Much of the fine details of the material in this article can be found in reference 2.  In this online article, I will simply summarize the main points.

The on-axis or axial image brightness (or more correctly, axial image irradiance) due to an imaging system with a circular stop may be written as pL(NA)², where L is the object radiance (Watts/m²sr).  The NA is calculated by the usual product n_isin(q_i).  The angle q_i is the image space marginal ray angle subtended by the system circular exit pupil. 

We have assumed that the refractive index in object space is unity.  Intuitively, the image irradiance due to the use of a non-circular stop must be described by an equation with exactly the same mathematical form except that the NA now has to be properly defined.  Why is this so?  Because for any system, no matter whether we use a circular or non-circular stop, the image irradiance must approach the value pL whenever the exit pupil is either infinitely large or when the image plane is infinitesimally close to the exit pupil. 

Why is that?  Because under both these limits, aperture shapes are indistinguishable from the point of view of the image.  This is similar to the problem in classical electro-statics where one is asked to calculate the total electric field amplitude at a point just above the surface of a very large thin conductor sheet.  The result is always equal to s/2e_o where s is the surface charge density and e_o is the permittivity of free space.  This result is independent of sheet conductor shape as long as the point of interest is very close to the conductor surface (or if the conductor is infinitely large). 

Thus, the electric field amplitude s/2e_o is analogous to the value pL for the case of image irradiance from a Lambertian source surface.  In other words, we note that the image irradiance for imaging systems with an arbitrary stop shape must always be given by pL times a number that is a function of aperture shape.  For a circular stop, this number is n_i²sin²(q_i).  For non-circular stops, this number would be a sort of effective NA,  "NA_eff", defined by noting (from classical radiometry) that the calculation for the axial image irradiance E_i due to an exit pupil with an arbitrary shape is determined by performing an integral over the entire  exit pupil surface³:

      [Eq. 1]

If we perform the integral in circular coordinates, we find the interesting result that NAeff may be approximated by the root mean square (RMS) average of the NA from a sample of X number of marginal rays around the edge of the entire exit pupil (Fig. 4):

      [Eq. 2]


Fig. 4 - Geometry of Eq. (2).

Now recall that F-Number may be defined by writing f/No. = 1/(2NA).  If we now write f/No._eff = 1/(2NA_eff), we arrive at our required definition for the effective F-Number.  But wait, this still doesn't look anything like the effective F-Number defined in the Zemax manual!  This leads us to the next section.