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 called the axial marginal rays.
Figure 1.8. 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 1.8. 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 scaffolded example below uses the same test before locating the entrance pupil.
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 1.8 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 1.9. 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.
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 and numerical aperture
For an on-axis object point, the entrance pupil’s diameter and its axial distance from that point determine the object-side marginal ray cone. Similarly, the exit pupil’s diameter and its distance from the corresponding on-axis image point determine the image-side cone.
We quantify each marginal ray cone by its numerical aperture, which combines the cone’s half-angle with the refractive index of the surrounding medium. If and are the half-angles of the object- and image-side marginal ray cones in media of refractive index and , then
These are distinct quantities. The two cone angles and refractive indices need not be equal, so the object- and image-side numerical apertures can differ. Figure 1.10 brings the two side-specific numerical apertures together in one centered system: one real aperture stop, viewed as two apparent pupils.
Figure 1.10. Schematic with transverse distances enlarged relative to axial distances. 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. This example lies away from the 4-f baseline: the rays form a real intermediate image between the lenses before converging to the final image. Lee’s equation still applies; here , , and . See Chapter 1.3’s “Beyond the baseline” discussion for the corresponding regime, illustrated there with different system parameters.
Virtual pupils describe real ray cones
A pupil is an image of the aperture stop, so its apparent location need not contain a physical opening. When that image is virtual, ray extensions locate the apparent stop image. Looking into the optics, an observer can see an opening that appears displaced or enlarged. The actual aperture stop still determines which rays pass.
For a virtual entrance pupil, extend the incident, object-side marginal-ray segments straight into the system. Their extensions reach the apparent pupil edges, although the actual rays bend at the lens and pass through the physical stop. Figure 1.11 shows both paths. The dashed lines describe the apparent opening; they are not additional paths taken by the light.
Figure 1.11. A virtual entrance pupil describes a real accepted cone. The lens images the 4 mm iris, as viewed from object space, into a 10 mm apparent opening at . Solid marginal rays pass through the physical iris; dashed extensions of their incident segments locate the apparent pupil edges. The lens opening is assumed large enough that the iris is the stop. Axial and transverse scales differ for readability.
The virtual pupil’s diameter and distance still determine the object-side cone angle. In this example, the on-axis object is at , so its distance to the apparent pupil is . In air, the paraxial object-side numerical aperture is
Direct ray tracing gives the same result. A ray starting on axis with initial angle reaches the lens at , leaves with angle , and reaches the iris at . Its 2 mm radius therefore permits . The apparent-pupil construction and the physical ray trace describe the same accepted bundle.
A virtual exit pupil has the corresponding interpretation on the image side: extend the outgoing ray segments back to the apparent stop image, then use its diameter and distance from the image to describe the image-side cone. A virtual pupil does not imply a virtual specimen image; these are images of different things. Nor should we treat the virtual pupil as a second physical aperture, because the actual rays need not follow their straight-line extensions at that location.
The pupil in this example has a finite apparent location. A pupil at infinity is a different limiting geometry: the directions toward its center become parallel. That case leads naturally to object-space telecentricity in the microscope design below.
Microscope design: infinity correction and object-space telecentricity
Chapter 1.3’s two-lens imaging system becomes a useful microscope model when the first lens represents the objective near the specimen and the second is a tube lens that forms the camera image. Infinity correction and object-space telecentricity are different properties with different benefits. One design can have both, as Figure 1.12 shows. Here the objective represents an equivalent idealized lens group; the drawing does not describe its internal lens elements.
Infinity correction leaves room between the lenses
Place the specimen in the objective’s front focal plane. Rays from each specimen point leave the objective as a parallel bundle. Different specimen points produce bundles traveling at different angles. The tube lens focuses each bundle into its back focal plane, giving the magnification
The objective–tube-lens separation does not need one exact value in this ideal model: the bundles remain parallel as they cross the space between the lenses. This is the same freedom seen in Chapter 1.3 when the object remains at the first lens’s front focal plane. If the tube lens moves, the sensor must move with it to remain at its back focal plane. Real systems allow a usable range of separation constrained by clear apertures, vignetting (clipping of off-axis ray bundles), and aberrations. We will discuss vignetting and aberrations in more detail later in the course.
