A. La Rosa
Lecture Notes
APPLIED OPTICS
Lecture-15 PROPAGATION of LIGHT
______________________________________________________________________
Here we explore different approaches used to describe the
propagation of light.
I. Huygens-Fresnel principle
Wavefront, Huygens principle, Huygens-Fresnel principle
II. Rayleigh scattering (Scattering from small particles)
II.1. Self-sustained propagation of electromagnetic fields
II.2. Elastic scattering from particles of size smaller than the wavelength
II.2.1 Propagation of light in tenuous media
Case: Lateral scattering
Case: Forward propagation
II.2.2 Propagation of light in dense media
III. Scattering from large particles
IV. Transmission and the Index of Refraction
V. Diffraction
Contrasting geometrical optics and diffraction. Difference between interference
and diffraction. Far- and near-field diffraction
V. 1 Fraunhofer Diffraction
V.2 Fresnel Diffraction
I. Huygens-Fresnel principle
I.1 Wavefront: It is the leading surface of a wave
disturbance
Wavefront ∑ will be distorted after passing the piece of
glass.
How to determine the new wavefront ∑ ′?
I.2 Huygens Principle
Ref: Hecht, “Optics”, Section 4.4.2
I.3 Huygens-Fresnel Principle
Ref: Hecht, “Optics”, Section 10.1
II. RAYLEIGH SCATTERING
Reference:
Eugene Hecht, "Optics," 4th Edition, Addison Wesley: Sections 4.1, 4.2
II.1 Self-sustained propagation of electromagnetic fields
In the empty space, light propagates forever.
E M P T Y
S P A C E
Light continues
propagating forward
forever
Beam can not be seen
from the side
Fig.1 The time-dependent electric and magnetic fields self-sustain each other,
resulting in an electromagnetic wave that propagates forever.
Ez and Bx sustain each other
and propagate with speed c
Z
B
E z
 x
y
t
Ez
c
Bx
X
 2 Ez 1  2 Ez

0
y 2 c 2 t 2
Ez ( y, t )  f ( y  ct )
Y
Bx
E z
   o o
y
t
where c 
1
μo ε o
propagating wave
II.2 Elastic scattering from particles of size smaller than the
wavelength
\When light propagate in the Earth atmosphere, it interacts with molecules of
nitrogen, oxygen,
These molecules have resonance in the uv-region; therefore
the incident visible light cannot cause resonance absorption.
The medium is therefore transparent.
Still, the electron clouds of the molecules can be driven by the incident radiation.
A photon is absorbed and re-emitted with the same frequency; i. e. light is
elastically scattered, randomly everywhere.
This happens particularly in a tenuous region of the atmosphere.
EARTH
ATMOSPHERE
Visible
light
Molecules
of air



Beam can be seen from the
side
Fig.2 The presence of small particles (size smaller than lambda) gives rise to elastic
scattering
Scattering from particles whose size is smaller than the
(1)
wavelength (let’s say size < /15) is referred as Rayleigh Scattering
Analytical description. One classical approach considers each molecule as a little
oscillator of resonance frequency 0 (where  0 is one of the resonance absorption
frequencies of the molecules.)

E  Eo e jω t

Incident
visible light
≠  0

Scattered
photons
x  [ Ae j ] e jω t
A  A( )
Molecule,
   ( )
Resonant absorption,  0

Fig. 3 Incident light of frequency  interacts with the molecules, and light of
the same frequency is (re-emitted) scattered in all directions.
Typical of resonant phenomena:
The closer  is to 0  the higher the amplitude A( ) , 
 the stronger the scattering
(2)
This implies:
Blue light scatters
strongly
 while red light scatters
comparatively weaker
(3)
Equivalently,
the forward propagating light is richer in red
the scattered out light will be richer in blue
Thus, Rayleigh scattering contributes to the blue color of the sky.
‘White’
light
EARTH
ATMOSPHERE
Forward beam
stronger
in red
Scattered light
stronger in blue
Fig. 4 The forward beam and the light scattered laterally present different coloration
Quantitatively, how much stronger is blue light scattered compared to red light?
To answer this question, let’s use the argument that an accelerated charge
produces an electromagnetic field.
From the figure above:
x  [ Ae j ] e jω t , v  j [ Ae j ] e jω t , a   2 [ Ae j ] e jω t
Electric field ~ acceleration
E ~ 2
Light intensity I ~ E2 ~ 4
(4)
~ (1/)4
Ired / Iblue = (  blue /  red)4 = (450 / )4 = ( 0.7 )4 = 0.22
II.2.1 Propagation of light in tenuous media
Lateral scattering and forward propagation
We want to understand why the denser the material the less the
lateral scattering
II.2.1.A Case: Lateral scattering
Molecules randomly distributed is space
Separation d between molecules greater than the wavelength
d > 

