I enjoy giving practical advice about close-up/macro, especially to people who are starting out. It is nice to do in its own right, but sometimes there are incidental benefits: explaining things encourages me to think things through more carefully to try to ensure what I'm saying is true, and relevant, and preferably with an easy to digest presentation. Sometimes other people will respond to correct what I've said, which is always very welcome, or add extra information I didn't know about. Sometimes I'll spot errors myself as I think things through. And sometimes I ramble off and do some practical experiments which teach me things I might not otherwise have stumbled upon. That happened yesterday.
A question came up in this thread at dpreview.com about the effective aperture of the Laowa 100mm macro, a lens that I use a lot. I thought I would show a practical example but as I was preparing it I realised that things were not working out the way I expected. There was something going on that I didn't understand about the way the effective aperture changed as the magnification changed.
See this post for an introduction to effective apertures. I have been using the simplified formula mentioned in that post
effective f-number = nominal f-number * ( 1 + magnification )
So when I photograph a springtail using 7X magnification with the lens set to f/45, the effective aperture I'm using is:
45 * ( 1 + 7 ) = around f/360
With my close-up lens setups I would use f-numbers equivalent to f/45 on the Sony full frame A7ii that I'm using at the moment. Since depth of field roughly doubles for each two stops decrease in aperture (which is a doubling of the f-number), using f/360 rather than f/45 gives me around 8 times greater depth of field than I could get previously.
Or so I thought.
Let's go back to that formula. That is the simplified version. The full version is this:
effective f-number = nominal f-number *
( 1 + magnification/pupil magnification )
Pupil magnification is the exit pupil divided by the entrance pupil. "The entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front (the object side) of the lens system. The corresponding image of the aperture as seen through the back of the lens system is called the exit pupil." (Wikipedia).
That is all very well, but manufacturers don't state the size of the entrance and exit pupils for their lenses and they can be difficult to measure, so pretty much everyone uses the simpler formula, which assumes that the pupil magnification is 1. I've always found the results I have got from the simplified formula good enough for my purposes.
Until yesterday.
I'll miss out the bit about how and why I got confused and what I did to grope my way out of the darkness to some sort of understanding. If you are interested you can read about that in the sub-thread starting here at dpreview. The upshot was that I came to think that, when using teleconverters, the formula should be
effective f-number = nominal f-number *
( 1 + magnification/ (pupil magnification * teleconverter power ) )
where teleconverter power is 1.4 for a 1.4X teleconverter, 2 for a 2X teleconverter, 2.8 for a 1.4X and a 2X teleconverter used together, and 1 if no teleconverter is used.
Not knowing what the pupil magnification is, we may as well as usual assume that it is 1. This simplifies the formula to
effective f-number = nominal f-number *
( 1 + magnification/ teleconverter power )
I did an experiment to test this formula. I used a Canon MPE-65 lens on a Canon 70D dSLR, and captured five images with 1X, 2X, 3X, 4X and 5X magnification set on the MPE-65, with all five using the same settings of f/11, 1/10 sec, ISO 3200. I photographed a light area on my computer screen and held the lens against the screen so there would not be an issue with the brightness changing because of different working distances from shot to shot. (The fact it would be out of focus would not matter).
I then added a 1.4X teleconverter and captured another five images with the same magnifications set on the MPE-65 and the same settings for aperture, shutter speed and ISO. Then the same again with a 2X teleconverter and lastly with both of the teleconverters together.
Before I did this, I did some calculations, to see what my proposed formula would predict I would see. To my surprise, as shown in last but one column of the following table, it predicted that with 1X magnification on the MPE-65 the four images using no teleconverter, 1.4X, 2X and 2.8X should all have the same effective aperture, so they should all be the same brightness. And it predicted the same would be the case for 2X, 3X, 4X and 5X magnification.
Those predictions are very different from what the usual formula predicts, as shown in the last column in the table below. It predicted that the images with 1X set on the MPE-65 would have a larger effective f-number and so would get darker when I added the 1.4X teleconverter, darker again with 2X and darker still with 2.8X. And it predicted the same would be true with 2X, 3X, 4X and 5X magnification set on the MPE-65. (It also predicted some pairs of images, shown by the colours in the table, where the brightness would match between images that had a different magnification set on the MPE-65 but which had the same overall magnification once the teleconverters were taken into account.)
Here is the table.
Here are the 20 images.
They exactly match the predictions from the formula which takes the teleconverters into account.
Is this a fluke, a special case, something to do with the MPE-65 in particular? That looks unlikely to me given that I did a different experiment with a Laowa 100mm macro that came up with similar conclusions.
With the Laowa 100mm macro on a Sony A7ii I captured images at 1X, 2X and 4X magnification, once with one 2X teleconverter and again with two 2X teleconverters. This time the magnification was the actual, overall magnification, so for example for the 1X magnification shot I set the magnification on the lens to 1:2 when using the 2X teleconverter and 1:4 when using both teleconverters. And similarly for the 2X and 4X shots.
Assuming once more that the pupil magnification is 1, using the usual formula so the effective aperture depends only on the magnification, then both 1X magnification shots should have the same effective aperture and therefore the same brightness, and the same for the pair of 2X magnification shots and the pair of 4X magnification shots.
In contrast, these are the calculations using the revised formula that takes the teleconverters into account. As shown by the colours, it predicts that the effective f-number, and hence the brightness, should match between the 1X magnification using the 2X teleconverter and the 2X magnification using 4X teleconversion, and also between 2X magnification using the 2X teleconverter and 4X magnification using 4X teleconversion.
Histograms from Raw Digger show that the two matches were in fact very, very close indeed.
I am persuaded that when using teleconverters, effective f-number can best be calculated using the formula
effective f-number = nominal f-number *
( 1 + magnification/ (pupil magnification * teleconverter power ) )
with the usual simplification of pupil magnification = 1 if the pupil magnification is unknown.
This means that the tiny apertures I am using are not as small as I thought they were, which in turn has implications for depth of field, diffraction softening and flash power requirements. That said, I don't think it makes much difference in practical terms, because I'm choosing my aperture settings based on experiments and experience as to what works for what I'm doing, how I like to go about and what I find acceptable as results. Knowing that the correct numbers to describe the apertures I'm using are are smaller than I thought makes no difference to that. Which makes this discovery, assuming it is true, more interesting than useful. (And the same will be the case if it eventually turns out to be wrong.)
There is a discussion about all this, with additional examples, in this thread at dpreview.com.
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