Spherical aberration

A spherical surface is the easiest shape to grind and polish, but it is not the shape that focuses a collimated beam to a point. Rays that cross a positive lens far from the axis are bent slightly too much and cross the axis before the paraxial focus, so the image of a point is a small disk with a bright core and a halo rather than a point. This is spherical aberration, the one Seidel aberration that is present on axis, and for a single lens focusing a laser beam it is usually the aberration that limits the spot size.

In wavefront terms it is a fourth-order error across the pupil, $W = W_{040}\rho^4$, where $\rho$ is the normalized pupil radius. Its size therefore grows as the fourth power of the aperture: halving the beam diameter on a given lens reduces the wavefront error sixteenfold. Refocusing to the plane of best focus balances part of the error with defocus; the balanced form is the Zernike term $Z_4^0$, and balancing reduces the rms wavefront error by a factor of four, from $0.298\,W_{040}$ to $0.0745\,W_{040}$.

Typical values

For a plano-convex N-BK7 lens of 50 mm focal length and 5.3 mm centre thickness, exact ray tracing of a 10 mm diameter collimated beam at 587.6 nm gives a longitudinal spherical aberration, the distance between the paraxial and marginal foci, of 0.54 mm when the curved side faces the collimated beam. With the lens reversed, flat side toward the beam, it is 2.19 mm, four times worse. In practice, the curved side of a plano-convex lens faces the collimated beam, or more generally the side where the rays are most nearly parallel, so that both surfaces share the bending.

Correction

Spherical aberration is reduced by splitting the bending between surfaces or elements, by choosing the lens shape (a best-form singlet), by combining positive and negative elements of different glasses, and above all by aspheric surfaces, which can remove it entirely for one pair of conjugates. Laser-diode collimators and fiber-coupling lenses are molded aspheres for this reason. A plane-parallel plate in a converging beam, such as a cover glass, a window or a cube beamsplitter, also adds spherical aberration, which is why high-NA microscope objectives specify the cover-glass thickness they are corrected for.

Measurement

On an interferometer, spherical aberration appears in the Zernike fit as the $Z_4^0$ coefficient. On a bench, it shows as a focus that shifts when the beam is apertured: the best-focus position measured with a small central beam and with the full aperture differ by about half the longitudinal aberration. A Foucault knife-edge or star test shows the characteristic difference between the patterns inside and outside focus.

References: W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); V. N. Mahajan, Optical Imaging and Aberrations, Part I (SPIE Press, 1998).