Mathematical Puzzle

Despite thinking of myself as pretty good at maths, I cannot for the life of me work this one out. I am sure that some gurus here will provide at least one correct answer....

I have a Canon 70-200mm f2.8 L IS lens which cost a bomb by my standards. I have also an extension tube set, 12mm, 20mm and 36mm.

Now, I don't want to pay £600+ for a f2.8 L IS macro lens just now (perhaps when I have won the lotto).

I appreciate that reduction in effective aperture is another part of the story, but can anyone tell me what extension would be required if the lens was set to 100mm, in order to be the same reproduction ratio as the macro lens.

Alternatively, can someone give me a good link to a theory site where the effects of extension tubes may be calculated.

Ta' in advance.

Hi,

Denbigh Gabbitas said:

Despite thinking of myself as pretty good at maths, I cannot for the life of me work this one out. I am sure that some gurus here will provide at least one correct answer....

I have a Canon 70-200mm f2.8 L IS lens which cost a bomb by my standards. I have also an extension tube set, 12mm, 20mm and 36mm.

Now, I don't want to pay £600+ for a f2.8 L IS macro lens just now (perhaps when I have won the lotto).

I appreciate that reduction in effective aperture is another part of the story, but can anyone tell me what extension would be required if the lens was set to 100mm, in order to be the same reproduction ratio as the macro lens.

Click to expand...

What image magnification do you want to achieve?

I'll send you some equations in a little bit - I just got up and I have to figger our where they are!

Best regards,

Doug

Denbigh Gabbitas said:

I appreciate that reduction in effective aperture is another part of the story, but can anyone tell me what extension would be required if the lens was set to 100mm, in order to be the same reproduction ratio as the macro lens.

Click to expand...

200mm.

Actually a bit less: 200mm minus the focussing length of the zoom when set to its minimum focus distance, so maybe 100-120mm. It is impossible to compute the exact value without knowing the exact lens design, focal length of your 200mm zoom is a lot less than 200mm at minimum focus distance.

I said 200mm, because of the following formula: when the reproduction ratio is 1:1 (which is the max for the macro lens), the distance lens to subject is the same as the distance lens to sensor. For a lens of focal length f, that distance is 2f.

The important point, however, is that the results will not be good. Zoom lenses are computed to be sharp at long distance, they get noticeably less sharp at short distances. They make poor macro lenses, generally.

Doug Kerr said:

What image magnification do you want to achieve?

Click to expand...

I supposed 1:1, since this is the maximum of the macro lens.

Doug Kerr said:

I'll send you some equations in a little bit - I just got up and I have to figger our where they are!

Click to expand...

But we can't use the equations without knowing the true focal length, which we don't for lenses with internal focus as is the 70-200mm zoom.

Hi, Denbigh,

The formula for reckoning the magnification of a lens with an extension is given by:

m' = m + (L/f)

where m' is the image magnification with the extension in place, m is the magnification the lens would give at the zoom and focus setting to be used with no extension, L is the amount of extension (the "length of the extension tube"), and f is the focal length of the lens.

It turns out that I have an technical article about this that I never quite finished and put up on my site. (That happens a lot to us old guys!) I'll try and do that today!

Do note that with regard to the aperture, the "reduction" in aperture that arises when using an extension tube is not a result of the extension tube itself but merely from the camera being focused at a relatively short distance (whether the lens is able to do that on its own or needs "help" from an extension tube).

In a simple case, the effective f-number of a lens is given by:

n' = n (1+m)

where n' is the effective f-number, n is the f-number of the lens for focus at infinity (the "rated" f-number), and m is the image magnification that results from the object/focus distance (whether or not an extension tube is involved in being able to do that).

Complications in lens design can make the precise result vary from that, but that is close enough for most planning purposes.

Hope that helps.

Best regards,

Doug

Hi Doug, Sorry to have got you out of bed too early...

I was being typically lazy when I posted my question but went on looking. I confirmed your first formula eventually - which of course suggests that a shorter focal length lens would be better than a longer - purely from the point of view of effectiveness of the extension tube. I had also found previously a Canon Museum Site which gives a lot of interesting info for lenses, including their maximum magnification http://www.canon.com/camera-museum/camera/lens/ef/telephoto_zoom.html
The maximum magnification for the £600 100mm macro is 1.0x. If I use, say, an 85mm lens (with MM = 0.11) with a 76mm extension, I should have my 1.0x.

Well, I have just tried that out. I only have 68mm of extension total, so I should have got 0.11 + 68/85 or 0.91; I got approx. 40mm of my rule, or 36/40 = 0.9 !! Magic. In addition, the closest focusing happened at about 120mm from front of lens, which gives plenty of room for lighting the target.

As far as the effective f-number was concerned, I used f1.8 and was shooting at 1/1000 with ISO 1000. I guess I was effectively between f2 and f2.8, since f1.8 steps outside the root2 progression. f numbers in this ball park are very encouraging for shooting in natural daylight.

