1027
one to safely view a partial solar eclipse in shadows and
images cast on the ground by a leafy tree.
All these phenomena can be explained based on the
nature of light and its behavior in different materials
and at material boundaries. If a material is internally
uniform in density and transparency, then light travels
through it in straight lines and only changes direction,
or refracts, at the boundary to another medium. At each
boundary the path direction changes, and the angle of
change (with respect to a perpendicular to the surface at
the penetration point) is called the angle of refraction. If
the surface is fl at, the light entering it at a given angle is
refracted the same amount everywhere on the surface.
If the surface is curved, then the angle of refraction
varies over the surface. If the shape is part of a sphere,
the piece is said to be a lens and it can focus or defocus
light, depending on whether the surfaces are convex or
concave. (It should also be noted that at each bound-
ary between different materials a fraction of the light
is refl ected: for glass and air it amounts to about 4%,
and can give rise to multiple internal refl ections inside
lenses or rear-surface mirrors.)
The angular amount of refraction of a material is
measured by a number unique to each material, called
the index of refraction, the ratio of bending in a material
to that of empty space, which is set equal to one. Air
and other low-density gases have indices of refraction
just a bit larger than one. Water at sea level has an index
of refraction of 1.33, natural crystals, glass and plastics
generally fall in the range 1.25 to 1.8. Some relatively
exotic composition glasses have indices of refraction
well over 2.5, which means lenses made out of them
can be of thinner material and still bend light as much
as thicker lenses made of lower index glass. (An aside:
Einstein’s relativity shows that, in the presence of mat-
ter, space itself curves and thus light’s path in space is
curved proportionate to the distance and density of that
matter. It shows up in astronomy in curved arc images
of distant galaxies seen around massive foreground
galaxies. This effect is ignored in classical optics and
so far has found no application in conventional pho-
tography!)
Lens making became a profession in the late renais-
sance in Europe. The Dutch optician, Snell (1580-1626),
discovered and published a short mathematical relation
for the bending of light in 1626, by sending light beams
through glass surfaces of varying shapes and indices
of refraction and carefully measuring the angle of in-
cidence of the light as it struck the glass, and then the
angle it was deviated to inside. He also measured the
angles at the light ray’s emergence on the far surface,
into the air.
Snell’s law is: n 1 sin θ 1 = n 2 sin θ 2.
In words that is: the index of refraction of light in
the fi rst medium times the sine of the angle of incidence
at the boundary is equal to the index of the light in the
second medium times the sine of the angle of refraction
in the second material. The unprimed numbers are the
numbers in the fi rst medium, the primed numbers refer
to the second medium. The Greek letter theta (θ) refers
to the angle in degrees. Recall that the sine of an angle is
a trigonometric property and can be found in mathemati-
cal reference tables or calculated by hand or computer.
Since by its defi nition, the sine of any angle between
zero degrees and 90 degrees is somewhere between 0
and 1, and since the index of any ordinary transparent
material is greater than one, a light ray entering glass
from air will be bent to a smaller angle than the angle of
incidence it had at entry. When a ray leaves the denser
material, passing from glass to air, the ray will bend
back to a larger angle.
Snell’s law was empirically derived. It can be derived
theoretically using the electromagnetic wave theory of
light founded by James Clerk Maxwell in the 1860s, and
also by Fermat’s principle of least time of travel. The
latter idea relies on the fact that the index of refraction
is not only a measure of bending strength but also the
ratio of the speed of light in a material to that in empty
space.
Snell’s law and a little further work led to a useful
formula, still applicable with certain restrictions, for
describing practical optical systems. It is called the lens
maker’s formula:
1/fl = (1/OD) + (1/ID).
The OD is the distance from the object in question
to the center of the lens, the ID is the distance from
the lens center to the focused image, and fl is the focal
length, the distance from the lens center to the focused
image when the object is at infi nity. The focal length is
also, by defi nition, one half the radius of curvature of
the lens surface. It should be noted that this formula can
be repeatedly applied to follow light through a series of
lenses. With proper observation of positive and negative
sign conventions it can also be used to study mirrors.
The restrictions are that the lens have a shallow curva-
ture, or equivalently that it has a very long focal length
compared to its diameter. And light rays are assumed to
travel in straight lines except when they cross material
boundaries. This is called the Thin Lens Approxima-
tion. The lens maker’s formula is a shortcut of use in
deciding where an image will focus for an object at a
given distance using a lens of known focal length, or for
a quick design of a simple optical system. This formula
also can be used to get the image magnifi cation, which
is the ratio: M= ID/OD.
Aside: Fresnel (1788–1827) in the early 1800s
garnered a large prize fund from the French Academy
of Science by designing a very thin lens, for use in
lighthouse lights. These were used successfully for
that purpose but now fi nd much larger use as lenses for