Reflection, Refraction, Diffraction, and Wave Interference
Reflection, refraction, diffraction, and wave interference are fundamental phenomena associated with light waves, each exhibiting unique characteristics and behaviors. **Reflection** occurs when light strikes a shiny surface, adhering to the Law of Reflection, where the angle of incidence equals the angle of reflection. **Refraction** involves the bending of light as it transitions between media of different densities, causing a change in speed and direction. This phenomenon is described by the index of refraction, with key implications for optics, including total internal reflection, which is essential for technologies such as fiber optics.
**Diffraction** is the bending of light around obstacles or through apertures, generating patterns of bright and dark spots due to the interference of light waves emanating from various points within the aperture. Finally, **wave interference** occurs when multiple light waves overlap, leading to constructive interference (intensifying the light) or destructive interference (diminishing the light), based on their phase relationships. These concepts are integral to understanding light behavior in various applications, from everyday phenomena to advanced optical technologies. Each phenomenon highlights the dual nature of light as both a particle and a wave, demonstrating intricate relationships that are essential in fields such as physics and engineering.
Reflection, Refraction, Diffraction, and Wave Interference
While wave phenomena properly belong to the field of physics, there are nevertheless interesting mathematical issues that arise in the study of waves. This article will be limited in scope to one particular type of waves: light waves. Depending on the phenomenon under consideration, light may be treated as a ray or as a wave.
Ray optics models the light emitted by a point as straight lines that travel radially outward. The light continues on this linear path unless it is obstructed by something, such as a mirror or an aperture. Ray optics provides an excellent description of the phenomena of reflection and refraction. There do exist phenomena involving light for which ray optics is inadequate: interference and diffraction. These phenomena require the acknowledgement of the wave nature of light.
Physical optics models light as a wave. Light is made up of two components: an electric field
and a magnetic field
, each of which satisfy a wave equation, which in one dimension is:
in these equations,
, which is the speed of light. The solutions of
and
are
where
and
are the wavelength and frequency of the wave, respectively, and
and
are the amplitudes of the oscillating electric and magnetic fields, respectively. The quantities
,
, and
are related by
In this article, all examples will take place at a single instant of time, so the
dependence of
and
can be ignored. Furthermore, since
and
are described by the same function, it will suffice to use a the following generic solution to describe either one.
When light waves of the same wavelength from different sources arrive at the same place at the same time, the waves undergo superposition (addition principle for waves). If they meet in phase, then they constructively interfere. If they meet a half-integer wavelength out of phase, then they destructively interfere. Otherwise the interference is partially destructive. See Figure 1.
Reflection
Simply put, reflection is what happens when light interacts with a shiny surface. See Figure 2.
In this Figure, a ray of light is incident on a plane mirror from the left. The incident ray makes an angle
relative to a line normal to the mirror. The light is reflected at the mirror, and the reflected ray departs to the right. The reflected ray makes an angle
relative to the normal. The Law of Reflection is this: The angle of incidence equals the angle of reflection. Stated in mathematical symbols, this law is
This law can be used to solve a practical problem: How long must a plane mirror be in order to see your entire body in it? To attack this question, a ray diagram is provided. See Figure 3.
A man stands in front of a plane mirror. Rays traveling to the mirror are shown in red, and rays traveling to the man's eye are shown in blue. At the bottom a ray is drawn from the man's foot to the bottom of the mirror, and at the top a ray is drawn from the top of the man's head to the top of the mirror. The reflected rays are both drawn to the man's eye using
. Since reflections from the whole length of his body reach his eye, he sees his entire body in the mirror. The length of the mirror is
, and so it is found that a mirror must be exactly half of your height in order for you to see your whole body in it.
Refraction
The speed of light is the speed with which light travels in a vacuum. It's speed in air is not much different, but when light enters media with greater density, such as water or glass, its speed decreases. The speed
with which light travels in a medium other than a vacuum is
where
is a constant that is characteristic of the medium. For vacuum
, but for any common medium
.
When light passes from one medium to another medium with a different
, it does not just slow down. It also changes direction. The results are summarized qualitatively in Figures 4 and 5.
This phenomenon is called refraction, and
in
is called the index of refraction. The Law of Refraction is
According to
as
increases,
decreases, and vice versa.
When light passes from an medium of higher index of refraction to one of lower index of refraction (as in Figure 5), there exists a critical angle
above which no light is transmitted into Medium 2. In this case,
. See Figure 6.
This phenomenon is called total internal reflection, and it is one of the physical principles on which fiber optic cables work.
Diffraction
Diffraction is the bending of light upon encountering an obstacle or aperture. When plane wave fronts of wavelength
arrive from a faraway source at an aperture of width
, every point in the gap of the aperture is a source of spherical wave fronts that continue onwards. See Figure 7.
A screen placed at a distance
from the aperture displays light whose intensity
is seen to obey the following function. See Figure 8.
Bright spots are observed wherever the intensity has a peak, and dark spots are observed wherever the intensity is zero. This happens because the spherical light waves from each point of the gap of the aperture travel a different distance to the screen, and this can result in a phase difference upon arrival. The condition for a dark spot on the screen is
There is no simple rule like
for bright spots.
Interference
Consider plane wave fronts of light on a two-slit aperture. See Figure 9.
If
is negligible compared to
, then the dominant effect is the interference due to the difference in distance traveled by light from the two slits, whose separation is
. The intensity observed on the screen is given by the following function, which has been plotted in Figure 9.
Bright spots appear on the screen whenever
has a peak, and dark spots appear wherever
is zero. Simple rules exist for finding bright and dark spots:
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