I might be telling everyone something they already knew, but here goes:-
I was thumbing through Ray's 'Applied photographic optics' the other day, and he points out something about enlarging lenses that should be obvious, but hadn't ever occured to me before. That is; that the resolution of enlarging lenses is pretty much always diffraction limited.

The reasoning runs: For enlarging lenses, the object and image planes are reversed, so effectively the image is normally at many times the focal length of the lens.
So far, so good; but this also means that the effective aperture of the lens is many times smaller than the marked numerical aperture of the lens.
Therefore the effect of diffraction is also multiplied by a factor of, the enlargement ratio, plus one.
If you stop your enlarging lens down to f/8 with a 10 times enlargement, you're effectively using f/88 as far as diffraction is concerned, and you'll only ever be able to get about 18 lppm resolution at the printing paper.
If you do a 20" by 16" print from 35mm, this drops to 12.5 lppm!

This all makes a complete nonsense of any arguments about printing paper resolving power, and it's pretty obvious that not even film grain is properly resolved in a print.

If you open the enlarging lens up a stop, then you'll gain resolution, -theoretically-, but paper flatness and lens imperfections are likely to take away as much as you gained.
Taken to its logical conclusion, this means that there's absolutely no point in making an enlargement greater than 25 diameters, because the print resolution can't be any greater than the resolution of the human eye, and the print just won't look sharp at close viewing.

The relevant formula for calculating the resolution is: Diffraction limited resolution (lppm) = 1/(1.22*lambda*N*(m+1)), where lambda is the light wavelength in mm, N is the numerical aperture of the lens and m is the magnification of the enlargement.
Luckily, printing paper is only sensitive to the shorter wavelengths of light, which improves things slightly.

-- Pete Andrews (p.l.andrews@bham.ac.uk), October 16, 2000

Answers

I tend to rely on how the grain looks through my grain focuser in setting my enlarger lens. I note that at some apertures, the grain looks sharper: so that's where I print. I don't usually look to see what that aperture is, and it may well conform to you theory. Actually, I don't think the aperture makes a visible difference in 8x10s, which are most of my output.

-- Keith Nichols (knichols@iopener.net), October 16, 2000.

>> the print just won't look sharp at close viewing.

Then again, who looks at a 40 cm by 50 cm print from less than half a metre?

Did you check any of the theoretical stuff you read? Maybe you should get a resolution testing negative and try to test the resolution of your entire system (i.e. enlarger and paper).

If, as you say, "it's pretty obvious that not even film grain is properly resolved in a print" I wonder why I see it.

About that formula: resolution (lppm) = 1/(1.22*lambda*N*(m+1))

Don't forget to enter lambda (such as 5e-5 mm for lambda = 500 nm, somewhere in the green region) in millimetres rather than metres, or you'll be off by a factor of 1000!

Also: "If you open the enlarging lens up a stop, then you'll gain resolution, -theoretically-, but paper flatness and lens imperfections are likely to take away as much as you gained."

Your formula says: If you open the lens one stop, you will gain a factor of approx. 1,4 in your resolution, or about a factor of almost 3 if you enlarge at f/4 when compared to the f/8 you suggested in your post. Knowing that my lens is best at approx. f/4, I almost always use this f-stop. I haven't noticed any problems with "paper flatness and lens imperfections". BTW: I would judge negative flatness to be much more critical.

-- Thomas Wollstein (thomas_wollstein@web.de), October 16, 2000.

Not my theory, but well proven physical limits on resolution.
New to me, because I'd never considered it from that point of view before.

What we see as grain in film is really clumps or clusters of individual grains. These clusters are about 3 to 10 microns across, appearing as ~0.05mm - 0.1mm blobs in a 10x enlargement.

I've given you the reference, and stated that lambda is expressed in millimetres. In fact the figure of 1.22*lambda*N Airy disc radius is referenced in most texts, and goes back to fundamental work on diffraction done by Lord Rayleigh. Look it all up, and work it out for yourselves.

-- Pete Andrews (p.l.andrews@bham.ac.uk), October 16, 2000.

Addendum: What is seen as grain in a print is actually the space between the grains. It's a negative remember.
The exact reference is 'Applied Photographic Optics' by Sidney F. Ray, Focal press, 1988 edition. (vis chapter entitled 'Enlarging lenses')

-- Pete Andrews (p.l.andrews@bham.ac.uk), October 16, 2000.

My first reaction was "something's been missed here", but on further thought, it makes sense. Think of the ray being diffracted by the aperture as having a longer "lever arm" to the paper, than it does in the reversed situation of the camera. The practical result is still the same and what we already knew- best enlarging aperture is just a bit down from wide open for most lenses.

But the heck with all of it- 8x10 contact prints are the only way to go!

-- Conrad Hoffman (choffman@rpa.net), October 16, 2000.