P
Fig. 5 Molecules in a tenuous media (average distance between molecules is greater
than the participating wavelengths) behave as independent oscillators, hence
producing a net intensity at P.
Let’s analyze the interference between the waves re-emitted by two arbitrary
molecules. For simplicity, let use two plane waves as a model (see also Fig.6).
E1 (at P)  E10COS (k r1   t  1 )
E 2 (at P)  E 20COS (k r2   t  2 )
(5)
Let’s assume also that both E10 and E 20 point in the direction out the plane of
the figure.
r1
r2
P
Fig. 6 Light emitted by the two sources modeled as two
plane waves
The intensity at P will be given by,
I (at P)   0 c  ( E1  E 2 )  ( E1  E 2 ) 
(6)
where the symbol   stands for a time average evaluation
 f 
1


 f (t ) dt
(7)
0
The integration time  varies in different situations, depending on the detector
integration time or the dimensions of the device used to detect the interference.
I (at P)   0 c  E1  E1  E 2  E 2  2 E1  E 2 
  0 c  E1  E1    0 c E 2  E 2   2 0 c E1  E 2 
 I1 (at P)  I 2 (at P)  2 0 c E1  E 2 



interference term
The interference term
2 E1  E2  2 E10  E20 COS (k r1   t  1 ) COS (k r2   t  2 )
Using Cos(A+B) + Cos(A-B) = 2Cos(A) Cos(B)
2 E1  E2  E10  E20 COS[ k (r1  r2 )  2 t  1  2 ] 

E10  E20 COS[ k (r1  r2 )  1  2 ]
(8)
The first terms changes so rapidly at optical frequencies , which should lead to an
average equal to zero.
 2 E1  E2   E10  E20  COS[ k (r1  r2 )  1  2 ] 
(9)
Because the two molecules under analysis,
not only
move randomly,
but also
their scattering emission (as well as excitation) occurs randomly.
Hence, their phase difference (1  2 ) changes randomly as a function of
time, which lead to a zero interference term.
2 E1  E2   0
Expression (8) becomes,
I (at P)  I1 (at P)  I 2 (at P)
 0
(10)
That is, randomly and widely spread molecules behave
as independent scattering centers of light.
The generalization is straightforward for N independent scattering centers,
N
I ( at P)   I n ( at P)
 0
(11)
n 1
Phasor analysis
In terms of phasors addition, finding the total electric field at P, at a given
instantaneous time t, consists of adding N phasors whose phases have no correlation
among them. In fact, the phase of the individual phasors are expected to differ
greatly, because scatterers are far apart in term of .
N
E (at P)   En0 e
j (k rn   t  n )
 SUM
n 1
I (at P)   0 c  SUM   0
2
(12)
lateral scattering
in tenuous media
Despite the random phase value of the different individual phasors, the total phasor
SUM cannot be zero all the time. (Otherwise the average intensity would be zero,
which would contradict expression (11).)
That is, the phasor SUM changes with time randomly (basically due to the random
value of (n (t ) ) but in such a way that its time-averaged amplitude-square is equal
to the value given in (11)
P
Illustration of the
phasors at a given
fixed time t.
3
N
Phasor
SUM
2
1
Overtime, the phasor SUM vaies randomly.
On average, its magnitude square is ≠ 0
Fig.7 Lateral scattering described in terms of phasors. The picture shows
schematically the different phasors at a given instantaneous time. The phase of the
individual phasors differ greatly (because scatterers are far apart in term of ).
In short, in lateral scattering,
The different phasors adding at the point P have uncorrelated phases.
Their phases differ greatly among them (because the scatterers are far
apart between them, compared to .)
Hence, the scatterers act independently, and the intensities from these
individual scatterers add up.
In conclusion, unimpeded by interference, light stream out of the
forward direction
(13)
II.2.1.B Case: Forward propagation
For the forward direction, the calculation of the light intensity reaching a point
like Q should in principle be similar to the case of lateral scattering (i.e. adding
phasors from randomly located scatterers).
Q
However, when the incident light and the scattered light occupy the same space,
there exists a peculiar behavior:
the re-emitted light from the scatterers (despite their random
location) tend to cooperate, marching in phase in the forward
direction.
(14)
This is illustrated in the figure below.
First, consider an incident lane wave, as sketched in the figure below. The profile at
the bottom of the diagram shows the amplitude of the electric filed.
Incident
plane wave
The following diagram shows the sequential excitation of two scatterers A and B.
The description is facilitated if the lateral separation between A and B is l/2, but this
specific separation is not necessary. (What matter is the assumption that the phase
separation between the excited filed and the scattered field is constant.)
Figure assumes emitted wave
rides in phase with the
excitation incident light.(But,
alternatively, a fixed phase
difference can also be
assumed).
A
Incident
plane wave
A
Molecule B shown located at a
lateral distance /2 from A, but
this is not crucial. Molecule B
could be located anywhere, still
the argument would be valid.
B
Scaterred waves
arrive in phase on
a planar wavefront
A
B
Fig. 8 The incident light is a plane wave ( Despite their random location of the
molecules, the light scattered in the forward direction march in phase on a planar
wavefront.
II.2.2 Propagation of light in dense media
In contrast to a tenuous media, as depicted in Fig. 5, consider a denser medium
where there are ~ 1 million of molecules in a cube of side . The average
separation between molecules is d ~ (500 nm)3 /106 = 5 nm. If the incident light is
in the visible region (~500 nm) then we are in the regime d << 
At a given time, the scattering from all the molecules in that hypothetical cube may
have some correlation. In the language of phasors, the phase of the different phasors
contributing to the field at point P may not be random; they will display some
interference since phasors from contiguous molecules will have similar phases, as
illustrated at the bottom in figure 9.
d << 
d is the separation
between molecules




