Use Huygens’s principle to explain the law of reflection
Use Huygens’s principle to explain the law of refraction
Use Huygens’s principle to explain diffraction
So far in this chapter, we have been discussing optical
phenomena using the ray model of light. However, some phenomena
require analysis and explanations based on the wave characteristics
of light. This is particularly true when the wavelength is not
negligible compared to the dimensions of an optical device, such as
a slit in the case of diffraction. Huygens’s principle is
an indispensable tool for this analysis.
Figure \(\PageIndex{1}\) shows how a transverse wave looks as
viewed from above and from the side. A light wave can be imagined
to propagate like this, although we do not actually see it wiggling
through space. From above, we view the wave fronts (or wave crests)
as if we were looking down on ocean waves. The side view would be a
graph of the electric or magnetic field. The view from above is
perhaps more useful in developing concepts about wave optics.
The Dutch scientist Christiaan Huygens (1629–1695) developed a
useful technique for determining in detail how and where waves
propagate. Starting from some known position, Huygens’s principle
states that every point on a wave front is a source of wavelets
that spread out in the forward direction at the same speed as the
wave itself. The new wave front is tangent to all of the
wavelets.
Figure \(\PageIndex{2}\) shows how Huygens’s principle is
applied. A wave front is the long edge that moves, for example,
with the crest or the trough. Each point on the wave front emits a
semicircular wave that moves at the propagation speed \(v\). We can
draw these wavelets at a time \(t\) later, so that they have moved
a distance \(s=vt\). The new wave front is a plane tangent to the
wavelets and is where we would expect the wave to be a time \(t\)
later. Huygens’s principle works for all types of waves, including
water waves, sound waves, and light waves. It is useful not only in
describing how light waves propagate but also in explaining the
laws of reflection and refraction. In addition, we will see that
Huygens’s principle tells us how and where light rays
interfere.
Reflection
Figure \(\PageIndex{3}\) shows how a mirror reflects an incoming
wave at an angle equal to the incident angle, verifying the law of
reflection. As the wave front strikes the mirror, wavelets are
first emitted from the left part of the mirror and then from the
right. The wavelets closer to the left have had time to travel
farther, producing a wave front traveling in the direction
shown.
Refraction
The law of refraction can be explained by applying Huygens’s
principle to a wave front passing from one medium to another
(Figure \(\PageIndex{4}\)). Each wavelet in the figure was emitted
when the wave front crossed the interface between the media. Since
the speed of light is smaller in the second medium, the waves do
not travel as far in a given time, and the new wave front changes
direction as shown. This explains why a ray changes direction to
become closer to the perpendicular when light slows down. Snell’s
law can be derived from the geometry in Figure \(\PageIndex{5}\)
(Example \(\PageIndex{1}\)).
Example \(\PageIndex{1}\): Deriving the Law of
Refraction
By examining the geometry of the wave fronts, derive the law of
refraction.
Strategy
Consider Figure \(\PageIndex{5}\), which expands upon Figure
\(\PageIndex{4}\). It shows the incident wave front just reaching
the surface at point A, while point B is still
well within medium 1. In the time \(Δt\) it takes for a wavelet
from \(B\) to reach \(B'\) on the surface at speed \(v_1=c/n_1\), a
wavelet from \(A\) travels into medium 2 a distance of
\(AA'=v_2Δt\), where \(v_2=c/n_2\). Note that in this example,
\(v_2\) is slower than \(v_1\) because \(n_1<n_2\).
Solution
The segment on the surface AB' is shared by both the triangle
ABB' inside medium 1 and the triangle AA′B′ inside medium 2. Note
that from the geometry, the angle ∠BAB' is equal to the angle of
incidence, \(θ_1\). Similarly, \(∠AB'A'\) is \(θ_2\).
Although the law of refraction was established experimentally by
Snell, its derivation here requires Huygens’s principle and the
understanding that the speed of light is different in different
media.
Exercise \(\PageIndex{1}\)
In Example \(\PageIndex{1}\), we had \(n_1<n_2\). If \(n_2\)
were decreased such that \(n_1>n_2\) and the speed of light in
medium 2 is faster than in medium 1, what would happen to the
length of AA'? What would happen to the wave front A'B' and the
direction of the refracted ray?
Answer
AA′ becomes longer, A'B' tilts further away from the surface,
and the refracted ray tilts away from the normal.
This applet
by Walter Fendt shows an animation of reflection and refraction
using Huygens’s wavelets while you control the parameters. Be sure
to click on “Next step” to display the wavelets. You can see the
reflected and refracted wave fronts forming.
Diffraction
What happens when a wave passes through an opening, such as
light shining through an open door into a dark room? For light, we
observe a sharp shadow of the doorway on the floor of the room, and
no visible light bends around corners into other parts of the room.
When sound passes through a door, we hear it everywhere in the room
and thus observe that sound spreads out when passing through such
an opening (Figure \(\PageIndex{6}\)). What is the difference
between the behavior of sound waves and light waves in this case?
The answer is that light has very short wavelengths and acts like a
ray. Sound has wavelengths on the order of the size of the door and
bends around corners (for frequency of 1000 Hz,
about three times smaller than the width of the doorway).
If we pass light through smaller openings such as slits, we can
use Huygens’s principle to see that light bends as sound does
(Figure \(\PageIndex{7}\)). The bending of a wave around the edges
of an opening or an obstacle is called diffraction.
Diffraction is a wave characteristic and occurs for all types
of waves. If diffraction is observed for some phenomenon, it is
evidence that the phenomenon is a wave. Thus, the horizontal
diffraction of the laser beam after it passes through the slits in
Figure \(\PageIndex{7}\) is evidence that light is a wave.