Optics 023 · Imaging, instruments, and visual systems

Compound Optics Workbench

A modular matrix-optics laboratory builds a Keplerian afocal telescope, a Galilean beam expander, and a physically folded 4f image relay from signed thin lenses, movable propagation distances, an explicit plane mirror, piecewise rays, and an independent ABCD solve.

Active design condition

C\ \text{or}\ B

$0.50$

Transverse transfer and carriage mismatch

A,\ \Gamma,\ \Delta d

$50\%$

Lossless matrix invariant

\det\mathbf M-1

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Compound Optics Workbench

BackgroundA compound optical train is not one effective-lens icon. Each free-space interval and each signed thin-lens power acts in travel order, so the complete ray map is $\begin{pmatrix}y_2\\\theta_2\end{pmatrix}=\mathbf M_N\cdots\mathbf M_2\mathbf M_1\begin{pmatrix}y_1\\\theta_1\end{pmatrix}$ . The colored piecewise rays receive the corresponding kick at every rendered lens, while the matrix is multiplied independently from the same prescription.

Why it mattersHow can one matrix predict imaging, magnification, and afocal behavior in a whole optical train?

Start with the essentials

Focus question: How can one matrix predict imaging, magnification, and afocal behavior in a whole optical train?
One-sentence intuition: The same four matrix entries answer different questions only after the reference planes are declared: $C=0\ \text{(afocal)},\qquad B=0\ \text{(conjugate planes)}$ . Afocality constrains output angle from parallel input; imaging constrains output height from launch angle. Confusing the two conditions creates a system that looks aligned but does not perform its task.

Core mathematical model

Elementary ray maps

\mathbf P(d)=\begin{pmatrix}1&d\\0&1\end{pmatrix},\qquad \mathbf L(f)=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix}

Propagation changes height through the incoming slope. An ideal lens changes slope at fixed height. Reversing their multiplication order changes the apparatus.

Separated-pair power

\Phi=-C=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2},\qquad f_{\mathrm e}=\frac{1}{\Phi}

Separation contributes optical power. The equivalent focal length diverges at the afocal setting instead of remaining equal to either component focal length.

Lossless first-order invariant

\det\mathbf M=AD-BC=1

Every displayed lossless propagation, thin lens, and unfolded plane-mirror fold has unit determinant. The residual is therefore an implementation check, not a fitted visual metric.

Common difficulties

Calling every parallel-looking train afocal

Typical misconceptionIf one central ray leaves nearly horizontal, the whole system is an afocal telescope or beam expander.

Better mental modelThe apparatus launches five independent parallel rays. Only a vanishing system-power coefficient makes every output angle independent of input height; one axial ray cannot test this condition.

Run the experiment

01
Scene 1: Afocal telescope
Start with the Keplerian scene. Drag the second carriage until the green and rose focal markers coincide, then verify that the five output slopes become parallel together.
What to observe: Moving either telescope group away from the focal-sum condition produces an output fan whose slope changes sign across the exact afocal station.
02
Scene 2: Beam expander
Switch to the Galilean expander. Match the virtual focus of the negative group to the front focus of the positive group and compare the measured height transfer with the signed focal-length ratio.
What to observe: The Galilean pair has no real internal crossing. Its negative first group creates a virtual focus that the positive group recollimates into a larger beam.
03
Scene 3: Relay imaging chain
Open the folded relay. Tune the optical separation to the four-focal-length path condition and confirm that all launch angles meet at one inverted point on the fixed image plane.
What to observe: The relay mirror changes laboratory direction but contributes no first-order power in the unfolded coordinate. Image failure comes from the lens spacing, not from drawing the bench around a corner.