Aspheric lens

An aspheric lens, or asphere, is a lens with one or both surfaces shaped as something other than a section of a sphere, usually a conic with small polynomial corrections. The extra shape freedom lets one element do what would otherwise take two or three spherical elements: bring every ray of a collimated beam to the same focus, removing spherical aberration at the design wavelength and conjugates. Aspheres with focal lengths of about 2–20 mm and numerical apertures up to about 0.5–0.6 are the usual choice for collimating laser diodes and coupling into single-mode fiber; larger aspheres appear in condensers, camera lenses and headlamps.

Surface description

Aspheric surfaces are specified by their sag $z(r)$, the axial depth of the surface at radial height $r$ from the vertex. The standard even-asphere form is

$$z(r) = \frac{c\,r^2}{1+\sqrt{1-(1+k)\,c^2 r^2}}$$ $$\quad + A_4 r^4 + A_6 r^6 + A_8 r^8 + \dots$$

where $c = 1/R$ is the vertex curvature, $k$ the conic constant and $A_{2i}$ the higher-order coefficients. With $k = 0$ and no polynomial terms the surface is a sphere; $k = -1$ gives a paraboloid, $k < -1$ a hyperboloid, and $-1 < k < 0$ a prolate ellipsoid.

The differences involved are small in absolute terms. For a vertex radius of 10 mm, a sphere reaches a sag of 1.340 mm at $r = 5$ mm, while a paraboloid with the same vertex radius reaches 1.250 mm; the 90 µm gap is more than a hundred wavelengths of visible light.

Why a single asphere can focus perfectly

Consider a plano-convex lens with its flat side toward a collimated beam. The rays travel parallel through the glass and refract only at the curved exit surface. Fermat's principle shows that this surface focuses them to one point exactly when it is a hyperboloid of eccentricity $n$, that is, conic constant

$$k = -n^2 .$$

For N-BK7 ($n_d = 1.5168$) this is $k = -2.30$. A spherical surface in the same place over-bends the marginal rays. A ray trace makes the size of the effect concrete: a 10 mm focal length N-BK7 plano-convex singlet, 3 mm thick, with the curved side toward a beam of 4 mm diameter (image-side NA about 0.20), brings the marginal ray to focus 0.43 mm in front of the paraxial focus. Turned the other way round it is worse, 1.86 mm. The diffraction-limited depth of focus at 632.8 nm, roughly $\lambda/\mathrm{NA}^2$, is about 16 µm, so the spherical singlet misses by a factor of about 27 even in its better orientation, whereas an asphere of the same focal length can be designed to be diffraction-limited over the full aperture.

Manufacture and measurement

Most small aspheres are precision glass-molded: a heated preform is pressed between polished mold inserts, which makes large volumes cheap once the tooling exists. Larger or high-precision aspheres are ground and polished with computer-controlled sub-aperture tools, infrared crystals and polymers are often diamond turned, and hybrid aspheres add a thin molded polymer layer to a spherical glass lens.

Form is checked with a profilometer against the design sag, and performance by measuring transmitted wavefront error at the design conjugates with an interferometer. A diffraction-limited part meets the Maréchal limit of about $\lambda/14$ RMS.

Where aspheres are used

• Laser diode collimation. A single molded asphere of a few millimetres focal length captures the fast-axis divergence of an edge emitter; many such lenses are designed to include the cover glass of the diode package in their correction.
• Fiber coupling. Aspheres focus a collimated beam onto a single-mode fiber core, and pairs of them form compact fiber collimators.
• Condensers and imaging. Aspheric condensers collect light from LEDs and lamps, and phone camera modules use stacks of strongly aspheric molded plastic elements.

Pitfalls

An asphere is corrected for one orientation, one pair of conjugates and one wavelength. Reversing it, using it at finite conjugates when it was designed for infinity, or inserting an uncompensated window changes the spherical aberration budget it was built to cancel. Because aspheres are usually used at high NA and their aspheric surface must share an axis with the other surface, tilt and decentre produce coma readily. A single asphere does nothing for chromatic aberration: the focal shift between the F and C lines is still about $f/V$, which is why broadband systems pair aspheric surfaces with an achromatic doublet or use aspheric achromats.

Common questions

What is the difference between an aspheric and a spherical lens?

A spherical surface has constant curvature and cannot focus all rays of a collimated beam to one point; an aspheric surface changes curvature with radius so that marginal and paraxial rays share a focus. The advantage of the asphere grows rapidly with aperture.

Which side of an aspheric lens should face the collimated beam?

The side the design specifies, shown on the lens drawing. In most catalog designs the more steeply curved surface faces the collimated beam, the same rule that applies to a plano-convex singlet, and the flatter side faces the diode or fiber.

Does an aspheric lens correct chromatic aberration?

No. It corrects spherical aberration at the design wavelength, but its focal length still varies with wavelength through the glass dispersion.

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. 1.