## MTF diffraction limit formula

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ray_parkhurst
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### MTF diffraction limit formula

What is the formula to calculate the % contrast vs aperture and freq in lp/mm? On the Rodenstock/Qioptic MTF curves they show circles/asterisks at the ends of each set of curves. These indicate the Nyquist limit in lp/mm for each spatial frequency. But I can't find a formula to allow me to calculate these. Any reference would be appreciated...Ray

rjlittlefield
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Transcribing from "Practical Optical System Layout / And Use of Stock Lenses", by Warren J. Smith (McGraw-Hill 1997), pages 120-121:
Line resolution is the ability to separate or recognize the elements of a pattern of alternating high and low brightness parallel lines. An optical system is a low-pass filter, in that it cannot transmit information at a spatial frequency higher than the cutoff frequency, given (in cycles per unit length) by

Equation (4.6): nu_0 = (2*NA)/lambda = 1/(lambda*f-number)

This frequency corresponds to a line spacing equal to the Sparrow criterion in Eq. (4.3). It is an absolute cutoff, with zero contrast between the light and dark lines in the image.
...
The modulation transfer function (MTF) describes the way that the optical system transfers contrast or modulation from object to image, as a function of spatial frequency. The modulation is defined as

Equation (4.8 ): M = (max-min)/(max+min)

where max and min are, respectively, the maximum and minimum values of brightness (in the object) or illumination (in the image) and the object is a pattern of parallel lines whose brightness varies according to a sine function. The modulation transfer factor for a specific frequency is the ratio of the modulation in the image to that in the object, or

Equation (4.9) MTF = M_i / M_o

For a perfect optical system the modulation transfer function is given by

Equation (4.10): MTF(nu) = 2/pi * (phi - cos(phi)*sin(phi))

and phi is defined as

Equation (4.11): phi = arccos((lambda*nu)/(2*NA))

where nu is the spatial frequency, lambda is the wavelength, and NA = n sin u is the numerical aperture. Note that the term within parentheses in Eq (4.11), being a cosine, cannot exceed unity; this then is the source of Eq. (4.6) for the cutoff frequency nu_0.
So for example with lambda = 0.00055 mm/cycle and NA = 0.125 (f/4), the cutoff frequency would be 454 cycles/mm. Cutoff at f/11 would be 165 cycles/mm, which agrees nicely with the experimental result shown at http://www.photomacrography.net/forum/v ... 164#101164. (Look at the f/11 image with a micrometer scale superimposed on it.)

The shape of the MTF curve for a perfect optical system (with a round aperture) is shown as the lower curve at http://www.photomacrography.net/forum/v ... 2691#42691. That posting also has some other comments about how the numbers vary depending on what assumptions you make and how you define what you're talking about.

I like to remember the curve as being a little below linear falloff from 0 to the cutoff frequency. 50% MTF is reached at about 40% of the cutoff frequency; 10% MTF is reached at about 80% of the cutoff frequency. Figure 1.25 at https://spie.org/x34304.xml illustrates that the shape of the curve has to do with the aperture shape: a circular aperture gives the form described above while a square aperture gives linear falloff.

One way to think about the cutoff frequency is that it scales linearly with NA and reaches its peak value at two cycles per wavelength (one for each lobe of a sine wave) when NA=1. (NA > 1 can be achieved only when you're working in dense media with refractive index > 1, and then I think it's actually a little confusing to keep track of what wavelengths and angles the formulas are really talking about.)

I'm hoping the answer to your question is in there somewhere. If it's not, let me know and I'll try to dig it out some other way.

--Rik

PS. The book I referenced is accessible in bits and pieces through Amazon. Visit http://www.amazon.com/Practical-Optical ... 0070592543 and use their Look Inside function. A good search function is "Modulation Transfer Function". I transcribed here from my hardcover copy.

rjlittlefield
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Just to be clear, the "NA" that's being talked about in the above formulas is on the sensor side, same as the f-number.

This contrasts with the "NA" that appears in other places such as my recent postings about DOF. In those, NA is on the subject side while f_eff is on the sensor side.

The spelling is certainly the same, but depending on how you think about them, the meanings may be either quite similar or much different.

