# Document

```Wave Optics
What is Light?
Light is a name for a range of electromagnetic radiation
that can be detected by the human eye. What is
Electromagnetic radiation has a dual nature as both
particles and waves. One way to look at it is as changing
electric and magnetic fields which propagate through
space, forming an electromagnetic wave. This wave has
amplitude, which is the brightness of the light,
wavelength, which is the color of the light, and direction at
which it is vibrating, called polarization. This is the
classical interpretation, described by Maxwell's Equations.
What is Light?
Than Planck, Einstein and others came along with
quantum theory. In terms of the modern quantum
theory, electromagnetic radiation consists of particles
called photons, which are packets ("quanta") of energy
which move at the speed of light. In this particle view
of light, the brightness of the light is the number of
photons, the color of the light is the energy contained
in each photon, and four quantum numbers determine
the polarization.
What is Light?
Which interpretation is correct? Both of them, actually. It turns out
electromagnetic radiation can have both wave-like and particle-like
properties. In this exploration of light, we will primarily take the
wave viewpoint as it is a more useful description of the everyday
properties of light, but keep in mind that both viewpoints are valid,
and sometimes we will use the quantum viewpoint too.
Light ranges from wavelengths of 7x10-5 cm (red) to 4x10-5 cm
(violet) and (like all electromagnetic radiation) travels at the speed of
light, 299,792,458 meters per second.
The frequency (number of wavelengths per second) of a light wave
may be calculated using the equation
c  
In quantum theory, a photon has energy equal to
h
Physical optics
Physical optics, or wave optics, is the branch of optics
which studies interference, diffraction, polarization, and
other phenomena for which the ray approximation of
geometric optics is not valid.
Geometric optics is an approximation of physical optics
which ignores wavelength (  0), and consequently wave
effects.
Huygens' principle
Huygens considered light to be a wave. He envisioned a wave crest
advancing by imagining each point along the wave crest to be source
point for small, circular, expanding wavelets, which expand with the
speed of the wave. The surface tangent to these wavelets determines
the contour of the advancing wave. Figure illustrates Huygens'
construction for a plane wave (a) and for a spherical wave (b)
Conditions for Interference
Coherent waves maintain constant phase in time for each
point of space
For interference of light waves the sources must be coherent that
is they maintain constant phase with respect to each other.
To get coherence, use a monochromatic source shining through 2
slits, or use laser, which is monochromatic, and send the beam
through 2 different paths (can use slits or other ways)
Don’t normally see interference in light, because:
Light waves have such small wavelengths
Light waves are usually not coherent
Ordinary sources produce well-phased light for only about 10-8
sec, effectively incoherent
Ordinary sources have random changing relative phase incoherent
Interference
Because light is a wave, the superposition principle is
valid to determine the constructive and destructive
interferences for light waves. Interference in light
waves is not easy to observe because the wavelengths
are so short. For constructive interference, two waves
must have the two contributing crests and the two
troughs arriving at the same time. For destructive
interference, a crest from one wave and a trough from
the other must arrive at a given point at the same time.
Young’s Double-Slit Experiment
Thomas Young first demonstrated interference from light waves with a
double slit. The schematic diagram for this experiment is shown in Figure.
The single light source is located at S0 (the
light waves must have identical frequency
and phase. The light beam is also
considered to be of one color), and the light
goes through two very narrow openings at
S1 and S2. Each of the slits act as a source
for circular expanding waves. The points of
intersection of two crests, one from each
slit, are points of constructive interference.
The point of intersection of a crest from one
slit and a trough from the other slit is a
point of destructive interference. Therefore,
the interference pattern called fringes,
consisting of alternating light and dark bars,
will be seen on the screen.
Figure illustrates the rays coming through two slits that are directed to
the point P on the screen.
The difference in path length of the two rays is given by d sin θ = l2 =
l1. If the path difference is a whole number of wavelengths, then
constructive interference takes place. If the paths differ by a half
number of wave lengths, destructive interference occurs. Using n to
represent any integer, the two cases may be written
Diffraction of light through a single slit.
Fresnel diffraction
A single slit yields an interference pattern due to diffraction and
interference. Imagine that the slit is wide enough to allow a number of
wavelets.
The rays from A and B interfere at P on a
distant screen. As shown, AP exceeds BP by
half a wavelength; therefore, the represented
waves destructively interfere. Also for every
wave originating between A and B, there is
another point between B and C with a wavelet
that will destructively interfere. The wavelets
cancel in pairs; thus, point P is a minimum or
dark point on the screen.
Whenever the path difference between AP and CP is a whole number
of wavelengths, a dark fringe will be produced on the screen because
the wavelets can be seen to completely cancel in pairs.
Figure illustrates the light rays traveling to another point on the screen.
The region of wavelets is divided into three.
Again, the waves through two regions
cancel in pairs, but now the waves from one
region constructively interfere to produce a
bright point on the screen. This is partial
reinforcement.
The positions of the light and dark
fringes formed by a single slit are
summarized in the intensity versus
angle sketch shown in Figure. The
center region of the pattern will be the
brightest band because the wavelets
completely, constructively interfere in
the middle.
Fraunhofer diffraction
If you replace a circular hole with a circular disk
of the same size, you get essentially the same
pattern, even with a bright spot in the center.
The Diffraction Grating
Polarization of light
Light is a transverse wave, meaning that it oscillates in
a direction perpendicular to the direction in which it is
traveling. However, a wave is free to oscillate right and
left or up and down or at any angle between the
vertical and horizontal.
Natural light is generally unpolarized, all planes of propagation
being equally probable.
Classification of Polarization. Linear polarization
Light in the form of a plane wave in space is said to be
linearly polarized. The transverse electric field wave is
accompanied by a magnetic field wave as illustrated.
Circular Polarization
Circularly polarized light consists of two perpendicular
electromagnetic plane waves of equal amplitude and 90° difference in
phase. The light illustrated is right- circularly polarized.
Elliptical Polarization
Elliptically polarized light consists of two perpendicular waves of
unequal amplitude which differ in phase by 90°. The illustration shows
right- elliptically polarized light.
Methods for achieving Polarization of light
Polarization may be achieved by reflection, scattering,
absorption or by the use of Birefringent Materials in Nicol prism.
Polarization by Absorption
Some kinds of crystals have a special property of polarizing
light, meaning that they force light to oscillate only in the
direction in which the crystals are aligned. They absorb more
light in one incident plane than another, so that light progressing
through the material become more and more polarized as they
proceed. This anisotropy in absorption is called dichroism. There
are several naturally occurring dichroic materials, and the
commercial material polaroid also polarizes by selective
absorption.
The scattering of light off air molecules
produces linearly polarized light in the
plane perpendicular to the incident light.
Polarization can be analized by crossed polarizers. Law of Malus.
Since the transmitted intensity
is proportional to the amplitude
squared, the intensity is given
by:
The Law of Malus gives the transmitted intensity through two ideal
polarizers. Note that it gives zero intensity for crossed polarizers.
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