How to calculate diffraction limited aperture at greater than 1:1 magnification?

[_T_]

Does anyone know the formula I would need to use to calculate diffraction at macro magnification?

I know that diffraction increases with magnification but I’ve been unable to find the actual formula for the relationship.

 

[xkaes]

The way to do it is to figure out the exposure factor for the amount of extension. For example, if the f-stop is set a f16, with an 4X f-stop/speed adjustment needed, the optical diffraction = Y. The formula might be on the web or in Blaker's book. I know it's out there -- or easy to compute. It's something like the f-stop * magnification. EX. 16 x 4 = 64

 

[ic-racer]

If you want to calculate the size of the Airy Disk on the film, then (approximately*):

Airy Disk Size(mm) = N / 750

Where "N" is your f-stop calculated from the "bellows factor" for your magnification.



*The Airy Disk is light and dark concentric circles, so only a rough approximation of size.

 

[wiltw]

Do not forget that the size of the visible/detectable blur disk assumes that the original image is blown up to 8x10 size in the assumption...the visibility of the blur is greater for larger prints and lesser for smaller prints. So always take such dimensions with the FINAL PRINT SIZE in mind...2x larger print means that blur disk must be 1/2 as large on the original film image in order to continue to fool the eye+brain...and THAT still carries an assumption about the viewing distance of the viewer to the print, as well!

 

[_T_]

Oh that makes sense because the area of the airy disks is being magnified with the square of the distance and the bellows extension factor is related to the intensity decreasing at the inverse of the square of the distance, so it’s all just the same as the inverse square law.

Thank you all. It’s very clear now.

 

[alkman]

Some of this might also depend upon the lens you are using. The SK literature for their makro symmar HM lenses recommends smaller apertures for greater magnification which seems somewhat counterintuitive.

 

[Mark J]

The diffraction limit is determined by the cone angle of the light - hence the f/number at the film side.
If you go as far as 1:1, the f/number at each side is now twice what it says on the lens scale.
Hence if you shoot at 1:1 at a marked f/number of f/16, it is like shooting at f/32 in terms of resolution and detail contrast ( MTF ) .

For a mag less than 1:1 then the bellows factor times the f/number gives you the effective f/number as mentioned by xkaes and ic-racer above.
Getting into Airy discs is not particularly useful for photography.

 

[Dan Fromm]

See:

Gibson, H. Lou. Close-Up Photography and Photomacrography. 1970. Publication N-
16. Eastman Kodak Co. Rochester, NY. 98+95+6 pp. The two sections were published
separately as Kodak Publications N-12A and N-12B respectively. Republished in 1977
with changes and without the 6 page analytic supplement, which was published
separately as Kodak Publication N-15. 1977 edition is ISBN 0-87985-206-2. Gibson is
very strong on lighting, exposure, and on what can and cannot be accomplished. His
books, although relatively weak on getting the magnification with lenses made for
modern SLR cameras, provide a very useful foundation for thinking about working at
magnifications above 1:10 and especially above 1:1. Extensive bibliography.

Lefkowitz, Lester. 1979. The Manual of Close-Up Photography. Amphoto. Garden
City, NY. 272 pp. ISBN 0-8174-2456-3 (hardbound) and 0-8174-2130-0 (softbound). A
thorough discussion of getting the magnification, lighting, and exposure. Especially good
on working above 1:1. Extensive bibliography

 

[wiltw]

It is the DISTANCE ON FILM that matters in the equation about the Airy disk, you do not see 'size of object' (object magnification) in this equation! It matters not if the object is recorded at 1/1000 of lifesize or at 1:1 (life size) size on film

...and because the film image is magnified by enlargement to larger print sizes, the visibility of the Airy disk is enlarged along with the apparent size of the object captured on film, which is why my earlier post on this topc pointed out the consideration of the final image size (and viewing distance)

 

[wiltw]

If you go as far as 1:1, the f/number at each side is now twice what it says on the lens scale.
Hence if you shoot at 1:1 at a marked f/number of f/16, it is like shooting at f/32 in terms of resolution and detail contrast ( MTF ) .

