f-number and focal plane illuminance

There has recently been considerable discussion here of the implication of what is sometimes called the "chimney effect" in digital cameras. This refers to the fact that, from a performance standpoint, we can think of the detectors of a digital sensor as lying at the bottom of tiny "chimneys". Thus, if we consider a tiny region of the object lying on the axis, and consider the "cone" of light emerging from the lens that will be converged at points across the corresponding region of the sensor, the light in the outermost portion of the cone will not be as effectual upon the photodetectors as the light in the central part of the cone. That's because the "more oblique" light is less able to "fully drop down the chimneys" and fall on the photodetectors.

One consequence of this is that as we increase the aperture of the lens (decrease the f-number), and consider a fixed luminance of a scene patch, the "impact" on the sensor increases more slowly than the area of the lens' entrance pupil - that is more slowly than the decreasing f-number would suggest. A recent report in Luminous Landscape presents some data developed by DxO Labs that shows the degree of this affect for various actual camera sensors. With an f/1.2 lens, the increase in exposure effect on the photodetectors over that from, say, an f/8 lens is about 1/2 stop less than the f-number would lead us to expect.

Now, those familiar with basic photometric theory may say:


In fact, that sounds credible. But it doesn't really work quite that way. Nevertheless, there is a real effect that story would seem to suggest.

Let's first look at the meaning of the thing about the cosine of the angle of incidence. Suppose we have a small "pencil" of light (a bundle of parallel rays within a tubular region of a certain diameter) and we consider it first striking the film "head on" and then we consider it striking the film at an angle of 30° from head on.

It will indeed cause an illuminance on the film less in the second case than the first, in the amount of the cosine of 30°.

And it also illuminates, with those two values of illuminance, a larger area in the second case than the first. (Imagine a circular wood dowel, first cut square across and then cut at an angle of 30° to straight across. The cut surface in the second case will be larger.)

In fact this is really why the illuminance is lower in the second case. There, the same amount of light (that contained in our hypothetical "pencil") lands over a larger area of the film. Since illuminance is the ratio of the amount of light to the area over which it lands, the illuminance is lower in the second (oblique arrival) case. (Note that this fundamental definition of illuminance contains no reference to a cosine; that has already done its work in our example by affecting what the area is.)

All this would fits well with the little story stated above.

But we might then ask, "what affect does this have on the behavior of a lens". More specifically:


In fact, answering that question directly is for more complicated than we might at first think. To do it, we must have in hand a model of what happens between the exit pupil and the film. This is very complicated. In fact, even respected optical textbooks often bungle analysis of this.

But we can get at the answer another way, not requiring us to have a model for "image space" photometrics.

Her is the basic plan of attack:

• We consider some tiny patch on the subject, having a certain luminance. (We cannot consider a point, since no light emerges from a point - it has zero area for any light to come from. We will use a point on the axis, since the off-axis situation introduces other complications, which will obscure the basic story here.)

• We consider how much light from that tiny patch enters the lens through the lens' entrance pupil. (We note that the f-number of the lens tells us, if we know the focal length) the diameter of the entrance pupil.)

• Assuming that the lens has 100% transmission (and if it doesn't, the final result will still come out the same way), then all the light from the subject patch entering ("captured by") the lens will be deposited on the film over a small region that is the image of the object region.

• We know the area of the "image patch"; its dimensions are just the dimensions of the assumed object patch times the magnification of the cameras in this situation.

• We know the amount of light that is deposited on this area, and the area of the region.
• Thus we can determine the illuminate on the film in that region.

Not a cosine of the angle of incidence in sight!

So, with this outlook, we might expect that, if we double the diameter of the entrance pupil, thus quadrupling its area (as would happen when we cut the f-number in half), then for an object region of a certain luminance we would get, on the image on the film, exactly twice the illuminance.

And that is very close to being true.

However, if we look at the matter in very fine detail, we find that this is not quite so.

Let's consider two small parts of the area of the entrance pupil, with the same area (perhaps 1 mm square), one at the center of the entrance pupil and one at the outer edge. We'll call these "windows".

