Optics I: Introduction

A short, arbitrary, condensed history of optics

Maxwell's equations

The wave equation

Cool things that happen to lightTotal 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 nearthe tomb of PharaohSesostris II (1900 BCE).

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.

Pyramid of Sesostris II(also known as Senusret II)

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 raysthemselves.

He also explained the lawsof 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.

Willibrord Snell (1591-1626)

n1

n2

1

2

1 1 2 2sin( ) sin( )n n =ni is the refractive index of each medium.

17th-century Optics

Robert Hooke (1635-1703) studied colored interference between thin films and developed the first wave theory of light.

Rene Descartes (1596-1659)

Descartes reasoned that light must be like sound. So he modeled light as pressure variations in a medium (aether).

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

James Clerk Maxwell (1831-1879)

Maxwell unified electricity and magnetism with his now famous equations and showed that light is an electromagnetic wave.

2

0

10

BE E

t

EB B

c t

∂∂∂∂

∇⋅ = ∇× =−

∇⋅ = ∇× =

rr r r r

rr r r r

where is the electric field, is the magnetic field, and c is the velocity of light.

rE

rB

Maxwell’s equations simplify to the wave equation for the electric field.

which has a simple sine-wave solution:

22

2 2

10

EE

c t

∂∂

∇ − =r

r

( , ) cos( )E r t t k rω∝ ± ⋅rr r r

/c kω=r

where

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. Edward Morley

(1838-1923) Albert Michelson

(1852-1931)

Albert Einstein

Einstein showed that light:

Albert Einstein (1879-1955)

is a phenomenon of empty space;

has a velocity that’s constant, independent of observer velocity;

is both a wave and a particle;

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.

The crucial issue is the relative phase of the incident light and this re-emitted 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.

Electric field at atom

Electron cloud

Emitted electric field

On resonance (the light frequency is the same as that of the atom)

( )ex tr

( )E tr

( )E tr +

=

Incident light

Emitted light

Transmitted light

Absorption of light varies massively.

Notice that the penetration depth varies by over ten orders of magnitude!

Visiblespectrum

Pe

netr

atio

n d

epth

into

wa

ter

Wavelength1 km 1 m 1 mm 1 µm 1 nm

UVX-ray

Radio Mic

row

ave

IR

1 mm

1 km

1 µm

1 m

Penetration depth into water vs. wavelength

Water is clear in the visible, but not in other spectral regions.

Variation of the refractive index with wavelength (dispersion) causes the beautiful prismatic effects we know and love.

Prisms disperse white light into its various colors.

Prism

Input white beam

Dispersed beam

Rainbows result from refraction andreflection 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 happensto 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.

If we then combine the two beams, their relative phase matters.The above Sagnac Interferometer measures rotation.

Using a partially reflecting mirror, we can split a beam into two.

Beam-splitter

Inputbeam

Mirror

Mirror

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.

It converts a function of time to one of frequency:

The Fourier transform is perhaps the most important equation in science.

and converting back uses almost the same formula:

The spectrum of a light wave will be given by:2

( )E ω%

%E(ω) = E(t)exp(−iωt)dt−∞

∞∫

E(t) = 12π

%E(ω)exp(+iωt)dω−∞

∞

∫

Diffraction

Light bends around corners. This is called diffraction.

The diffraction pattern far away is the (2D) Fourier transform of the slit transmission vs. position.

The light pattern emerging from a

single small rectangular

opening

Light is not only a wave, but also a particle.

Photographs taken in dimmer light look grainier.

Very very dim Very dim Dim

Bright Very bright 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 aresult, 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 theshortest 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.