How to calculate magnification with close-up lenses

[Thomas]

Close-ups are real fun to shoot and you can produce some stunning results.
The tools of trade to get close enough to your subject are macro lenses, close-up filters (synonym for close-up lenses*) and extension tubes or bellows. If you want to know what sort of magnification you can achieve with these contraptions you have to do some math unless you bought a macro lens, which normally gives you the max magnification as one of its technical data (normally 1:1).

If you are in for extension tubes or a bellow max magnification is very easy to calculate. If d is the extension length of the tube or bellow in mm and f is the focal length just divide d by f to obtain the magnification. A 25mm extension with a 50mm lens yields a magnification of 25/50 = 0.5 or 1:2 as people normally write. A 100mm extension yields 100/50 = 2 (a.k.a. 2:1). Easy!**

Now there is the third alternative of close-up filters. These filters just screw into the filter thread of your lens. Thus it is important to buy a close-up filter that matches the diameter of the filter-thread (or you have to use step-down rings). This can be done with any lens you already have (if the filter-thread matches), is a cheap alternative to a dedicated macro lens (max. 150$/EUR for a high quality close-up filter) and is very easy to carry around all the time. So you never miss that fantastic close-up opportunity! Be warned though that image quality with cheap close-up filters can be disappointing and they are optically matched best to longer lenses or tele-zooms (>=70mm). If you want the best, get yourself the Canon 500D close-up filter which is an apochromatic design effectively built from two lenses that guaranty a high degree of colour correction and has a focal length of 500mm (2 dioptres). Buy a 77mm version so that you only need one close-up filter in your life and get a cheap step-down ring (or two) to match the filter to your lens(es) if those have smaller filter-threads than 77mm.

Now the 1,000$ question is: What sort of magnification can you expect, when you screw this close-up filter to your lens? As you will shortly see there is some very easy math involved to get a good approximation of this question. And some lengthier explanations if you want to know more.

Well, here’s the easy answer exemplified with two lenses, a 70-200mm zoom and a 400mm fixed focal plus a 500mm close-up filter (remember: f is the focal length of the lens):
The magnification is f/500mm or 1:(500mm/f).
This formula gives
- 1:7.1 at 70mm focal length
- 1:2.5 at 200mm focal length
- 1:1.25 at 400mm focal length

I personally find a magnification in the range of 1:2.5 up to 1:7 quite useful as a nature shooter and seldom require going below that.
You should also understand that the distance from your front-lens (which is now the close-up filter) to your subject for these magnifications is 50cm (= the focal length of your close-up filter). This is quite convenient if you shoot shy insects or need to keep a distance from your subjects. But beware: as long as you have the close-up filter screwed to your lens you are not able to get anything sharp that is further away than 50cm

Now there is a caveat: These figures hold true only if you have focused your lens to infinity. As soon as you focus your lens closer greater magnifications can be achieved. How much so, will be left to a follow-up article. There will be also some thoughts about image quality in general and some real life tests of the my trusty old Sigma 400mm/5.6 lens with close-up filter in direct comparison to a dedicated macro lens

So stay tuned!

If you have any comments, additions or questions please feel free to ask/post in this thread.
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* I'll refer to close-up lenses as “close-up filters” throughout this article to avoid any confusion with the lens that is attached to your camera-body
** I’m not digging into the math you need to calculate the magnification if you focus the lens in front of the extension tube (or the bellow) to closer than infinity

[Citruspers]

Great article!

How about lens reversing? Up for a followup article?

[Victor Hoffmann]

Thanks Thomas

[Thomas]

How to calc magnification with close-up lenses: Part 2

Now onto the mysteries of calculating magnification if you focus the lens with the close-up filter closer than infinity. Unfortunately we now have to distinguish between lenses with IF-design and lenses with a "normal" non-IF design. Now what’s the catch about IF-design? IF (= internal focusing) designs tend to shorten their focal length when focusing close, while normal non-IF designs simply move the whole lens further away from the sensor without shifting any internal lens-groups. Those normal lenses keep their focal length when focusing.
Now that you have understood from the first part that the focal length is the important factor to calculate the magnification with a close-up filter it is clear that IF-design with an effectively shrinking focal length tends to reduce the effect of the close-up filter. So to calculate the magnification at closest focus distance we have to calculate the "shrink-factor" of an IF-lens first. But we'll get to this later.

Let’s have a look at non-IF design lenses first.
What you need to know for this calculation is the maximum magnification that the lens alone is capable of. This is normally part of the technical data or you can test this for yourself easily by photographing a ruler and dividing the mm on the ruler by the width of your sensor.
In our prototypical case our 70-200mm zoom has a max magnification of 1:8 (throughout the zoom range) and the 400mm lens a magnification of 1:3. So the zoom has quite mediocre close-up capabilities whereas the fixed focal is pretty good for a “non-macro” lens. Now let’s follow the necessary calculations with the 400mm lens:
1. We calculate how far the lens has to be extended to get a magnification of 1:3. That is 400mm/3 = 133mm. I.e. the front element of the lens is now 133mm+400mm = 533mm away from the sensor
2. We calculate what the focal length of the 400mm combined with the 500mm close-up filter is. That is 1/(1/400mm+1/500mm)=222mm
3. For this combined focal length of 222mm and the distance of the front element 533mm we calculate the magnification as 533mm/222mm-1 = 1.4 or 1:0.71 if you like.
4. If you feel so inclined you can also calculate the distance from the front-element to the subject as 533mm/1.4 and get 381mm
Phew, that's it! This is our first calculation to this subject and you can now easily calculate the resp. Numbers for the 70-200mm zoom.

