Entrance pupil determines depth of field

[Tom Axford]

Bill Claff has recently argued that the exit pupil of a lens is what really matters when determining the depth of field.

However, object field analysis of depth of field for an ideal lens clearly shows that the entrance pupil is the important variable, not the exit pupil. Most discussions of DoF seem to concentrate on the image field and ignore the object field. Yet object field analysis of DoF has been known for years and has been discussed several times in these forums. I first became aware of it from contributions by bobn2.

My understanding of object field analysis is set out below (sorry about it being in jpeg images, but that was easier for me to produce). I have tried to make explicit all the key assumptions used as I know that unstated assumptions are a frequent source of misunderstanding and confusion.

Unless there is a fundamental flaw in this argument, it is clearly the entrance pupil that matters.

 

[Detail Man]

... object field analysis of depth of field for an ideal lens clearly shows that the entrance pupil is the important variable, not the exit pupil.

Unless there is a fundamental flaw in this argument, it is clearly the entrance pupil that matters.

It is not that the size of Entrance Pupil does not matter. It is that the ratio of the diameter of the Exit Pupil over diameter of the Entrance Pupil (Pupil Magnification, "p") also matters.

.

The full Depth of Field identity for the case of an asymmetrical lens-system provided:


Page 24

View: original size

The numerical relationships are relatively simple - in that the right-hand multiplicative factor in the numerator [in the potentially asymmetrical case, when expressed in terms of Image Magnification (m)] is the sum of the reciprocals of "m" as well as "p" (if not equal to 1, as in thin-lens). Or restated, the arithmetic sum of m and p divided by the product of m and p.

m is normally rather small in value, increasing up to 1; and p is 1, or of a decreased value.

The smaller (in numerical value) of the two variables will dominate in determining the DOF.

Source: "Depth of Field in Depth", Jeff Conrad, 2006 (Erratum at the end added in 2018)

.

Source: https://www.dpreview.com/forums/post/64012402

 

[Tom Axford]

Please read what I wrote. The formula for DoF requires the entrance pupil, subject distance, field diameter and CoC ratio to determine the DoF. Nothing else is needed, not the lens focal length, not the pupil magnification, etc.

Of course, it is always possible to come up with different formulae for the DoF that make use of different sets of independent variables, but that is irrelevant.

 

[Detail Man]

Sorry, I missed the fact that your treatise evidently only applies to a (symmetrical) thin-lens - a description that does not apply to any of the camera lenses that you or I have ever used.

 

[Sympa]

Would it be possible to construct extreme lens designs and simulate them, to prove these points?

Perhaps using bclaff's ray tracer?

 

[ThrillaMozilla]

Bill Claff has recently argued that the exit pupil of a lens is what really matters when determining the depth of field.

Did he really? I missed that.

In a recent discussion Claff also pointed out that it's the image of the entrance pupil, not the physical aperture, that matters. So I think he understands it, but he may have approached it from a consideration of the lens design.

However, object field analysis of depth of field for an ideal lens clearly shows that the entrance pupil is the important variable, not the exit pupil. Most discussions of DoF seem to concentrate on the image field and ignore the object field.

Yes, and for many purposes the analysis is much, much simpler in the object field. Sorry, I didn't read your analysis, but I agree with your conclusion.

It has indeed been discussed many times. I posted a drawing recently that I think makes it very clear, but sorry I can't find it. Maybe it will turn up.

 

[alanr0]

Please read what I wrote. The formula for DoF requires the entrance pupil, subject distance, field diameter and CoC ratio to determine the DoF.

As far as I can tell, your derivation is correct, and is commendably simple.

I question how useful it is in practice, at least for most photographers.

Nothing else is needed, not the lens focal length, not the pupil magnification, etc.

You need the entrance pupil diameter. It could be measured with a travelling microscope, or with a second camera at known magnification. More likely it will be calculated from lens focal length and relative aperture (f-number).

You also need the field diagonal at the subject plane. You can walk over to the subject and use a tape measure, or insert a calibrated scale into a macro shot. Arguably, this effectively becomes a measurement of image magnification.

