Defects of Images (Aberrations)

Our neat formulas assumed thin lenses, small apertures and rays close to the axis. Real instruments break these rules, and their images show it as blur, smears, coloured fringes and bent lines. These defects are called aberrations.

Spherical aberration#

A spherical mirror or lens brings only paraxial rays to a single focus. Rays striking further out, the marginal rays, focus at a slightly different point (for a concave mirror, closer to it). The image of a point becomes a blurred patch.

The rays crowd along a bright curve, the caustic (you can see one inside a coffee cup in sunlight). On a screen the image is a disc; moving the screen finds where the disc is smallest, the circle of least confusion. The distance between the paraxial and marginal foci measures the aberration.

Reducing it

• Parabolic mirrors focus all parallel rays at one point, and a lamp at the focus sends out a parallel beam: car headlights use them.
• Stops: an opaque sheet with a small hole lets only a narrow pencil of rays through. Sharper, but dimmer.
• Share the bending between both lens surfaces. A plano-convex lens imaging a distant object does much better with its curved side toward the object than its flat side.
• Combine convex and concave lenses, whose spherical aberrations are opposite, so they partly cancel.

For a single lens forming a real image of a real object, spherical aberration can be reduced but never removed completely.

Coma#

For a point object off the axis, rays through different rings of the lens form circles of different sizes whose centres are shifted. Together they make a comet-shaped smear: coma. Careful choice of radii and stops reduces it.

Astigmatism#

For an off-axis point, the image is also spread along the axis. Move a screen and you first get a short line image, then a different shape, staying in rough focus over a range. (This lens defect shares its name with, but is not the same as, the eye defect.)

Curvature of field#

The best image of a flat object is often formed not on a flat plane but on a curved surface. It is closely linked to astigmatism; both are reduced by stops in the right places.

Distortion#

If the magnification changes with distance from the axis, an extended object is imaged out of shape: straight lines come out curved.

Chromatic aberration#

The refractive index of glass varies with wavelength, so a lens has a slightly different focal length for each colour: longest for red, shortest for violet. In white light each colour forms its own image.

• Axial (longitudinal) chromatic aberration: the separation along the axis between the images for the extreme colours. A convex lens puts violet nearer than red (positive); a concave lens does the opposite (negative).
• Lateral chromatic aberration: since m depends on v, the images for different colours differ in size.