Re: Noise levels as a function of pixel size

In article <docn5o$19ua$1@xxxxxxxxxxxxxxxxxx>, Ilya Zakharevich <nospam-abuse@xxxxxxxxx> writes
[A complimentary Cc of this posting was NOT [per weedlist] sent to
Kennedy McEwen
<rkm@xxxxxxxxxxxxxxxxxxxx>], who wrote in article <eu4tEzFbXLqDFw2h@xxxxxxxxxxxxxxxxxxxx>: 
Photographically it doesn't make any sense at all to reduce the pixel
below about 3um pitch.


There is a physical limit, but it is much more than an order of
magnitude below the size you claim.

The wavelength of the spacial cut-off frequency of a lens with
incidence angle phi is

 lambda/(2*sin(phi)*n)

here lambda is wavelength, n is the refraction coefficient at the
focus plane.  The sensor which will collect ALL the information
available in the focal plane should have the step 1/2 of this.  This gives

 lambda/(4*sin(phi)*n)

for the limit sensor pitch. 

 I do not know n for silicon; however, if
there is an air gap between the last optical element and the sensor,
then n should be replaced by 1.  Anyway, assume n=1.6 for silicon
(this is closer to one for SiO2).  For lambda=0.55 micron, and phi
close to 90 degrees, one gets the absolute limit for pitch of 0.086
microns.

NO!!! That would only be valid if the entire gap between the lens and the sensor had a refractive index of 1.6. You are applying the optical formulae for an immersion lens to a non-immersed system!

The full formula for diffraction limited lens of circular aperture in uncollimated light is:

MTF(u) = (2A - sin(2A))/pi      (1)
where cos(A) = W.u.f            (2)
where
u is the spatial frequency
W is the wavelength of the light and
f is the f/# of the lens.

You will find the derivation of this formula in any good textbook dealing with optical MTF and fourier mathematics,
eg. J. W. Goodman's "Introduction to Fourier Optics", which I happen to have in front of me right now.

As you can see from the above, the MTF falls to zero in the case where A=0, hence
u=1/(W.f) (3)

All that is necessary is for the pixels to have small enough pitch such that the Nyquist limit of their sampling frequency equals this cut-off. Any smaller is simply wasting potential pixel area and sensitivity. The sampling frequency is simply 1/p, so the limiting resolved spatial frequency is u=1/2p, which can be applied back into (3) above to give
p(min) = W.f/2 (4)

Introducing realistic values into (4) gives examples such as:
f/11    > 3um
f/5.6   > 1.5um
f/2.8   > 0.7um

Note that these figures are for the diffractive optical MTF limit - the useful optical resolution of a diffraction limited lens depends on the SNR of the sensor behind it, but is rarely more than 70% of this, resulting in a 50% increase in these numbers.

You will also notice that these values also align pretty well with the actual maximum apertures available on typical small pixel (and large pixel for that matter) digital cameras, so it certainly isn't the order of magnitude out that you claim.

Sticking a slab of silicon between the lens doesn't change the numbers in any way shape or form. Immersing *both* lens and digital sensor in a high refractive index medium does, but someone already has the IP on that. (and he isn't too far away from this conversation right now!)

Your errors are that

 a) you consider Airy disk which has very little to do with
    intensively post-processed digital photography (as opposed to
    Nyquist limit of the cut-off frequency), and

On the contrary, I have used both spatial (Airy disc) and frequency analyses in this thread, repeating the latter in more detail for you above. Both give roughly the same numbers and I have also explained where the difference arises (Airy disc is merely an approximation, the frequency analysis is exact but overestimates the pixel size because the useful MTF is greater than the zero).

 b) take estimates for useful f-numbers from larger format lenses.
    Since a price of a lens element is about 4th or 5th degree of its
    size, much better lenses become economically feasible for smaller
    formfactors.

As stated in a previous post, I believe that to be your error - assuming that the same tradeoffs can be applied to the small format as to the larger one. If you are arguing minimum pixel size limits there is little point in making the same resolution/SNR trade-offs in the optic that may be acceptable in the large film format when the simplest solution is just to accept larger, high SNR pixels in the first place.
--
Kennedy
Yes, Socrates himself is particularly missed;
A lovely little thinker, but a bugger when he's pissed.
Python Philosophers (replace 'nospam' with 'kennedym' when replying)
.

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