You need the distance from subject to the entrance pupil. You can figure out where this is for each lens you use, but location will typically move around as you change focal length of zoom lenses, and may vary with subject distance for internally focussing lenses. You can usually ignore this for landscape and moderate subject distances. The exact entrance pupil position becomes important for macro photography. It can be calculated from the location of the front principal surface and the pupil magnification.

Of course, it is always possible to come up with different formulae for the DoF that make use of different sets of independent variables, but that is irrelevant.

What is relevant is the extent to which your selected variables are both independent, and known with acceptable accuracy.

You may find that your formula meets your own needs, but I am not convinced that "Nothing else is needed".

As you point out: "... the hyperfocal field diameter is Ra. It is not possible to work out the hyperfocal distance without further information."

 

[AiryDiscus]

Bill Claff has recently argued that the exit pupil of a lens is what really matters when determining the depth of field.

Did he really? I missed that.

In a recent discussion Claff also pointed out that it's the image of the entrance pupil, not the physical aperture, that matters. So I think he understands it, but he may have approached it from a consideration of the lens design.

The entrance pupil is the image of the aperture stop as viewed through the front member. It would be unnatural to think of an image of that image.

However, object field analysis of depth of field for an ideal lens clearly shows that the entrance pupil is the important variable, not the exit pupil. Most discussions of DoF seem to concentrate on the image field and ignore the object field.

Yes, and for many purposes the analysis is much, much simpler in the object field. Sorry, I didn't read your analysis, but I agree with your conclusion.

It has indeed been discussed many times. I posted a drawing recently that I think makes it very clear, but sorry I can't find it. Maybe it will turn up.

 

[alanr0]

Would it be possible to construct extreme lens designs and simulate them, to prove these points?

Perhaps using bclaff's ray tracer?

Sure. But you can also consider very simple cases, where a pencil and the back of an envelope is good enough.

Consider a symmetrical macro lens, with entrance pupil coincident with the front principal surface, and exit pupil of identical size at the rear principal surface. For a thin lens, all are in the same plane.

Pupil diameter A = focal length / Relative aperture = f / N

For 1:1 imaging, subject distance from first principal point is 2f, and image distance from rear principal point is also 2f.

Working f-number is Nw = 2f / A = 2 N.

Depth of field = depth of focus = 2 Nw c = 4 N c, where c is the diameter of the circle of confusion. Here image space and object space CoC are identical.

Now consider the same lens with the aperture located at the front focal plane of the lens, distance f in front of the previous location. The lens is now image-space telecentric with infinite pupil magnification.

Distances from lens to subject and to image are unchanged, and the image magnification is still 1:1. However the distance from the subject to the entrance pupil has halved, the tangent of the half-angle subtended by the aperture at the subject has doubled, and so we have only half the depth of field.

The image plane depth of focus has also halved. The exit pupil is now at infinity and the working f-number is the same as the f-number at infinity focus. Depth of focus is half that of the previous example.

Depth of field = depth of focus = 2 Nw c = 2 N c

The object-space and image-space descriptions give the same answer.

To my mind the object-space description appears simpler, but I find the image-space description uses parameters which are easier to quantify.

YMMV.

 

[ThrillaMozilla]

Bill Claff has recently argued that the exit pupil of a lens is what really matters when determining the depth of field.

Did he really? I missed that.

In a recent discussion Claff also pointed out that it's the image of the entrance pupil, not the physical aperture, that matters. So I think he understands it, but he may have approached it from a consideration of the lens design.

The entrance pupil is the image of the aperture stop as viewed through the front member.

Of course it is.

It would be unnatural to think of an image of that image.

What you are saying, I believe, is that a pupil is an image, but the stop is the physical aperture. OK, fine.

 

[ThrillaMozilla]

What is relevant is the extent to which your selected variables are both independent, and known with acceptable accuracy.

You may find that your formula meets your own needs, but I am not convinced that "Nothing else is needed".

The equations are not as I would have written them, but the OP's geometrical model is spot-on. It's also the basis for some useful theorems (or lemmas) that can be used in the field. Of course, the calculation is so simple that they used to put it on lens barrels.