This space accommodates filters, beam splitters, and other intermediate components compatible with the optical design. “Infinity” describes where the objective alone would form the specimen’s image without the tube lens. It does not mean that the physical lens separation can be unlimited.
The chief ray from an object point is the ray whose path passes through the center of the aperture stop. The marginal rays pass through its edges and bound the accepted bundle. In Figure 1.12, the stop is at the objective’s back focal plane. Its image viewed from specimen space—the entrance pupil—is at infinity. Consequently, the object-side chief rays from different specimen points are parallel to the optical axis. This property is called object-space telecentricity.
Figure 1.12. A paraxial schematic of an idealized microscope with both properties; axial and transverse scales differ for readability. Solid chief rays are parallel to the optical axis in specimen space and pass through the stop center; dashed marginal rays meet the stop edges. Between the objective and tube lens, the rays from each specimen point are parallel to one another, while the two bundles have different directions. The tube lens brings each bundle to its own image point. Lens openings other than the indicated stop are assumed large enough to transmit the drawn rays. The chief rays from the two specimen points are not parallel after the tube lens because the illustrated separation is 280 mm, rather than . At a separation of 250 mm, both would emerge parallel to the optical axis. Explore the effect of lens separation in Chapter 1.3’s interactive explorer.
The distinction is about which rays are parallel and where: infinity correction makes rays from one point parallel after the objective; object-space telecentricity makes chief rays from different points parallel before it. Infinity correction does not automatically imply telecentricity: the aperture-stop location must also satisfy the telecentric condition.
In ordinary imaging, otherwise identical objects at different distances can appear at different sizes; this distance-dependent change in apparent size is a perspective effect. Object-space telecentricity suppresses that scale change. With the lenses and sensor fixed, an axial shift of the specimen leaves the chief-ray image positions unchanged in the ideal model, stabilizing image scale—for example, while adjusting specimen position to find focus. The marginal rays can still spread into a defocus blur around those positions. Telecentricity does not prevent that blur or guarantee constant magnification when the sensor is moved arbitrarily.
We will revisit telecentricity in scanning microscopy in Chapter 2 and in OCT in Chapter 3. Telecentricity on the sample side makes the central beam directions approximately parallel across lateral scan positions. This simplifies the spatial interpretation of a scan, but it does not keep the beam focused at every depth.
Finding NA when the entrance pupil is at infinity
Object-side numerical aperture describes the actual accepted marginal-ray cone leaving the on-axis object point: . A finite entrance pupil lets us find that angle from its apparent radius and distance. In an object-space telecentric system, however, the entrance pupil is at infinity and has no finite diameter; we cannot insert these into the finite-distance pupil formula directly. The accepted cone still has a finite angle. Telecentricity makes the chief rays parallel to the axis, not all the rays within each object point’s bundle.
For the ideal microscope above, the stop is one objective focal length behind the objective. Paraxial propagation gives . Setting the ray height equal to the stop radius therefore gives, in air,
For several lenses before the stop, use one matrix. Combine propagation and refraction from the chosen object plane to the aperture-stop plane, using the ray state :
For the on-axis point, , so . The marginal ray reaches a stop edge. Thus, for ,
This paraxial shortcut works for telecentric and nontelecentric systems, provided the other openings transmit the specified bundle. Here has units of length; in the simple microscope above it equals . Outside the paraxial regime, trace real marginal rays to obtain their actual object-side angle.
The chief ray passes through the stop center, so . Object-space telecentricity requires for all object heights, which means . This is the object-to-stop matrix. Its coefficients are not those of the object-to-image matrix whose imaging condition is . The reference planes determine which condition we are testing.
F-number: the working image cone and the nominal lens specification
Image-side numerical aperture describes the angular cone reaching a specified image point. We can describe the same cone using an image-side working f-number: a smaller working f-number means a wider cone and a larger image-side NA. In air and for paraxial rays,
Here is the exit-pupil diameter and is the positive axial distance between the exit pupil and the image. The ratio uses the exit pupil’s apparent size and location, with ray extensions when needed. This describes the actual imaging configuration, whether the object is nearby or at infinity. The reciprocal relation is a small-angle approximation in air; in a non-air image space, retain the refractive index explicitly when calculating NA.
