Identify which rays a system accepts, distinguish a physical stop from its apparent pupils, and describe the object- and image-side ray cones.
The engineering question
Chapter 1.2 answered where ideal paraxial rays meet. A real instrument also selects which ray directions can reach its sensor. Which physical opening sets that accepted cone, and how should a camera or eye describe the opening from either side?
Model
Geometric apertures in a paraxial system
Keep the centered, paraxial lenses of Chapters 1.1 and 1.2, and represent each physical aperture as a hard geometric mask: a ray either passes through its clear opening or is blocked. The mask does not redirect a ray that passes.
Useful for: deciding which paraxial ray cone is geometrically accepted and locating its apparent entrance and exit pupils.
Not predicted here: diffraction, radiometry or brightness, aberrations, scattering, and the detailed transmission of a real iris.
Focus tells where; an aperture tells which rays survive
At an ideal image plane, Chapter 1.2's condition makes the final height independent of a ray's initial slope. If an aperture blocks some of those rays, the remaining ideal rays still meet at that same ideal paraxial image plane. In this geometric model, aperture clipping changes the accepted cone, not the location of ideal focus.
That distinction matters for a sensor: at the ideal image plane, a focused object point maps to one sensor coordinate even though an aperture selects only part of its ray bundle.
The aperture stop limits the on-axis ray cone
A physical aperture is any clear opening: a lens edge, an iris, a tube, or a detector window. For a specified on-axis object point, the aperture stop is the physical aperture that imposes the tightest limit on the allowed bundle of ray directions. Its boundary rays are the axial marginal rays.
Figure 1.5. The aperture stop is identified by the ray cone it limits, not by physical diameter alone. For this on-axis object point, the 8 mm lens opening is the stop; the downstream 6 mm opening is nonlimiting.
Ray trace behind Figure 1.5. The two limits come from direct ray tracing. Starting at with variable slope ,
The lens radius gives , while the downstream-aperture radius gives . The smaller permitted angle identifies the lens as the stop. The independent example below uses the same test step by step.
Do not simply choose the smallest visible hole. Its position, the object point, and any optics before it determine the allowed initial slope. A later physical opening can be smaller yet nonlimiting for the specified on-axis object point.
Explore: which aperture is the stop?
Start from the Figure 1.5 system, then change an opening or its downstream position. Compare the two direct-trace bounds on : the smaller permitted initial slope identifies the aperture stop for this on-axis object point.
Pupils are the apparent aperture stop
The physical aperture stop may be buried inside an optical system. What an observer or object point effectively sees is its image through the other optics.
Figure 1.6. The entrance and exit pupils are two views of the same aperture stop. They can differ in location and diameter because different optical groups form their images. The dashed colored paths are side-specific stop-image constructions, not ordinary left-to-right imaging rays.
Viewpoint
Apparent aperture
Construction
Object side
Entrance pupil
Image of the aperture stop through the optical elements before the stop, viewed from object space.
Image or sensor side
Exit pupil
Image of the aperture stop through the optical elements after the stop, viewed from image space.
A pupil can be magnified, displaced, or virtual. It is an apparent image of the stop, not necessarily a plane through which actual rays physically pass. For a specified on-axis object point, reference the object-side marginal cone to the entrance pupil; reference the corresponding image-side cone to the exit pupil. Those are side-specific geometric views of the same physical stop.
Explore: where are the entrance and exit pupils?
Hold one physical stop between two thin lenses, then switch viewpoints. The explorer constructs the object-side entrance pupil or image-side exit pupil of that same stop and reports its position, diameter, and whether the apparent image is real or virtual.
Cone angle, f-number, and numerical aperture
Let be the entrance-pupil diameter and the exit-pupil diameter. The nominal f-number uses and the effective focal length :
Numerical aperture names a cone on a specified side of the system. If and are the object- and image-side half-angles in media of refractive index and , then
These are distinct quantities. They need not be equal because the refractive indices and pupil magnification can differ between the two sides. The f-number convention uses , while the image-side cone is geometrically referenced to the exit pupil.