The diameter of the aperture and the FL of the optic does not usually change (for discussion purposes, ignoring AF macro lenses whose FL *does* change for macro distances (like the Canon 50mm EF macro) so the f/stop is NOT altered. The reason for apparent loss of light at macro distance is simply that the photon falling within the image circle are SPREAD OUT over a LARGER AREA when the image is focused for macro distances, thus the image at the film plane is dimmed in intensity by that greater area, rendering the f/stop to be APPARENTLY smaller than the set number.

 

[Mark J]

The diameter of the aperture and the FL of the optic does not usually change

The lens focal length is still 'f' but 1/f = 1/u + 1/v

 

[wiltw]

The lens focal length is still 'f' but 1/f = 1/u + 1/v

If a 50mm FL f/2 aperture lens is at Infinity focus, FL= 50mm and aperture diameter is 25mm....let us start with that basic fact.

When the same 50mm f/2 aperture lens is afixed onto a 50mm extension tube, it focuses on a subject whose image is lifesize on the film plane...

1. the lens distance to subject is 2*50mm, the lens distance to focal plane is 2*50mm and the subject is rendered at 1:1 at the film plane
2. the 50mm FL f/2 aperture lens is at macro distance focus for 1:1, its FL = still 50mm and its aperture diameter is still 25mm
3. the size of the image circle at the focal plane is made larger by the longer distance from lens node to focal plane; that is all that is happening
   the actual aperture is determined by the mechanical opening of the aperture and that CANNOT change. The 'effective f-number' is simply a conceptual thing to explain the fixed quantity of photos spread acrossed much larger image area at the focal plane so the image at the focal plane is made dimmer (than if the lens were focused at Infinity) and there is a need for exposure correction (2x) because the indicated f-number (f/2) is effectively giving the apparent f/stop reduced transmission of light of an f/4 f-number...but the APERTURE diameter is NOT CHANGED in using a lens at macro distance.

Per the book written by members of the faculty of the Rochester Institute of Photography (1990): Basic Photographic Materials and Processes, p 35-36

"...it is necessary to make an exposure adjustement when the image distance is larger than one focal length -- that is, when the camera is focused on objects that are closer than infinity....There are various methods for making the adjustment when photographic objects withing the 10-focal-length range. One methid is to calculate the effective f-number, and then determine the exposure time for that number rather than for the f-number marked on the lens. The effective f-number is found by multiplying the marked f/number by the ratio of the image distance to the focal length, or​

​

Effective f-number = f-fumber x (Image distance/focal length)​

 

[Mark J]

Yes, I agree with you on all that.
My point was that once you operate at finite distance, you now have two focal distances, 'u' and 'v' , each of which has its own f/number.
In the simple example of 1:1 , both 'u' and 'v' are now 100mm.
If your lens is still set at f/2, then the aperture is 25mm, and the f/number at each side if f/4.
This goes with the reduction in brightness, as you point out.

The lens conjugate equation ( above ) can help give you the f/number at each side, at any other magnification.

If you know the f/number then you can also predict the diffraction cut-off frequency, if you want.
Otherwise just refer to what you know about what minimum f/number is acceptable on your format.

 

[wiltw]

Yes, I agree with you on all that.
My point was that once you operate at finite distance, you now have two focal distances, 'u' and 'v' , each of which has its own f/number.
In the simple example of 1:1 , both 'u' and 'v' are now 100mm.
If your lens is still set at f/2, then the aperture is 25mm, and the f/number at each side if f/4.
This goes with the reduction in brightness, as you point out.

The lens conjugate equation ( above ) can help give you the f/number at each side, at any other magnification.

If you know the f/number then you can also predict the diffraction cut-off frequency, if you want.
Otherwise just refer to what you know about what minimum f/number is acceptable on your format.

Mark,
My replie was not 'disagreement'...I was merely trying to provide additional explanation behind exposure adjustment in macro shooting (vs. shooting at infinity) created by the greater distance from lens node to focal plane (for macro shooting). The internet is notorious for spreading urban myth, and keeping things factual and not erroneous is more necessary than before.