Read the working f-number from Figure 1.10. Follow the image-side marginal rays from the exit-pupil edges to the on-axis image point. The axial separation of that pupil plane and the image plane is ; their diameter-to-distance ratio determines the cone. It is the exit pupil, rather than the physical stop buried between the lenses, that supplies this image-side geometry. For the model underlying the schematic, , , and , so
Nominal f-number is a lens specification
Commercial photographic lenses commonly specify aperture using a nominal f-number, while microscope objectives commonly specify object-side numerical aperture (often called objective NA). The nominal f-number compares a focusing system’s effective focal length with its entrance-pupil diameter . For a system in air with positive effective focal length,
This ratio is dimensionless. A nominal f-number of 8 is written ; at a fixed focal length, a smaller f-number means a larger entrance pupil. It characterizes the lens system at a given aperture setting. Working f-number describes the actual image cone at the chosen object and image distances.
A distant scene makes the nominal ratio especially useful: the entrance pupil determines the width of the accepted parallel bundle, while effective focal length determines the image scale for a given scene angle. Nominal f-number therefore compares collecting aperture with that image scale.
What is effective focal length?
The effective focal length describes the focusing power of the entire lens system, including optics on both sides of the aperture stop. We define it using Chapter 1.2’s ray-transfer matrix, with the ray vector and air on both sides:
To see its meaning, send in a ray parallel to the optical axis, so . The matrix gives
For , a ray entering above the axis acquires a downward tilt: the system has positive focusing power. If , an incoming parallel bundle remains parallel and there is no finite effective focal length; such a system is called afocal. For one ideal thin lens, , so . Effective focal length is not the object-to-entrance-pupil distance.
At infinity focus, nominal and working f-numbers agree
For a centered focusing system in air with the object at infinity, the on-axis input bundle is parallel. Its marginal ray enters at height ; this is the same height at any plane before the optics because the incident ray is parallel to the axis. The magnitude of its output angle is the image-side cone half-angle. Thus, in the paraxial approximation,
Under these assumptions, nominal and working f-numbers agree: . This result also holds for compound lens systems. From the image side, the same cone gives , where is the exit-pupil diameter and is its positive axial distance to the image. The optics after the stop form its apparent image—the exit pupil. The exit pupil’s diameter and distance from the image describe the cone formed by the actual image-side marginal rays.
Figure 1.13 illustrates the geometry with 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 1.13. 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 .
At finite distances, use the actual image cone
When the object is at a finite distance, the incoming rays are no longer parallel, and nominal f-number alone does not generally determine the image-side NA. The two f-numbers do not generally agree.
Return to Figure 1.11. Its single thin lens has and , giving nominal . The object at is imaged at . With no optics after the iris, the exit pupil is the 4 mm iris at . Thus,
This configuration is nominally , but its actual image cone corresponds to a working f-number of approximately 35. Substituting nominal into would give 0.05, which is not the image-side NA here: the actual image distance and exit-pupil geometry matter.
The complete 4-f relay is a revealing exception
A relay system images one plane onto another. For Chapter 1.3’s two-lens relay, multiplying the matrices for two thin lenses separated by gives
At Chapter 1.3’s baseline spacing, , the two-lens system has : it is afocal, meaning a parallel input bundle leaves as a parallel output bundle. It has no finite effective focal length, even though it can form an image with finite object and image distances.
The complete relay therefore has no useful finite nominal f-number. Formally, an infinite effective focal length divided by a finite entrance-pupil diameter gives an infinite value. This does not mean that the relay’s image-side cone is zero or that it collects no light; the nominal definition is unsuitable for describing this relay with finite object and image distances. Use the object- and image-side NA, or the image-side working f-number.
For example, use from Figure 1.5 in Chapter 1.3, and assume a clear diameter for lens 2. This diameter is an additional assumption for this example. Suppose lens 2 is the aperture stop and there are no optics after it. Its exit pupil is then at lens 2, and the baseline image lies behind it, so
Thus this afocal relay has an image-side working f-number of approximately 6. This assumes the preceding optics transmit the entire cone admitted by lens 2.
Worked examples
Guided
Connect f-number and numerical aperture
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. What happens to both quantities if the entrance-pupil diameter doubles while the focal length stays fixed?
Hint / setup
Use .
For the stated air and infinity-focus conditions, .
Keep both lengths in millimetres, and check which quantity grows when the pupil widens.
