256
7 Light in Biology and Medicine
7.26 Optics
Geometric optics is a study of the properties of light in interaction with matter under
conditions in which the wavelength of the light can be assumed much smaller than
the dimensions of macroscopic structures with which the light is interacting. It then
becomes possible to follow the behavior of light by how light rays 42 are affected by
substances. When one assumes small wavelengths (in the sense above), Maxwell’s
equations become the “eikonal equation”. 43
There are several useful and general properties of light propagation in the
geometric optical limit, all derivable from the eikonal equation. (In the following
statements, angles are measured between a light ray and the normal to an involved
surface.)
Law of Reflection When light reflects from a smooth surface, 44 the incident ray,
the normal to the surface, and the reflected ray are all in one plane. In addition, the
incident angle and the reflected angle are the same.
When light enters at an angle into a transparent medium, we say the light is
refracted (bent) if the direction of light rays changes on entering.
Law of Refraction As a ray crosses the boundary between two materials, the
incident ray, the normal, and the refracted ray are in the same plane. In addition,
the sine of the incident angle and the sine of the refracted angle are in the same
ratio as the average speed of light in the first material in which the incident light
originated to the second material into which the light entered. This statement is
‘Snell’s Law’, and can be expressed as n i sin θ i = n r sin θ r , where n is called the
index of refraction, defined to be the ratio of the speed of light in a vacuum to that
in the material.
With the law of reflection and refraction, image formation by mirrors and lenses
follows. A mirror is a cylindrically-symmetric opaque device with a reflective
surface that causes rays arriving parallel to the axis of symmetry to the surface
to be reflected and then converged toward or diverged from a focal point. A lens
is cylindrically-symmetric transparent device that refracts rays and that causes rays
arriving parallel to the axis of symmetry to either converge or diverge toward or
from a focal point. The distance from the mirror or lens to the focal point is called
42 Light rays are directed lines constructed perpendicular to light wave fronts and point the way the
wave fronts are moving.
43 Eikonal comes from the Greek word ` ικ ´
ων, meaning image. The eikonal equation from which
all of geometric optics follows, is derived from Maxwell’s equations, and can be expressed as
(∇S)
2 = n 2 . Given the behavior of the index of refraction n(x, y, z) (i.e. the speed of light in a
vacuum divided by its speed in the material at (x, y, z)), one can solve for the ‘eikonal’ S. Then
the equation S(x, y, z) = constant will define a surface in space that is everywhere perpendicular
to optical rays. A light ray, specified by r (x, y, z), will be a solution of n dr/ds = ∇S, where ds
is an infinitesimal distance along the ray. For more details, see Sect. A.1.
44 In this context, smooth means that any bumps on the surface are much smaller than the
wavelength of the light.
7 Light in Biology and Medicine
7.26 Optics
Geometric optics is a study of the properties of light in interaction with matter under
conditions in which the wavelength of the light can be assumed much smaller than
the dimensions of macroscopic structures with which the light is interacting. It then
becomes possible to follow the behavior of light by how light rays 42 are affected by
substances. When one assumes small wavelengths (in the sense above), Maxwell’s
equations become the “eikonal equation”. 43
There are several useful and general properties of light propagation in the
geometric optical limit, all derivable from the eikonal equation. (In the following
statements, angles are measured between a light ray and the normal to an involved
surface.)
Law of Reflection When light reflects from a smooth surface, 44 the incident ray,
the normal to the surface, and the reflected ray are all in one plane. In addition, the
incident angle and the reflected angle are the same.
When light enters at an angle into a transparent medium, we say the light is
refracted (bent) if the direction of light rays changes on entering.
Law of Refraction As a ray crosses the boundary between two materials, the
incident ray, the normal, and the refracted ray are in the same plane. In addition,
the sine of the incident angle and the sine of the refracted angle are in the same
ratio as the average speed of light in the first material in which the incident light
originated to the second material into which the light entered. This statement is
‘Snell’s Law’, and can be expressed as n i sin θ i = n r sin θ r , where n is called the
index of refraction, defined to be the ratio of the speed of light in a vacuum to that
in the material.
With the law of reflection and refraction, image formation by mirrors and lenses
follows. A mirror is a cylindrically-symmetric opaque device with a reflective
surface that causes rays arriving parallel to the axis of symmetry to the surface
to be reflected and then converged toward or diverged from a focal point. A lens
is cylindrically-symmetric transparent device that refracts rays and that causes rays
arriving parallel to the axis of symmetry to either converge or diverge toward or
from a focal point. The distance from the mirror or lens to the focal point is called
42 Light rays are directed lines constructed perpendicular to light wave fronts and point the way the
wave fronts are moving.
43 Eikonal comes from the Greek word ` ικ ´
ων, meaning image. The eikonal equation from which
all of geometric optics follows, is derived from Maxwell’s equations, and can be expressed as
(∇S)
2 = n 2 . Given the behavior of the index of refraction n(x, y, z) (i.e. the speed of light in a
vacuum divided by its speed in the material at (x, y, z)), one can solve for the ‘eikonal’ S. Then
the equation S(x, y, z) = constant will define a surface in space that is everywhere perpendicular
to optical rays. A light ray, specified by r (x, y, z), will be a solution of n dr/ds = ∇S, where ds
is an infinitesimal distance along the ray. For more details, see Sect. A.1.
44 In this context, smooth means that any bumps on the surface are much smaller than the
wavelength of the light.