Optics I: Introduction
A short, arbitrary, condensed history of optics
Maxwell's equations
The wave equation
Cool things that happen to light
Total internal reflection
Interference
Diffraction
The laser
Nonlinear optics
Ultrafast optics
The Fourier transform and its key role in optics
Optics in
Ancient History
A mirror was discovered
in workers' quarters near
the tomb of Pharaoh
Sesostris II (1900 BCE).
Pyramid of Sesostris II
(also known as Senusret II)
Ancient Greeks (500-300 BCE)
Burning glass mentioned by Aristophanes (424 BCE)
Law of reflection: “Catoptrics” by Euclid (300 BCE)
Refraction in water mentioned by Plato in “The Republic”
But they thought that the eye emits rays that reflect off objects.
Ancient Greeks: Ancient light weapons
Early Greek and
Roman historians
report that Archimedes
equipped several
hundred people with
metal mirrors to focus
sunlight onto Roman
warships in the battle
of Syracuse (213 -211
BCE).
This story is probably apocryphal.
Optics in the Middle Ages: Alhazen
Arab scientist Alhazen (~1000 AD) studied spherical
and parabolic mirrors.
Alhazen correctly proposed that
the eyes passively receive light
reflected from objects, rather
than emanating light rays
themselves.
He also explained the laws
of reflection and refraction
by the slower movement of
light through denser substances.
Optics in early 17th-century Europe
Hans Lippershey applied
for a patent on the Galilean
telescope in 1608.
Galileo (1564-1642) used
one to look at our moon,
Jupiter and its moons, and
the sun.
Two of Galileo’s telescopes
Galileo’s drawings of the moon
Willibrord Snell
Willibrord Snell discovered
the Law of Refraction, now
named after him.
q1
n1
q2
Willibrord Snell
(1591-1626)
n2
ni is the refractive index of each
medium.
n1 sin(q1 )  n2 sin(q2 )
17th-century Optics
Descartes reasoned that
light must be like sound.
So he modeled light as
pressure variations in a
medium (aether).
Rene Descartes (1596-1659)
Robert Hooke (1635-1703) studied colored
interference between thin films and developed the
first wave theory of light.
Isaac Newton
"I procured me a triangular glass
prism to try therewith the
celebrated phenomena of colours."
(Newton, 1665)
Isaac Newton
(1642-1727)
After remaining ambivalent for many years, he eventually
concluded that it was evidence for a particle theory of light.
James Clerk Maxwell
Maxwell unified electricity and
magnetism with his now famous
equations and showed that light is
an electromagnetic wave.
E  0
B  0
B
 E  
t
1 E
 B  2
c t
James Clerk Maxwell
(1831-1879)
where E is the electric field, B is the magnetic field,
and c is the velocity of light.
Maxwell’s equations simplify to the
wave equation for the electric field.
2
1

E
2
 E 2
0
2
c t
which has a simple sine-wave solution:
E (r , t )  cos(t  k  r )
where
c / k
The same is true for the magnetic field.
Light is an electromagnetic wave.
The electric (E) and magnetic (B) fields are in phase.
The electric field, the magnetic field, and the
propagation direction are all perpendicular.
Michelson & Morley
Michelson and Morley then
attempted to measure the
earth's velocity with respect
to the aether and found it to
be zero, effectively disproving
the existence of the aether.
Albert Michelson Edward Morley
(1852-1931)
(1838-1923)
Albert Einstein
Einstein showed that light:
is a phenomenon of empty space;
has a velocity that’s constant,
independent of observer velocity;
is both a wave and a particle;
Albert Einstein (1879-1955)
Excited medium
and undergoes
stimulated emission,
the basis of the laser.
The interaction of light and matter
Light excites atoms, which then emit more light.
Electric field
at atom
Electron
cloud
Emitted
electric field
Incident light
+
E (t )
xe (t )
E (t )
Emitted light
=
On resonance (the light frequency is the
same as that of the atom)
Transmitted light
The crucial issue is the relative phase of the incident light and this reemitted light. If these two waves are ~180° out of phase, destructive
interference occurs, and the beam will be attenuated—absorption.
If they’re ~±90° out of phase: the speed of light changes—refraction.
Absorption of light varies massively.
Penetration depth into water vs. wavelength
1m
Radio
Microwave
Penetration depth into water
1 km
IR
UV
X-ray
Water is clear in the
visible, but not in
other spectral
regions.
1 mm
Notice that the
penetration depth varies
by over ten orders of
magnitude!
1 µm
1 km
1m
1 mm
1 µm
Wavelength
Visible
spectrum
1 nm
Variation of the refractive index with
wavelength (dispersion) causes the beautiful
prismatic effects we know and love.
Input
white
beam
Dispersed beam
Prism
Prisms disperse white light into its various colors.
Rainbows result from refraction and
reflection of sunlight in water droplets.
Note that there can be two rainbows, and the top one is inverted.
There are many interesting optics effects in real life…
An interesting question is what happens
to light when it encounters a surface.
At an oblique angle, light can be completely transmitted
or completely reflected.
Total internal reflection is the basis of optical fibers,
a billion dollar industry.
Light beams can interfere with each
other: Two point sources…
Different separations. Note the different patterns.
Constructive vs. destructive interference…
The idea is central to many laser techniques, such as holography,
ultrafast photography, and acousto-optic modulators.
Tests of quantum mechanics also use it.
Light beams can be intentionally made
to interfere with each other.
Using a partially reflecting mirror, we can split a beam into two.
Mirror
Beamsplitter
Mirror
Input
beam
If we then combine the two beams, their relative phase matters.
The above Sagnac Interferometer measures rotation.
Often, they do so by themselves.
Fourier decomposing
functions plays a big
role in optics.
Here, we write a square wave as a
sum of sine waves of different
frequency.
The Fourier transform is perhaps the
most important equation in science.
It converts a function of time to one of frequency:
E( ) 



E(t)exp(i t)dt
and converting back uses almost the same formula:

E(t)  1
2

E( )exp(i t)d

The spectrum of a light wave will be given by:
E ( )
2
Diffraction
Light bends around corners. This is called diffraction.
The light pattern
emerging from a
single small
rectangular
opening
The diffraction pattern far away is the (2D) Fourier transform of the
slit transmission vs. position.
Light is not only a wave, but also a particle.
Photographs taken in dimmer light look grainier.
Very very dim
Bright
Very dim
Very bright
Dim
Very very bright
When we detect very weak light, we find that it’s made up of
particles. We call them photons.
The Laser
A laser is a medium that stores energy, surrounded by two mirrors.
Photons entering the medium undergo stimulated emission. As a
result, the intensity exiting from the medium exceeds that entering it.
A partially reflecting output mirror lets some light out.
A laser will lase if the beam increases in intensity during a round trip:
that is, if I3 > I0.
Electromagnetism is linear: The
principle of Superposition holds.
If E1(x,y,z,t) and E2(x,y,z,t) are solutions to Maxwell’s
equations,
then E1(x,y,z,t) + E2(x,y,z,t) is also a solution.
This means that light beams pass through each
other without affecting each other.
Nonlinear Optics produces many exotic
effects.
Sending high-intensity infrared
laser light into a crystal yielded
this display of green light:
Nonlinear optics allows us to
change the color of a light
beam, to change its shape in
space and time, and to test the
fundamental principles of
quantum mechanics.
Ultrashort laser pulses are the
shortest events ever created.
This pulse is only
4.5 x 10-15 seconds
long, that is, 4.5
femtoseconds:
How do we measure such a short event?
We must use the event
to measure itself.