A. La Rosa Lecture Notes

APPLIED OPTICS

Lecture-1: EVOLUTION of our UNDERSTANDING of LIGHT

_______________________________________________________________

What is light? Is it a wave?

Is it a stream of particles?

A. Light as a particle

NEWTON (1642 – 1727) was the

most prominent advocate of this theory

Ray of light conceptualized as a stream of

very small particles emitted from a source

of light and traveling in straight lines. Fig.1

This view was based on the fact that light

can cast sharp shadows.

But cannot explain well what is now known

as Newton’s rings

Fig.2 Newton’s rings.1

B. Light as a WAVE

CHRISTIAN HUYGENS (1629 – 1695) contemporary of Newton, championed

this view

When two beams of light intersect they emerge unmodified (different

than the case when ‘particles’ collide.)

1801 YOUNG’s Double-Slit experiment

The complex shadows formed by

the two slits (in the form of

seem to demand an interpretation

of light as a wave.

Fig.3 Two-slit experiment

1 We know that the contemporary interpretation of light is that it is constituted by photon

‘particles’. Can you envision a way to interpret these Newton’s rings under this ‘photon” particles

view?

1821 FRESNEL

Light is a transverse wave

Light is polarized

Explained the phenomenon of double

refraction in calcite. Fig. 4

Independent of this progress in optics, the study of electricity and magnetism was

also flourishing

JAMES C. MAXWELL (1831-1839) is a genius who condensed the phenomena

of electromagnetism into a set of four equations

Predicts EM-waves travel at speed = 00

1

It turns out S

km000,300

1

00

!!

Light must be an electromagnetic wave !

1887 HEINRICH HERTZ confirms experimentally

the existence of electromagnetic waves

1887 A. MICHELSON and E. MORLEY

if the speed of light is constant in the aether and the earth presumably moves

in relation to the aether (at ~67,000 mi/h)

then the speed of light with respect to the

earth should be affected by the planet’s

motion. BUT no motion of the earth with

respect to the aether was detected.

A. EINSTEIN, special theory of relativity

Rejected the aether hypothesis

Light always propagates with a definite velocity c (in empty

space) which is independent of the motion of the light source

Light was then envisaged as a self-sustained electromagnetic wave

The 19th century: served to place the wave theory of light on a firm

foundation.

C. Interpretation of Light as a WAVE is inconsistent with some

experiments

However, by the end of 19th century - beginning of 20th century:

It became evident that the wave theory of light could

not explain certain experiments (the blackbody

radiation, the photoelectric effect for example.)

Indeed, the wave theory of light began to crumble.

The ultraviolet catastrophe Predicting the spectral density )(I of light inside a cavity at a temperature T.

Scattered (re-emitted) light

Atom (modeled as an oscillator)

Incident radiation

I()

Reflecting walls

qe, me

xo is the electron’s amplitude of os- cillations.

me, qe: electron’s mass

and charge.

: angular freq of os- cillations (electron and light).

xo

Fig.5 Light intensity existent inside the cavity is absorbed by the atom and re-emitted in all directions. At equilibrium, the rate at which light energy is absorbed and the rate at which the atom re-emit the light must be equal, which requires a particular value of the spectral light intensity I()

Classical prediction

Number of modes of frequency : 22

2

6 cf~

Average energy of each mode : kT

Spectral density

kT fI

22

2

6)(

c

(1)

However, the experimental results were quite different in the high frequency

regime. Only at low frequencies there was an agreement between

the classical prediction and the experimental results.

I()

Frequency

Classical prediction Experimental

results

Fig.2 The serious discrepancy between the experimental results and the theoretical prediction of the spectral light intensity at high frequencies is called the ultraviolet catastrophe.

The photoelectric effect

Fig.3 Einstein proposed an explanation based on quantized electromagnetic

fields (1905), corroborated by Milikan in 1914.

The Compton effect

Scattered light

Incident x-ray radiation

Compton, 1923 X-rays incident on a graphite target.

electron

'

Fig. 4 Scattered light contains frequencies different than the incident one.

