Center of perspective and nodal point

[Helen Bach]

...I did a google search on "Zeiss entrance pupil" and came up with several examples of published data, including positions and sizes of the entrance and exit pupils. Lens theory says that the displacements from the corresponding principal planes should be f*(1-p) and f*(1 - 1/p) for the exit and entrance pupil respectively. I checked the consistency of the published data with theory and found some significant discrepencies. Jeff Conrad did a more extended search and found that sometimes the lens data fits theory quite well and sometimes it doesn't.

Neither of us understands why there should be such differences. One conjecture is that all the gaussian optics used in the theory depends on the paraxial approximation, and for real lenses it is not valid. Do you have any thoughts on the matter? More generally, it seems clear for many large format lenses that we use rays which are far from the lens axis, but eveything still seems to work properly. Do you know anything about this, and do you have a reference which someone like me could understand without in essence becoming a lens designer myself?

I hope Jeff doesn't mind, but I will attach a zip file containing his spreadsheet of results. I doubt a problem will arise, but just in case remember it is his intellectural property.

Leonard,

Just a quick note.

I've had a look at your data, and some more Zeiss data. The apparent error appears to be caused by the lack of precision of the given data: it is only to three sig figs. A small change to the pupil diameters, for example, could remove the discrepancy.

Were you looking for a derivation of the f*(1-p) and f*(1 - 1/p) displacements? I'll post them if you wish.

Best,
Helen

[Jerry Fusselman]

Were you looking for a derivation of the f*(1-p) and f*(1 - 1/p) displacements? I'll post them if you wish.

I would love to see them!

[Helen Bach]

Here are the two derivations. They are drawn roughly to scale for a Zeiss 60 mm f/3.5 CFi lens. They show the bundle of rays for an object at infinity (to show the relationship for the exit pupil) and for an image at infinity (for the entrance pupil). I hope that they are legible.

Best,
Helen

[Leonard Evens]

The arguments are also contained in Jeff Conrad's excellent Large Format Photography page article "DOF in Depth". It is basicaly the same as what Helen sketched in her response above, but he fills in all the details, and it is easy to read.

As to Helen's explanation of why the data is off for some of the lenses, it does appear that it has to do with the sizes of the entrance and exit pupils, as she says. The values given are not consistent in those cases with the rest of the data, which does appear to be consistent. I suspect that there was some error, either in measuring the sizes directly or in computing their sizes or even in listing the results. In the last example in the spreadsheeet, for example, the actual pupil magnification appears to be something like 0.346 rather than 0.369. This seems to me to be too large to be explained by the number of signficant digits in the values given for entrance and exit pupils.

[Emmanuel BIGLER]

Emmanuel...Is the point that for keystoning and similar matters you do need to use the nodal point?

Yes. Definitely.

We can describe the situation as follows.

Consider an object plane and an image plane, parallel and reasonably located so that it does make sense to try and find an image within the limits of a reasonable Depth of Field (DOF). For example, the object will be located quite far, the image close to the lens focal plane.

When the lens, even a very complex, thick, compound lens, has a unit pupillar magnification, in those conditions, the image is immune against residual keystone effects when you tilt the lens provided that the object and the film plane are parallel.

This is the basic rule of all photographic textbooks. In fact it appears that it is a particular property of lenses with unit pupillar magnification.

Of course ordinary, basic, keystone effects would occur as soon as the object and film plane are not parallel.

When the lens has a non-unit pupillar magnification, and with the object and film plane parallel, the image will exhibit residual, non-conventional, keystone effects when the lens is tilted.

This effect occurs with the Canon 24mm TS-E lens for which (I have double checked with what was measured by the photographer who discovered the effect) the pupillar magnification ratio is above 2, someting like 2.5, which is unusual for a non fisheye photographic lens.

On another but related matter:

I did a google search on "Zeiss entrance pupil" and came up with several examples of published data, including positions and sizes of the entrance and exit pupils. Lens theory says that the displacements from the corresponding principal planes should be f*(1-p) and f*(1 - 1/p) for the exit and entrance pupil respectively. I checked the consistency of the published data with theory and found some significant discrepencies. Jeff Conrad did a more extended search and found that sometimes the lens data fits theory quite well and sometimes it doesn't.

