Optics 028 · Imaging, instruments, and visual systems

Anamorphic & Cylindrical Optics

A three-dimensional anamorphic bench exposes the two orthogonal meridians of a parallel-axis cylindrical Kepler pair. Gaussian cross-section rings pass through the internal line waist, become an elliptical collimated pupil, and drive a reciprocal angular-grid squeeze plus a separately labeled geometric bokeh footprint.

Line waist or mapped aspect

z_w, 2w_w, S_p

$0.50$

Orthogonal extent or angular squeeze

\ell_{\mathrm{line}}, A_\theta

$50\%$

Matrix residual

C_x, \Delta_{\det}

$0.00\pi$

Model regime $\text{valid model regime}$

The powered axis uses ideal lossless paraxial thin lenses and a 532 nm fundamental Gaussian beam; the orthogonal axis is unpowered. Glass shape and glow are display encodings. Thick-surface prescription, anamorphic aberrations, vignetting, flare, coatings, polarization, sensor sampling, and diffraction PSF are outside this model.

Physics tutorial

Audit the physics of Anamorphic & Cylindrical Optics

BackgroundA cylindrical lens has optical power in one meridian and no power in its orthogonal meridian. This bench places two positive, parallel-axis cylindrical lenses at their afocal separation, giving the powered-axis transfer $\mathbf M_x=\mathbf L(f_2)\mathbf P(f_1+f_2)\mathbf L(f_1)=\begin{pmatrix}-f_2/f_1&f_1+f_2\\0&-f_1/f_2\end{pmatrix}$ . The zero lower-left element is the visible promise: a collimated input leaves collimated even though its cross-section has changed.

Why it mattersHow can orthogonal optical power reshape one image axis without focusing the other?

Start with the essentials

Focus question: How can orthogonal optical power reshape one image axis without focusing the other?
One-sentence intuition: The pair cannot enlarge pupil width and field angle independently. Its reciprocal first-order scales are $S_p=\left|\frac{f_2}{f_1}\right|,\qquad A_\theta=\left|\frac{f_1}{f_2}\right|,\qquad S_pA_\theta=1$ . This reciprocal relation is the quantitative core of anamorphic squeeze and desqueeze, while the internal powered-axis waist explains laser-line generation.

Core mathematical model

Powered and unpowered meridians

\mathbf L_x(f)=\begin{pmatrix}1&0\\-1/f&1\end{pmatrix},\qquad \mathbf L_y=\mathbf I

The cyan ray family receives the thin-lens angular kick; the gold family crosses the same glass without first-order lens power. A surface prescription would add thickness and aberration terms that this model does not claim.

Afocal cylindrical pair

d=f_1+f_2,\qquad C_x=0,\qquad \det\mathbf M_x=1

The first lens drives each collimated marginal ray through the internal line focus. At the focal-length sum, the second lens removes the outgoing powered-axis angle. Unit determinant is the lossless paraxial phase-space check.

Gaussian line-waist regularization

q_{\mathrm{out}}=\frac{Aq_{\mathrm{in}}+B}{Cq_{\mathrm{in}}+D},\qquad \frac{1}{q}=\frac{1}{R}-\mathrm i\frac{\lambda}{\pi w^2}

Geometric rays collapse to zero powered-axis width. The complex beam parameter replaces that singularity with a finite fundamental-Gaussian waist while the unpowered axis remains broad, producing a real line rather than an infinitely bright mathematical object.

Geometric oval bokeh

\mathcal A_{\mathrm{bokeh}}=\max\!\left(S_p,S_p^{-1}\right)

For deliberate geometric defocus, the blur footprint inherits the mapped pupil aspect. It is not a diffraction point-spread function and does not include aberration, clipping, coatings, or cinematic flare streaks.

Common difficulties

Treating every oval flare as anamorphic bokeh

Typical misconceptionAny horizontal oval or streak in a movie frame directly measures the lens squeeze ratio.

Better mental modelA mapped defocus footprint can follow pupil aspect, but flare streaks also depend on reflections, coatings, stops, clipping, aberrations, and sensor response. This Lab displays only the ideal first-order pupil inheritance and labels it separately from a diffraction PSF.

Run the experiment

01
Scene 1: Cylindrical line focus
Start at cylindrical line focus. Change the first focal length and watch the cyan powered-meridian rays move their waist while the gold orthogonal extent remains broad. Verify that the second lens stays at the focal-length-sum separation.
What to observe: Only the cyan meridian changes slope at each cylindrical lens. The gold rays cross both elements without a first-order angular kick, so the internal focus is a line rather than a point.
02
Scene 2: Anamorphic squeeze
Switch to anamorphic squeeze. Compare the circular input grid, the elliptical output pupil rings, and the compressed angular grid. Reverse the focal-length ratio and verify that expansion and compression exchange roles.
What to observe: Increasing the second-to-first focal-length ratio widens the collimated output pupil along the powered axis but narrows the corresponding field angle by precisely the reciprocal factor.
03
Scene 3: Elliptic bokeh and laser line
Enter oval bokeh and laser line. Compare the physically tiny powered-axis Gaussian waist with the long unpowered-axis line, then check that the separately labeled geometric bokeh card follows the same pupil ratio.
What to observe: The rendered glass curvature changes with focal length for legibility, but it remains a display representation of an ideal thin-lens power. The model boundary is reached first when the marginal powered-axis angle is no longer small.