, N is f-number, C is circle of confusion, f is focal length.

Let's assume the subject is pretty far away, like >10 focal lengths. Then the magnification is small, bellows factor is negligible, and the magnification is simply m = f / u. Meaning if your subject is say 1 meter tall, the height on sensor is 1 meter * f/u.

Substitute m into the DOF equation and we get DOF = 2 * N * C / m^2.
f and u are subsumed into the magnification.

If you change focal length and then move your feet so as to keep magnification (image size) constant, then the DOF remains constant.

That’s the derivation.

What I’m not sure about (but think it’s true) is if focal length completely falls out or is just negligible at short distances, where magnification and pupil magnification have to be considered.

Thank you, that is the step that I could not remember.

 

[cowanw]

I always have a spasm when we talk of these things when people talk of a smaller capture size yielding more DOF. Is it the magnification of the print or the negative that counts?

 

[wiltw]

I always have a spasm when we talk of these things when people talk of a smaller capture size yielding more DOF. Is it the magnification of the print or the negative that counts?

BOTH count...DOF with an 8x10 print is the 'standard' for programs and the scales on fixed focal length lenses...if you make a 16x20" print instead of 8x10, the DOF of the larger print is proportionately shallower.

 

[cowanw]

So why do people talk about a smaller sensor having more Depth of field.
To fill the image with a same size image the smaller the sensor the shorter the focal length and the more enlargement is required.

 

[reddesert]

Comparing the normal lenses for a small and large format, at the same print size, and the same f-number: the smaller format does in fact have more depth of field.

If you stand in the same place with a 35mm camera, 50mm lens, at f/8, and a 4x5 camera, 200mm lens, at f/8, and take the same picture, and enlarge both to the same print size, you'll get basically the same framing and perspective, but the 4x5 will have less DOF. This is because you are using an entrance aperture of 50/8 = 6.25mm on the 35mm camera, and 200/8 = 25mm on the 4x5. So out of focus subjects have a wider cone of light proceeding from subject to aperture.

This is one reason why photographers using large format stop down a lot, and using f/8 is unusual. f/32 would give you the same DOF as the 35mm, at the cost of a longer exposure of course.

 

[Jim Jones]

Reddesert makes a strong point. Comparing DOF on prints of identical size is more practical than comparing DOF on negatives. Also, rather than calculate the hyperfocal distance, consider that it is maybe 1000 or 2000 times (depending on the photographer's preference) the entrance pupil of the lens at the shooting aperture.

 

[Ko.Fe.]

I use online dof calculator and dof scale on my RF lenses. It helps in fast picture taking process a.k.a. street photography.
But I’m sure chats like op did are useful for LF. Everything which is easier in 135 needs to be done special way with LF.

 

[Nodda Duma]

It barely scratches the surface of what I worry about in lens design. I whipped the spreadsheet together in about 20 minutes at work so for me this is second nature.

Lens designers don't use rules of thumb for this basic stuff.

 

[_T_]

I was turned on to this resource on apug back before photrio existed:
https://lenspire.zeiss.com/photo/app/uploads/2018/04/Article-Bokeh-2010-EN.pdf

This is the newest version of the article, and it does a pretty good job of going into as much detail as a photographer might ever need to know about the effects of the basic elements of the design and operation of a lens on depth of field and quality of bokeh.

Nicely and concisely confirms everything that Nodda Duma has stated in this thread, while also going into a bit more detail of the descriptive how and why for those who's learning style is so inclined.