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Depth of field and focal length

MGrayson

Subscriber and Workshop Member
:lecture:

We all know two things about DoF: a) it gets shallower with longer lenses and b) it gets shallower as you get closer to the subject. Three things! We know three things! c) DoF gets shallower with lower f-number.

NOTE: I will NOT be talking about different sensor sizes and equivalence. You can all relax now.

One curious claim, though, is that a) and b) almost exactly cancel out if you make the *subject* the same size in each picture. Take a head shot with a 250mm lens at f/4 and you'll have the same DoF as the same headshot taken with a 120mm lens at f/4 on the same size sensor. Of course, the 120mm lens will have to be half the distance from the subject to get the same head size.

This can be seen from the formulas, or from geometric optics (ray tracing), but there's nothing like a demonstration! Between my own face and a Fuji lens, I picked the more attractive subject. Here are two shots at f/4 with the 250/4 and the 120/4, both wide open, and both cropped the same size.

250/4


120/4


Of course, the pictures are different, and the subject geometry is different, but it's about the same size in both pics, and the DoF, visible on the tape measure, is roughly equal.

Here are the same at f/8:

250/4 @f/8


120/4 @f/8


Now modern lenses are not classical thin lenses, so I wasn't sure that this would hold at all. I just always wanted to do this experiment. It looks pretty good.

Yes, there is nothing MF about this, except that I used a GFX 100 for the shots. I would have posted it to the Boring Lecture forum, but there isn't one. I should probably start it. Tired of Jim Kasson's careful measurements? Come to the Boring Lecture forum! :grin:

Thanks,

Matt
 
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Photon42

Well-known member
There is a difference in background blur, but maybe I had already one Pastis too much ...

IIRC there was some discussion on how quickly the sharpness falloff is between lenses, when the new SL lenses were introduced. Leica was sort of claiming their f2 is the new f1.4. Just like Fuji claims their 44x33 is the new large format :ROTFL:
 

Shashin

Well-known member
One curious claim, though, is that a) and b) almost exactly cancel out if you make the *subject* the same size in each picture. Take a head shot with a 250mm lens at f/4 and you'll have the same DoF as the same headshot taken with a 120mm lens at f/4 on the same size sensor. Of course, the 120mm lens will have to be half the distance from the subject to get the same head size.
I see your lecture, and raise you a tutorial.

Or to put it another way, if you maintain the angular size of the entrance pupil as viewed from the object, DoF is maintained--more or less. Just as depth of focus (the space in front and behind the image plane) is dependent on the f-number, not the focal length, as the f-number is proportional to the angular size of the exit pupil from the image plane. (Extreme magnification and how a lens focuses can change this (lenses can focus by changing lens to image plane distance or changing focal length).)
 

MGrayson

Subscriber and Workshop Member
There is a difference in background blur, but maybe I had already one Pastis too much ...

IIRC there was some discussion on how quickly the sharpness falloff is between lenses, when the new SL lenses were introduced. Leica was sort of claiming their f2 is the new f1.4. Just like Fuji claims their 44x33 is the new large format :ROTFL:
There IS a big difference in background blur. What's surprising is that that is unrelated to Depth of Field. A distant streetlight will make a bigger bokeh with the longer lens, because there isn't a big proportional difference between the distance to the two lenses. The effects only balance out when the 120 is half the distance to the subject as the 250. (Yes, I know that 120 isn't half of 250, but then 1.4 isn't the square root of 2.)

I'm making no claims about HOW lens sharpness falls off. It clearly differs between lenses. I'm talking about very coarse effects.

I see your lecture, and raise you a tutorial.

Or to put it another way, if you maintain the angular size of the entrance pupil as viewed from the object, DoF is maintained--more or less. Just as depth of focus (the space in front and behind the image plane) is dependent on the f-number, not the focal length, as the f-number is proportional to the angular size of the exit pupil from the image plane. (Extreme magnification and how a lens focuses can change this (lenses can focus by changing lens to image plane distance or changing focal length).)
You are, as usual, exactly right.