--Rik

ray_parkhurst
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Thanks Rik, exactly what I was looking for.

Do the typical test patterns used for lens measurements have sinusoidally-varying brightness? I have always thought they were 100% bright followed by 100% dark (square wave).

ray_parkhurst
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Ray

Ancient1
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The full download of "Practical Optical System Layout / And Use of Stock Lenses", by Warren J. Smith (McGraw-Hill 1997) can be found here.
http://ebookbrowse.com/practical-optica ... d132673438
Eugene Cisneros

ChrisR
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(Link needs an edit, it's http://ebookbrowse.com/practical-optica ... d132673438)

rjlittlefield
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Thanks for the reference. (I fixed the link in Eugene's posting.)

--Rik

DQE
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Does anyone know how to bypass the forced installer and just get the PDF ebook? Needless to say I am very uninterested in inflicting malware on my PC!

BTW, there are other ebooks of interest linked here, including things like using Fresnel lenses in optical design.
-Phil

"Diffraction never sleeps"

ChrisR
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I rejected all the ".exe"s.
Scroll down a screen or so and there's a list of PDFs, the book you want is the top one. I tried that and it said I wasn't allowed, and gave an alternative link. That just downloaded the .pdf.

DQE
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ChrisR wrote:I rejected all the ".exe"s.
Scroll down a screen or so and there's a list of PDFs, the book you want is the top one. I tried that and it said I wasn't allowed, and gave an alternative link. That just downloaded the .pdf.
Thanks for the tips.

I couldn't get any of the links to work, including the one you recommended. I tried two browsers (Firefox with a lot of ad-blocking plugins, and IE), with the same problems.

-Phil

"Diffraction never sleeps"

ChrisR
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I just tried again. It's no longer diverting to the alternate link. Perhaps it needs to be overloaded to do that. I found links to links all over, but they're all executables.

ChrisR
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Chris S.
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I was curious to see what that Website's executable would do, so I ran it in a "sandbox" (a virtual environment where suspect software can be safely tested without making actual changes to your computer).

The first thing it did was try to get administrator privileges (within the sandbox, I complied). Next, it asked me to accept the terms of an ad-supported download program (within the sandbox, I did so). Then it tried to install a number of other programs and change my default browser--nine separate items across three pages (I dismissed all of these). Even so, it proceeded to install some 1,300 files in 400 folders before the installer failed (which likely was because of its being sandboxed). None of these files included the pdf of the book. It also had a number of processes up and running that I don't recognize.

Of course, all this junk was confined to the sandbox, so making it disappear was the matter of couple of mouse clicks. But I suspect it might be very difficult or even impossible to clean up a machine on which the executable had really been allowed to run, and that it would greatly compromise the performance of the computer.

I would not trust that Website in the least. Bully for ChrisR for figuring out how to retrieve the bait from the trap--even if the feat is not repeatable.

--Chris
Last edited by Chris S. on Mon Mar 25, 2013 3:04 pm, edited 1 time in total.

DQE
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The "PDF installer exe" file also damaged my PC to some extent, including disabling my keyboard. Malwarebytes picked up a couple of things it tried to install. Unfortunately, all the major anti-malware/anti-virus packages have now become unable to defeat the latest malware.

It took quite a bit of effort to restore my PC to normal functionality. Thankfully I keep several independent, full hard drive backups on external hard drives. They are turned off and unplugged, so I can carefully restore my hard drives. I still worry about malware that can defeat such simple restoration efforts, though.

I guess there's a lesson here - if it sounds too good to be true, it DEFINITELY is!

That's a good idea about installing such high-risk downloads only do within a sandbox or within a throwaway PC. I need to look into setting up a cheap PC for such purposes. But what about malware that silently installs itself onto one's PC? This stuff made itself obvious upon casual inspection to some extent.

It's of course also easy to install malware within PDF files. I wish Adobe had better malware protection and weren't so vulnerable to scripts, etc. One of the tempting PDF ebook files at this web page was hosted from Russia and I am guessing likely contains malware.

Being somewhat of a cynic/realist, I fully expect to wake up some day and find that *nothing* works any more due to a malware invasion...or maybe this has already happened.
-Phil

"Diffraction never sleeps"