Full solution
The camera operates at . At infinity focus in air, the working and nominal f-numbers agree in the paraxial approximation: . Both f-number and numerical aperture are dimensionless. Doubling the entrance-pupil diameter to 6 mm gives
The wider pupil halves the f-number and approximately doubles the image-side NA: it admits a wider cone at the image. The shortcut also applies to compound focusing systems in air at infinity focus, but finite object distances require the actual image-side cone.
Scaffolded
Find the stop and use its virtual entrance pupil
An on-axis object point in air lies at . A thin lens at has and a 12 mm clear diameter. A 4 mm iris lies at ; these are the only openings. Identify the aperture stop, locate its entrance pupil and find the pupil diameter. Then calculate the paraxial object-side NA and check it against a direct ray trace. Finally, find the image position, image-side NA, and working f-number, and compare the working and nominal f-numbers.
Hint / setup
Trace a ray from with variable slope . Compare the slope bounds imposed by the lens radius and iris radius.
Once the stop is known, image it through the optics before it. For the reverse viewing construction, use and .
Use , then measure the distance from the object point to that apparent pupil.
Find the image coordinate using the thin-lens equation. With no optics after the iris, the iris itself is the exit pupil. Use and compare it with .
Full solution
Identify the stop. At the lens and just after it,
At the iris,
The two clear radii impose
The iris permits the smaller slope range, so it is the aperture stop. At its limiting angle, the ray height at the lens is only , safely inside the lens’s 6 mm radius.
Locate the entrance pupil. Viewed backward through the lens, the iris is an object at . Thus,
Therefore , or . The negative image coordinate in the reverse viewing construction makes this pupil virtual. Its diameter is
Recover the accepted cone. The object-to-pupil distance is , so
This agrees with the limiting initial angle from the direct trace in air. The virtual pupil describes the same accepted cone; it does not add a physical opening at .
Compare nominal and working f-numbers. The thin-lens equation gives
The exit pupil is the iris itself: and . Thus,
Meanwhile, the nominal f-number is
The system is nominally , but its working f-number is approximately 37.5 at these finite conjugates. Using would overestimate the actual image-side NA by a factor of three. The working f-number follows the actual exit-pupil-to-image cone.
Independent
Analyze an infinity-corrected, telecentric microscope
In air, an idealized objective at has . The specimen is at , and a 4 mm aperture stop is at the objective’s back focal plane, . A tube lens with is at , with the sensor at . Assume paraxial rays and that all other apertures transmit the bundles.
Find the magnification. If the tube lens moves to , where should the sensor move, and does the magnification change?
Return to the original lens and sensor positions. If the specimen shifts slightly along the axis, what happens to the image scale and focus on the fixed sensor? Identify which behavior follows from infinity correction and which from object-space telecentricity.
Calculate the paraxial object-side NA for the original specimen plane.
Hint / setup
Separate the role of the specimen’s focal-plane position from the role of the stop’s focal-plane position. For the NA calculation, trace a ray from the on-axis specimen point to the stop edge: propagation from the objective’s front focal plane to its back focal plane gives .
Full solution
Magnification and tube-lens motion. The specimen is at the objective’s front focal plane, so each point produces a parallel bundle after the objective. The tube lens focuses it into its back focal plane:
When the tube lens moves to , the sensor must move to . The ideal focused magnification stays . This spacing flexibility follows from infinity correction, subject to the stated nonlimiting-aperture assumption.
Specimen motion with a fixed sensor. The stop is at the objective’s back focal plane, so the entrance pupil is at infinity and object-side chief rays are parallel to the axis. An axial specimen shift therefore leaves the chief-ray intercepts on the fixed sensor unchanged in this ideal model: the image scale stays stable, suppressing perspective size changes. Marginal rays generally no longer meet there, so defocus blur appears. This scale stability follows from object-space telecentricity; it does not guarantee sharp focus or constant scale under arbitrary sensor movement. Infinity correction alone does not imply telecentricity.
Object-side NA. For the on-axis specimen point, the stop’s 2 mm radius sets
An entrance pupil at infinity does not mean a zero cone angle. Here the finite stop radius and objective focal length set the accepted marginal-ray angle, while the stop center sets the chief-ray direction.
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 ?
Have you distinguished nominal from image-side working ? In air and for paraxial rays, use ; substitute nominal only when the imaging conditions justify their equality, as at infinity focus.
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?
