Lens

A lens is a piece of transparent material, usually glass, fused silica, a crystal or a polymer, with curved surfaces that refract light so that rays from one point are redirected toward another. A converging (positive) lens brings a collimated beam to a focus at a distance equal to its focal length $f$; a diverging (negative) lens spreads it as if from a virtual point at $-f$. Laboratory lenses have focal lengths from a few millimeters (aspheres for collimating laser diodes) to a meter or more, with diameters of 6–50 mm; the optical power $1/f$ is quoted in diopters (m$^{-1}$), so a 50 mm lens is a 20 D lens.

Image formation

For a thin lens in air, an object at distance $s$ in front of the lens is imaged at distance $s'$ behind it, with

$$\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}, \qquad m = -\frac{s'}{s} .$$

With $f = 50$ mm and an object 150 mm away, the image forms at 75 mm with magnification $m = -0.5$: inverted and half size. An object at $2f$ images at $2f$ at unit magnification; an object inside $f$ gives a virtual, upright, magnified image, the magnifying-glass case. The focal length itself follows from the surface radii and index through the lensmaker's equation, covered in the focal length entry. For a plano-convex lens it reduces to $f = R/(n-1)$, so N-BK7 ($n_d = 1.5168$) needs $R = 25.84$ mm for $f = 50$ mm.

Focusing a laser beam

For a collimated Gaussian beam of waist radius $w$ at the lens, the focused waist radius is approximately

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

A HeNe beam at 633 nm with $w = 1$ mm focused by the 50 mm lens gives $w_0 \approx 10\ \mu\text{m}$. A shorter focal length or a larger input beam gives a smaller spot, until aberrations of the lens, rather than diffraction, set the size. The ratio of focal length to clear aperture is the f-number, and the cone half-angle on the image side is expressed as the numerical aperture.

Lens types

• Singlets. Plano-convex, biconvex, meniscus and their negative counterparts; inexpensive and adequate at modest apertures.
• Achromatic doublets. A crown positive element cemented to a flint negative element, bringing two wavelengths to a common focus and reducing spherical aberration.
• Aspheres. A non-spherical surface that corrects spherical aberration in a single element; standard for collimating laser diodes and coupling into fiber.
• Cylindrical lenses. Focus in one axis only; used to shape elliptical diode beams or form light sheets.
• Gradient-index (GRIN) lenses. Flat-faced rods whose index varies radially; common in fiber components.
• Objectives. Multi-element assemblies corrected over a field and waveband, specified by magnification and numerical aperture.

Chromatic behavior

Because the index depends on wavelength, so does the focal length. For a singlet the spread between the blue F line and the red C line is roughly $f/V$, where $V$ is the Abbe number: for the 50 mm N-BK7 lens, $f$ is 49.47 mm at 486.1 nm and 50.24 mm at 656.3 nm, a shift of 0.77 mm, close to $50/64.2 = 0.78$ mm. This matters whenever a lens is used at a wavelength other than its design wavelength, for instance aligning an infrared system with a visible laser.

Practical points

A plano-convex lens focusing a collimated beam performs best with the curved side toward the beam, which shares the bending between the two surfaces and reduces spherical aberration by a large factor compared with the reverse orientation. Each uncoated surface reflects about 4%, so lenses for laser work carry anti-reflection coatings matched to the wavelength band. Catalog focal lengths are usually quoted at 587.6 nm or at a design laser wavelength, and the stated "effective focal length" is measured from the principal plane, which for a thick or meniscus lens can lie outside the glass; the back focal length, from the last surface, is what sets the physical spacing.

Common questions

What is the difference between a convex and a concave lens?

A convex lens is thicker at the center than at the edge and converges light (positive focal length) when its index exceeds that of the surroundings. A concave lens is thinner at the center and diverges light (negative focal length). The sign flips if the lens has a lower index than the medium around it, as for an air bubble in water.

How is the focal length of an unknown lens found?

For a positive lens, focus a distant object, such as a window across a room, onto a card and measure the lens-to-card distance; this gives $f$ to within a few percent when the object is more than about 50 focal lengths away. More accurate methods use a collimated laser and a shear plate or autocollimation.

Why do lenses have aberrations?

Spherical surfaces are easy to make but do not bring all rays from a point to exactly one point, and the refractive index varies with wavelength. The resulting departures from perfect imaging are corrected by combining elements, adding aspheric surfaces, and stopping down the aperture.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 5 and 6; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 1 and 3; W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).