Pupil magnification ratios and macro lens DoF

More knowledgeable users of macro lenses will be aware that depth of

field in the macro regime depends on aperture, magnification (M), and

the pupil magnification ratio (P) - there is a factor of (1+M/P) in

the equation - see here, for example:

 

http://www.vanwalree.com/optics/dofderivation.html#eq12

 

However, there is a complete dearth of information about this

parameter of macro lenses, which could be an important factor in

choosing between different lenses particularly when trying to

photograph 3 dimensional natural subjects. Observing my Tamron 90mm

f/2.5 macro, P is close to 1 at infinity focus, but I estimate it's

around 0.7 at closest focus, giving a handy boost to DoF. I also

noted that if I could reverse mount my 17-35mm zoom (82mm thread, so

not possible on my body with its protruding popup flash) I would have

a P of something like 0.2 which would make a huge difference to DoF.

 

For those who don't know what the pupil magnification factor is, it is

the ratio of the apparent size of the aperture viewed from the rear of

the lens (the "exit pupil") divided by the apparent size of the

aperture viewed from the front of the lens (the "entrance pupil"). In

order to observe it more clearly it can help to stop the lens down a

couple of stops from wide open. For lenses that lack manual control

over aperture, you may be able to do this by selecting the aperture

via your camera and using depth of field preview to stop the lens

down, removing the lens while DoF preview is still active.

 

Perhaps others might like to contribute their findings with their own

lenses to build a resource that all can refer to.

I believe the "boost to DOF" comes from a change in actual working aperture, thus although you might set you lens to f8, do the calculations including pupil magnification and find you get "extra" DOF, what's really happening is that you're not working at f8 as you might think you are, you're optically working at f11.

 

So I don't believe you're actually getting any more DOF for a given lens at a given aperture, you're just doing a calculation that takes factors into account that are normally not significant.

 

I'd be happy to be corrected if I'm wrong.

Not only is the diameter of the exit pupil larger, but it is also farther away. This will yield the same relative aperture. Pupillary magnification buys you no "free lunch".

When the pupillary magnification is less than unity, you indeed obtain

greater DoF for a given marked aperture, but it appears that, for a

constant shutter speed, you actually get less DoF.

<p>

When the object distance is small in comparison with the hyperfocal

distance,

<blockquote>

DoF ~ 2Nc/m(1/m + 1/p)

</blockquote>

where<br>

 N = Lens f-number<br>

 c = Circle of confusion<br>

 m = Image magnification<br>

 p = Pupillary magnification<br>

<p>

For a given magnification m, the ratio of the DoF for a lens with pupillary

magnification p to that for a lens with unity pupillary magnification then

is

<blockquote>

R_DoF ~ (1 + m/p)/(1 + m)

</blockquote>

In each case, the exposure is the nominal value multiplied by the

Exposure Correction Factor; the compensation for lens extension is

<blockquote>

ECF = (1 + m/p)^2

</blockquote>

For a given magnification m, the ratio of the ECF for a lens with pupillary

magnification p to that for a lens with unity pupillary magnification then

is

<blockquote>

R_ECF = (1 + m/p)^2/(1 + m)^2

</blockquote>

For constant exposure time, the DoF ratio is

<blockquote>

R_DoF = (1 + m)/(1 + m/p)

</blockquote>

Conversely, for constant DoF, the exposure time ratio is

<blockquote>

R_ECF = (1 + m/p)/(1 + m)

</blockquote>

Pupillary magnification is no magic bullet for increased DoF; in one sense,

it is precisely the opposite. In any event, it doesn't even appear on my

list of criteria for choosing a macro lens.

<p>

Nonetheless, once you've chosen a lens, it sometimes does help to recognize

the effect of pupillary magnification on DoF. If you are given to

calculating DoF (which in most cases, I think, is merely an academic

exercise), it may help to know that the DoF that obtains from a typical

long-focus macro lens is slightly greater than would be calculated when

ignoring pupillary magnification.

I don't think I was suggesting that the factor which adjusts from nominal aperture to effective aperture (you can think of it as an adjusted bellows factor) provides a "free lunch". Of course, a corollary of a smaller effective aperture is a longer exposure time (or the need for a more powerful flash mutatis mutandis). Equally, a camera body will adjust the exposure based on the light actually impingeing on its metering system. However, (and correct me if I am wrong), I think that macro lenses tend to report aperture to camera bodies based on aperture at infinity focus, and do not adjust for these factors that apply in close focus. Therefore, I consider that the data are potentially useful.

Perhaps we could now return to the purpose of the thread?

Modern lenses do apply the "bellows factor" corrections to the aperture and report it to the camera electronically.

 

Unfortunatly, they don't report the corrected focal length. Most modern macro lenses, including your Tamron, change focal length as they focus. It won't help you too much, being a Canon guy with a Tamron zoom, but I've a table on my website that has the front and rear node locations for three Nikon macros (the 60, 105, and 200mm) at 8 macro ratios each, which gives up all sorts of interesting insights about this...

 

http://www.swissarmyfork.com/lens_table_1.htm

The Tamron 29-300 "macro" undergoes a drastic shift in focal length when close focused. I think at 300mm and closest foucus it probably drops to something like 100mm.

 

All these things make calculation of actual DOF quite difficult. However the bottom line is that at a given macro magnification for a given true effective working aperture, you get the same DOF to within normal limits of measurement.