M
 

Shashin

Well-known member
There IS a big difference in background blur. What's surprising is that that is unrelated to Depth of Field. A distant streetlight will make a bigger bokeh with the longer lens, because there isn't a big proportional difference between the distance to the two lenses. The effects only balance out when the 120 is half the distance to the subject as the 250. (Yes, I know that 120 isn't half of 250, but then 1.4 isn't the square root of 2.)

I'm making no claims about HOW lens sharpness falls off. It clearly differs between lenses, but I'm talking about very coarse effects.

M
I have never really studied that, but I suspect it is related to linear perspective. The ratio between the distance of the camera to subject and camera to background object means the difference in the degree of blur changes with those distances. I guess that would be hard to quantify as there is no blur at the object plane. Or this can be explained by relative magnification--but perhaps inversely proportional, which would account for less blur with the longer focal lengths.

I miss geometry...
 

MGrayson

Subscriber and Workshop Member
I have never really studied that, but I suspect it is related to linear perspective. The ratio between the distance of the camera to subject and camera to background object means the difference in the degree of blur changes with those distances. I guess that would be hard to quantify as there is no blur at the object plane. Or this can be explained by relative magnification--but perhaps inversely proportional, which would account for less blur with the longer focal lengths.

I miss geometry...
The size of the point spread is the intersection of the cone based on the aperture and the focal plane. When the two apertures "look" the same from the focal plane, these cone intersections are fairly similar for nearby points. A distant vertex (the streetlight) will make a cone that looks like a cylinder based on the apertures, and will thus be twice as large intersecting the focal plane (which is still at the subject).

Here are the two (coincident) cones on the larger and smaller apertures, both the same f-stop. An OOF point makes cones that intersect the focal plane in small circles. Remember that the two lenses are capturing the same amount of the focal plane, so circles on the focal plane will look the same size (head shots) in both captures.



If we zoom way in, we see that the two circles are almost the same size.



But for a distant object, the cones become cylinders, and the circles are twice as big for the larger opening.



M
 
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Jack

Sr. Administrator
Staff member
While DoF is similar, the 120 makes your lens look fat. Just sayin...


:grin::grin::grin:
 

Jack

Sr. Administrator
Staff member
Perhaps of interest to your discussion, a somewhat weird but older recognized phenomena will be noticed when shooting larger formats; and that is lens extension required to focus on the subject, and then along with the associated increase in focal length to accomplish it, you in turn further reduce DoF. For example, a "normal" lens for a 16x20 large format view camera is 600mm. However, to make a "head and shoulders" portrait, you need to focus nearly to 1:1 with that camera, and in so doing, your 600mm lens has become effectively 1200mm in length. So that one lens can give you both a normal view at infinity focus, and the roughly ideal doubling of focal length to make the portrait...
 

Shashin

Well-known member
The size of the point spread is the intersection of the cone based on the aperture and the focal plane. When the two apertures "look" the same from the focal plane, these cone intersections are fairly similar for nearby points. A distant vertex (the streetlight) will make a cone that looks like a cylinder based on the apertures, and will thus be twice as large intersecting the focal plane (which is still at the subject).
Matt, thanks for all that. I certainly agree with your analysis. I am uncertain it explains some of the difference in the Bokeh/CoC. While I think the geometric representation you have is right, for me, there is still the issue of the appearance of the image. What your diagrams don't address is the rendering of the scale of the objects behind the subject--the magnification in both setups are not the same. What does the image of the CoC look like in the image?

(I am just thinking through this so bare with me.)

The ratio of image size is equal to the ratio of the camera to the objects in an image. In this case, we are using two focal lengths to create the same magnification at the object plane. The distance in that case, just to keep the math simple, is proportional to the focal length--so the focal length and (lens) object distance between the two cases could be x and 2x for a 125mm lens and a 250mm lens. However, the far object in both cases is 2ft. For simplicity, lets assume the 125mm lens is also 2ft from the lens so x = 2ft. So the ratio in size between the lens and ruler width is the ratio between the camera and lens distance and the camera and ruler distance or x:2x (the 125mm case) and 2x:3x (the 250mm case)(forgive my math notation, I am not a mathematician, just in case you have not worked that out).

So here is the question: does the ratio of object sizes based in camera to object distances apply equally to the image of the CoC?
 

