We’ve seen in the last article that the job of an ideal photographic lens is simple: to receive photons from a set of directions bounded by a spherical cone with its apex at a point on the object; and to concentrate them in directions bounded  by a spherical cone with its apex at the corresponding point on the image.   In photography both cones are assumed to be in air.

In this article we will distill the photon collecting and distributing function of a complex lens down to its terminal properties, the Entrance and Exit Pupils, allowing us to deal with any lens simply and consistently.

Since photons travel in straight lines and bend at media interfaces just like rays do, the pupils as defined for photons will work just as well with ray tracing and vice versa.   Therefore we will achieve our objective by relying entirely on geometrical optics (paraxial, Gaussian), which maps points to points, lines to lines and planes to planes, from object to image space, subject to lens magnification.   In this we will be aided by Bill Claff’s excellent Optical Bench at photonstophotos.net.

On the way we will introduce the pupils’ critical role in aberrations and wave optics, which will be discussed in the next article.   This is not meant to be an exhaustive treatment of the subject but merely an introduction to give keen photographers simple intuitions about the factors at play.

Sphere to Sphere

Let’s assume photographic distances and a very short period in time.   Photons can then be thought of as pellets from a shotgun fired from a spot on the object towards the lens, all traveling in a straight line and at the same speed, in a certain set of directions.  The set of directions interacting with the lens is limited by its aperture, resulting in the acceptance cone described in the previous article in terms of the relative solid angle.  Photons arrive at the lens randomly spread out on the area defined by the cone’s spherical cap

Assuming we observed the arriving photons on the spherical cap and averaged their count locally, we would see that they wouldn’t all interact with the lens at the same time, per the simplified two dimensional diagram in Figure 1.

Wavefronts and the Gaussian Reference Sphere

The first photons to arrive are those that have had to travel the shortest distance, those on the cone/optical axis.  The instant they enter glass they slow down and change direction according to Snell’s law, because the speed of light is inversely proportional to the index of refraction of the medium – and glass’ (about 1.5) is higher than air’s (about 1)

(1)  

with the speed of light in a vacuum.  As a result of the typical shape of a lens, photons off-axis travel relatively more in air and get slowed less so that, ideally, there is a moment in space/time when photons are all aligned in a single plane inside the lens (the blue dots above).  The opposite happens as they leave the lens, photons assembling spherically on the cap of a cone of directions all pointing  to the single spot on the image plane predicted by geometrical optics.

However, in a tip of the hat to wave/particle duality, photons laid out spherically on the cap are said to be part of a wavefront, in phase with the Gaussian reference spheres, centered on the relevant Gaussian spots at the object and in the image, with radii and respectively.

Lenses as Black Boxes

Lenses in practice are much more complex than the fictional single-element version in Figure 1, since they have to deal with aberrations, focusing, zooming etc.  Here for instance is the sketch of a current well-corrected 50mm prime, the Nikkor Z 50mm f/1.2 S, lifted with permission from Bill Claff’s excellent Optical Bench at his photonstophotos.net site.

Quite a collection of elements.  However, in the end the job of both simple and complex lenses is essentially the same: accept photons from a point at the scene – in the form of a diverging spherical wavefront – and produce photons converging to a point on the image plane, ideally also in the form of a spherical wavefront.

So as often seen in the literature,^[1] it is useful to think of a complex lens as a black box, no matter how complicated its insides.  Whatever happens within the lens, the end result is distilled to a photon distribution at its entry and exit ports: a spherical wavefront entering the box through a hole (an aperture) in the front surface of the box – and ideally another spherical wavefront exiting from a hole in the back one (pardon the badly botched attempt at perspective below).

In photography the apertures are mostly circular or approximately so and they represent the stop which limits the extent of the incoming photon wavefront, hence its solid angle.

Pupils

Finite apertures in optics are referred to as pupils, the one at the front of the lens is the Entrance Pupil and the one at the back the Exit Pupil.  They are significant because any complex optical system can be simplified down to these two surfaces, assuming that they completely describe the photons diverging from the object and converging onto the conjugate image point.

This is also true of ray tracing, a key technique in geometric optics.  For instance on the image side we can pretend to a good approximation that the photons/ray bundles converging onto a point in the image all come from a filled Exit Pupil – so if we know the photon/ray distribution within it we can determine how the image will turn out.  Superposition applies, so simply add the contribution of all object points spatially to create the geometric image according to raytracing rules.

What happens between the two pupils depends on the physical makeup of the lens system and it is also assumed to be fully described by geometrical optics.