D. Planck’s Hypothesis of quantized energy

In dealing with the ultraviolet catastrophe problem, it turns out, the difficulty

brought by classical physics was that, in general, it assigned an average energy of

the oscillator equal to kT, regardless of the natural frequency (o) of the oscillator.

Planck (1900) realized that he could obtain an agreement with the experimental

results if,

rather than treating the energy of an oscillator (of natural frequency

) as a continuous variable, the energy states of the oscillator have

only discrete values: 0 , , 2, 3, …

the energy steps would be different for each frequency

=(

where the specific dependence of in terms of had to be

determined.(Notice we have dropped the use of the sun-index zero when

indicating the natural frequency).

q

I n c i d e n t r a d i a t i o n

P l a n c k p o s t u l a t e d t h a t t h e en e r g y o f

t h e o s c i l l a t o r ( o f n a t u r a l f r e q u e n c y ) i s q u a n t i z e d

Fig.5 An atom (of natural frequency w) takes up energy

from the incident radiation in the form of lumps having

energy values , 2, 3, …

Planck’s calculation of average energy of the oscillator:

0

0

/

/

)(

n

Bn

Bn

n

n

PlancklTk

TkE

EωE

E

e

e

W

where nE = n(); n= 1 2, 3, …

1

1)(

kT

Planckl

e

ωW

average energy of an oscillator (atom) of natural frequency

. (Notice, it is different than the classical prediction kT)

Planck’s prediction

Number of modes of frequency : 22

2

6 cf~

Average energy of each mode :

1

1)(

kT

Planckl

e

ωW

Spectral density

PlanckW fI

22

2

6)(

c

(2)

I()

Classical prediction

Experimental

results

Particle’s and wave’s energy quantization

Historically. Planck initially (1900) postulated only that the energy of the oscillating particle (electrons in the walls of the blackbody) is quantized. The electromagnetic energy, once radiated, would spread as a continuous.

It was not until later that Plank accepted that the oscillating electromagnetic waves were themselves quantized. The latter hypothesis was introduced by Einstein

(1905) in the context of explaining the photoelectric effect, which was corroborated later by Millikan (1914).

E. Quantum Mechanics

Schrodinger Equation

)( 2

2

22

tx,Vxmt

i

“This equation marked a historic moment constituting the birth of the quantum

mechanical description of matter. The great historical moment marking the birth of

the quantum mechanical description of matter occurred when Schrodinger first

wrote down his equation in 1926.

For many years the internal atomic structure of the matter had been a great

mystery. No one had been able to understand what held matter together, why

there was chemical binding, and especially how it could be that atoms could be

stable. (Although Bohr had been able to give a description of the internal motion

of an electron in a hydrogen atom which seemed to explain the observed

spectrum of light emitted by this atom, the reason that electrons moved this way

remained a mystery.)

Schrodinger’s discovery of the proper equations of motion for electrons on an

atomic scale provided a theory from which atomic phenomena could be

calculated quantitatively, accurately and in detail.” Feynman’s Lectures, Vol III,

page 16-13

F. Quantum Electrodynamics (QED)

Reference: The following description is taken from Feynman, “QED, The strange theory of light and

matter,” Princeton University Press (1985).

Quantum mechanics was a tremendous success (could explain chemistry).

However, the description of light-matter interaction still faced difficulties. [Maxwell

theory of electricity and magnetism had to be changed to be in accord with the new

properties of quantum mechanics. The theory of light-matter interaction, called

“quantum electrodynamics” was finally developed in 1929].

But the theory was troubled.

Right after Schrodinger, Dirac developed a relativistic theory of the electron that did

not take into account the effects of the electron’s interaction with light. But it was

expected to provide a good starting point.

Quantum electrodynamics was straightened out by Julian Schwinger, Sin-Itiro

Tomonaga and Feynman in 1948. This is the theory that Feynman describes in his

“QED, The strange theory of light and matter,” Princeton University Press (1985). Such

theory has been tested over a wide range on conditions. Aside from gravitation and

nuclear physics, QED can explain every phenomenon accurately.