MGrayson

Subscriber and Workshop Member
Matt, thanks for all that. I certainly agree with your analysis. I am uncertain it explains some of the difference in the Bokeh/CoC. While I think the geometric representation you have is right, for me, there is still the issue of the appearance of the image. What your diagrams don't address is the rendering of the scale of the objects behind the subject--the magnification in both setups are not the same. What does the image of the CoC look like in the image?

(I am just thinking through this so bare with me.)

The ratio of image size is equal to the ratio of the camera to the objects in an image. In this case, we are using two focal lengths to create the same magnification at the object plane. The distance in that case, just to keep the math simple, is proportional to the focal length--so the focal length and (lens) object distance between the two cases could be x and 2x for a 125mm lens and a 250mm lens. However, the far object in both cases is 2ft. For simplicity, lets assume the 125mm lens is also 2ft from the lens so x = 2ft. So the ratio in size between the lens and ruler width is the ratio between the camera and lens distance and the camera and ruler distance or x:2x (the 125mm case) and 2x:3x (the 250mm case)(forgive my math notation, I am not a mathematician, just in case you have not worked that out).

So here is the question: does the ratio of object sizes based in camera to object distances apply equally to the image of the CoC?
If I understand your question correctly, the answer is "sorta". Here's a picture of a more distant object with CoC of its two endpoints. Then I'll work out the formulas...



The Blue intervals on the focal plane are the Bokeh of the ends of the arrows in the 120/4 capture. The Red intervals (almost completely covering the Blue ones) are for the 250/4. Both the magnifications (distances between the intervals of the same color) and the CoC's are different.

OK.. to be exact. Lenses are distance 2 and 4 from the focal plane. their physical apertures are .5 and 1, respectively. I'll let x be distance from the focal plane. Then the bokeh for the 120/4 satisfies (similar triangles) .5/(2+x)=b/x, so b = .5 x/(2+x) = x/(4+2x). For the 250/4, it is x/(4+x). When x is near zero, these are close because the 4 in the denominator overwhelms the x vs. 2x. When x is large, the disk sizes converge to 1/2 and 1 respectively. For magnification, let's go to a pinhole camera. Then an arrow (it's always an arrow) of length 1 at position x on the central axis... well, it's tail goes to the center of each image. It's head goes to ... similar triangles again... for the 120mm, t/2=1/(2+x), or t=2/(2+x). For the 250mm, t/4=1/(4+x) or t=4/(4+x). Again, when x is zero, we have the same magnification (wrt the final image). As x grows, t decreases for both lenses (as it should), but for large x, the t for the 120 goes as 2/x, and for the 250, it goes as 4/x... Exactly what you'd expect from their focal lengths!

Here are plots as functions of distance from the focal plane.



Note! While the bokeh increases the same to first order with both lenses (in this particular setup), the magnifications agree only to zeroth order - they start changing linearly with x. This is why the geometry (appearance) of your subject changes a lot, but the DoF doesn't.

M
 
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Shashin

Well-known member
Matt, thank you. I appreciate you taking so much time and effort! (I hope the COVID isolation is not getting to you.)

Geometric optics is fairly straightforward, but the visual implications are more complex. The last plot is particularly interesting in this regards.
 

MGrayson

Subscriber and Workshop Member
Weirdly, the Magnification and Bokeh curves cross at the same distance (x=4) for both lenses. The object looks smaller with the 120, but it is also blurred less, so at THAT distance from the focal plane, the objects will look exactly the same, but with a magnification difference! I'll have to check that one with an actual camera and see...



The 120/4 can focus at 2 feet, but the 250/4 can't quite do 4 feet. I'll have to change the numbers a bit. I can do 6 feet and 3 feet, but then the critical object distance will be 6 feet further out, or 12 feet from the 250, 9 feet from the 120.

To be continued...

Matt

Oh, another prediction of geometric optics is that if you focus at infinity, and take pictures of Bob at 10 feet and 100 feet, and blow up the 100 foot image so that the two Bob's are the same size, then they will look identically blurry. Have to test that one, too.

And in all the stuff above, I am, of course, ignoring the effect that Jack mentioned above about lens "focal length" changing as you extend the bellows.
 