I do however think that the concept stated,

"If your lens is still set at f/2, then the aperture is 25mm, and the f/number at each side if f/4"​

... is not a sound principle...there is not an 'f-number at each side', so if you can point to any academic evidence as proof of that concept, I would be happy to reconsider.

 

[Mark J]

I hadn't really considered this question very deeply. To me it seems natural that because you can always calculate an NA for the image side and object side, then it follows you have an image f/number and an object f/number.
These are standard quantities provided in the first-order listing of a lens ( used at non-infinity case ) in optical design software.
My feeling ( and that of Goodman, below ) is that the normal f/number given on a lens is only the infinity case - which is not the unique definition.

This link here shows how messy the definition of f/number can get - and is not an ideal answer to your question . However you can see he talks on page 1 already about "For an object not at infinity, by analogy, the F-number is often taken to be the half of the inverse of the tangent of the marginal ray angle." so he has no problem with the concept.

https://wp.optics.arizona.edu/jpalmer/wp-content/uploads/sites/65/2018/11/f_word.pdf

I did check Welford's book for how he tackles the subject, but it was annoyingly lacking any discussion of f/number.
I'll see if i can dig out a better commentary on the subject in the next day or two while I'm 'on the job'.

 

[reddesert]

The f-number is directly related to the taper ratio (opening angle) of a cone of light whose base is the circular-ish aperture and whose apex is a point on the film. There is another cone of light from a point on the subject to the aperture, and that beam also has an f-number, it's just less common to speak about it.

The effective f-number of the beam on the image side is a real thing, not simply a concept, it's the actual ratio of the beam. This is much easier to explain with a picture, so I drew one. I drew this for a 50mm f/2 lens at magnification 1:2, so the asymmetry between object and image planes would be clear. On the right hand side of this drawing, there is a beam, a cone of light onto the film that is 75mm high and has a base of 25mm (the lens aperture). So it is an f/3 beam.

 

[Mark J]

Mark,
... so if you can point to any academic evidence as proof of that concept, I would be happy to reconsider.

It's surprising how little is written in the main textbooks about f-number. It supports what Goodman says above.
Today I checked Longhurst's book, then Warren Smith's 'Modern Optical Engineering' then Kingslake's 'Lens Design Fundamentals'.
Longhurst only gives the infinity definition of f-number in a section about the Lagrange invariant.
Warren Smith says that f-number is 'typically used for the infinity case whereas NA is more useful for finite conjugates'
Kingslake has the comment copied below, which is the way I have always considered it over the years :

 

[wiltw]

thx to both for the contribution of information about f-number 'in front' vs 'behind'

I have to wonder, that when 1:1 is the object size in front of the lens and behind the lens, the distance to subject vs. distance to focal plane is IDENTICAL...2*FL, the total distance subject-to-focal plane = 4*FL

So in that case, it would seem that the computed f-number in front and behind the lens is the SAME, wouldn't ti?!

 

[Mark J]

Yes. As in my post from Sunday.

 

[Dan Fromm]

Mark, by and large photographers don't use or think about NA. Photographic objectives rarely, if ever, have NA engraved on them or have apertures scaled in NA. Similarly, light meters don't seem to have NA scales. Microscope objectives, nearly all fixed aperture, have NA engraved on them. And microscopists think in terms of NA, not f number.

 

[Mark J]

Yes, which is why I was talking about effective f-number until wiltw questioned it.

 

[RalphLambrecht]

Some of this might also depend upon the lens you are using. The SK literature for their makro symmar HM lenses recommends smaller apertures for greater magnification which seems somewhat counterintuitive.

There is also a simplephysical limit for aperture where the aperture should not be smaller than 3mm or 1/8 of an inch regardless of other calulations. This is typically the N limit of a lens, as in a 50mm lens (50/3 ~f/16 is thesmallest practical aperture).

 

[xkaes]

 

[Dan Fromm]

For a discussion of the conditions under which stopping down reduces depth of field, with photos showing the effect, see Gibson (1970), mentioned in post #8 above. Terrifying.