rn
P
SUM
Fig. 9 Scattering by molecules in a dense medium. Since the average
separation d between molecules is much smaller than , the phase of
phasors originated from contiguous molecules do not differ greatly. On
average the contribution from the molecules in a cube of side  would be
close to zero. .
N
E (at P)   En0 e
j (k rn   t  n )
 SUM
n 1
Summation carried over all the molecules
inside the cube of side .
Further, if we considered that the incident beam has a plane wave wavefront profile,
for a given phasor associated to the A source, given the compact arrangement of the
molecules there will be another source B located /2 apart so that these two phasors
will interfere destructively. Given the larger than  lateral extension of the bean,
one can find pairs of molecules that interfere destructively. Hence, we expect then
that the net interference at P will be very small, if not zero.
> 

A
B
P
Fig. 10 Due to the dens medium, for a given scaterer A, there will be
another scatter B such that their electric field at P will differ p radians in
phase, hence given a net zero field.
The argument above would be valid also for analyzing wave scatered in the
backward direction.
It is plausible to conclude then that, different than the case of lateral scattering in
tenuous media,
Almost no light is scattered laterally or backwards
(15)
when light propagates in a dense homogeneous medium
“The more dense, uniform, and ordered the medium, is(the more nearly
homogeneous), the more complete will be the lateral destructive interference and
the smaller amount of non forward scattering. Thus, most of the energy will go into
the forward direction, and the beam will advance essentially undiminished.” E.
Hecht. Optics 4th edition, p. 91
III. Scattering from large particles
When the size of the particle  is smaller than the wavelength 
the scattered light from the different compacted atoms interfere
almost constructively, thus the scattering is strong.
As the particle increases in size approaching the wavelength ()
The molecules located on the extremes scatter radiation that is
not any more in phase; hence the overall coherence decreases;
this happens first (as the particle size increases) at the short
wavelengths (blue).
Accordingly, for incident white light, the particle scatters more
coherently in the red than in the blue
Mie scattering Gustav Mie in 1908 published a theoretical analysis of scattering
from spherical particles of any size.
Why do clouds look white (sometimes)?
Although water is essentially transparent, water vapor appears white. The reason:
If the grain size (drop of water)  is small but larger than the
wavelength , light will enter each transparent particle and (after
reflection and refraction) emerge. This will happen regardless of the
wavelength, so the emerging light will appear “white”.
This is the mechanism why few things appear white, like, for example,
sugar, salt, paper, clouds, snow, paint.
Busted concept:
 Even though we tend to consider paper, talcum powder, and sugar as
“opaque” white substances, that is not correct. (Spread some sugar
grains on a sheet of paper, and you will see through them without
difficulty.
 White paint results from suspending colorless transparent particles
(zinc oxide, titanium, or lead) in an equally transparent medium
(linseed oil, acrylics)
Indeed, if the index of refraction of the particles np is different than the index of
refraction of the medium nm ( np ≠ nm ), then there will be reflections (at all
wavelengths) and the paint will appear white and opaque. To color paint, one needs
only to dye the particles so they absorb all wavelength radiation, except the desired
color.
But if np ~ nm, then there will be no reflection from the grain boundaries. The
