Optics 033 · Imaging, instruments, and visual systems

Telecentric Metrology Camera

Can a camera measure an object's size correctly when the image is out of focus? Yes, with a telecentric lens. A stop at the shared focal plane makes the central ray of every light bundle run parallel to the axis, so defocus widens each spot without moving its center, and the measured size stays right. If a lens rim cuts off part of a bundle, the spot becomes lopsided, its center shifts, and the measurement goes wrong.

Compare ordinary, object-space, image-space and double-telecentric imaging on two live optical benches. Move the gauge and the sensor, shift the aperture stop and clip finite pupils; computed sensor images at equal scale show 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 $\text{valid model regime}$

Model checked against analytic results

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 shows an ordinary camera and cyan shows the selected system; both image the same moving 20 mm gauge. Compare the two sensor images at the same physical scale. A stop offset of zero puts the stop at the correct focal plane. Object shift is positive toward the objective; sensor shift is positive away from the last lens. Clear radius changes the aperture itself, not the curvature of the glass.

The model uses ideal paraxial lenses and integrates geometrically over the pupil. The estimate uses centroids of edge points and a fixed calibration, not a threshold fit to the ring. Diffraction, noise, aberrations and the accuracy of commercial metrology are not included.

Physics tutorial

Audit the physics of Telecentric Metrology Camera

BackgroundA telecentric camera does not keep every depth in sharp focus. Its aperture stop picks out the chief ray, the ray through the center of the stop. Carrying the ray height and slope from the object to the sensor gives $m_c=A-B\frac{A_s}{B_s}$ . The apparent size is set by where the chief ray hits the sensor. The finite bundle of rays around it can still spread into a blur spot.

Why it mattersCan a camera measure an object's size correctly when the image is out of focus?

Start with the essentials

Focus question: Can a camera measure an object's size correctly when the image is out of focus?
One-sentence intuition: Put the stop at the focal plane shared by two lens groups, and the chief rays on both sides run parallel to the optical axis. The ideal relay obeys $m_{\mathrm{double}}=-\frac{f_2}{f_1},\qquad b_g=2a_s\left|\frac{B}{B_s}\right|$ . A constant scale is not the same as staying in focus. If an aperture clips the beam, even the centroid measurement can fail.

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 center of the stop, so its height at the stop is zero. Each ray drawn is solved for a chosen point inside the circular stop, and its slope changes at each ideal lens plane. The finite clear apertures are checked in the same order.

Chief-ray scale and 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|

When the pupil is symmetric and nothing clips it, the blur spot stays centered on the chief ray. The spot can grow without moving that center. The diameter shown comes from the centroids of two separate edge points and a fixed nominal calibration, not from thresholding 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, moving the object leaves the chief-ray scale on the sensor unchanged, as long as the sensor distance v stays fixed. With the stop at the front focus, moving the sensor leaves the scale unchanged, as long as the object distance s stays fixed. Each one-sided design still responds to motion on its 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 sits where the rear focus of the first group meets the front focus of the second. When the object moves, the best image plane moves by the longitudinal magnification times that displacement. Moving the sensor to the new plane restores geometric focus; leaving the sensor in place keeps the scale but blurs the image. Diffraction and noise are not modeled, so the Lab claims neither practical measurement accuracy nor unlimited depth of field.

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|}

Once the pupil is clipped, its mean position need not sit at the center. Shrinking the lens clear radius can then bias the measured diameter, even with the telecentric stop in the right place. Pink rays end at the opaque rim. If no light from an edge point gets through, the measurement is reported as invalid instead of showing a 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 fixes the chief-ray geometry, but it does not control the whole point-spread function. Defocus widens the spot, diffraction and noise limit how well a real edge can be located, and clipping breaks the symmetry of the pupil. The ideal centroid result holds only while the needed edge rays reach the sensor.

Run the experiment

01
Scene 1: Object-space depth sweep
Start in Object-space. Keep sensor shift and stop offset at zero and sweep the object shift. Compare both rings with their identical dashed calibration circles, then move the sensor.
What to observe: With the object shifted by 40 mm, the ordinary camera measures a 20 mm gauge as 24 mm. An object-side telecentric camera with its stop in the right place still reads 20 mm, although the image blurs.
02
Scene 2: Image-space sensor sweep
Open Image-space and sweep the sensor shift. The selected ring keeps its scale, while the ring from the ordinary camera changes size. Then move the object to see that the scale still depends on the object side.
What to observe: Away from focus, a larger stop gives a larger geometric blur. Stopping down does not change the nominal chief-ray scale. Diffraction is left out, so this does not show that resolution can improve without limit.
03
Scene 3: Double telecentric relay
Open Double telecentric. Move both planes, then offset the stop and shrink the lens clear radius. Check the chief-ray angles, the geometric blur and the part of the edge pupil that gets through, one at a time.
What to observe: The lens clear-radius slider changes the actual clear disk and the opening in the opaque mount. It sets an aperture radius, not the radius of curvature of the drawn glass.