Will those each collect the same amount of light from our hypothetical small region on the object. Not quite. Why?

• The distance from the subject patch to these two windows is slightly different (the "path" to the outer window is at a slight angle). Thus the luminous flux density from the object patch will be less at the outer window than at the central one (yes, its that old inverse square law). My God, is it very much smaller? No. Just smaller.

• The amount of light that will pass through a window bathed in light of a certain luminous flux density depends on the area of the window as observed from the direction of arrival of the light of interest (the "projected area" of the window in that direction). In the case of our outer window, at a (slight) angle to the path of the arriving light, its projected area is less than that of the central window (which the light strikes head-on). My God, is it very much smaller? No. Just smaller.

Thus the light collected from a patch on the object by a windows of a certain area in the entrance pupil is less as that area is farther from the center of the pupil.

So, as we add area to the pupil by decreasing the f-number and thus increasing the diameter of the pupil, the amount of light collected from some patch on the object does not increase quite as fast as the area increases. If we cut the f-number in half, the amount of light collected from the patch does not (quite) quadruple.

Thus, as we cut the f-number in half, the illuminant on the film for that object patch does not (quite) quadruple. By very much? Certainly not, except perhaps for high-magnification macro work, where the rim of the entrance pupil falls a substantial distance off the axis as seen from the object. In most cases, the effect is so small that it is masked by other complications in "real" lens behavior.

But note that in explaining this, I made no mention of the angle at which rays from the outer regions of the exit pupil strike the lens, or the cosines of those angles. I didn't have to. Nor do I know, precisely, how to do that. Nor, apparently, really do the authors of respected optical text books.

Best regards,

Doug

Those pesky cosines

Having mentioned what I consider to me an error in many optical textbooks, I thought I would take a minute to discuss the matter.

The issue came before me as I was looking into the matter of the theoretical falloff in illuminance on the focal plane (for a given object luminance) as the object moved off axis.

It is often said that, in an idealized situation, the falloff goes as the fourth power of the cosine of the angle by which the object was off the axis.

In a lens in which the pupils were collocated with the principal points, the angle by which the image of an object is off axis in (image space) is the same as the angle by which the object is off axis (in "object space"). Thus what angle it is whose cosine is involved isn't really an issue - they are the same angle.

But many authors call attention to the fact that in many real camera lenses, the pupils are displaced from the principal points (often intentionally, for various reasons, including minimizing the angle of incidence of marginal rays à propos mitigating the "chimney effect" on digital sensors ). In such a case, the angle by which the image is off-axis will not be the same as the angle by which the object is off-axis.

Thus, the question of which angle's cosine gets into the act becomes pertinent. Different authors come to different conclusions about that. The most popular results are:

• The falloff is proportional to the third power of the object's angle off-axis times the cosine of the image's angle off axis.

• The falloff is proportional to the fourth cosine of the image's angle off axis.

These both differ from the traditional expression of the relationship.

Seeing this, I undertook an independent derivation of the relationship, following a different tack than that of the other authors - a more direct one. It is essentially the one I discussed in the note above, having to do with keeping track of the light (luminous flux) "collected" by the lens.

In my analysis, I had to articulate the specific assumptions I had adopted about my "ideal lens). One of these was that the lens was rectilinear; that is, it had no geometric distortion. You might wonder what that has to do with the issue, but it is very critical. It affects the ratio of the size (and thus the area) of the image of a patch on the object to the area of the patch itself, as we move from the center of the image. That ratio is (in my approach) the very centerpiece of the matter of the illuminance upon the image patch.

My conclusion was (for the ideal lens I assumed, albeit one with displaced pupils):

• The falloff is proportional to the fourth power of the cosine of the object's angle off axis.

That is, it was just the traditional relationship, often claimed to be naïve in the light of the more detailed analysis of various authors!

Now, why did I not get the same result as the other authors? I think it is this.

