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.

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.

The previous section gave the theoretical basis for deriving an effective F-Number definition.  Now, the integral in Eq. (1) is also known as the image space "projected solid angle" or PSA in classical radiometry.  As such, we can write:

   [Eq. 3]

If we again write f/No._eff = 1/(2NA_eff), and substitute Eq. (3) into this, we arrive at the defined effective F-Number as written in the Zemax manual.  The PSA of an imaging system is routinely calculated in Zemax to determine the relative illumination of the optical design. 

Since the PSA may be calculated for all off axis field points, the effective F-Number may also be defined for all off axis points on the image plane.  Try it! - Open up your favorite lens design in Zemax and click on the drop down menu for relative illumination under Analysis > Illumination > Relative Illumination.  A window appears displaying the system relative illumination.  Then click on the text button of that window and you see a list of effective F-Number values as a function of field (you must of course define your field points first). 

For other details, consult the Zemax user's manual⁴.  Thus, we now have a very practical way of calculating the effective F-Number of an optical imaging system with arbitrary stop shape, based on very fundamental radiometric principles.

Summary

This article has discussed :

• The meaning of the f/# in optical systems with non-circular stops
• The theoretical basis of the defined effective f/# calculation in ZEMAX

External References

1.  For an explanation of what a Pechan prism is, see, for example, Design and Mounting of Prisms and Small Mirrors in Optical Instruments, by Paul R. Yoder, Jr., (SPIE Press, Bellingham, WA, 1998), pp. 41-42.
2.  Ronian H. Siew, "f/No. and the radiometry of image forming optical systems with non-circular aperture stops," in Optical Modeling and Performance Predictions II, Mark A. Kahan, editor, Proceedings of SPIE Vol. 5867 (2005).
3.  V.N. Mahajan, Optical Imaging and Aberrations (SPIE Press, Bellingham, WA, 1998), pp. 108-118.
4.  Zemax Optical Design Program User's Guide, ZEMAX Development Corporation.