As the title would indicate, I'm having some difficulty with equations.

Here's a few questions that I have.

1) Say you have an objective on a lens, then bellows attached to the lens. How would one calculate this magnification? Or F number? DOF? How would one calculate resolution? I understand that these are ultimately theoretical considerations that assume all lenses/objectives are of equal quality, but still.

2) I've read that an objective's useful magnification range is 500 to 1000 times its N.A. How therefore can people use for instance, a mitutoyo 2x plan apo? It's N.A is .055 which would give it a useful magnification range of 27.5x - 55x. Since it has a focal length of 100mm, I'm clearly missing something; to reach 27.5x the bellows or camera lens would need a focal length of 2750mm... since I'm not seeing any palm-tree sized optical setups, I again, can only assume that I'm missing something.

I'm trying to create an all-encompassing spreadsheet to display to me the relevant information needed to calculate optimal step size, among other things.

To be honest, I'm realizing while constructing this sheet that I'm missing quite a few things. I would greatly appreciate all help you guys can give me. Thanks!

## Difficulty With Equations

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### Difficulty With Equations

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- rjlittlefield
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You've mentioned in another thread that you're thinking of Mitutoyo objectives, so I'm going to presume this current thread comes from the same line of thinking.

In that case, let me see if I can simplify your life.

In this same case (rear lens focused at infinity), the effective f-number of the combination will be just magnification/(2*NA). For example, a Mitutoyo 10X NA 0.28 objective, used with a 200 mm rear lens focused at infinity, gives an effective f-number equal to 10/(2*0.28), about f/17.86 . If the rear lens were 135 mm, then the magnification would only be 6.75 (as discussed above), and the effective f-number would be 6.75/(2*0.28), about f/12.05 .

The nominal DOF of any objective, used as intended, and calculated according to the diffraction-limited criterion of quarter-lambda wavefront error for green light, will be just 0.00055/(NA^2), measured in mm. For example that 10X NA 0.28 objective will have DOF calculated as 0.00055/(0.28^2) = 0.0070 mm (7.0 microns). Notice that this number does not depend on overall magnification, only on the NA of the objective.

As for resolution, the cutoff frequency of an objective can be calculated as nu_0 = (2*NA)/lambda . For example, any objective with NA 0.28 has a cutoff frequency of (2*0.28)/0.00055 = 1018 line pairs per mm, for green light, measured on subject. Note that this is the cutoff frequency -- the lowest frequency for which the MTF of the objective will drop all the way to zero, so that target with sine intensity would render as uniform gray. The frequency at which MTF = 50% will be at about 40% of cutoff, roughly 400 line pairs per mm.

For the case where the rear lens is focused at infinity, all these calculations are straightforward.

But you said "then bellows attached to the lens". This suggests that you have in mind a setup where the rear lens is not focused at infinity. In that case (rear lens not focused at infinity), the behavior of the combination becomes very difficult to model accurately. The governing equations are much more complicated, require that you know characteristics of the lenses that are never published (in particular the locations of their "principal planes"), and are very difficult to apply correctly.

I have done those calculations -- and matched the results against experiment -- mostly as an academic exercise to convince myself that the equations actually do work when all the required values are finally known and plugged in correctly.

But I have never used that approach in practice, and my best recommendation is to just not go there.

Some further insight may be gained by putting together a couple of these equations. Think again about that NA 0.28 objective, which will have a diffraction-limited cutoff frequency of roughly 1000 line pairs per mm. Using the "500 to 1000" guideline, which gives 140 to 280, we can see that "maximum usable magnification" according to the guideline must mean cutoff at about 7 to 3.5 line pairs per mm. This compares to normal human acuity of around 6 line pairs per mm, so the "maximum usable magnification" guideline is basically identifying the range where diffraction limits the lens resolution to about the same resolution as the human eye.

--Rik

In that case, let me see if I can simplify your life.