I'll be back in a minute and give you the results. And then we go on to the dreaded shrink-factor

Have fun!

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In answer to your question Citruspers, unfortunately I'm not into lens reversing. So I cannot rely on personal experience, but perhaps the math is easy enough for me. If that's the case I might dip into this water after the current thread comes to a close...

[Thomas]

How to calc magnification with close-up lenses: Part 3

Well, here’re the results for the 70-200mm with a max magnification of 1:8 when you add the 500mm close-up filter. Assuming the zoom is non IF-design!
At 70mm you can get a max magnification of 1:3.54 with a front-element-to-subject distance of 279mm
At 200mm you can get a max magnification of 1:1.74 with a front-element-to-subject distance of 391mm.
This would get you smoothly from 1:7.1 (at 70mm and infinity) down to 1:1.7 (at 200mm and closest focus distance) with the investment of a simple close-up lens. Not bad, huh?

The only bad news is, that most modern zooms are of the IF-design variant and thus have a shrink-factor of <1. This adds another step to the calculations because NO manufacturer reports the shrink-factor of its lenses

[Thomas]

How to calculate magnification with close-up lenses: Part 4

Now we need to deal with the dreaded shrink-factor. As this factor happens to turn up time and again, I've devoted a separate article to this beast: How to calculate the "shrink-factor" of a lens.

You don't need to read it if you just like to proceed with this magnification thread, but the formulas are really clean and simple to calculate this value from the technical data published for most lenses. The results from the calculation for our two prototypical lenses came out as follows:
- The Sigma 400/5.6 has a shrink factor of 0.75x. That is, it behaves like a 300mm lens at its closest focusing distance of 1.6m.
- The brand-new Nikkor 70-200/2.8 VRII has a shrink-factor of 0.67x. That is, it behaves like a 133mm lens at its nominal focal length of 200mm and its closest focusing distance of 1.4m.

Now, that's all we need to know to go ahead with our calculation of magnifications. We just substitute the nominal focal length that were used for calculation in parts 2+3 of the lens with the effective focal lengths.
Let's do the 400mm first...

We'll be back after a short break. Stay tuned!

[Thomas]

How to calculate magnification with close-up lenses: Part 5

1. We calculate how far the lens has to be extended to get a magnification of 1:3. That is 300mm/3 = 100mm. I.e. the front element of the lens is now 100mm+300mm = 400mm away from the sensor
2. We calculate what the focal length of the 300mm combined with the 500mm close-up filter is. That is 1/(1/300mm+1/500mm)=188mm
3. For this combined focal length of 188mm and the distance of the front element 400mm we calculate the magnification as 400mm/188mm-1 = 1.13 or 1:0.89 if you like.
4. If you feel so inclined you can also calculate the distance from the front-element to the subject as 400mm/1.13 and get 354mm

So the IF-designed Sigma 400/5.6 should boast a max magnification of 1:0.89 when focusing to 1.6m and using a 500mm close-up filter. Remember that the magnification at infinity was calculated as 1:1.25.
So that's all you've got when you screw the close-up filter to this lens: a slight variation of magnifications between 1.13x and 0.8x.

The range of magnifications you get with the zoom is much greater, but you'll see in our next installment that the shrink-factor of this specific Nikkor zoom prevents any additional flexibility that might have come from focusing.

But this is a calculation for another day...

[Bjorn van Sinttruije]

Just finished watching Numb3rs, with David Krumholtz as world-famous mathematician. You remind me of his character a little Thomas. I'll just stick with my "Essential Mathematics for Business and Economics / T. Bradley" book...

- Bjorn -

[Thomas]

Well, Bjorn: You will soon find out that "Essential Mathematics for Business and Economics" is much tougher than my simple and elegant formulas for shrink factor and magnification of IF-designs
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Well now onto part 6: How to calculate magnification with close-up lenses in case of a zoom with a 0,67x shrink factor

I'll just do it for the case of the lens zoomed to it's longest focal length, a nominal 200mm. Remember: if you focus this lens to infinity the magnification that can be achieved with the aid of a 500mm close-up filter is 1:2.5. We also calculated already that a non-IF zoom would go down to 1:1.74 when you focus the lens to 1.4m. But this lens is an IF-design so we expect the max magnification that can be achieved is somewhere between 1:1.74 and 1:2.5. The exact value depends on the effective focal length of this zoom when focused to 1.4m which we already calculated to be only 133mm (see part 4). And here we go:

1. We calculate how far the lens has to be extended to get to its maximum magnification of 1:8 (w/o close-up filter). That is 133mm/8 = 17mm. I.e. the front element of the lens is now 17mm+133mm = 150mm away from the sensor
2. We calculate what the focal length of the 133mm combined with the 500mm close-up filter is. That is 1/(1/133mm+1/500mm)=105mm
3. For this combined focal length of 105mm and the distance of the front element 150mm we calculate the magnification as 150mm/105mm-1 = 0.43 or 1:2.3 if you prefer this notation.
4. The distance from the front-element to the subject is 150mm/0.43 = 349mm

Now, if you compare the magnification of 1:2.5 when the lens is focused to infinity with the 1:2.3 when the lens is focused to 1.4m you can see that there is not much to be gained magnification-wise through focusing. And this is due to the really heavy shrinking of this particular zoom.