transparent medium will remain transparent, whereby decreasing the whitness of the
object.
Ref: E. Hecht, 4th Ed., p.132.
IV. Transmission and the Index of Refraction
Photons exists only at speed c, still the transmission of light through a
homogeneous medium appears to occur at a lower speed v = c/n, where n is the
identified as the index of refraction.
This apparent contradiction was explicitly addressed in Lecture-3. We found that
there is no need to postulate that light slows down when traveling through a
medium. Rather, the introduction of the index of refraction (i. e. the concept that
light slows down) simply offers an alternative convenient and compact way to
express the forward net electric field that results from adding the field of the light
source Es and the field E’ generated from mediu’s molecules excited by the light
source. In fact the net electric field at a point located in the forward direction was
calculated without invoking a slowdown of the speed of light.
Es
Y
Light
source

Es (at P) + E’ (at P)
P
q
z
Z
Es (z,t)  Eoei(ωt - kz )
d
Fig. 11 Primary wave Es propagate though the medium and drive the electric dipole
of the molecules in the medium. The accelerated molecules radiate a secondary wave
Es. Primary and secondary waves interfere at P.
We found that the electric field E’ (produced by the oscillating charges in the
material) is given by,
E' (z, t)  - Nd
 Nd
q
2 o c
q 
i xoei (t - z/c )
xo e- i / 2 ei (t - z/c )
2 o c
Here xo is the amplitude of oscillations of the individual charges q, which is
obtained through the solution of the equation, m (
That gives xo 
d2
dt
x 
d
dt
x  o x )  qEo ei t
2
q
1
Eo .
2
2
m ( o -  )  i  
E' ( z, t )  Nd
q
2

1
2m o c ( o -  )  i  
2
2
e- i / 2 E ei (t - z/c )
o
(16)
We notice that the lagging of the field E’ with respect to the field of the source
E ( z,t )  E ei(ωt - kz) has two origins:
s
o
a) Within the elementary treatment that models the atom as an oscillator, the factor
1
( o -  )  i  
2
2
is the typical frequency dependent lagging caused by the fact
that the oscillator cannot respond instantaneously to the driving force.
b) The factor e- i / 2 accounts for the accumulated contribution from the different
field produced by the individual oscillator in the medium
The electric field given in (16) can also be obtained if we postulate a slowing
down of the light speed by a factor of n when travelling in a medium, provided that
n is given by,
n
1
N
q
2
1
2m o ( o -  )  i  
2
2
(17)
In passing, notice the phase of the complex index of refraction is identical to the
phase difference between the dipole moment p  - ex of in the individual atoms in
the medium and the applied external field Es (as discussed in Lecture-5).
p
e2
1
Es
2
m ( o -  2  i   )
(18)
It is very illustrative to compare i) the lagging phase of E’ with respect to Es
(expression (16), and ii) the phase displayed by the complex index
refraction
Eof
P) + or
E’ (at P)
s (at
Y
the lagging phase of the electric dipole relative also to Es.
 << o
Es 
p
E’
Y
d
 ~ o
E’
Es

Es

p
Y
d
 >> o
E’
p
Fig. 12 The three phasors propagate counterclockwise at angular frequency .
Notice that at resonance, when  = o, E’ and Es are 180 degrees out of phase;
that is, they are just opposite to each other. The net electric field is just weaker and,
equivalently, there would be no need to invoke a time delay due to the slowdown of
the wave while travelling to the medium. This is compatible with the fact that at 
= o the index of refraction is 1.
│xo │
o

n
o

Fig. 13 Top: Dipole amplitude. Bottom: Real part of
the index of refraction.
V. DIFFRACTION
Contrasting diffraction and geometrical optics
Diffraction is any deviation from geometrical optics that results from the
obstruction of a wavefront of light.
Geometrical Optics
Diffraction effects

