Understanding aperture stops and pupils is fundamental to lens design. This page breaks down the optical and geometric principles that govern how stops limit light and how pupils—formed as images of the stop—affect system performance. Whether you’re designing for imaging efficiency, field of view, or telecentricity, a clear grasp of these concepts is essential.
In an optical system, the aperture stop refers to the physical aperture that limits the bundle of rays going through the system. Figure 1 shows an exaggerated image of a Cooke Triplet lens with an aperture stop behind the 2nd component in the lens. The stop can be a lens diameter, iris or other mechanical aperture in the system. This is where the terms “f-stops” and “stopping down” come from in photography. They refer to the F-number of the lens — see our page on Numerical Aperture and F/# for more information.
In camera lenses, f-stops are often set to reduce or expand the diameter of the aperture stop. Each f-stop changes the amount of light by a factor of 2 from the previous f-stop. This is why most camera lenses seem to be labeled with peculiar F-numbers (F/1, F1.4, F/2, F/2.8, etc.). These numbers make a lot more sense and are very useful when you understand that moving up or down by an f-stop changes the amount of light, and therefore the exposure time, by a factor of 2.
Figure 1: Aperture Stop in a Cooke Triplet
Now that we have a good understanding of the aperture stop, what is a pupil? The simple definition is that a pupil is an image of the aperture stop made by the lenses before or after it. There are 2 common names for pupils in an optical system: The entrance pupil (EP) and the exit pupil (XP).
The entrance pupil (EP) is the image of the aperture stop formed by the lens elements in front of it—those to the left of the stop in Figures 1 and 3. Conversely, the exit pupil (XP) is the image of the stop formed by the lens elements behind it, shown to the right of the stop in the same figures.
When a real lens with an internal stop is viewed from the front or rear, the pupil can be seen as an illuminated disc that can appear near or far away depending on the lens elements between the stop and the viewer.
Figure 2: Exit Pupil of a Real Lens
The location of a pupil is determined by the angle of the chief ray. In Figure 3, the entrance pupil (EP) is located by tracing the chief ray—ignoring refraction—forward until it intersects the optical axis, shown as the magenta ray. The exit pupil (XP) is found similarly, by tracing the exiting chief ray backward to its intersection with the axis, shown as the center blue ray. Unlike the aperture stop, which is physically located within the lens, pupil positions depend entirely on the chief ray angle and can, in theory, lie anywhere—even at infinity.
Visit our page, Chief Ray Angle and Telecentricity for more information.
Not only will the pupil locations vary wildly between different lens types, but the pupil diameters will also vary. To determine the pupil diameters a similar process to finding the pupil locations can be followed. Instead of tracing the chief ray back through the lens elements, the marginal rays can be traced back to the pupil locations to determine the pupil diameter. This is shown in Figure 3 by tracing the top and bottom marginal rays exiting the lens to the XP location. The EP diameter is not shown in the image for simplicity. As can be seen in Figure 3, the EP and XP diameters can be larger, smaller or the same size as the aperture stop depending on the marginal ray angles entering or exiting the lens, respectively.
Figure 3: Entrance and exit pupil locations in a Cooke Triplet lens system. The entrance pupil (magenta) is the forward-projected image of the aperture stop through the front lens group. The exit pupil (blue) is the backward-projected image through the rear group. Chief and marginal rays (red) are shown for reference, illustrating how pupil positions are determined by ray geometry rather than physical surfaces.
Our page on Etendue introduces the optical invariant. Both etendue and the optical invariant are conserved quantities and their conservation is tied to the fundamental law of physics – the conservation of energy. Because the optical invariant is conserved, it is the same at any point in a lens (See Figure 4).
Figure 4: Optical Invariant in a Cooke Triplet
As can be seen in Figure 4, the invariant at the stop is:
The optical invariant holds true at any pupil location in the system, since the chief ray height is zero at those points. This explains why pupil size changes with lens geometry: if the chief ray angle at the exit pupil is smaller than at the stop—as shown in Figure 3—the exit pupil diameter must increase to conserve the invariant. This outcome aligns perfectly with the result you’d get by tracing rays backward.
If you know the angle at the stop, the stop size, and the chief ray angle at the image, then the exit pupil diameter and its position aren’t design choices—they’re set by physics!
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