- Huygens' Principle
- Law of Reflection from Huygens' Principle
- Law of Refraction using Huygens' Principle
- Fermat's Principle
- Optical Path Length
- Optical Reversibility

Within the approximation represented by geometrical
optics, light is understood to travel out from its source along straight lines
or **rays**. The ray is simply the path along which energy is transmitted
from one point to another in an optical system. The ray is a useful, although
abstract, construct; perhaps the best approximation to a ray of light is a
pencil-like laser beam. When a light ray traverses an optical system consisting
of several homogeneous media in sequence, the optical path is a sequence of
straight-line segments. The laws of geometrical optics that describe the
subsequent direction of the rays are succinctly stated as:

**Law of Reflection**: When a ray of light is
reflected at an interface dividing two uniform media, the reflected ray remains
within the **plane of incidence**, and the angle of reflection equals the
angle of incidence. The plane of incidence includes the incident ray and the
normal to the point of incidence.

**Law of Refraction (Snell's Law)**: When a ray of
light is refracted at an interface dividing two uniform media, the transmitted
ray remains within the plane of incidence and the sine of the angle of
refraction is directly proportional to the sine of the angle of
incidence.

These laws can be visually seen in the following figure

The figure illustrates Huygens' construction for a
narrow, parallel beam of light to prove the law of reflection. Huygens'
principle must be modified to accommodate the case in which a wavefront, such as
*AC*, encounters a plane interface, such as *XY*, at an angle. Here
the angle of incidence of the rays *AD*, *BE*, and *CF* relative
to the perpendicular *PD* is * _{i}*. Since points along the plane
wavefront do not arrive at the interface simultaneously, allowance is made for
these differences in constructing the wavelets that determine the reflected
wavefront. If the interface

Here we must take into account a different speed of
light in the upper and lower media. If the speed of light in vacuum is *c*,
we express the speed in the upper medium by the ratio *c/n _{i}*,
where

Similarly, a wavelet of radius
(*n _{i}/n_{t}*)

(6.1)

Three possible paths from *A* to *B* are
shown. Let's look at the arbitrary path *ACB*. If point *A'* is
constructed on the perpendicular *AO* such that *AO* = *A'O*, the
right triangles *AOC* and *A'OC* are equal. Thus *AC = A'C* and
the distance traveled by the ray of light from *A* to *B* via *C*
is the same distance from *A'* to *B* via *C*. The shortest
distance from *A'* to *B* is obviously the straight line *A'DB*,
so the path *ADB* is the correct choice taken by the actual light ray.
Geometry shows that for this path, * _{i }= _{r}*. Also note that to maintain

We can also prove the law of refraction. If the light
travels more slowly in the second medium, light bends at the interface so as to
take a path that favors a shorter time in the second medium, thereby minimizing
the overall transit time from *A* to *B*.

Mathematically, we are required to minimize the total time

(6.2)

Since other choices of path change the position of the
point *O* and therefore the distance *x*, we can minimize the time by
setting :

(6.3)

where the last step used the relationships shown in the figure. Introducing the refractive indices of the media, we arrive at Snell's law

(6.4)

Fermat's principle, like that of Huygens, required
refinement to achieve more general applicability. Situations exist where the
actual path taken by a light ray may represent a maximum time or even one of
many possible paths, all requiring equal time. As an example of the latter case,
consider light propagating from one focus to the other inside an ellipsoidal
mirror, along any of an infinite number of possible paths. Since the ellipse is
the locus of all points whose combined distances from the two foci remain
constant, all paths are indeed of equal time. A more precise statement of
Fermat's principle, which requires merely an extremum relative to neighboring
paths, may be given as follows: **The actual path taken by a light ray in its
propagation between two given points in an optical system is such as to make its
optical path equal, in the first approximation, to other paths closely adjacent
to the actual one**.

With this formulation, Fermat's principle falls in the class of problems called variational calculus, a technique which determines the form of a function that minimizes a definite integral. In optics, the definite integral is the integral of the time required for the transit of a light ray from starting to finishing points.

(6.5)

where the summation is called the **optical path length
**traversed by the ray. Clearly for an inhomogeneous medium where *n* is
a function of position, the summation must be changed to an integral

Since the optical path length is related to the time, we
can restate Fermat's principle again as **a light ray in going from point
A to point B must traverse an optical path length that is
stationary with respect to variations of that path**.

Consider the specular reflection of a single light ray
from the *x-y* plane. By the law of reflection, the reflected ray remains
within the plane of incidence, making equal angles with the normal at the point
of contact. If the path is resolved into components, it is clear that the
direction of the incident ray is altered only by reflection along the *z*
direction, and then in such a way that its *z* component is simply
reversed. If the direction of the incident ray is described by its unit
vector ,
then the reflection causes

(6.6)

It follows that if a ray is incident from such a direction as to reflect sequentially from all three coordinate planes, then

(6.7)

and the ray returns precisely parallel to the line of its original approach. A network of such corner reflectors ensures the exact return of a beam of light.

Last updated: July 13, 1997

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