Irradiance calculation

[wandell]

n some simulations I had increased scene irradiance to compensate for changes in pupil size so as
to keep retinal irradiance constant. Debugging that was what led to the fix to wvf2oi mentioned earlier.

There was still about a four percent difference in retinal irradiance, though, over the change from 2mm
to 7mm pupil diameter. My compensation was in direct proportion to the change in pupil area with
pupil diameter.

I tracked the effect down to the formula used in oiCaculateIrradiance, very nicely commented:

Description:
% The scene spectral radiance (photons/s/m2/sr/nm) is turned into
% optical image irradiance (photons/s/m2/nm) based on information in the
% optics. The formula for converting radiance to irradiance is:
%
% irradiance = pi /(1 + 4 * fN ^ 2 * (1 + abs(m)) ^ 2) * radiance;
%
% where m is the magnification and fN is the f-number of the lens.
% Frequently, in online references one sees the simpler formula:
%
% irradiance = pi / (4 * fN ^ 2 * (1 + abs(m)) ^ 2) * radiance;
%
% (e.g., Gerald C. Holst, CCD Arrays, Cameras and Displays, 2nd
% Edition, pp. 33-34 (1998))
%
% This second formula is accurate for small angles, say when the sensor
% sees only the paraxial rays. The formula used here is more general
% and includes the non-paraxial rays.
%
% On the web one even finds simpler formulae, such as
%
% irradiance = pi / (4 * FN ^ 2) * radiance
%
% For example, this formula is used in these online notes
% http://www.ece.arizona.edu/~dial/ece425/notes7.pdf;
% <http://www.coe.montana.edu/ee/jshaw/teaching/RSS_S04/
% Radiometry_geometry_RSS.pdf>

The pi /(1 + 4 * fN ^ 2 * (1 + abs(m)) ^ 2) doesn't go exactly as the
ratio of the pupil areas, which leads to the difference.

I remember this formula from earlier comparisons between isetbio and
PTB calculations, which were based on the ratio of pupil area. I
didn't think too hard about it at the time.

But now I'm wondering. The number of photons that enter the
eye across changes in pupil size really does scale with pupil area.
At least I'd think so. That would lead to the conclusion that
the number of photons in the retinal irradiance should scale
with pupil area.

But with the formula above, that's not what happens.

So in the full formula above, how do we explain where the
missing photons go across changes in pupil size.