## Magnification Calculations

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Kevin Childress
Posts: 24
Joined: Fri Aug 14, 2015 2:19 pm
Location: Lowell, North Carolina
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### Magnification Calculations

Greetings from a forum newbie! I'm hoping to find a macro-mathematician to help me validate the total lens magnification involved with a particular lens arrangement. I hate to ask anyone to crunch numbers for me but admittedly I'm having trouble with some of this. My only understanding of the math is to break it down to granular pieces. My advance apologies if I misuse a term or phrase - please set me straight if I misunderstand or mis-speak any of this.

The scenario is this: 27.5mm extension tube (plus) Nikon AF-S DX 1:1 Macro 85mm f/3.5 (plus) a reversed Nikon 50mm AF-S f/1.8.

I consistently find reference to the same calculations used to determine magnification power for these components. Taking the reversed 50mm lens standalone for example, everything I find says the reversed 50 should act as a +20 diopter (the formula being: 1000mm / focal length of lens = +20 diopter). Then, translating a +20 diopter is supposed to be 6x magnification (the formula being: +20 diopter / 4 + 1 = 6x power). So my first question is do you agree the reversed 50mm lens alone provides 6x magnification?

And then we have the 85mm 1:1 macro lens with 27.5mm of extension tube. According to an extension calculator found at Cambridge In Color, this combination should provide 1.32x magnification (the formula being: 27.5mm extension tube / 85mm focal length + 1.0 lens reproduction ratio = 1.32x). Question 2: Does this sound accurate or can this be refined?

What I really don't understand is how we combine the magnification powers of the separate components. Is it a simple matter of multiplying one by the other (6x * 1.32x = 7.92x)?

The thing is I don't believe for a second that I'm getting anywhere close to 7x with the combination described above. I tend to believe what I can see, touch, and feel. And since we can measure pixels very precisely in Photoshop, what I'm seeing is consistently between 3.46x and 3.49x over a known 1:1 baseline. I could round that down to 3.4x and be quite happy.

Kevin

Pau
Posts: 5368
Joined: Wed Jan 20, 2010 8:57 am
Location: Valencia, Spain
Kevin, welcome aboard!

I let the maths for more skilled people but the simpler and often most accurate way if you have the equipment is to directly measure it: Just take a picture of a rule parallel to the sensor longer dimension and calculate the ratio with the actual sensor width, no need to count pixels en PS.

Many modern macro lenses actually change their focal lengt at close focus
I tend to believe what I can see, touch, and feel
Me too
Pau

rjlittlefield
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Joined: Tue Aug 01, 2006 8:34 am
Location: Richland, Washington State, USA
Contact:
Kevin, welcome aboard!

I have a degree in math and I've spent much of my life struggling to make calculations match observations. Let's see if I can help to explain the discrepancy in this case.

First, I think you have good instincts. It is far simpler to make an accurate measurement, than it is to do some calculation that is both correct and applicable.

1. "do you agree the reversed 50mm lens alone provides 6x magnification?"

No, that number is nonsense. I do not recognize the formula that you're using. It's probably valid in some circumstance, but certainly not here. 50 mm will give 6X if you reverse it in front of a 300 mm telephoto that is focused at infinity. The formula for that case is both simple and accurate: magnification = rear_FL / front_FL = 300/50 = 6. You can use the same formula to consider reversing your 50 in front of your 85 focused at infinity. The number will be 1.7X, and you might want to confirm that just to get some confidence.

2. And then we have the 85mm 1:1 macro lens with 27.5mm of extension tube. According to an extension calculator found at Cambridge In Color, this combination should provide 1.32x magnification (the formula being: 27.5mm extension tube / 85mm focal length + 1.0 lens reproduction ratio = 1.32x). Question 2: Does this sound accurate or can this be refined?

I do recognize that formula. It would be accurate if the 85 mm lens had a fixed focal length, but it probably doesn't. Modern macro lenses typically shorten their focal length as you turn the ring to make them focus closer. Of course none of the lens elements change shape when you do this. What's really happening is that turning the ring changes the distance between various elements inside the lens. It's that change in distance between elements that causes the change in effective focal length. (We'll hear more about distance between elements in a moment.)

Now, a bit of relevant theory...

It is always true that if you add extension, you "add magnification". The amount of added magnification is simple to compute, if you know the focal length:

This is the basis for the formula at Cambridge In Colour. (Note the spelling; it's a UK site.)