Neither of us understands why there should be such differences. One conjecture is that all the gaussian optics used in the theory depends on the paraxial approximation, and for real lenses it is not valid. Do you have any thoughts on the matter? More generally, it seems clear for many large format lenses that we use rays which are far from the lens axis, but eveything still seems to work properly. Do you know anything about this, and do you have a reference which someone like me could understand without in essence becoming a lens designer myself?

I hope Jeff doesn't mind, but I will attach a zip file containing his spreadsheet of results. I doubt a problem will arise, but just in case remember it is his intellectural property.

Leonard I have no clue for such discrepancies.

I have looked at what Prof. Kingslake says about pupils in 'Fundementals of Lens Design'. He does not says much details, however he clearly says that the actual position of pupils and therefore the pupillar magnification should be understood as paraxial-gaussian only.

So if we follow Kingslake, and why should'nt we, the only fuzziness I can imagine for the position of pupils in a lens with fixed groups are the dependance vs. wavelength and temperature dependance of gaussian parameters !!

As far as the formulae for displacements are concerned, may be we could write them algebraically : f(1-p) and f(1/p - 1). When p>1, both pupils move to the entrance of the lens (negative displacement), when p<1 both pupils move to the film direction (positive displacement). We could also note that there is no mystery, we would get the same formulae for any couple of conjugated images, it is a direct consequence of the regular object-image equations written algebraically : 1/x'= 1/x + 1/f ; with p =x'/x.

Since it is unlikely that the pupill cross the foci, in fact those displacements are of the same sign and there is no upside/down inversion between the entrance and the exit pupil. Hence p =x'/x algebraically as well and p is positive in any reasonable photographic lens ;-)


Another consequence of the displacement of the pupils is that the f-number of an asymmetric lens has to be defined w/respect to the entrance pupil. I let our readers do the derivation, simply start from the fundamental photometric equation, basically the squared sine of the angle under which the radius of the exit pupil is seen from film in the focal plane, and combine with the displacements of the pupils w/respect to the principal planes, displacements expressed as a sole function of f and the pupillar magnidication ratio p !! And a miracle happens, everything simplifies when expressed as a function of the entrance pupil diameter !

[Helen Bach]

Emmanuel,

I don't think that there are any significant discrepancies. If you do a quick analysis of the potential error produced by using imprecise data you will see that the numbers are consistent with the formulae. The pupil diameters are the most critical. It's that old problem of subtracting a number close to unity from unity.

Furthermore, we don't know which of the data are taken directly from Zeiss information and which are calculated (I assume that the pupil magnification has been calculated, by the way). If any numbers have been calculated, even by simple addition or subtraction, then the potential error from using rounded or truncated numbers is increased.

Best,
Helen

[Leonard Evens]

Helen,

Look again at Jeff Conrad's spreadsheet, which I attached previously as a zip file. Look at the last column, the Tele-Apotessar 2.8 300 mm lens. It is true that H16 and H17 are calculated by "subtracting" other values which were taken directly from the Zeiss data. But in some cases that "subtraction" actually amounted to addition because of the signs, and in any case, the quantities being subtracted were not particularly close together. So I don't see how your explanation works in that case, unless you assume very large errors in the measurements.

For the next to last column, the 50 mm f/1.4 Planar, the two values being subtracted are closer together, but the larger is still roughly 2 to 3 times the smaller. Some quick error analysis suggest that the relative error in the difference could be 4 times the relative error in the larger for G17 and 6 times it for G16. Errors in the last significant digit wouldn't be large enough to account for the actual discrepency, since it could produce in each case at worst a 2 percent error in the final difference. On the other hand, given the relatively small values, it is possible that the measurements were off by enough to account for the discrepancy, but the measurement errors would have to be quite large, more than a mm, to account for the differences.

I still think the errors for those two lenses are probably in the sizes of the entrance pupils. Either they were measured or calculated incorrectly, or else they were listed incorrectly. But there may also be errors in some of the other listed values.