The other authors' analyses drew upon a useful concept in image photometrics. It says that the exit pupil (for a lens regarding a small object patch of some luminance) appears to be a luminous disk of uniform luminance, which luminance is just the assumed luminance of the object patch (discounted for transmission loss in the lens). We can then treat this as the source for what happens from that point on.

In the work on image falloff, the authors drew upon this and proceeded from that point. In doing so, they assumed that not only was the luminance of this "disk" uniform across it, it was also the same regardless of the angle from which it was viewed (that is, it acted like a Lambertian emitter). I have not seen an insightful analysis of this, but I suspect it is not so. Or perhaps it is so, but the analysis failed to taken into account the distortion situation for the lens.

(I have corresponded with one of the authors, now retired, about this and he in effect replied, "Hmm. Maybe so.")

Now, in any actual lens, various factors will make the falloff substantially depart from the relationship for an "ideal" lens, so the distinction between the various "ideal" theoretical relationships is of no real consequence. Still, when someone says, "well, of course an actual lens would not behave the same as an ideal one, but your relationship for an ideal lens is not precise", the issue is on the table.

By the way, the role played in this by geometric distortion in the lens is well recognized (if not in the midst of analyses of which cosines are which). A known "trick" to reduce the degree of falloff is for the lens designer to intentionally introduce a degree of geometric distortion that will reduce the degree of falloff (we suspect this is only done in "cheap" cameras).

Best regards,

Doug

I knew that, didn't I?
But thanks for asking, Doug.
Now where'the 'on' switch?
Cheers
Tom

Hi, Tom,

Tom dinning said:

I knew that, didn't I?
But thanks for asking, Doug.
Now where'the 'on' switch?

Click to expand...

Yes, always the most important real question!

Best regards,

Doug

Doug, an excellent dissertation!

I wonder if two minor factors in limiting effective exposure of film to off axis parts of the image are:

1. Film is not perfectly matt so light arriving obliquely is more reflected from the film surface than light arriving at normal incidence. If some of the light never gets into the film it can't generate exposure. Cosine to the fourth power fall-off might be worsened?

2. Retrofocus wideangle lenses show "lateral magnification of the exit pupil", the exit pupil looks bigger off-axis than it does on-axis, so more light gets sent to the edges of the image than one would expect. Maybe image fall-off can be limited to a cosine cubed function?

Hi, Maris,

Maris Rusis said:

I wonder if two minor factors in limiting effective exposure of film to off axis parts of the image are:

1. Film is not perfectly matt so light arriving obliquely is more reflected from the film surface than light arriving at normal incidence. If some of the light never gets into the film it can't generate exposure. Cosine to the fourth power fall-off might be worsened?
An interesting point. This might be looked at as a matter of the film being non-lambertian as a receptor of light (we most often think of that matter in terms of an emitting surface, whether self-luminous or illuminated, but it applies equally to a receiving surface).

2. Retrofocus wideangle lenses show "lateral magnification of the exit pupil", the exit pupil looks bigger off-axis than it does on-axis, so more light gets sent to the edges of the image than one would expect. Maybe image fall-off can be limited to a cosine cubed function?

Click to expand...

As well, in may lens designs, the entrance pupil has a greater projected area from off-axis than if it were a true static circular aperture. This is often encouraged intentionally in the design as a way of mitigating off-axis falloff.

Note, that as I point out in my article, we need not really be concerned with the behavior of the exit pupil in this matter (rather just following the conservation of luminous flux).


But what is of considerable importance is the matter of geometric distortion (since that causes variation, over the image, of the ratio of the area of some element of the scene to the area of portrayal in the image).

I think that geometric distortion might be a corollary of lateral pupil magnification.

Thanks for your insights.

Best regards,

Doug

Hi, Maris,

In my previous note in this thread I mentioned by belief that the models of exit pupil behavior found in many textbook derivations of the theoretical function of off-axis falloff must be faulty. Let be discuss that a little.

