7.26 Optics
257
the focal length f , taken negative if the affected rays are sent away from the focal
point. We will call lenses and mirrors optical devices. Note that a plane mirror or a
flat piece of glass will have an infinite focal length.
Plane mirrors and spherically-surfaced mirrors are relatively easy to make, using
silver deposited on glass. (This is best done on the front of the glass, but for
protection of the coating, non-technical mirrors have the reflective surface on the
back side of the glass, with the disadvantage that reflection off the glass itself
produces faint secondary images.) A mirror with a negative f has a convex surface,
and is called diverging.
A piece of bulk glass spherically ground on opposite sides, with the center of
the spheres along one axis, becomes a lens. Either side may be convex, planar or
concave. Whether a lens is converging or diverging depends on the curvature of
both sides of the lens, and on the index of refraction of the glass relative to the index
of refraction of the substance surrounding the glass. 45
The inverse of a lens’ focal length f is called its ‘optical power’, and is given
a unit name ‘diopter’ when the focal length is given in meters. Note that the word
power here is easily misconstrued, as optical power refers neither to an energy rate
nor to the lens’ magnification. If a lens is diverging, f is negative, and so is the
optical power.
Eyeglass lenses are usually constructed to be concave on the inside and convex on
the outside, to accommodate the eyelash movements. Both diverging and converging
lenses can be constructed this way. In the case of the diverging eyeglass lenses, the
glass is thinner nearer the optical axis. An eye which is astigmatic (i.e. its focal
length along the horizontal axis differs from that along the vertical axis) requires an
eyeglass lens with a different curvature in each of the two axis directions. Eyeglass
prescriptions usually give three numbers, the ‘spherical power’, the ‘cylindrical
power’, and the axis of astigmatism, where the spherical power is the optical
power needed along the axis (‘principle meridian’) with the least optical power;
the cylindrical power is the added optical power needed along the perpendicular
meridian; and the angular orientation of the principal meridian, measured in degrees
starting at zero for a horizontal meridian, 90 ◦ for a vertical meridian measured from
zero counterclockwise looking at the eye.
With the laws of optics above, it is straightforward to construction light rays from
an object, intersect a mirror or lens, and then leave it. Rays of light from each point
on the object, after reflection or refraction by the optical device, converge toward
a point, producing a real image point, or divergence from a point, making a virtual
image point. It is relatively easy to show geometrically that for a thin optical device,
45 The focal length of a thin lens obeys the ‘lens maker’s equation’, 1/f = (n/n 0 − 1)(1/R 1 −
1/R 2 ), where n is the index of refraction of the glass, n 0 the index of refraction of the surrounding
material (e.g. air), R 1 the radius of curvature of the lens on the ray entrance side, and R 2 the radius
of curvature of the lens on the ray exit side. If the center of curvature is further along the optical
axis in the direction a ray is traveling after intersecting the lens, then the radius of curvature is
positive; if the center of curvature is behind from the direction of travel of the ray at refraction,
then the radius of curvature is negative.
257
the focal length f , taken negative if the affected rays are sent away from the focal
point. We will call lenses and mirrors optical devices. Note that a plane mirror or a
flat piece of glass will have an infinite focal length.
Plane mirrors and spherically-surfaced mirrors are relatively easy to make, using
silver deposited on glass. (This is best done on the front of the glass, but for
protection of the coating, non-technical mirrors have the reflective surface on the
back side of the glass, with the disadvantage that reflection off the glass itself
produces faint secondary images.) A mirror with a negative f has a convex surface,
and is called diverging.
A piece of bulk glass spherically ground on opposite sides, with the center of
the spheres along one axis, becomes a lens. Either side may be convex, planar or
concave. Whether a lens is converging or diverging depends on the curvature of
both sides of the lens, and on the index of refraction of the glass relative to the index
of refraction of the substance surrounding the glass. 45
The inverse of a lens’ focal length f is called its ‘optical power’, and is given
a unit name ‘diopter’ when the focal length is given in meters. Note that the word
power here is easily misconstrued, as optical power refers neither to an energy rate
nor to the lens’ magnification. If a lens is diverging, f is negative, and so is the
optical power.
Eyeglass lenses are usually constructed to be concave on the inside and convex on
the outside, to accommodate the eyelash movements. Both diverging and converging
lenses can be constructed this way. In the case of the diverging eyeglass lenses, the
glass is thinner nearer the optical axis. An eye which is astigmatic (i.e. its focal
length along the horizontal axis differs from that along the vertical axis) requires an
eyeglass lens with a different curvature in each of the two axis directions. Eyeglass
prescriptions usually give three numbers, the ‘spherical power’, the ‘cylindrical
power’, and the axis of astigmatism, where the spherical power is the optical
power needed along the axis (‘principle meridian’) with the least optical power;
the cylindrical power is the added optical power needed along the perpendicular
meridian; and the angular orientation of the principal meridian, measured in degrees
starting at zero for a horizontal meridian, 90 ◦ for a vertical meridian measured from
zero counterclockwise looking at the eye.
With the laws of optics above, it is straightforward to construction light rays from
an object, intersect a mirror or lens, and then leave it. Rays of light from each point
on the object, after reflection or refraction by the optical device, converge toward
a point, producing a real image point, or divergence from a point, making a virtual
image point. It is relatively easy to show geometrically that for a thin optical device,
45 The focal length of a thin lens obeys the ‘lens maker’s equation’, 1/f = (n/n 0 − 1)(1/R 1 −
1/R 2 ), where n is the index of refraction of the glass, n 0 the index of refraction of the surrounding
material (e.g. air), R 1 the radius of curvature of the lens on the ray entrance side, and R 2 the radius
of curvature of the lens on the ray exit side. If the center of curvature is further along the optical
axis in the direction a ray is traveling after intersecting the lens, then the radius of curvature is
positive; if the center of curvature is behind from the direction of travel of the ray at refraction,
then the radius of curvature is negative.