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A. La Rosa
Lecture Notes
APPLIED OPTICS
Lecture-1: EVOLUTION of our UNDERSTANDING of LIGHT
_______________________________________________________________
What is light?
A.
Is it a wave?
Is it a stream of particles?
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.
 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.1
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
1
 Predicts EM-waves travel at speed =
 0 0
 It turns out
1
 0
0
 300,000
km
!!
S
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.
Reflecting
walls

I() 
Incident
radiation
Scattered (re-emitted)
light

xo
qe, me
Atom (modeled as
an oscillator)
xo is the electron’s
amplitude of oscillations.
me, qe: electron’s mass
and charge.
: angular freq of oscillations (electron
and light).
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 :
Average energy of each mode :
2
~ f
6 2 c 2
kT


2  
I
(

)

f
kT




Spectral density
2 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()
Classical
prediction
Experimental
results
Frequency 
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
'


Incident
x-ray radiation
Scattered
light

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

Incident
radiation
P l a n c k p o s t u l a tneedr gt yh aot f t h e e
t h e o s c (i lol fa tn oa rt u r a l f )r e q u e n c y
is quantized
Fig.5 An atom (of natural frequency w) takes up energy
from the incident radiation in the form of lumps having
, 2, 3, …
energy values
Planck’s
calculation
of
average

WPlanckl (ω)  E 

En e
n0

e
 En / kBT
 En / kBT
n0
where
En
= n();
n= 1 2, 3, …
energy
of
the
oscillator:
assical
ediction
WPlanckl (ω)  
1

e kT  1
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 :
Average energy of each mode :
2
~ f
6 2 c 2
WPlanckl (ω)  
1

e kT  1


2  
I
(

)

f
W
Planck




Spectral density
2 2

 6 c  
(2)
I()
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
i

 2  2
 
 V ( x,t )
t
2m  x 2
“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.
n=1
D
S
1m
x
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) = eit 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 
probability A.
QED RULE-2
eit X)
(a complex number) is assigned to the amplitude
(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
≡
 A(SXP)
all X
=
i t X)
 e 
(2)
X
(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 = A 2
(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
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.
H
J
M
B
C
G
A
D
I
E
L
K
F
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.
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.