Aperture stop, pupils, chief and marginal rays

Every optical system has one opening that limits the size of the cone of rays it accepts from an on-axis object point: the aperture stop. The entrance pupil is the image of the aperture stop formed by the elements in front of it, as seen from the object; the exit pupil is its image formed by the elements behind it, as seen from the image. The marginal ray goes from the on-axis object point to the edge of the aperture stop, so it sets the cone angle and with it the f-number and numerical aperture. The chief ray goes from an off-axis object point through the center of the aperture stop, and therefore through the centers of both pupils; it sets the image height and the field angle. A camera lens's iris, the eye's pupil (2–8 mm across) and the back focal plane of a microscope objective are all aperture stops.

Locating the stop and the pupils

The aperture stop is found by imaging every candidate opening into object space and choosing the one that subtends the smallest angle from the on-axis object point; that image is the entrance pupil. The other openings set the field: the one whose image subtends the smallest angle from the center of the entrance pupil is the field stop, which limits the field of view, and openings that clip off-axis bundles partly cause vignetting. The stop is a separate concept from clear aperture, which is the usable diameter of each individual element.

A worked example shows why the pupil, and not the physical stop, sets the f-number. A 10 mm iris placed 20 mm behind a thin lens of 50 mm focal length is seen from the front through the lens. The thin-lens equation gives a virtual image 33.3 mm behind the lens, magnified 1.67 times, so the entrance pupil is 16.7 mm in diameter and the lens works at

$$N = \frac{f}{D_\text{EP}} = \frac{50\ \text{mm}}{16.7\ \text{mm}} = 3.0,$$

where dividing by the iris diameter would give f/5. The marginal-ray numerical aperture in image space for an object at infinity is about $1/(2N) = 0.17$.

Where the stop position matters

Moving the stop along the axis, with its diameter adjusted to keep the entrance pupil the same size, leaves the on-axis cone unchanged but changes which part of each lens the off-axis bundles pass through. Distortion, coma, astigmatism and lateral color all depend on stop position, which is one reason symmetric designs place the stop at the center. Wavefront error is defined over the exit pupil, and Zernike coefficients and Strehl ratios refer to it.

Placing the stop at the front focal point of the elements behind it puts the exit pupil at infinity: the system is image-space telecentric, and every chief ray reaches the image parallel to the axis. With the stop at the back focal point of the elements in front, the entrance pupil is at infinity and the system is object-space telecentric. An infinity-corrected microscope objective has its stop at or near its back focal plane and is close to object-space telecentric. For an objective that satisfies the Abbe sine condition, the pupil diameter is $2 f\,\mathrm{NA}$: a 60× objective designed for a 200 mm tube lens has $f = 3.33$ mm, and at NA 1.4 its back aperture is 9.3 mm across.

The exit pupil of a visual instrument is where the observer's eye belongs. A 10×50 binocular has a 50 mm entrance pupil and a 50/10 = 5 mm exit pupil, a good match to the dark-adapted eye; the distance from the last lens to the exit pupil is the eye relief.

Pupils in laser systems

Scanning and wavefront-sensing systems are designed around pupil conjugates. A galvo scanner mirror is placed at a pupil, or relayed to one, so that tilting it moves the focused spot without moving the beam off the objective's back aperture; two-axis scanners relay one mirror onto the other for the same reason (beam steering). A wavefront sensor is placed at an image of the pupil so that each part of its lenslet array samples a fixed part of the aperture.

Pitfalls

• Using the iris or front-element diameter in place of the entrance pupil when computing the f-number.
• Treating principal planes and pupils as the same surfaces. Principal planes govern the image position and magnification; pupils govern ray bundles, and they coincide only when the stop is at a thin lens.
• Overfilling or underfilling a pupil. A laser beam narrower than the objective's back aperture reduces the effective NA and enlarges the focal spot; a wider one is clipped and loses power.

Common questions

What is the difference between the aperture stop and the field stop?

The aperture stop limits how much light from each point gets through, and so sets the brightness and resolution. The field stop limits which points are imaged, and so sets the field of view. In a camera the iris is the aperture stop and the sensor edge is the field stop.

What are the chief ray and the marginal ray used for?

They are the two rays traced in a first-order layout. The marginal ray gives the pupil and image positions and the cone angle; the chief ray gives the field angle and the image height. Together they define the Lagrange invariant, which is proportional to the étendue of the system in one dimension.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017). M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).