Well, that's enough now with the theory! And before we dip into some real-life testing of all the mathematical wizardry with my soon-to-arrive Canon 500D close-up filter please read the disclaimer carefully:
All calculations are based on a simplified model of a lens, even for non-IF-design "normal" lenses.
So take all calculations with a grain of salt and don't delve into too many decimal places.
Apply at your own risk

[Thomas]

Just a quick update: The expected delivery date of the Canon close-up filter was shifted to end of October
So you have to wait until everything about the validity of my calculations and the IQ of such a solution will be revealed.

In the meantime: Anybody who has bought a Canon 500D close-up filter, feel free to post your experience here!

[Thomas]

Some additional remarks:
- As you can see from the calculations a close-up filter is more effective on long lenses/zooms than on short lenses/zooms. E.g. a 500mm close-up filter on a 50mm lens will only get you to 1:10 magnification at infinity and around 1:4 at closest focus distance. But this is still 2x the magnification you normally get with a standard 50mm lens. With extension tubes and bellows the reverse is true: a 50mm extension tube brings you down to 1:1~1:0.9 with your 50mm lens but only to 1:10~1:4 with your 500mm super-tele.
- As you focus a non-IF lens closer by extending it (either using the focus ring, extension tubes, or a bellow) the brightness of the lens will decrease proportional to the amount of extension. Thus the effective aperture of the lens will increase. Here is the calculation: If you extend a 60mm f/2.8 macro lens by 30mm to achieve 1:2 magnification the aperture is effectively stopped down to
2.8x(60mm+30mm)/60mm = 4.2 which is a little over 1 stop. At 1:1 magnification you lose 2 stops!
- The situation is a little better with IF-designs: You can read about the effective aperture-shrink-effects with the micro-Nikkor 105/2.8 VR and the Sigma 150/2.8 macro lens here.
- When you are using a close-up filter the lens will keep its designated aperture (when focused to infinity). So a 400mm f/5.6 lens with an added 500mm close-up filter gets down to 1:1.25 magnification and still has an effective aperture of f/5.6. While if you use a 320mm bellow to achieve the same magnification with this lens, you’d end up with an effective aperture of 5.6x(400mm+325mm)/400mm that is f/10.1
- Keep in mind though that the effective aperture is also influencing dof. So when using a close-up filter you can achieve an even shallower dof than with other solutions. Unfortunately this is not normally your goal in macro photography as most of us struggle to get a decent dof by stopping down (even beyond the diffraction limit).
- There was also the question of calculating magnifications with lens reversing. Lens reversing means that you buy a special tube or bellow where you can mount the lens in reverse orientation, i.e. the front lens towards the camera and the rear lens pointing to the subject. Now that is easy to calculate as you simply use the same formulas as with normal orientation. I.e. if you want 2:1 magnification (larger than life) you need to extend the lens by 2x its focal length (e.g. a 50mm lens by 100mm). The reason you reverse-mount the lens has nothing to do with the magnification but only with the image quality of the setup.

[popo]

On the effective aperture at relatively higher magnifications, there is a simple formula that can be used which I first saw in the Canon MP-E65 manual:

effective aperture = (1 + magnification) * nominal aperture

e.g. if you use a 1:1 macro lens of nominal aperture f/2.8 at its minimum focus distance, the effective aperture double the value = f/5.6 or two stops less. Things get dark very fast as you go to even higher magnifications and smaller apertures... this formula's results seem to agree with the examples in Thomas' last post.

On using close up filters, I believe it too would suffer from light loss as the system magnification is increased, although I couldn't say how or why.

One part I'm not sure about is if the nominal aperture or effective aperture value determines other characteristics such as DoF and diffraction limiting. At high magnifications it can be around f/100 and I still get practically no DoF! But is the closeness of the subject the stronger influence? On diffraction limiting, I've seen one page on the Canon website before that suggested this was based on the effective aperture.

[Thomas]

Yes, our formulas for "aperture-shrink" of non-IF-design lenses concur!
With close-up filters you don't have aperture-shrink because you neither change the absolute aperture (in mm diameter of the lens) nor the lens-to-sensor distance.
And btw.: dof decreases by the square of the increase in magnification

[Thomas]

The Canon 500D close-up filter is due this week.
So expect my real-life tests of magnification and IQ at the weekend.

Any idea, what a good test-target with minute structures is for close-ups around 1:1? Feathers?

[Citruspers]

Anything organic should do for 1:1 magnification? Preferrably less fuzzy than feathers to judge sharpness?