Opaque
object
Screen
Opaque
object
Screen
Fig. 14 Edges of an optical image are blurred due to diffraction. It is a consequence of the
wave character of light. (More often, however, the sharpness of an optical image is more
seriously degraded by the optical lens’ aberrations. Diffraction limited is actually
acceptable optics in a variety of applications.)
Contrasting diffraction and interference phenomena
Actually, they describe the same on fundamental principle; both phenomena
result from the addition of waves. Still is customary to find the following
classification:
Interference: The addition of waves from a discrete number sources
Diffraction: The addition of waves from a continuum sources of light
Aperture
size “b”
P
P
Screen
Opaque object
with an aperture
of finite size
Interference
E ( at P ) 
N

n 1
En ( at P )
Screen
Diffraction
E ( at P) 
b
y0  (at P)(y)dy
# of sources in a
segment dy located
at the aperture
Fig. 15 Interference and diffraction are both based on the addition of waves.
Classification of diffraction according to the
far-field or near-field excitation/detection distances
 Fraunhofer, or far-field, diffraction:
Both the source of light and observation screen are
far enough from the diffraction aperture,
so that the wavefronts arriving at the aperture and observation screen may be
considered planes.
Q
P
p
Fig. 16 Fraunhofer diffraction. The source of light “P” is so far away such that the
wavefronts arriving to the aperture can be considered to be plane waves.
b

Fig. 17 In Franhoufer diffraction the viewing screen is far away from the
aperture, such that as the viewing screen is moved relative to the aperture, the
size of the diffraction pattern scales uniformly, but the shape of the diffraction
pattern does not change.
 Fresnel, or near-field, diffraction
The source of light and/or the observation screen are
are so close that the curvature of the wavefront must be taken into account.
In this approximation, both the shape and size of the diffraction pattern
depend on the distance between the aperture and the screen.
Q
P
p
Criterion for evaluating the far- and near- field condition
In this section we establish a criterion to evaluate whether or not we have
Fraunhofer or Fresnel diffraction
What is the required separation distance between the source
and the aperture, as well as between the aperture and the
detector, in order to have Fraunhofer or Fresnel diffraction?
For simplicity, let’s consider the case in which the source P and the detector Q are
located on the axis that passes through the center of the aperture.
The objective is to find out when
this spherical wavefront can be replaced
by
the plane wavefront
without causing significant changes on the
calculation of the fields at Q.
r’
h
P

Q
p
Notice in the figure below that the field at Q will depend on whether we use optical
paths like PAQ (A on the plane wavefront) or PBQ (B on the spherical wavefront).
The key parameter to consider in our analysis is , for,
 if “p” were sufficiently large such that  becomes much smaller than the
wavelength ,
then the optical path lengths PAQ and PBQ cannot produce too much difference
in phase.
Hence no major difference in the calculation of the field at Q will result when
using either optical path.
This condition would lead to Fraunhufer diffraction.
nd out when
ont can be
or changes

Otherwise, if P were too close to the aperture such that makes  (or ),
then whether we use SAP or SBP will make a difference (the phases of the
corresponding phasors eiOPL would differ significantly).
This condition would correspond to Fesnel diffraction.
More straightforward will the be to find the condition under which the Fraunhofer
diffraction occurs. With that in mind, let’s find out a relationship involving  and
the other parameters involved in the diffraction.
Notice, whether we use ray AQ or
ray BQ will be irrelevant if <<
r’
A B
h
P

Q
p
  r ' p
 r '
(r ' ) 2  h 2
 r ' r ' 1  (h / r ' ) 2
Consider the case where ( h / r ' )  1
  r ' r ' (1 - 12
h2
1h
)

2
2
r'
2
r'
Hence, in addition to ( h / r ' )  1 , Fraunhofer diffraction occurs when
h2
h2
 r ' .
  , or,

r'
Since h2 scales with the area of the aperture, the last result can be put in more
general terms:
Fraunhofer diffraction: r ' 
Fresnel diffraction:
Alternative derivation:

r2

h

r1

r' 
aperture area

aperture area
