A. La Rosa
PSU-Physics
Lecture Notes
PH 464/564 ECE 594
APPLIED OPTICS
________________________________________________________________________
How does light "really" decides which path to follow?
An elementary Quantum Dynamics point of view
References: Richard Feynman, “QED, The strange theory of light and matter,” Princeton University
Press (1985).
Feynman Lectures, Vol-I, Chapter 26, "Optics: The Principle of Least Time."
Section 26-6.
Objective
This section aims to describe the connection between
i) the quantum mechanics concept of amplitude probabilities (to be described
below); and
ii) the classical variational principle (described already in the previous sections)
Amplitude Probability
Tracing light from point source S to point P via reflection on a mirror:
Light does not seem to behave as a wave.
Instead, the rays seem to be made out of photons
(which are detected in discrete quantities by a photon counter).
S
P
X
Mirror
Fig.1 Particular geometrical rectilinear path available for a photon to
travel from, S to P via reflection on the mirror. (X is an arbitrary point
on the mirror.)
How does light really propagates? How can we predict the dynamics involved in its
propagation?
Below we present a version of what is believed it really happens, from a quantum
dynamics (in an elementary version) point of view:
The arrival of a photon starting at S and ending at P
is probabilistic.
The calculation of the probability is as follows.
CASE: Tracing light from point source S to point P via reflection on a mirror
This time we consider that light does not seem to behave as a wave (in the sense
described by the Huygens or the double slit experiments). Instead, light rays seem to be
made out of photons (as revealed by the photoelectric effect experiment), which are
detected in discrete quantities by a photon counter.
A new model to predict the way in which light propagates. Feynman presents a version
of what is believed it really happens, from an elementary quantum dynamics point of
view:
The arrival of a photon starting at S and ending at P
is probabilistic.
That is, in the situation shown in Fig. 1 below, there are many paths possible for a
photon leaving S to arrive P, via one reflection on the surface. One of those possible
paths is shown in Fig. 1
S
P
X
Mirror
Fig. 1 Particular geometrical rectilinear path available for a photon to travel
from S to P, via reflection on the mirror. (X is an arbitrary point on the
mirror.)
The calculation of the probability is as follows.

All arbitrary paths “S mirror P” are possible.
(Figure shows only straight line paths, but, in general, any curved path can be
considered.)

For a particular path SXP (where X is an arbitrary point at the mirror)
an AMPLITUDE PROBABILITY A(SXP) is assigned to it.
A(SXP) = ei
where t(X)

is the angular frequency of the light
(1)

t(X)
is the time taken by the light if it were to follow the particular
path SXP.

is determined by the path traversed (we will assume it does not
have a major effect in calculating A (SXP)
We will argue below that the phase  plays a more preponderant role.
Since all the possible events are characterized by the same
initial state (photon leaving S), and
final state (photon arriving at P),
the total amplitude probability is obtained by adding all the individual amplitudes,
Total amplitude probability =
(photon starts at S
and arrives at P)
=
 X A(SXP)
 X eiX
(2)
The summation is
over all the points X
on the mirror
Expression (2) is then an addition of phasors whose phase (x) will be determined
by the corresponding time of travel. A graphical construction of this total phasor is
given in the figure below.
S
P
X
 X A(SXP)
Mirror
e X
i
Fig. 2 A complex number is attached to each path. The addition of all these complex
numbers (corresponding to all the possible paths) gives the total amplitude probability.

The probability that a photon starting at S ends at P is given by the magnitude
square of the phasor given in expression (2).
Pprob(SP) =  A ( SXP )
2
(3)
X
Amplitude Probability and the Variational Principle
How does this ultimate picture of photons and amplitude probability relate to the
variational principle?
As an example let’s evaluate graphically the total phasor indicated by expression (2) but
take just a few representative paths.
First, from our experience of using the classical Fermat’s principle, we know that the
time of travels varies like this,
t
A
B
S
C
D
E
S
P
A
B
C
D
E
P
F
For paths neighbor to SAP
the corresponding phasors
will change dramatically from
one path to the next because the
time of travel vary substantially
even for neighbor paths
A
A
B
C
D
E
F
For paths neighbor to SEP
(similarly)
E
For paths closer to the one that makes the optical path
stationary, the time does not change drastically; so the
corresponding phasors align each other
S
P
A
B
C
D
E
F
C
The paths that are neighbor to the “classic” path are the ones that contribute the most
to the (complex) amplitude probability. We see here, therefore, the connections
between the quantum mechanics concept of amplitude probabilities and the classical
variational principle.
A
C
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)
(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.
(1)
That is, a phasor 
probability A.
eit X) (a complex number) is assigned to the amplitude
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 ≡
 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 different path that has different
length.
H
B
C
A
D
J
I
G
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.