Figure 1.7 is a deliberate special case. It has one thin lens whose clear aperture is the only physical aperture stop, with no optical elements before or after it. Viewed without intervening optics, each apparent pupil is the physical opening: at the lens plane, the physical stop, entrance pupil, and exit pupil coincide.
Figure 1.7. In this special one-lens system, the physical stop and both pupils coincide at the lens plane, so . For its image-side cone in air at infinity focus, the paraxial approximation gives .
The shortcut is conditional, not a definition. It describes Figure 1.7's simple image-side, air, infinity-focus geometry. At finite conjugates, a working f-number may differ; pupil magnification can make object- and image-side cones different; and a non-air image space changes the relevant refractive index.
Figure 1.8 brings the two side-specific numerical apertures together in one centered system: one real aperture stop, viewed as two apparent pupils.
Figure 1.8. Schematic, not to scale. In a general centered system, the physical aperture stop is real; its entrance and exit pupils are apparent images made by the optics on either side. Solid lines trace physical marginal rays through the real stop; dashed outlines and lines are pupil-image constructions. The object-side marginal cone is referenced to the entrance pupil, and the image-side marginal cone to the exit pupil, so and can differ.
Worked examples
Name a simple cone, image a physical iris into an entrance pupil, then identify a stop with ray transfer.
Guided
Name a simple lens cone
A simple one-lens camera in air is focused at infinity. Its effective focal length is and its entrance-pupil diameter is . Find the nominal f-number and the paraxial image-side numerical aperture .
Hint / setup
Use the entrance pupil, not an arbitrary internal opening: .
The stated air, infinity-focus, simple-lens geometry permits .
Keep both lengths in millimetres before cancelling their units.
Full solution
The lens is therefore described as . Under the stated paraxial, image-side, air conditions,
Numerical aperture is dimensionless. Do not apply this shortcut automatically to a finite-conjugate system, a non-air image space, or a system whose relevant pupil is magnified away from the lens plane.
Scaffolded
Locate a virtual entrance pupil
A clear, nonlimiting lens with lies at laboratory coordinate . A 4 mm physical iris lies at . Viewed from the object side, where is the entrance pupil and what is its diameter?
Hint / setup
For the reverse viewing construction only, introduce , positive toward the observer. The iris is then a real object at .
Use the same signed thin-lens equation in the coordinate: .
Convert back with , then use .
Full solution
In the reverse viewing coordinate,
Thus , or in the laboratory coordinate. The magnification is
Therefore . Because , this is a virtual pupil on the laboratory side of the lens; no screen belongs at that location. The lens was assumed clear enough that it does not become the aperture stop instead.
Independent
Find the stop with ray transfer
An on-axis object point is at . A lens at has and an 8 mm clear diameter. A downstream aperture at has a 6 mm diameter. Which is the aperture stop for this object point?
Hint / setup
Start with the on-axis ray state and variable initial slope .
Find the height at the lens, apply the thin-lens slope change, then propagate to the downstream aperture.
Compare the largest permitted magnitude of at each clear radius: 4 mm at the lens and 3 mm downstream.
Full solution
Free propagation from to the lens gives
The lens clear radius requires
After the positive lens,
One hundred millimetres later, the downstream-aperture height is
Its 3 mm clear radius would allow
The lens permits the smaller initial-slope range, so the lens is the aperture stop. Its 8 mm opening is physically larger than the downstream 6 mm opening, yet the downstream opening is nonlimiting for this specified on-axis object point.
Before you trust the answer
Model selection and engineering checks
Have you separated the ideal paraxial image plane (where ideal rays meet), the aperture stop (which on-axis ray cone passes), and the pupils (apparent views of that stop)?
Did you name the pupil from the correct side: entrance from object space and exit from image or sensor space?
For an aperture-stop test, did you trace the same on-axis object point through every candidate opening and compare allowed initial slopes rather than physical diameters?
Are the length units consistent in , and have you identified the medium and the side of the cone before quoting or ?
Is the paraxial approximation plausible for the marginal rays, and have you avoided treating a geometric aperture model as a diffraction, brightness, or aberration prediction?
