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nielsgeode
Joined: 31 Jul 2012 Posts: 246 Location: Groningen, Netherlands

Posted: Sun Feb 16, 2014 2:12 am Post subject: Overview of Microscope objectives DoF 


There are different ways to calculate the depth of field, and they give different results. Therefore, I think it may be useful to have an overview of common microsope objectives in a table together with their DoF's and "recommended" step sizes at different magnifications.
I still find it difficult to 'find' the best step size for my objectives, so any input and experience is welcome
This is a list of the objectives that I own:
Mitutoyo 5x NA 0.14 WD 34
Mitutoyo 7.5x NA 0.21 WD 35 (will be ordered next week and I will publish test results)
Mitutoyo 10x NA 0.28 WD 33.5
Mitutoyo 20x NA 0.42 WD 20
Mitutoyo 50x NA 0.55 WD 13
Nikon CF Plan 20x NA 0.40 WD 11.0
Nikon 4x BE Plan NA 0.10 WD 25
Nikon 4x Plan NA 0.10 WD 30
Nikon 5x LU Plan NA 0.15 WD 23.5
Nikon 4x Plan Apo NA 0.20 WD 15.7
Nikon Plan 10x NA 0.25 WD 10.5
Nikon 50x NA 0.45 WD 17
Olympus SPlan 80x NA 0.75 WD 4.10
Nikon M Plan 60x NA 0.70 210/0 ELWD (finite)
Nikon M Plan 40x NA 0.40 210/0 ELWD (finite) 

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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 18576 Location: Richland, Washington State, USA

Posted: Sun Feb 16, 2014 10:23 am Post subject: 


The standard conservative formula for total DOF is this:
Quote:  total DOF = lambda / (NA*NA) 
where lambda is the wavelength and NA is the numerical aperture.
For common values of NA, using green light lambda = 550 nm, this gives the following values:
Code:  NA DOF (microns)
0.10 55.0
0.14 28.1
0.15 24.4
0.20 13.8
0.21 12.5
0.24 9.55
0.25 8.80
0.28 7.02
0.40 3.44
0.42 3.12
0.50 2.20
0.55 1.82
0.60 1.53
0.65 1.30
0.70 1.12
0.75 0.98

What these numbers give is the distance between the top and bottom of the zone where the wavefront error is no more than 1/4 lambda across the entire aperture, for a "perfect" lens that has no aberrations at perfect focus.
It represents the depth of field within which (theoretically) there should be no more than 20% loss of contrast, compared to the image at perfect focus.
In other words, if you think in terms of 80% > 100% > 80% as you focus through the subject, it's the distance between the two 80's. Sometimes I call this the "twosided DOF".
This formula corresponds to the first term in the more complex formula given by Nikon at http://www.microscopyu.com/articles/formulas/formulasfielddepth.html. Nikon's formula gives a bigger number, attempting to account for finite sensor resolution.
It also corresponds to the entire formula lambda / (2*NA*NA) used by Mitutoyo, except that Mitutoyo's formula gives the "onesided DOF" that corresponds to 100% > 80%.

I think that there is still some legitimate debate about the validity of those formulas for our applications. It's well known that under some conditions, you can stack with a larger step size and still get a good image. No problem there. The question is whether a smaller step size is required for some subjects, and if so, why?
As far as I can tell, the theoretical formulas were worked out for perfectly flat smooth subjects. They tell how the appearance of that flat smooth subject will change depending on focus.
But for focus stacking the problem is often a little different from that. Our subjects are not necessarily smooth, not even in small areas. Instead they may be nanoscopically rough, with many small bumps and valleys at about the same size as light waves. It is not clear to me exactly how the appearance of those rough subjects may change depending on focus. It's conceivable to me that a rough subject might change appearance more quickly due to interference between neighboring features, but I am not aware of any investigations that address this issue.
In the tests that I have run, the formulas do seem to give pretty good estimates of DOF. See for example the results and methods at http://www.photomacrography.net/forum/viewtopic.php?p=103997#103997 and especially the followup at http://www.photomacrography.net/forum/viewtopic.php?p=104108#104108.
But perhaps things are different with other subjects. It would be nice to see that carefully investigated and documented.
Rik 

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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 18576 Location: Richland, Washington State, USA

Posted: Sun Feb 16, 2014 1:50 pm Post subject: 


For completeness and crossreferencing, let me mention that the lambda / (NA*NA) formula is an asymptotic expansion of a more accurate formula that looks like lambda/(2*n*(1sqrt(1(NA/n)^2))). The relative error between the two formulas grows only a little faster than NA^2. It is about 1.6% at NA 0.25, 7.2% at NA=0.50, 9% at NA 0.55, increasing to 25% at NA 0.80 and larger beyond that.
Recalculating DOF using the more accurate formula gives these numbers:
Code:  NA DOF DOF
(simple (more accurate
formula) formula)
0.10 55.0 54.9
0.14 28.1 27.9
0.15 24.4 24.3
0.20 13.8 13.6
0.21 12.5 12.3
0.24 9.55 9.41
0.25 8.80 8.66
0.28 7.02 6.87
0.40 3.44 3.29
0.42 3.12 2.97
0.50 2.20 2.05
0.55 1.82 1.67
0.60 1.53 1.38
0.65 1.30 1.15
0.70 1.12 0.96
0.75 0.98 0.81

See http://www.photomacrography.net/forum/viewtopic.php?p=124722#124722 for more discussion and reference to the original publication, which includes experimental validation under their test conditions (again, an essentially flat smooth subject).
Rik 

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nielsgeode
Joined: 31 Jul 2012 Posts: 246 Location: Groningen, Netherlands

Posted: Sun Feb 16, 2014 11:35 pm Post subject: 


Thank you Rik!
I have the feeling that magnification also influcences the DoF. At least, when you hook up a 10x objective on a 100mm tube lens, you get more DoF, then with the same objective on a 200mm tube lens, and I think that the NA does not change. Am I right? 

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rjlittlefield Site Admin
Joined: 01 Aug 2006 Posts: 18576 Location: Richland, Washington State, USA

Posted: Sun Feb 16, 2014 11:56 pm Post subject: 


nielsgeode wrote:  I have the feeling that magnification also influcences the DoF. At least, when you hook up a 10x objective on a 100mm tube lens, you get more DoF, then with the same objective on a 200mm tube lens, and I think that the NA does not change. Am I right? 
You are correct that the NA does not change.
You're also correct that in the particular situation you've described, you will quite possibly get more DOF from the 5X configuration (10X on 100 mm). What's happening in that case is that you've gotten into the awkward territory where image quality is limited by sensor resolution in addition to diffraction. It's like shooting a landscape at f/10 on an APSC sensor. The circleofconfusion model isn't right because it ignores diffraction, and the pure diffraction model isn't right either because it ignores sensor resolution. This is why Nikon adds that second term in their formula. It's an attempt to incorporate sensor resolution.
When the sensor  and the rest of the imaging system, including the monitor and the viewer's eyes  has sufficient resolution to capture everything that's in the optical image, then DOF depends only on NA. The folks at Delft specifically addressed that in their Figure 6, which demonstrates no change in DOF despite over 2X change in magnification by changing the relay lens.
Rik 

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