Optics 035 · Imaging, instruments, and visual systems

Pupils, Stops & Vignetting

What do the different openings inside a lens system each limit? The aperture stop is the opening that limits the cone of light from a point on the axis. Seen through the lenses in front of it, it appears as the entrance pupil; seen through the lenses behind it, as the exit pupil. The field stop limits how much of the scene reaches the image. Any other opening that clips slanted rays darkens the corners, which is vignetting.

Image a movable physical aperture stop through two finite lenses and watch the real and virtual pupils pass through infinity. Compare the exact pupil regions that accept light with unnormalized geometric vignetting and field cropping at the image plane.

Entrance / exit pupil coordinates

x_E,x_X

$0.50$

Entrance / exit pupil diameters

D_E,D_X

$50\%$

Accepted fraction / area throughput

T_a,G_a

$0.00\pi$

Model regime $\text{valid model regime}$

The model is a paraxial focal relay, and the overlaps of circular footprints are integrated exactly. Image brightness is scaled to one fixed pupil-area reference, not calibrated radiometry. Diffraction, cosine-fourth falloff, aberrations and stray-light paths are not included.

Physics tutorial

Audit the physics of Pupils, Stops & Vignetting

BackgroundThe aperture stop sets how wide a cone of rays from an on-axis point can pass. The entrance and exit pupils are images of that stop formed by the optics in front of it and behind it; they are not extra hardware. The geometric pupil transmission is $T_a=\frac{A_{\mathrm{accepted}}}{\pi a^2}$ . A bundle from an off-axis point can fit through the iris and still be clipped by the edge of another lens.

Why it mattersWhat do the different openings inside a lens system each limit?

Start with the essentials

Focus question: What do the different openings inside a lens system each limit?
One-sentence intuition: In this focal relay, axial positions are measured from the front lens and the iris position is signed. The pupil centers then sit at $x_E=-\frac{f_1d}{d-f_1},\qquad x_X=L+\frac{f_2(L-d)}{f_1-d}$ . With the iris at the shared focal plane, both pupils move off to infinity while the real iris keeps its finite size. A virtual pupil cannot emit light, and it cannot block light the way a stop does.

Core mathematical model

Ray height and slope through free space and a thin lens

y_2=y_1+\Delta x\,u_1,\qquad u_2=u_1-\frac{y_2}{f}

The two transverse coordinates propagate independently. At each real opening the ray is checked first, before the next lens bends it; a ray that is blocked stops at that plane.

Pupil size from the magnification of the stop

m_E=-\frac{f_1}{d-f_1},\quad m_X=-\frac{f_2}{f_1-d},\quad D_E=2a|m_E|,\quad D_X=2a|m_X|

The sign of the magnification tells you whether the image is upright or inverted; the displayed diameter is never negative. Whether a pupil is real or virtual depends on the direction light travels when it images the stop, not on whether the pupil's position coordinate is positive.

The pupil area that every opening lets through

A_{\mathrm{accepted}}=\iint_{\mathcal P_a\cap\mathcal P_1\cap\mathcal P_2\cap\mathcal P_F}\mathrm{d}A,\quad G_a=\frac{A_{\mathrm{accepted}}}{\pi a_{\mathrm{ref}}^2}

Each real circular opening is mapped onto the iris plane, and the area common to all of them is computed exactly from circular arcs. Throughput is referred to a fixed radius of fourteen millimetres. Closing the iris can raise the accepted fraction while the total geometric throughput falls.

Field stop versus aperture stop

m=-\frac{f_2}{f_1}=-0.8,\qquad |m|h\le r_F

At the image plane, all rays from one object point arrive at the same spot. A centered field stop placed there cuts off the field sharply and never shrinks only part of a point's pupil. Move the stop in front of the image and it clips bundles gradually instead. The image card is turned upright for easier viewing.

Common difficulties

A pupil is a physical hole, and every dark edge is the same effect

Typical misconceptionA translucent disc at the pupil image can block rays, and one radial darkening filter can stand in for every kind of vignetting.

Better mental modelOnly real hardware clips rays. Mechanical vignetting changes the part of the pupil that passes light as the field point moves. A field stop at the image plane cuts off the field. Natural radiometric roll-off and diffraction are separate effects, and this model leaves them out. It does not model stray light or ghost images either.

Run the experiment

01
Scene 1: Pupil conjugates
Sweep the aperture-stop position through the common focal plane and watch the signed pupil positions and sizes. When a pupil goes to infinity, the readout says so instead of drawing a circle clamped at a finite size.
What to observe: The marker is the object point you selected. Moving it updates the ray bundle and the pupil mask, while the image of the full sensor field stays the same.
02
Scene 2: Image-plane field stop
In the field-stop scene, compare object heights just inside and just outside the cutoff. Then move the stop away from the image plane: the sharp cutoff of the field turns into partial clipping of the pupil.
What to observe: For an on-axis point, the iris is the aperture stop at every allowed combination of settings. Within this range, the field stop near the image never takes over as the stop that limits the on-axis cone.
03
Scene 3: Mechanical vignetting
In the mechanical-vignetting scene, close the iris and compare the accepted fraction with the throughput referred to a fixed area. Neither card rescales itself to its own new maximum.
What to observe: The exact geometric area does not predict diffraction, calibrated exposure, aberrations, coating losses or the natural roll-off of illumination.