Infinity objectives, including the Mitutoyo M Plan Apo's, are designed to be used only in conjunction with a rear lens that is focused at infinity. In that case, the objective gives its rated rated magnification when used with a 200 mm focal length rear lens. For any other focal length, the magnification just scales in proportion to that. For example, if you use the objective with a rear lens that is 135 mm, focused at infinity, then the combination gives the rated magnification of the objective, multiplied by 135/200 = 0.675 .1) Say you have an objective on a lens, then bellows attached to the lens. How would one calculate this magnification? Or F number? DOF? How would one calculate resolution?

In this same case (rear lens focused at infinity), the effective f-number of the combination will be just magnification/(2*NA). For example, a Mitutoyo 10X NA 0.28 objective, used with a 200 mm rear lens focused at infinity, gives an effective f-number equal to 10/(2*0.28), about f/17.86 . If the rear lens were 135 mm, then the magnification would only be 6.75 (as discussed above), and the effective f-number would be 6.75/(2*0.28), about f/12.05 .

The nominal DOF of any objective, used as intended, and calculated according to the diffraction-limited criterion of quarter-lambda wavefront error for green light, will be just 0.00055/(NA^2), measured in mm. For example that 10X NA 0.28 objective will have DOF calculated as 0.00055/(0.28^2) = 0.0070 mm (7.0 microns). Notice that this number does not depend on overall magnification, only on the NA of the objective.

As for resolution, the cutoff frequency of an objective can be calculated as nu_0 = (2*NA)/lambda . For example, any objective with NA 0.28 has a cutoff frequency of (2*0.28)/0.00055 = 1018 line pairs per mm, for green light, measured on subject. Note that this is the cutoff frequency -- the lowest frequency for which the MTF of the objective will drop all the way to zero, so that target with sine intensity would render as uniform gray. The frequency at which MTF = 50% will be at about 40% of cutoff, roughly 400 line pairs per mm.

For the case where the rear lens is focused at infinity, all these calculations are straightforward.

But you said "then bellows attached to the lens". This suggests that you have in mind a setup where the rear lens is not focused at infinity. In that case (rear lens not focused at infinity), the behavior of the combination becomes very difficult to model accurately. The governing equations are much more complicated, require that you know characteristics of the lenses that are never published (in particular the locations of their "principal planes"), and are very difficult to apply correctly.

I have done those calculations -- and matched the results against experiment -- mostly as an academic exercise to convince myself that the equations actually do work when all the required values are finally known and plugged in correctly.

But I have never used that approach in practice, and my best recommendation is to just not go there.

**It's much better to use the objectives as intended, with the rear lens focused at infinity or very close to that, and just use the simple calculations discussed above.**True. What you're missing is that this spec for "useful magnification" is talking about direct view through the eyepieces of a microscope, not the magnification onto a densely packed image sensor whose output will be further enlarged before viewing. If you use that 2X objective with its standard 200 mm tube lens, and add 10X eyepieces for direct view, you get to 20X which is still well within the range of useful magnifications. But if you were to swap in 25X eyepieces, then you'd be at 50X which is getting pretty dicey. Similarly, if you image onto an APS-C sized sensor, say 22 mm wide, then make a print 220 mm wide from the captured image, the final magnification would be 2*(220/22) = 20X, well within useful. But if you were to make a print 550 mm wide, then you'd be at 50X, again pretty dicey if you tried to view that print from close up. If you really want a 50X print that looks good, then you need to use an objective with larger aperture, such as Mitutoyo's 5X NA 0.14 objective, for which the guideline would say "maximum usable magnification" would be 0.14 * 500 to 1000 = 70X to 140X.2) I've read that an objective's useful magnification range is 500 to 1000 times its N.A. How therefore can people use for instance, a mitutoyo 2x plan apo? It's N.A is .055 which would give it a useful magnification range of 27.5x - 55x. Since it has a focal length of 100mm, I'm clearly missing something;

Some further insight may be gained by putting together a couple of these equations. Think again about that NA 0.28 objective, which will have a diffraction-limited cutoff frequency of roughly 1000 line pairs per mm. Using the "500 to 1000" guideline, which gives 140 to 280, we can see that "maximum usable magnification" according to the guideline must mean cutoff at about 7 to 3.5 line pairs per mm. This compares to normal human acuity of around 6 line pairs per mm, so the "maximum usable magnification" guideline is basically identifying the range where diffraction limits the lens resolution to about the same resolution as the human eye.

