Optics 028 · Imaging, instruments, and visual systems

Anamorphic & Cylindrical Optics

How can a lens stretch or squeeze an image in one direction only? A cylindrical lens focuses light in one direction and leaves the other alone. A pair of them can widen or narrow a beam along that one axis, but the field angle changes the opposite way: double the beam width and the field angle halves. That trade is how anamorphic film squeezes a wide scene onto the frame and unsqueezes it for projection, and how a laser line is made.

A three-dimensional anamorphic bench shows the two orthogonal meridians of a Kepler pair of cylindrical lenses with parallel axes. Rings tracing the Gaussian cross-section pass through the internal line waist and become an elliptical collimated pupil. That pupil sets the reciprocal squeeze of an angular grid and 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 has no optical power. The glass shapes and glow are only for display. Thick-surface prescriptions, anamorphic aberrations, vignetting, flare, coatings, polarization, sensor sampling and the diffraction PSF are not included.

Physics tutorial

Audit the physics of Anamorphic & Cylindrical Optics

BackgroundA cylindrical lens has optical power in one meridian and none in the perpendicular one. This bench puts two positive cylindrical lenses, with parallel axes, at their afocal spacing. Along the powered axis the ray transfer is $\mathbf M_x=\mathbf L(f_2)\mathbf P(f_1+f_2)\mathbf L(f_1)=\begin{pmatrix}-f_2/f_1&f_1+f_2\\0&-f_1/f_2\end{pmatrix}$ . The zero in the lower-left corner is what the scene lets you check: a collimated beam leaves collimated, though its cross-section has changed shape.

Why it mattersHow can a lens stretch or squeeze an image in one direction only?

Start with the essentials

Focus question: How can a lens stretch or squeeze an image in one direction only?
One-sentence intuition: The pair cannot widen the beam and the field angle independently. To first order, the two scale factors are reciprocal: $S_p=\left|\frac{f_2}{f_1}\right|,\qquad A_\theta=\left|\frac{f_1}{f_2}\right|,\qquad S_pA_\theta=1$ . Anamorphic squeeze and desqueeze rest on this reciprocal relation. The narrow waist between the lenses, along the powered axis, is how a laser line is made.

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 rays get the thin-lens change in angle; the gold rays pass through the same glass with no first-order power. A real surface prescription would add thickness and aberration terms, which this model does not include.

Afocal cylindrical pair

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

The first lens bends each collimated edge ray through the line focus between the lenses. Placed at the sum of the focal lengths, the second lens removes the angle along the powered axis. A determinant of one checks that the paraxial transfer is lossless and conserves phase-space area.

A Gaussian beam gives the line focus a finite width

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 squeeze to zero width along the powered axis. The complex beam parameter replaces that singularity with the finite waist of a fundamental Gaussian beam, while the unpowered axis stays broad. The result is a real line of light with finite brightness, not an infinitely bright mathematical line.

Geometric oval bokeh

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

With deliberate defocus, the geometric blur takes on the aspect ratio of the mapped pupil. It is not a diffraction point-spread function, and it leaves out aberrations, clipping, coatings and 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 defocused blur can follow the pupil's aspect ratio, but flare streaks also depend on reflections, coatings, stops, clipping, aberrations and the sensor's response. This Lab shows only the ideal first-order effect of the pupil shape 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, not 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
Open oval bokeh and laser line. Compare the tiny Gaussian waist along the powered axis with the long line along the unpowered axis, then check that the separately labeled geometric bokeh card follows the same pupil ratio.
What to observe: The drawn glass curves more or less with focal length so the scene is easy to read, but it only stands for an ideal thin-lens power. The model first breaks down when the edge-ray angle along the powered axis is no longer small.