Optics 035 · Imaging, instruments, and visual systems

Pupils, Stops & Vignetting

Image a movable physical aperture stop through two finite lenses, watch real and virtual pupils cross infinity, and compare exact accepted pupil regions with unnormalized geometric vignetting and image-plane field cropping.

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 $\text{valid model regime}$

Paraxial focal relay and exactly integrated circular footprint intersections. Image brightness uses one fixed pupil-area reference, not calibrated radiometry. Diffraction, cosine-fourth roll-off, aberrations and stray-light paths are excluded.

Physics tutorial

Audit the physics of Pupils, Stops & Vignetting

BackgroundThe aperture stop limits the on-axis ray cone. Entrance and exit pupils are its images through the front and rear optics, not extra pieces of hardware. Geometric pupil transmission is $T_a=\frac{A_{\mathrm{accepted}}}{\pi a^2}$ . An off-axis bundle can be clipped by another lens edge even though its rays fit through the iris.

Why it mattersWhich stop limits aperture, which limits field, and where do their pupils appear?

Start with the essentials

Focus question: Which stop limits aperture, which limits field, and where do their pupils appear?
One-sentence intuition: For this focal relay, using the front lens as axial zero and the iris position as the signed coordinate, the pupil centers are $x_E=-\frac{f_1d}{d-f_1},\qquad x_X=L+\frac{f_2(L-d)}{f_1-d}$ . At the common focal plane both pupils go to infinity; the physical iris stays finite. A virtual pupil cannot emit light or act as an opaque stop.

Core mathematical model

Propagation and thin-lens slope

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

Both transverse coordinates propagate independently. The ray is checked against each finite opening before it can receive the next lens kick; rejected paths terminate at that plane.

Aperture images and signed size

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 describes image orientation; the displayed diameter is nonnegative. Real versus virtual is determined by the propagation direction used to image the stop, not by whether its global coordinate is positive.

Geometric accepted area

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 to the iris plane. Their intersection is integrated by exact circular arcs. The common reference radius is fourteen millimetres; closing the iris can improve the accepted fraction while reducing total geometric throughput.

Field stop versus aperture stop

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

At the image plane every ray from one object point has the same position. A centered field stop there cuts the field sharply, without partially shrinking that point’s pupil. Moving the stop before the image can instead cause gradual bundle clipping. The image card is rotated upright for viewing.

Common difficulties

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

Typical misconceptionA translucent circle may block rays at the pupil image, and a generic radial darkening filter can stand in for all vignetting.

Better mental modelOnly real hardware clips rays. Mechanical vignetting changes the pupil region with field; an image-plane field stop cuts the field; natural radiometric roll-off and diffraction are separate phenomena, omitted here. No stray-light or ghost model is claimed.

Run the experiment

01
Scene 1: Pupil conjugates
Sweep the aperture-stop position through the common focal plane. Watch the signed pupil positions and sizes; infinity is reported instead of drawing a clamped circle.
What to observe: The marker is the actual selected object point. Changing it updates the ray bundle and pupil mask; the full sensor-field image remains fixed.
02
Scene 2: Image-plane field stop
In Field stop, compare object heights below and above its cutoff. Move the stop away from the image plane to turn sharp field rejection into partial pupil clipping.
What to observe: At zero field the iris is the aperture stop across every permitted parameter combination. The near-image field stop cannot secretly become the on-axis aperture limiter in this range.
03
Scene 3: Mechanical vignetting
In Mechanical vignetting, close the iris and compare accepted fraction with area-referenced throughput. Neither card normalizes itself to a new maximum.
What to observe: The exact geometric area does not predict diffraction, calibrated exposure, aberration, coating loss or natural illumination roll-off.