Cylindrical lens

A cylindrical lens has a surface shaped as part of a cylinder: it is curved along one transverse axis (the power axis) and straight along the other. It therefore focuses or diverges light in one plane while leaving the perpendicular plane untouched, turning a collimated round beam into a line focus rather than a point. Catalog cylindrical lenses come in plano-convex and plano-concave forms, as rectangular or round elements, with focal lengths from a few millimetres to several hundred millimetres; they are the standard tool for anamorphic beam shaping, for producing laser lines and light sheets, and for correcting astigmatism.

Focal length and line focus

Along its power axis a cylindrical lens behaves like an ordinary lens of the same cross-section, so a thin plano-convex cylinder has

$$f = \frac{R}{n-1} .$$

An N-BK7 cylinder ($n_d = 1.5168$) with $R = 25.84$ mm therefore has $f = 50$ mm. Focusing a collimated Gaussian beam of radius $w$, the line has a half-width at the focus of

$$w_0 \approx \frac{\lambda f}{\pi w},$$

set only by the power axis. For a 632.8 nm beam of 1 mm radius and $f = 50$ mm this is 10.1 µm, a line about 20 µm wide (1/e² full width). Along the unpowered axis the beam keeps its original 2 mm diameter, so the line is 2 mm long and slowly diverging with the beam. Longer lines are made by expanding or fanning out the beam in that axis with a second, crossed cylinder.

Two identical cylindrical lenses crossed at 90° and placed close together act approximately as one spherical lens; separated by a distance, they form an anamorphic system with different effective focal lengths and focal positions in the two axes.

Anamorphic beam shaping

Edge-emitting laser diodes emit an elliptical, astigmatic beam, with a fast-axis divergence several times the slow-axis divergence. Taking, as an example, full angles of 30° and 8°, the beam after a spherical collimating lens is elliptical with an axis ratio of $\tan 15°/\tan 4° \approx 3.8$. A cylindrical telescope acting on the narrow axis, for instance a $-20$ mm and a $+75$ mm cylinder spaced 55 mm apart, expands that axis by 3.75 and leaves the beam round to within about 2%. This is the cylindrical counterpart of a beam expander; an anamorphic prism pair performs the same task without adding focusing power, at the cost of a lateral beam offset.

Because diode beams also have different apparent source positions in the two axes, a weak cylindrical lens can cancel the residual astigmatism after collimation, so both axes come to a waist at the same plane.

Other uses

• Light-sheet microscopy. A cylindrical lens focuses an expanded laser beam into a thin sheet that illuminates one plane of a sample from the side.
• Line generators and scanning. Machine vision, barcode readers and laser triangulation sensors project lines with cylindrical optics. A Powell lens, an acylindrical element with a rounded roof, spreads light into a line of more uniform intensity than the Gaussian profile a simple cylinder gives.
• Slits and linear arrays. Cylindrical lenses match a round source to a spectrometer slit or a line-scan sensor.

Alignment and pitfalls

The critical alignment is rotation about the optical axis, often called clocking. If the lens axis is rotated by an angle $\theta$ relative to the intended axis, part of its power appears in the perpendicular plane and a cross term couples the two axes: to first order the unwanted power scales as $\sin^2\theta$ and the coupling term as $\sin\theta\cos\theta$. At $\theta = 1°$ these are about $3 \times 10^{-4}$ and 0.017 of the lens power. The coupling is the larger effect: the line focus is rotated by $\theta$, and in combination with other cylindrical elements or an astigmatic input it also twists through focus. In practice the lens is rotated while watching the line on a camera until it is aligned with the pixel rows and at its narrowest.

Cylindrical lenses have the same spherical aberration in their power axis as spherical singlets of equal cross-section, so the curved side should face the collimated beam, and fast line foci use acylindrical lenses, the one-axis counterpart of an aspheric lens. Decentring the lens along its power axis steers the line, as with any lens, while decentring along the flat axis has no effect, which makes the translation alignment forgiving in one direction. Catalog focal lengths are quoted at a design wavelength, usually 587.6 nm, and shift with wavelength as for a spherical singlet.

Common questions

What is a cylindrical lens used for?

To focus or expand a beam in one direction only: making laser lines and light sheets, circularizing elliptical diode beams, matching a beam to a slit or linear detector, and correcting astigmatism.

Can two cylindrical lenses replace a spherical lens?

Approximately, when two lenses of equal focal length are crossed at 90° and placed in contact. With any separation the two axes focus at different distances, which is useful for anamorphic shaping but is a source of astigmatism when a round focus is wanted.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 3.