Optics 033 · Imaging, instruments, and visual systems

Telecentric Metrology Camera

Compare ordinary, object-space, image-space and double-telecentric imaging on two live optical benches. Move the gauge and sensor, displace a real aperture stop and clip finite pupils while equal-scale computed sensor images expose perspective, blur and centroid bias.

Edge-centroid estimate / chief scale

\hat d,m_c

$0.50$

Object-side / image-side chief angle

\alpha_o,\alpha_i

$50\%$

Geometric blur / accepted edge pupil

b_g,A_{\mathrm{edge}}/A

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Telecentric Metrology Camera

Object depth offset

\Delta z_o

40\,\mathrm{mm}

Metrology sensor shift

\Delta z_i

0\,\mathrm{mm}

Stop offset

\Delta s_a

0\,\mathrm{mm}

Stop radius

a_s

3\,\mathrm{mm}

Metrology clear radius

a_L

26\,\mathrm{mm}

Amber is an ordinary camera; cyan is the selected system, imaging the same moving 20 mm gauge. Compare both sensor images at the identical physical scale. Zero stop offset gives the correct focal-plane placement. Object shift is positive toward the objective; sensor shift is positive away from the last lens. Clear radius changes the real aperture, not glass curvature.

Paraxial ideal lenses and geometric pupil integration. The estimate uses edge-point centroids and fixed calibration, not a ring threshold fit. Diffraction, noise, aberrations and commercial metrology accuracy are outside the model.

Physics tutorial

Audit the physics of Telecentric Metrology Camera

BackgroundA telecentric camera does not freeze a photograph at every depth. Its aperture stop selects the chief ray that passes through the stop center. Propagating height and slope from object to sensor gives $m_c=A-B\frac{A_s}{B_s}$ . The apparent scale follows the chief-ray intercept, while a finite bundle can still spread into a blur spot.

Why it mattersCan a blurred image still keep the right size—and when does clipping break that promise?

Start with the essentials

Focus question: Can a blurred image still keep the right size—and when does clipping break that promise?
One-sentence intuition: A common focal-plane stop between two lens groups makes both chief-ray directions parallel to the optical axis. The ideal relay obeys $m_{\mathrm{double}}=-\frac{f_2}{f_1},\qquad b_g=2a_s\left|\frac{B}{B_s}\right|$ . Constant scale and zero defocus are different properties; aperture clipping can invalidate even the centroid measurement.

Core mathematical model

Transfer to a real aperture stop

\begin{pmatrix}y_s\\u_s\end{pmatrix}=\begin{pmatrix}A_s&B_s\\C_s&D_s\end{pmatrix}\begin{pmatrix}y_o\\u_o\end{pmatrix},\qquad u_{o,c}=-\frac{A_s}{B_s}y_o

The chief ray passes through the stop center, so its stop height is zero. Every displayed ray is solved for a chosen point inside the circular stop and receives a slope change at each ideal lens plane. The finite clear apertures are checked in the same order.

Chief scale versus geometric blur

y_i=m_cy_o+\frac{B}{B_s}y_s,\qquad m_c=A-B\frac{A_s}{B_s},\qquad b_g=2a_s\left|\frac{B}{B_s}\right|

A symmetric unvignetted pupil centers the spot on the chief ray. Its diameter can grow without moving that center. The displayed diameter is inferred from two independent edge-point centroids using a fixed nominal calibration, not from fitting a threshold to a blurred ring.

Object-side and image-side are not interchangeable

m_{\mathrm{obj}}=1-\frac{v}{f},\qquad m_{\mathrm{img}}=-\frac{f}{s-f}

With the stop at the rear focus, object motion leaves the sensor chief-ray scale unchanged at fixed sensor distance v. With the stop at the front focus, sensor motion leaves it unchanged at fixed object distance s. Each single-sided system still responds to motion on the other side.

Double telecentric, but still finite depth of field

d_{12}=f_1+f_2,\qquad m_c=-\frac{f_2}{f_1},\qquad \Delta z_{i,\mathrm{focus}}=m_c^2\Delta z_o

The stop belongs at the shared rear/front focus. The best image plane moves by longitudinal magnification times the object displacement. Moving the sensor to that plane restores geometric focus; holding it fixed preserves scale but produces blur. There is no diffraction or noise model here, so no practical measurement accuracy or unlimited depth-of-field claim is made.

Vignetting shifts a spot centroid

\bar y_i=m_cy_o+\frac{B}{B_s}\langle y_s\rangle_{\mathrm{accepted}},\qquad\hat d=\frac{|\bar y_{i,+}-\bar y_{i,-}|}{|m_0|}

A clipped pupil need not have zero mean. Shrinking the lens clear radius can therefore bias the inferred diameter even with a correctly positioned telecentric stop. Pink rays end at the opaque rim; a missing edge pupil is reported as invalid rather than a fabricated zero diameter.

Common difficulties

Telecentric means sharp and measurable at every depth

Typical misconceptionIf the displayed magnification is constant, blur and aperture size cannot affect measurement.

Better mental modelTelecentricity controls chief-ray geometry, not the full point-spread function. Defocus broadens the spot; diffraction and noise limit real edge estimation; clipping breaks pupil symmetry. The ideal centroid result is only valid while the necessary edge rays are delivered.

Run the experiment

01
Scene 1: Object-space depth sweep
Start in Object-space. Sweep object shift while leaving sensor shift and stop offset at zero. Compare the two rings against the identical dashed calibration circles; then move the sensor.
What to observe: At an object shift of 40 mm, the ordinary camera estimates 24 mm for a 20 mm gauge. A correctly stopped object-side camera retains 20 mm while becoming blurred.
02
Scene 2: Image-space sensor sweep
Open Image-space. Sweep sensor shift: the selected ring keeps its scale, whereas the ordinary camera changes scale. Now move the object to reveal the remaining object-side dependence.
What to observe: A larger stop produces a larger geometric blur away from focus. Stopping down does not change nominal chief scale; omitted diffraction prevents claiming arbitrarily good resolution.
03
Scene 3: Double telecentric relay
Open Double telecentric. Move both planes, then offset the stop and shrink the lens clear radius. Inspect the chief angles, geometric blur and accepted edge pupil separately.
What to observe: The lens clear-radius slider changes the actual transparent disk and opaque mount opening. It is an aperture radius, not the radius of curvature of the illustrative glass.