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Shashin

Well-known member
And in all the stuff above, I am, of course, ignoring the effect that Jack mentioned above about lens "focal length" changing as you extend the bellows.
You know more about the Fuji optics, but the Pentax the 120mm macro works by changing the lens to image plane distance. But the longer focal length lenses are internal focal (IF), which changes focal length to focus--the Fuji 250mm looks like it is IF. That difference will influence results, especially at shorter object distances. Not sure how much those matter in this case, but I imagine the mathematical modelling will be close to actual use.

And thanks again for doing this work. I am enjoying this thread.
 

Abstraction

Active member
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
 

MGrayson

Subscriber and Workshop Member
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
Yes, that's what the first set of images was supposed to demonstrate. The size of the OOF disk at the focal plane is determined by the angle at the vertex of the "light cone" and distance from the focal plane. Magnification tells you how big THAT disk will be in the final image. For nearby points, the angle won't change much.

But looking at the pictures brought up the question about how that relationship changes when you move further from the plane of focus. Different focal length lenses will change both the magnification and the angular width of the cone differently with distance. Formulas are great, and necessary, but I always want pictures, either diagrams or photographs, to go with them.

Matt
 

Godfrey

Well-known member
Very interesting discussion! I follow .... :)

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics... :cool:

G
 

MGrayson

Subscriber and Workshop Member
Very interesting discussion! I follow .... :)

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics... :cool:

G
I’m ignoring everything that happens once the light hits the entrance pupil ( where the aperture appears to be from outside ), and I’m assuming that it doesn’t move when focusing. These assumptions are wildly false, especially when either a) the subject to lens distance is comparable to the lens to sensor distance or b) the lens is complex with many and/or floating elements. The experiments are to see how good the simplest approximations work in non-macro photography, and to illustrate some of the less obvious relationships. Jack’s example counts as macro. A head shot on an 16x20 camera is definitely around 1:1, and, as Jack said, the distance from lens to film can double, throwing the simple arguments off. So yes, it is more like a pinhole with the longer focal length. If you stop down to f/256, or f/32,768 it would BE a pinhole camera of that longer focal length.

I’m not mistaking this for photography! The best photographer I know is unaware of any optics theory. He knows what camera settings work and he concentrates on his lighting and subject. Since I’m 90% mathematician and 10% photographer, I find this stuff fun.

Matt
 

Shashin

Well-known member
Depth of field is a function of image magnification and aperture. If both are the same, the DoF will be the same regardless of the focal length, film/sensor format, etc.
Actually, that would not be true (I assume you are using the term magnification as linear magnification at the image plane, which is the usual meaning of the term). DoF is a perceptual/subjective quality of an image (which is why it needs to define a permissible circle of confusion based on human perception). Format will impact DoF because format effects viewing conditions, which is why DoF scales change with format--you could also display images based on the the size of their relative format, where DoF would appear the same, but no one does that.

And by aperture, that is not the same as f-number. If you mean the angular size of the entrance pupil, then that would be more accurate, but format is still going to influence the results across formats.

And this is were the conversation become complex--how do we perceive DoF?
 

Shashin

Well-known member
Very interesting discussion! I follow .... :)

I know that when you're doing pinhole photography, the distance of the pinhole from the receiver plane determines the FoV or "effective focal length". And reading Jack's discussion of the 16x20 camera lens, I begin to wonder if a lens on such a large capture mechanism begins to act like a pinhole due to the relative sizes of the format and the size of the iris diameter...? And how this relates to the other issues you're calculating on DoF and out-of-plane focus qualities...?

Any thoughts on that stuff? Or am I conflating things in an inappropriate way? Although I have a degree in Mathematics, my specialization was statistics and statistical theory, nowhere near the world of optics... :cool:

G
I doubt you would ever get to such small apertures that a lens would have f-number as high as those of a pinhole camera--one definition of a pinhole is that it has so much DoF it does not need optics. But you are right that the effective aperture can be significantly different from the marked aperture. Basically, if you draw out a lens at twice its focal length, for example, having a 300mm lens 600mm from the image plane, the effective aperture will be two stops less--it will go from f/11 to f/22, for example, explaining why exposures can be significantly impacted in macro photography. Qualities such as diffraction and magnification will change by the same degree as well.
 
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