The main criticism of this model for calculation is that the distance to the entrance pupil is not known in general. But distance is conventionally measured from the sensor plane, and that's also a problem for all depth-of-field calculators. One can find wonderful calculators on the web, but they don't have the required lens parameters either. One wonders what cloud they pulled the calculations from, unless Bill Claff's equations magically solve this problem. Sorry, I haven't looked.

Forget that last sentence. I don't have time to litigate this stuff.

 

[bclaff]

Would it be possible to construct extreme lens designs and simulate them, to prove these points?

Perhaps using bclaff's ray tracer?

Yes, and I already proved (with the ray tracer) that it's the exit pupil.

See this .

 

[bclaff]

...

You need the distance from subject to the entrance pupil. ...

Will that's (1 + 1/m) * f but we also have (1 + m * f) from exit pupil to image.
So there's nothing special about that.

You may have noticed that in my derivation I use magnification not distance.

 

[ThrillaMozilla]

It is not that the size of Entrance Pupil does not matter. It is that the ratio of the diameter of the Exit Pupil over diameter of the Entrance Pupil (Pupil Magnification, "p") also matters.

The diameter of the entrance pupil is sufficient without any knowledge of the exit pupil.

Sorry, I missed the fact that your treatise evidently only applies to a (symmetrical) thin-lens....

No it doesn't. His geometrical analysis is spot-on. Look again and read his analysis carefully, up to the first equation.

I don't know how Claff gets to the same point, but object-space calculations are much simpler in my opinion. I assume that somehow they are both right.

 

[bclaff]

It is not that the size of Entrance Pupil does not matter. It is that the ratio of the diameter of the Exit Pupil over diameter of the Entrance Pupil (Pupil Magnification, "p") also matters.

The diameter of the entrance pupil is sufficient without any knowledge of the exit pupil.

Sorry, I missed the fact that your treatise evidently only applies to a (symmetrical) thin-lens....

No it doesn't. His geometrical analysis is spot-on. Look again and read his analysis carefully, up to the first equation.

I don't know how Claff gets to the same point, but object-space calculations are much simpler in my opinion. I assume that somehow they are both right.

If object space means using only the entrance pupil and disregarding pupil magnification then object space is only "right" if pupil magnification is unity.
We are often forced to make that assumption but it's not a real world scenario.

I'm happy with something along the lines of DOF depends on the exit pupil but without knowing pupil magnification we substitute entrance pupil as a matter of practice.
I simply think that knowledgeable people ought to know and state their assumptions.

But denying the role of the exit pupil is simply denial.

--
Bill ( Your trusted source for independent sensor data at PhotonsToPhotos )

 

[Tom Axford]

It is not that the size of Entrance Pupil does not matter. It is that the ratio of the diameter of the Exit Pupil over diameter of the Entrance Pupil (Pupil Magnification, "p") also matters.

The diameter of the entrance pupil is sufficient without any knowledge of the exit pupil.

Sorry, I missed the fact that your treatise evidently only applies to a (symmetrical) thin-lens....

No it doesn't. His geometrical analysis is spot-on. Look again and read his analysis carefully, up to the first equation.

I don't know how Claff gets to the same point, but object-space calculations are much simpler in my opinion. I assume that somehow they are both right.

If object space means using only the entrance pupil and disregarding pupil magnification then object space is only "right" if pupil magnification is unity.
We are often forced to make that assumption but it's not a real world scenario.

I'm happy with something along the lines of DOF depends on the exit pupil but without knowing pupil magnification we substitute entrance pupil as a matter of practice.
I simply think that knowledgeable people ought to know and state their assumptions.

But denying the role of the exit pupil is simply denial.

I did not simply deny the role of the exit pupil. I derived a formula for DoF in terms of the entrance pupil, distance to the entrance pupil, field of view at the subject and the circle of confusion ratio. That formula does not involve the exit pupil, hence my conclusion that the exit pupil is not necessary to work out DoF.