All this because someone wants me to go out with his mycological group and take photos of funghi for their meetings. Nothing like a nice well-defined challenge! And not having to buy a £600 macro lens is such a relief. (Not that there ever was a chance.)
Thanks for all replies.

Denbigh Gabbitas said:

Despite thinking of myself as pretty good at maths, I cannot for the life of me work this one out. I am sure that some gurus here will provide at least one correct answer....

I have a Canon 70-200mm f2.8 L IS lens which cost a bomb by my standards. I have also an extension tube set, 12mm, 20mm and 36mm.

Click to expand...

Hi Denbigh,

Is that the Canon EF 70-200mm f/2.8L IS II USM, so the mark II version? If so, then your minimum focus distance is 1.2 metres. When you plug that value in into my DOF Output Quality planning tool, as:
2.1) Focus distance: 1.2 m, and you set
2.2) Focal length: 100 mm, then you'll get
2.6.1) Magnification factor (M): 0.09091, and behind that it tells you that that is equal to an extension of 9.09 mm.

If you want to achieve a 1:1 magnification factor, then you need 100 mm extension, so an additional 90.91 mm, which you don't have in the form of stacked extension tubes. You do have 77.09 mm (9.09 + 12 + 20 + 36mm). So by iteratively trying a few magnification factors until you get 77.09mm, it turns out that 0.7709:1 produces the desired (rounded) maximum achievable extension, at a 2.1) focus distance of 0.2297 m.

That looks predictable, at 100mm, an extension of 77.09 mm achieves a magnification factor of 0.7709, and an extension of 100 mm would give a magnification factor of 1.0 . At 200mm zoom you'll get half of the magnification factor, while at 70mm zoom you'll be able to get slightly larger than 1.1 magnification on the sensor of your camera.

My tool will also give you the depth of field consequences for different apertures, but you'd also need to start from the beginning by specifying your output quality goals, and work your way down to filling in the details.

Alternatively, can someone give me a good link to a theory site where the effects of extension tubes may be calculated.

Click to expand...

For the theoretical backgrounds, there is hardly a better reference than Paul van Walree's excellent page on the subject, here, and here. Another equally highly acclaimed site is David Jacobson's Lens tutorial page, and the associated Lens FAQ.

Doug Kerr also has published some good essays on the depth of field and COCDL aspect of things.

Cheers,
Bart

Hi, Denbigh,

Denbigh Gabbitas said:

Hi Doug, Sorry to have got you out of bed too early...

Click to expand...

Oh, I was up - just not awake!

I was being typically lazy when I posted my question but went on looking. I confirmed your first formula eventually - which of course suggests that a shorter focal length lens would be better than a longer - purely from the point of view of effectiveness of the extension tube.

Click to expand...

Indeed.

I had also found previously a Canon Museum Site which gives a lot of interesting info for lenses, including their maximum magnification http://www.canon.com/camera-museum/camera/lens/ef/telephoto_zoom.html

Click to expand...

Nice reference. Thanks.

The maximum magnification for the £600 100mm macro is 1.0x. If I use, say, an 85mm lens (with MM = 0.11) with a 76mm extension, I should have my 1.0x.

Well, I have just tried that out. I only have 68mm of extension total, so I should have got 0.11 + 68/85 or 0.91; I got approx. 40mm of my rule, or 36/40 = 0.9 !! Magic. In addition, the closest focusing happened at about 120mm from front of lens, which gives plenty of room for lighting the target.

As far as the effective f-number was concerned, I used f1.8 and was shooting at 1/1000 with ISO 1000. I guess I was effectively between f2 and f2.8, since f1.8 steps outside the root2 progression. f numbers in this ball park are very encouraging for shooting in natural daylight.

All this because someone wants me to go out with his mycological group and take photos of funghi for their meetings. Nothing like a nice well-defined challenge! And not having to buy a £600 macro lens is such a relief. (Not that there ever was a chance.)
Thanks for all replies.

Click to expand...

That all sounds excellent. Glad to have such nice agreement with the theory! I hope the fungi are able to pose properly!

Best regards,

Doug

Denbigh Gabbitas said:

The maximum magnification for the £600 100mm macro is 1.0x. If I use, say, an 85mm lens (with MM = 0.11) with a 76mm extension, I should have my 1.0x.

Well, I have just tried that out. I only have 68mm of extension total, so I should have got 0.11 + 68/85 or 0.91; I got approx. 40mm of my rule, or 36/40 = 0.9 !! Magic. In addition, the closest focusing happened at about 120mm from front of lens, which gives plenty of room for lighting the target.

As far as the effective f-number was concerned, I used f1.8 and was shooting at 1/1000 with ISO 1000. I guess I was effectively between f2 and f2.8, since f1.8 steps outside the root2 progression. f numbers in this ball park are very encouraging for shooting in natural daylight.

Click to expand...

Stop down that 85mm prime or you will get very fuzzy corners... It is not designed to be used that close.

As to the zoom: focal length of modern 70-200 zooms decreases considerably when focussed close. When you dial 100mm on your 70-200, you have probably a lot less than 100mm, possibly even less than your 85mm prime.