In passing let me note that for the Tele-Apotessar, the values in rows 16 and 17 for entrance and exit pupil positions relative to the principal planes are very close to satisfying the lens equation, i.e they are conjugate positions, which suggests they are in fact accurate. For the Planar, the sum of the reciprocals gives a focal length closer to 49 mm, so it is possible something else is wrong.

[Helen Bach]

Leonard,

I looked at the spreadsheet you attached, and also at all the current data on the Zeiss website. Only the Tele-Apo-Tessar and Planar on Jeff's spreadsheet show a difference in the calculated values that cannot be explained by simple rounding error. What is interesting is that, for both lenses, there is a value for the pupil magnification that does satisfy both the two displacement formulae, and it requires less than 2% change in the pupil diameters to acheive.

Best,
Helen

[Leonard Evens]

Helen,

I think we are now on the same page. It was only the last two columns that Jeff and I considered anomalous. The others were so close that we considered them to be in agreement with theory. As you say, in those cases, the small differences are explainable through observational erros or errors in the computation such as rounding error.

[Emmanuel BIGLER]

From Michael Briggs:
But unless this lens has a huge displacement between the entrance pupil and front nodal point, his test doesn't experimentally show that the entrance pupil is better than the nodal point.

News from the front after staying silent for a while.

I did some tests on a 360 Schneider Tele-Arton telephoto lens.
Anybody who owns a telephoto any format can do the same, it is extremely easy to show.

Combining my own measurements with the published Schneider data,

http://www.schneiderkreuznach.com/ar...ele_obj_67.pdf

this is what I find

- from my estimation the entrance pupil (EP) is located approx 130 mm behind the front filter mount i.e. well inside the bellows at about 6 mm behind the last lens vertex

I determined (roughly) the position of the PE simply by plotting on a piece of paper a couple of rays aiming at the iris stopped down to the maximum allowed = f/45

- from Schneider's specs the principal/nodal points are located in air in front of the front filter mount
H=N = at 137 mm in front ; H'=N' at 63 mm in front ; effective focal length = 353

I have also checked for the position of H and H' by finding the position of the front focal point and what I found, no surprise, matched Schneider's specs.
Reversing the lens, I found about 500 mm between the (now reversed) front filter ring and the focal plane. The image should have been terrible, but you can actually focus and see the image without problem except that my bellows was not long enough to shield from stray light.

So anybody can redo those experiments even without any original manufacturer's specs. Only the actual value for the focal length is difficult to determine within a few millimetres of precision.
For example I tried to use the x = f tan(theta) approach, by measuring the displacement "x" of the image of a distant object for a given angle of rotation "theta", estimated on a pan/tilt head with graduations every 5 degrees. I found a value of the f.l. between 330 and 380 mm which is a bit rough for an interval of confidence ;-) the nice thing with the f-theta approach aiming a distant object is that you are not annoyed by the position of the principal planes. They can be anywhere.

So with this good ol' telephoto lens we have everything in hand to get a robust separation between the nodal points and the entrance pupil.

Combining both data, I find that the distance between H and the PE is approx 267 mm ; according to basic formulae this yields a pupillar magnification factor PM equal to approx 0.57

Well to date I have no images to show you but I rotated the camera around H, H', the PE and the focal plane (this is easy with a monorail camera). The object was a foreground at 3 metres and the background, a building at 200 metres.

So it worked exactly as expected.
Rotating around H (137 mm ahead of the filter ring) or H' (63 mm ahead of the filtre ring) yielded visible parallax effects when rotating the full image width of 120 mm in 4"x5",
Rotating around the focal plane (290mm behind the front filter ring) yielded visible parallax effects in the opposite direction,

Rotating around the PE (130 mm behind the front filter ring, approx 6mm behind the last lens vertex), minimized the parallax effects.

As expected, no mystery and no surprise.

I must confess that I was a bit dissapointed ;-). I would have preferred to find something exotic or unexpected ;-)

A perfectly useless experiment ;-) just for fun.

I was doing the experiment yesterday evening while listening to the Soccer World Cup Final in Berlin. I found the result before our national team lost the final against Italy. ;-)

In order to see the foreground and the background together I had of course to stop down to f/22 which is the normal working aperture for this kind of lens. At full aperture 5.5 DOF is not enough to see the foreground @3 metres. But no need to stop @f/45 to see the effect.