Optics 024 · Imaging, instruments, and visual systems

Parfocal Zoom Challenge

A four-group mechanical zoom track exposes the curved variator-compensator locus required to hold a fixed sensor in focus while effective focal length changes by more than threefold. Wide, trajectory, and tele scenes share one signed ABCD train, field-ray fans, spot clouds, collision checks, and live focus residuals.

Effective focal length and half field

f_{\mathrm e},\ \theta_{1/2}

$0.50$

Fixed-sensor blur and focus coefficient

r_{\mathrm{blur}},\ A_s

$50\%$

Compensator tracking and invariant

\Delta x_c,\ \det\mathbf M-1

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Parfocal Zoom Challenge

BackgroundA true parfocal zoom changes system power while a fixed sensor remains conjugate to the same distant scene. For a collimated input bundle, the fixed-plane condition is $A\!\left(\mathbf M_{\infty\to\mathrm s}(x_v,x_c)\right)=0$ . The variator and compensator therefore follow different mechanical trajectories; adding their positions into one separation variable cannot represent the mechanism.

Why it mattersCan moving lens groups change focal length while the image plane stays fixed?

Start with the essentials

Focus question: Can moving lens groups change focal length while the image plane stays fixed?
One-sentence intuition: The green station is the compensator root solved for the current variator coordinate. Departing from it makes the entrance pupil paint a sensor blur radius $r_{\mathrm{blur}}=\lvert A_s\rvert a,\qquad x_c=x_c^{\star}(x_v)$ . This is why a varifocal focal-length slider and a parfocal multi-group trajectory are physically different products.

Core mathematical model

Four-group sensor matrix

\mathbf M_{\mathrm s}=\mathbf P(z_s-x_4)\mathbf L_4\mathbf P(x_4-x_c)\mathbf L_c\mathbf P(x_c-x_v)\mathbf L_v\mathbf P(x_v-x_1)\mathbf L_1

Two groups stay fixed and two move independently. Every displayed ray visits these stations in the same order as this product.

Focus and focal-length identity

A_s=0,\quad \det\mathbf M_s=1\ \Longrightarrow\ B_s=-\frac{1}{C_s}=f_{\mathrm e}

At the parfocal root, sensor height no longer depends on entrance-pupil height. The remaining angle coefficient maps field angle to image height and equals effective focal length.

Paraxial field of view

\theta_{1/2}=\arctan\!\left(\frac{h_s}{\lvert f_{\mathrm e}\rvert}\right)

The sensor half-height is fixed. Increasing effective focal length therefore narrows the accepted field while enlarging the angular image scale.

Common difficulties

Calling a focal-length slider parfocal zoom

Typical misconceptionChanging one effective focal-length number is enough; the sensor may be redrawn wherever the new focus appears.

Better mental modelHere the sensor never moves. The variator changes effective focal length, the compensator must find a separate root, and the three entrance heights expose any remaining focus error as a real spot spread.

Run the experiment

01
Scene 1: Wide-angle endpoint
Begin at the wide endpoint. Move the compensator away from the green station and watch each colored field bundle split into three sensor spots while the sensor itself remains fixed.
What to observe: The compensator locus is nearly but not exactly a constant offset from the variator. The small curvature is the mechanical information a simple separation slider discards.
02
Scene 2: Parfocal trajectory
Enter the trajectory challenge. Drag the variator, then use the independent compensator control to recover the residual minimum and the smallest blur radius.
What to observe: At the root, all entrance-pupil heights for one field angle meet at one sensor height. Different field angles remain separated because they encode different object directions, not defocus.
03
Scene 3: Telephoto endpoint
Compare the wide and tele endpoints. Verify that the lens groups occupy different stations, the fixed sensor remains sharp, and the displayed field angle narrows as effective focal length grows.
What to observe: The tele endpoint uses a larger effective focal length and a smaller half field. Focus preservation and field-of-view change occur simultaneously, but they are controlled by different matrix entries.