QED is also the prototype for new theories that attempt to explain things going on

inside the nuclei of atoms. It turns out, quarks, gluons, …, etc. all behave in a certain

style, the “quantum” style.

F1. QED analysis of light phenomena

Photons: Particles of Light

QED considers light to be made of particles (as Newton originally thought), but the

price of this great advancement of science is a retreat by physics to the position of

being able to calculate only the probabilities that a photon will hit a detector.

Event: Light travels from the source S, reflects from the surface at X, and

arrives to the detector at D

We assign an amplitude probability (a complex number) that such an event will occur.

S

1 m

x

Consider 10 paths of

increasing length

n=1 D

X

Fig. Light reflected from a mirror

QED RULE-1 (Assignment of amplitude probability)

How to do such an amplitude probability assignment?

i) Pictorial view We assign to the amplitude probability an arrow.

To obtain the arrow we do the following:

Let’s imagine that we have a stopwatch that can time a photon as it

moves. It has a handle that moves rapidly. When the photon leaves

the source, we start the watch. As long as the photon moves, the

handle turns. When the photon arrives at the detector we stop the

watch.

The handle ends up pointing in a certain direction.

We then draw a corresponding arrow of magnitude 1 (blue arrow in

the figure).

ii) More formal view

Amplitude probability ≡ A(SXP) = eit

X) (1)

(photon starts at S and

arrives at D via the path S X P)

where is the angular frequency of the light and

t(X) is the time the light takes to travel via SXP.

That is, a phasor eit

X) (a complex number) is assigned to the amplitude

probability A.

QED RULE-2 (For events that have the same initial and final states)

Since the photon has many optional paths available to go P from X, the total

amplitude probability is given by,

Total amplitude probability A ≡ Xall

A(SXP) = X

eit

X) (2)

(photon starts at S and arrives at P via any path joining S and P)

where is the angular frequency of the light and t(X) is the time light takes to travel

from S to P passing through X (the latter located at the interface).

QED RULE-3 The probability for an event to occur is given by the square

of the final amplitude: P = 2 A

(3)

Consider the reflection of light coming from a surce S and reaching a detector via

reflection from a mirror.

We want to calculate the chance that the detector will make (4)

a “click” after a photon has been emitted by the source

Classical view: The mirror will reflect light where the angle of incidence is

equal to the angle of reflection

Classical way: In the graph above, it would appear that the ends of the mirror

contribute to nothing to the reflection phenomena.

QED view: Every possible path contributes to the amplitude probability.

Should the reflections path involving the center of the mirror have more

weight than the once reflecting from the edges? Answer: No. All the

path have equal chance.

QED view: QED view assigns an equal amplitude probability to

each possible path

For the analysis of a mirror, an infinite number of path would have to be

considered. To simplify the problem, let’s divide the mirror into a number of

smaller discrete strips.

S P

Each amplitude will be represented by an arrow of a standard (arbitrary) size Consider 10 paths of

increasing length

While the size of the arrow will be essentially the same, its orientation will be

different for the different reflection point selected. This is because it takes a

different time for as photon to go through a differente path that have different

length.

Top: Figure shows each possible path the photon could take to go from the source

to that point in the mirror and then to the detector. Middle: A plot of the

corresponding time for each possible path. Below the graph is the direction of each

amplitude probability (arrow). Bottom: The result of adding all the arrows. Notice

the major contribution to the total arrow comes from arrows E through I, whose

directions are nearly the same because the timing of their path is nearly the same.

This also happens to be where the total time is least. It is therefore approximately

right to say that light goes where the time is least.

G

H

M

J

K

L

I

E

A

B

F D

C

Why do the edges appear to make no contribution?

We zoom in to see in more detail the contribution to the total amplitude probability from

the edges of the mirror. Notice when the arrows are added, they go in a circle, hence

adding up nearly to nothing.

The above gives us a clue to engineer a way to get contribution from th e edges.

As we move from left to right, we notice the arrows have a bias orientation to

the right then to the left, and so on. If only the section with arrows biased to

the right are kept (etching away the sections with arrows to the left), then a

substantial amount of light will be reflected. Such a mirror with preferentially

etched regions is called a diffraction grating.