Pupil Pictures

Pupils in photography do not usually correspond to a specific lens element.   They are images of the most limiting physical stop, normally deep inside a lens: you can see the two small vertical bars representing a mechanical iris before the focusing elements in Figure 2.  Since the stop is on a plane perpendicular to the optical axis so are its images, according to Gaussian rules.

In photographic lenses pupils are typically virtual images, meaning that we are not physically able to insert a sensor at their location and see an image there.  The apparent size of the pupils needs to take into account all of the light bending caused by glass before and after the stop, depending on whether the stop is being looked at from the front or from the back.  If the relative amount  or type of glass is different before and after the mechanical stop, as it often is, the apertures will look to be different in size.

Pupil size and relative diameter is what photographers colloquially refer to as ‘the aperture’ of a lens.

Pupil Related Variables

We typically assume circular pupils, but they do not need to be.  The ratio of their diameters is referred to as Pupil Magnification ()

(2)  

with Entrance and Exit pupil diameters respectively as shown in Figure 4.  The pupils are images of each other, meaning for instance that any property ascribed to the photon distribution in one can just as easily be ascribed to the distribution in the other, subject of course to magnification .  Pupil magnification is important because it defines amongst others the distance to the ideal image plane ^[2]

(3)  

with geometric image magnification as previously discussed; and the published focal length, with the lens focused at infinity.  The farther the lens is focused, the smaller is, the more tends to .

We saw in the previous article how the Effective or working f-number (), critical for Exposure, diffraction and aberration theory, is expressed in terms of Exit Pupil variables.  With paraxial approximation in air

(4)  

the last ratio taking into account the Abbe Sine Condition useful to keep aberrations like coma under control near the optical axis.  Working f-number can also be expressed in terms of Entrance Pupil variables.  Substituting Equations (2) and (3) into (4) we have

(5)  

with ,  the published f-number with the lens focused at infinity,  preceded by the bellows factor well known to macro photographers.   It is obvious that here if tends to zero tends to  .

Pupil Positions

With the recipe for a lens in hand there are many ways to estimate the position of its pupils.  For instance since all rays/photons are supposed to be emanating from the Exit pupil on their way to a point on the image plane, perhaps the most intuitive is to backtrace the chief ray, which crosses the optical axis in the plane of the mechanical stop, to where it intersects the optical axis.  Alas, aberrations get in the way and make this an unreliable way to find the location of the Exit pupil.

There are better and more involved methods to determine pupil location – and lens recipes are not easy to find or  interpret – but fortunately Bill Claff has already worked them out for us for the lenses in his database, just head over to the Optical Bench section of his photonstophotos.net site.  Pressing the button will produce lines representing the pupils and their extent ( for the Entrance Pupil and for the Exit Pupil) thank you very much Bill.

If lens elements to the right of the physical stop move, say to focus, so will the location of the Exit pupil.  On the other hand, in many prime lenses the elements to the left of the physical stop stay put during focusing and therefore so does the position of the Entrance pupil.

Why They Are Useful

In practice it is useful to know where the pupils’ virtual images form in the physical lens.   They allow us to simplify a complex optical system down to two planes into and out of which photons and rays enter and exit the lens.

For instance, where the optical axis pierces the Entrance Pupil is the center of perspective and the reference point for Field of View – therefore when doing a panorama the camera should ideally be pivoted around it.

Also lens pupils are an effective interface between geometrical and wave optics, since they relate both to rays and photons – and photons can be interpreted both as particles and waves.  The rays filling the pupils determine  the geometric image via ray tracing; and the photon field filling the pupils determines wave effects via Fourier Optics, as discussed in a dedicated article.  In fact the image on the sensing plane can be thought of as a convolution of the geometric image predicted by geometrical optics – with the intensity Point Spread Function predicted by Fourier Optics.

So for instance when taking into account the effects of aberrations or the wave nature of light on the captured image, the size and distance of the Exit Pupil relative to the image plane is a key factor in image quality dear to photographers: working f-number from Equation (4) is a determinant of Exposure, it is related to the amount of aberrations and it affects the resolution of the image, influencing derivative parameters like Depth of Field.   See the articles on DoF and  Fourier Optics for a lot more on these subjects.

Next: an Introduction to Pupil Aberrations

Notes and References

<sub>1. Much inspiration for this article came from Introduction to Fourier Optics, 3rd Edition. Joseph W Goodman.
2. “Depth of Field from Working F-number and Image magnification“, A Geometric Optics Calculation and comparison with alternative approaches, Alan Robinson (Unpublished, 2020)
3. Basic Wavefront Aberration Theory for Optical Metrology, James C. Wyant and Katherine Creath.</sub>