Now, it may be that the role of the exit pupil is somehow subsumed into one of the other variables, such as the distance to the entrance pupil. I do not understand fully how those things are inter-related.

But we have two approaches that both appear to be correct: one involves only the entrance pupil and its position, while the other involves the exit pupil and other variables. It would be good to be able to reconcile these two different results.

 

[bclaff]

....

If object space means using only the entrance pupil and disregarding pupil magnification then object space is only "right" if pupil magnification is unity.
We are often forced to make that assumption but it's not a real world scenario.

I'm happy with something along the lines of DOF depends on the exit pupil but without knowing pupil magnification we substitute entrance pupil as a matter of practice.
I simply think that knowledgeable people ought to know and state their assumptions.

But denying the role of the exit pupil is simply denial.

I did not simply deny the role of the exit pupil. I derived a formula for DoF in terms of the entrance pupil, distance to the entrance pupil, field of view at the subject and the circle of confusion ratio. That formula does not involve the exit pupil, hence my conclusion that the exit pupil is not necessary to work out DoF.

Now, it may be that the role of the exit pupil is somehow subsumed into one of the other variables, such as the distance to the entrance pupil. I do not understand fully how those things are inter-related.

But we have two approaches that both appear to be correct: one involves only the entrance pupil and its position, while the other involves the exit pupil and other variables. It would be good to be able to reconcile these two different results.

The circle of confusion relates to the size of the blur disc which in turn depends on the exit pupil. This is the piece you are glossing over.

 

[Tom Axford]

The circle of confusion relates to the size of the blur disc which in turn depends on the exit pupil. This is the piece you are glossing over.

Please explain.

 

[alanr0]

...

You need the distance from subject to the entrance pupil. ...

Will that's (1 + 1/m) * f but we also have (1 + m * f) from exit pupil to image.
So there's nothing special about that.

That is only true for the symmetric case where the entrance pupil coincides with the principal surface of the lens.

You may feel that this is implicit, but it is not one of the three explicit assumptions Tom makes in deriving the DoF formula in his OP.

You may have noticed that in my derivation I use magnification not distance.

As did I, in my post to your "Geometric treatment" thread. I agree that using magnification simplifies the analysis for the asymmetric case. Whether one prefers an approach based on pupil magnification or a working f-number / NA is a matter of taste.

In practical terms, magnification is of direct practical significance in macro and close-up photography -- the regime where the various treatments tend to diverge. Arguably, it is less intuitive for very small magnifications (larger subject distances), but here many alternative formulations reduce to perfectly adequate simple forms.

 

[alanr0]

What is relevant is the extent to which your selected variables are both independent, and known with acceptable accuracy.

You may find that your formula meets your own needs, but I am not convinced that "Nothing else is needed".

The equations are not as I would have written them, but the OP's geometrical model is spot-on. It's also the basis for some useful theorems (or lemmas) that can be used in the field. Of course, the calculation is so simple that they used to put it on lens barrels.

I don't dispute the accuracy of Tom's derivation. It is the "Nothing else is needed" that I am kicking back against.

The main criticism of this model for calculation is that the distance to the entrance pupil is not known in general. But distance is conventionally measured from the sensor plane, and that's also a problem for all depth-of-field calculators. One can find wonderful calculators on the web, but they don't have the required lens parameters either. One wonders what cloud they pulled the calculations from, unless Bill Claff's equations magically solve this problem. Sorry, I haven't looked.

Forget that last sentence. I don't have time to litigate this stuff.

But you do raise a valid point.

Bill's last two equations here give a formula for the conventional distance from sensor plane to subject. It is the sum of 3 terms:

1. Subject distance measured from first principal plane: u = f (1 + m)/m
2. Image distance measured from rear principal plane: v = f (1 + m)
3. Distance between principal planes: i

So for an accurate subject distance for macro photography, you need the distance between principal plane locations. Alternatively, if you can figure out the image magnification, then you don't need i, for depth of field calculations. The working f-number is sufficient. If needed, you can calculate this from the pupil magnification or the exit pupil location. Nw = N (1 + m/p)