The problem in using this formula is that you have to know the focal length, and as mentioned, the effective focal length of a macro lens is often not what's printed on the lens barrel.

Further, the total_focal_length that goes into the formula is the combined effective focal length of all the lenses in front of the extension. In your case that's the 85 mm combined with the 50 mm. But as mentioned, the 85 mm probably isn't really 85 mm once you're turned its focus ring, and in any case the focal length for the combination depends on the separation between the lenses.

The effect of separation would be easily calculated, once you know the right formula, except that the "distance" needed by the formula depends on lens characteristics that are seldom published and are not simple to measure. So, you're pretty much stuck.

For completeness, the relevant formula is given in Wikipedia:
1/f = 1/f1 + 1/f2 - d/(f1*f2)
for the combined focal length f, given two lenses with focal lengths f1 and f2, separated by a distance d.

To be accurate, the distance d must be measured between corresponding "principal planes" of the two lenses. The locations of those principal planes are the seldom published lens characteristics that I mentioned above.

At this point, I hope you're becoming comfortable with the idea that doing an accurate calculation for your combo is difficult.

I could tell you exactly how to do it, but in order to get some of the numbers you need for the calculation, I would have to start by telling you to measure some magnifications!

Probably that process would be educational, but mostly what it's educational about is convincing you that the formulas really are correct if only you can find the right numbers to plug into them.

For determining the actual magnification, it's far simpler to just photograph a ruler, measure field width and do the obvious division:
magnification = sensor_width / field_width

Is this helping?

--Rik

Kevin Childress
Posts: 24
Joined: Fri Aug 14, 2015 2:19 pm
Location: Lowell, North Carolina
Contact:
Rik,

Yes, your reply helps tremendously - lots of light bulbs turning on over here! And thank you so much for taking the time to write such a detailed reply - it is very much appreciated.

I can see the difficulty in finding the 'correct' number through the mathematical approach. And as you and Pau suggest, I've also found it quite simple (and seemingly quite accurate) to compare sensor width and image field width. I've been using the 85mm/50mm reversed combination for about a year but I just threw the extension tube into the mix about a week ago. As best as I can tell so far, the extension tube is gaining me ~17% greater magnification over the 85/50 reversed without the tube. Its not earth shattering but measurable.

I had also calculated the 1.7x (without the added extension) and have verified to as much certainty that I can that number is valid when comparing images.

Also completely understood what you said about the focal lengths changing inside the lenses. I do indeed adjust the two lenses' focus rings independently at times for the sake of composition. If ever I'm after 'the closest I can possible get', then I set both rings to infinity and move the subject into the focal plane from there.

I want to thank you again for taking the time to help me; your effort is more helpful than you could realize.

All the best,

Kevin

rjlittlefield
Posts: 21400
Joined: Tue Aug 01, 2006 8:34 am
Location: Richland, Washington State, USA
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Kevin,

I'm glad to hear that helped.

Another reference you might find helpful is FAQ: Stopping down a lens combo, especially the 5th post that talks about inverted versus normal perspective. Combos can be surprising beasts!

Another aspect of how odd combos can be, if you push them to extreme cases, is hinted by my comment that It is always true that if you add extension, you "add magnification". There are two reasons for the quote marks around "add magnification".

The first reason is to distinguish between adding magnification and multiplying magnification. Suppose you have a combo whose combined focal length is 50 mm, and you add 100 mm of extension, then you add 2X (=100/50). If the base combo were giving say 1.5X, then the extended combo gives 2X + 1.5X = 3.5X. This is adding magnification, in the literal sense of adding the numbers. But suppose that instead of extending the combo, you stick a 2X teleconverter behind it. In that case the magnifications multiply, 2X * 1.5X = 3X. In this case the final numbers are not much different, but in other cases they can be quite far apart.

The second reason is that in some unusual lens systems, adding extension actually reduces the magnification. This happens when d > f1+f2, which causes the combined focal length to become a negative number, despite the fact that the combo continues to serve as an ordinary imaging lens that forms a real image some relatively short distance in back of the lens. With that sort of a system, when you stand back and look at the object, the lens, and the image, you think to yourself that the lens must have a pretty short focal length. But then when you extend it in hopes of getting more magnification, you actually end up with less! The governing equation is still that