If we were designing an optical imaging system that uses prisms (either to fold the optical path or to rotate the image or both), we'd want to consider sources of stray light.  For instance, if we used a Pechan¹ prism (see Fig. 1) we'd soon notice that off axis ray bundles beyond a certain field angle will not undergo total internal reflection (TIR) at the air gap of the prism (Fig. 2).  These rays continue through the prism and can do serious damage to the image contrast.  They may even create the appearance of ghosting at the image plane. 

To eliminate these troublesome non-TIR off axis rays, a rectangular aperture may be used at the exit face of the Pechan prism. This blocks off the non-TIR rays in the dimension of the non-TIR ray field angle (Fig. 3).  If this rectangular aperture vignettes the on axis ray bundle, then this aperture becomes the aperture stop of the whole optical system.  Perhaps we'd do a little re-optimization for this new stop position.

Fig. 1 - Pechan prism (created in Zemax sequential mode using coordinate breaks).

Fig. 2 - Pechan prism (created in Zemax hybrid non-sequential mode to analyze stray light).

Fig. 3 - Pechan prism with rectangular aperture to clip off the non-TIR off axis rays.

So far so good.  But the optical system probably is still required to satisfy an F-Number specification.  That is, the image must certainly be required to be as bright as the customer would want, no matter whether we use a circular stop or a rectangular stop.  In fact, we'd somehow want the image to have the correct brightness we need, independent of stop shape. 

We might find, for instance, that a triangular aperture did the best job in eliminating the prism stray light.  The question then is, "What's the F-Number of this system if I have a non-circular aperture stop?"  While it is true that the numerical aperture (NA), as we usually understand it, would seem to be a function of the lateral dimension at the plane of a non-circular stop, it must also be true that there is a certain "effective" NA associated with the image since the image has a certain "brightness", and since both F-Number and NA are known as quantities that specify the brightness of an image. 

How do we calculate this "effective" NA or F-Number?  At first glance, one might be tempted to convert a non-circular aperture into a circular aperture of the equivalent surface area, and then to calculate the NA or F-Number in the usual way.  But an in-depth review of basic radiometry will remind us that two Lambertian sources of the same surface area but with different shapes will give rise to different axial irradiances. 

Now, the vast majority of lens design tasks involved in industry is to image objects that appear as Lambertian source surfaces with respect to the imaging optics.  This condition allows us to treat the exit pupil of a lens system as a Lambertian source surface with respect to the image plane.  Thus, if Lambertian source surfaces of the same area but different shape give rise to different image irradiance, then exit pupils of different shapes but the same surface area will indeed give rise to different image brightness.  Since the exit pupil is the image of the aperture stop, it follows naturally that aperture stops of the same area but with different shapes will yield different image brightness. 

Aha! We discover that what we need is a proper definition for an overall "effective" NA or F-Number which is a universal indicator for image brightness, independent of stop shape.