 

It's better (and much easier) to use magnification as the basis of DOF, rather than trying to figure out what the focal length is and where the nodal points are.

 

So when photographing small subjects, it really doesn't matter which lens you pick (other than from a perspective viewpoint). At the same magnification and at the same shutter speed for correct exposure (which means they're working at the same effective aperture), you get essentially the same DOF.

I completely agree with Bob on using magnification when figuring macro DoF.

In addition to eliminating the need to know changes in focal length, the

magnification usually is what is of most interest, anyway.

 

That said, for a given image magnification, the DoF IS affected by the

pupillary magnification, so it's important to know the pupillary

magnification if you're calculating DoF. For example, at 1:1, the Canon

180 mm f/3.5 macro lens appears to have a p of about 0.5, so the DoF is 50%

greater than would obtain from a lens with a p of unity. However, it would

be difficult to find a 180 mm lens for a 35 mm camera with a p anywhere near

unity; almost any 35-mm lens of similar focal length is a telephoto design,

so by definition, p is less than one. The small value of p then is more a

consequence of the focal length than of any magic on the part of the Canon

180 mm lens.

 

I still don't think the pupillary magnification would have much influence

on my choice of a macro lens. I'd look at perspective (i.e., focal

length), along with cost and weight. Most current long-focus macro lenses

are internal focusing, and consequently reduce the focal length at close

focus.

 

Canon lens/body combinations report the marked lens aperture. In a prior

life, I had the 60 mm and 105 mm Nikon AF micro lenses, and as I recall,

they reported effective aperture. Perhaps Joseph can confirm this.

The only problem with trying to do the macro calculations based on "true effective working aperture" and maginfication is that you have a rough approximation for the working aperture on the true macro lenses (like the 90mm Tamron that the original poster, Mark, mentioned, or the 60mm, 105mm, and 200mm Nikons that I mentioned) but you don't have even a decent wild guess on something like Bob's 28-300mm.

 

And, of course, you'll have little idea of the magnification or "true effective working aperture" if you put the lens on a bellows, extension tube, or reverse it (like Mark mentioned). This even applies to one of the true macro lenses, let alone Bob's consumer zoom, because the aperture and focal length are varying all over the place when you adjust focus on the lens.

 

So, you're right back to needing nodal points. Sorry kids, there ain't no such thing as a free lunch.

Canon macro lenses report marked f-number and indicate reproduction ratio,

so there is no problem doing the calculations based on these values. You

do, of course, need to estimate the pupillary magnification at various

reproduction ratios. It's sad that manufacturers cannot provide these

data.

 

When a lens is reversed or used with extension tubes or bellows, or

combination thereof, there is no question about needing nodal points. It's

similar to the tiresome problem of trying to measure absolute image

distances on a view camera. The hand-camera user need not bother with

swings and tilts, but must deal with the additional complication of

shifting nodal points and changing focal lengths with many current lenses.

 

I've generally been quite satisfied with what I can do with the Canon 180

mm macro: 1:1 (2:1 with a 2x extender) at 250 mm working distance. I

suspect the 200 mm micro-Nikkor would be comparable, though I don't know if

it will accept an extender. With either focal length, it isn't practical

to rely on additional extension to increase magnification. If I wanted

greater magnification, I'd get the 65 mm macro, which provides up to 5x on

the camera.

 

I don't mean to suggest that there is anything wrong with extension tubes,

bellows, or reverse mounting a lens--I've done all of these in years past.

In those days, every lens I used was unit focusing, so I didn't need to

deal with changing lens geometry. At least with Pentax, I usually was able

to get the pupillary magnifications simply by asking. I computed exposure

correction factors for every lens with every combination of tubes and

carried these charts for reference.

 

I still carry such a chart for the Canon 90 mm TS-E with various

combinations of extension tubes in case I take exposure readings with a

handheld meter. Fortunately, that lens is unit focusing, so the

calculations are fairly simple.

 

For what I do, bellows and extension tubes don't seem to offer any benefit

over using a macro lens on camera, and their use entails considerably more

effort than I care to expend. I have no doubt that there are situations in

which a more complicated setup might have capabilities that mine does not.

In those cases, a table such as Joseph has developed is quite helpful.

I think we've established that Nikon lenses provide a reasonable approximation to true effective apertures (at least when not mounted on a tube), but Canon mount ones do not. Hopefully those who find this thread while researching will find the erudite contributions useful. For those who own the lenses in Joseph's table and the will to persevere with the calculations from the data it will be a useful resource. Jeff's discussion of the issue is thorough and valuable. However, thus far we have little data for Canon or other lenses. Perhaps this leaves the technique of using camera metering to observe changes in exposure of an evenly and constantly lit subject (e.g. a grey card) for a given marked aperture as focus distance is racked in from infinity to 1:1 to infer effective apertures at different magnifications.

<p>It's an old thread, but I might as well put the data here as this comes up with Google.<br>

<br />I measured (or tried to) the pupil ratio of Canon EF 100/2.8 L at five magnification settings. It seemes that 0.94 -0.61 * m is a reasonable approximation. The "accurate" curve levels at ends and is a bit steeper in the middle, but not enough to matter for DoF calculation. I wouldn't mind if somebody checked the end values. If they are about right, the others likely are as well.</p>