--Rik

First off, thanks again for your quick insight.

Another question:

How would one calculate the FOV of the images resulting from a setup? Lets say for continuity's sake that I again used a mitutoyo 5x. I attached it to a Pentax K1 (have you had any experience with them at high magnification?) with, Idk, a 200mm lens.

Now lets say I used the same equipment but used in lieu of a traditional lens, a raynox dcr-250 coupled to bellows. How would I calculate it then? Would anything change?

Part of the issue I'm having is that I'm trying to come up with the best known-good setup for my application (I know, best is a) subjective up to a point, and b) people spend years looking for it), but cannot find equations (probably about a 50-50 shot that I'm also not understanding the available ones correctly) to describe them, which makes planning them nearly impossible.

One of my primary concerns is finding the optimal stacking increment. A distance at which I will not be needlessly capturing the same data, but without leaving gaps in my coverage of the specimen along its z axis. To that end, I have yet another question. DOF decreases as magnification increases. So lets say I use again, a mitutoyo 5x. I have it set at 27.5x. What would the depth of field be at 27.5x vs 55x? I assume it scales linearly??

I will have a bunch more questions, I hope that's not a problem. I do greatly appreciate your comments, you're quite knowledgeable and the fact that you lend your time to helping people with their questions is admirable. If you don't mind my asking, what's your background?

In any case, thanks a bunch for your help, and I look forward to your reply.

Another question:

How would one calculate the FOV of the images resulting from a setup? Lets say for continuity's sake that I again used a mitutoyo 5x. I attached it to a Pentax K1 (have you had any experience with them at high magnification?) with, Idk, a 200mm lens.

Now lets say I used the same equipment but used in lieu of a traditional lens, a raynox dcr-250 coupled to bellows. How would I calculate it then? Would anything change?

Part of the issue I'm having is that I'm trying to come up with the best known-good setup for my application (I know, best is a) subjective up to a point, and b) people spend years looking for it), but cannot find equations (probably about a 50-50 shot that I'm also not understanding the available ones correctly) to describe them, which makes planning them nearly impossible.

One of my primary concerns is finding the optimal stacking increment. A distance at which I will not be needlessly capturing the same data, but without leaving gaps in my coverage of the specimen along its z axis. To that end, I have yet another question. DOF decreases as magnification increases. So lets say I use again, a mitutoyo 5x. I have it set at 27.5x. What would the depth of field be at 27.5x vs 55x? I assume it scales linearly??

I will have a bunch more questions, I hope that's not a problem. I do greatly appreciate your comments, you're quite knowledgeable and the fact that you lend your time to helping people with their questions is admirable. If you don't mind my asking, what's your background?

In any case, thanks a bunch for your help, and I look forward to your reply.

The tsumeb mine isn't just hallowed ground, it's hollowed ground.

- rjlittlefield
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**Posts:**21823**Joined:**Tue Aug 01, 2006 8:34 am**Location:**Richland, Washington State, USA-
**Contact:**

Sorry for the brief reply here. I won't have time to answer questions for the next couple of days.

But in the meantime, I have a question of my own that I'm hoping will help me figure out how to help you better.

You wrote

In other words, how could that number 27.5 be measured, or how could it be computed based on other numbers that you could measure?

--Rik

But in the meantime, I have a question of my own that I'm hoping will help me figure out how to help you better.

You wrote

Can you explain very carefully what "set at 27.5x" means to you?So lets say I use again, a mitutoyo 5x. I have it set at 27.5x.

In other words, how could that number 27.5 be measured, or how could it be computed based on other numbers that you could measure?

--Rik

- rjlittlefield
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**Posts:**21823**Joined:**Tue Aug 01, 2006 8:34 am**Location:**Richland, Washington State, USA-
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OK, that's very helpful.I'm used to dealing with microscopes where the magnification is simply calculated.