Many textbook discussions of off-axis falloff draw upon the handy concept that the exit pupil of a lens behaves as a luminous disk, its luminance being uniform across its area if the pupil is viewed from a particular point on the focal plane. That luminance in fact is equal to the luminance of the scene at the point corresponding to the point of observation on the focal plane (that is, the point on the scene whose image falls at that point on the focal plane), discounted for non-unity transmission through the lens.

The textbooks often give a proof of this model of the exit pupil. But in fact those proofs invariably assume a point of observation on-axis (simplifies the derivation).

Then, in the derivations of the off-axis falloff function, the author often assumes that this "luminous disk" model of the exit pupil in every case holds for points of observation off-axis.

But in fact, the derivations proceeding on that basis give a result for the falloff function that is not consistent with consideration of the conservation of luminous flux.


Thus, I conclude that the classical luminous disk model of the exit pupil described above must not be (necessarily) valid for points of observation off-axis.

I have not however attempted to develop a corresponding luminous disk model of the exit pupil for the general case (for observation from a point on the focal plane that is not necessarily on-axis). Thus my conclusion about lack of generality of the model is merely "circumstantial".

I raised this issue some years ago via private correspondence with Warren J. Smith, the author of Modern Optical Engineering. He agreed that it was a very interesting thesis.

Derivations based on the generalized validity of the luminous disk model of the exit pupil include the result that the falloff follows the fourth power of the cosine of the off-axis angle in image space. I consider that result questionable for the reason I discussed.

Best regards,

Doug

I thought I would talk a little about pupil magnification as this morning's pre-breakfast entertainment.

The entrance pupil of a lens is the virtual image of its aperture stop from in front of the lens. That is, it is what we apparently see as the aperture stop (in diameter and longitudinal position) looking from in front of the lens through the lens elements that lie physically in front of the aperture stop.

Similarly, the exit pupil of a lens is the virtual image of its aperture stop from behind the lens. That is, it is what we apparently see as the aperture stop (in diameter and longitudinal position) looking from behind the lens through the lens elements that lie physically behind the aperture stop.

In addition to their being virtual images of the aperture stop, the two pupils are conjugate; that is, one is the image of the other (and vice-versa), as considered "through the entire lens".


In a symmetrical lens, the two pupils are coincident with the two principal points of the lens. In this case, the two pupils will invariably have the same diameter. We say that the pupil magnification is unity. ("Magnification" here is in the sense of lateral image magnification.)

In asymmetrical lenses, the two pupils may not fall at the two principal points, a situation we may describe as pupil displacement. If one pupil is displaced from its "home" principal point,. the other pupil will always be displaced from its "home" principal point. There is a fixed relationship between the two displacements (with focal length as a parameter). When there is pupil displacement, the diameters of the two pupils are not equal; the pupil magnification is no longer unity. And there is a fixed relationship between the two pupil displacements and the image magnification (focal length again being a parameter).

Thus in a lens design, rather than stating the two pupil displacements, we may more concisely state the image magnification (which then pins down the two displacements).

Note that none of this has anything to do with the possibility that in an actual lens, a pupil, viewed from a point off-axis, may not have the same elliptical projected outline we would see when observing a simple circular outline from that same angle.

*********

The applicable parts of breakfast this morning will (hopefully) be cooked for the first time on the new induction cooktop we installed yesterday. Some rework of the countertop opening was required, complicated by the fact that in the prior history of cooktops in this house a physically-large one was in place, and when its smaller successor was emplaced, a wood escutcheon was overlaid to accommodate it. But after about an hour of measuring, an hour of engineering, and a half hour of sawing with a saber saw, we had an handsome (if unusual) opening that gladly received the new machine. (The new machine, although apparently having a smaller footprint that the second earlier cooktop, is evidently deeper, so it was a close fit within the cabinetry.)

Carla spent last evening in "ground school" with the rather elaborate manual for the machine.

It even has a "sabbath mode", in which its functionalities and user interface details are restricted to conform with some accepted (I am tempted to say, sacrilegiously, "tortured") interpretation of Jewish law regarding activities on the sabbath, without preventing use altogether for cooking. (Our fancy refrigerator in Weatherford had a comparable thing.)

Best regards,

Doug