For you, the first key thing to learn is that the "magnification" of a direct view microscope is completely different from the "magnification" that appears in photographic DOF formulas. They have the same spelling, but they are different concepts. The numbers are also quite different, typically by a ratio of around 10:1, so it can be pretty important to keep straight which one is being talked about.

With a microscope, "magnification" describes a ratio: how many times larger the object appears to be when viewed through the scope, versus viewing the same object by unaided eye at a distance of 10 inches (25 cm). This number is usually calculated by just multiplying the nominal magnification of the objective, times the magnification of the eyepiece, times the magnification of a zoom control if the scope has one of those. The "magnification" of each of those parts actually means something different, but they're chosen in such a way that the calculation ends up working to give the overall ratio I mentioned.

In a photographic DOF formula, "magnification" describes a different ratio: how big the image is on sensor, versus the real physical size of the object. When using an infinity objective in the proper way (with the tube lens focused at infinity), this magnification can be calculated by multiplying the nominal magnification of the objective, times a scaling factor that is equal to the actual focal length of the tube lens divided by the focal length of the tube lens that the objective was designed for. With a Mitutoyo M Plan Apo objective or any Nikon CFI objective, the formula would be:

actual_magnification = objective_nominal_magnification * tube_lens_focal_length / 200

As an example, if you use a Mitutoyo 5X objective in front of a 200 mm tube lens, then you get a magnification of 5X (=5*200/200). But if you use the same objective in front of a 125 mm tube lens, then you get a magnification of only 3.125 (=5*125/200).

Using the photographic magnification, it is very simple to calculate the FOV -- just divide the sensor size by the magnification. For example, if you're working at 5X onto an APS-C size sensor that is 22 mm wide, then your FOV on subject is 22/5 = 4.4 mm wide.

The second key thing to realize is that classic DOF formulas have two deficiencies: 1) they do not incorporate the effects of diffraction, and 2) they assume that it's acceptable to lose a certain amount of detail, which is represented by a number called "circle of confusion".

Both of these assumptions become problematic when working with microscope objectives, modern digital imaging, and focus stacking. On the one hand, if image resolution is not limited by diffraction, then stepping at the DOF computed from a classic value for circle of confusion will cause obvious focus banding when zoomed in to view actual pixels at full camera resolution. On the other hand, if image resolution is limited by diffraction, then the classic formulas may suggest a smaller step size than is really needed.

As an alternative to the classic formulas which are based on geometric optics, I have come to prefer an alternative approach that is based directly on wave optics. It computes DOF based on limiting the amount of sag in the MTF curve at maximum focus error. The formula is very simple:

TDF = lambda/(NA^2)

where TDF is the total depth of field (front limit to back limit), lambda is the wavelength of light (typically 0.00055 mm for green), and NA is the subject-side numerical aperture. Earlier, I gave the example that 10X NA 0.28 objective will have DOF calculated as 0.00055/(0.28^2) = 0.0070 mm (7.0 microns).

When using a microscope objective in the proper way, NA is just the number stamped on the objective. This immediately implies that the DOF calculated by wave optics does not depend on magnification in this case.

When used with other optics, such as an enlarging lens mounted on bellows, you have to use some second formula to calculate subject-side NA based on lens settings and magnification. The appropriate formula varies depending on the situation. In these cases, DOF does depend on magnification, because the subject-side NA depends on magnification.

For what it's worth, this is the approach used to compute the DOF tables on the Zerene Stacker website at http://zerenesystems.com/cms/stacker/do ... romicrodof.

In recent years, I'm the fellow who wrote Zerene Stacker. I've also spent a lot of time studying DOF and trying to figure out the best ways to think about it. Degrees in math and computer science, self-taught in optics, prior employment doing scientific software development for contract R&D, part-time teacher of math at the local branch of our state university. Over 50 years fascinated by photographing small things.If you don't mind my asking, what's your background?

--Rik