Transcript lecture_three_2016

Chap. 3 Propagation of Light
3.1 Laws of reflection and refraction
3.1.1 Huygens’s principle
Consider a source of electromagnetic waves and these waves
propagate out in a spherical way. Recall that a wavefront is a
surface over which an optical disturbance has a constant phase.
Huygens postulated that every point on a primary wavefront
serves as the source of spherical secondary waves also called
wavelets, such that the primary wavefront at some later time is
the envelope of these wavelets. Moreover, the wavelets advance
with a speed and frequency equal to those of the primary wave
at each point in space. This has since become known as the
Huygens’ Principle.

'
Fig. 3.1 Propagation of a wavefront
via Huygens’s principle.
Fig. 3.1 shows a view of a wavefront  , as well as a number of spherical secondary waves
(wavelets), which after a time t, have propagated out to a radius of vt. The envelope of all
'

these wavelets is then asserted to correspond to the advanced primary wave
.
3.1.2 Snell’s Law and Law of Reflection and refraction
We have seen previously that light is an electromagnetic phenomenon and already in the 16th
century way before Maxwell a lot of studies were done of the interaction of light with water
and with glass. The kind of experiment that were is depicted Figure 3.2.
Incident
Air
Water
θ1
!
1!
1
!
2!
1
θ3
reflected
θ2
refracted
Fig. 3.2 Light at the interface of air and water
Suppose a beam of light strikes the surface between air and glass at an angle of incidence
Some light striking the surface is reflected with and angle of reflection
.
. Some of that light
goes into the water and is refracted with an angle of refraction
There was a Dutch man called Snellys who in the seventheen century found three rules that
govern the relationship between these three beams of light.
1. The first one is that the incident, reflected and refracted beams are in the same plane, which
is the plane of the board.
2. The second thing that he found is that the angle
which is called the angle of reflection is
the same as the angle of incidence. That means
the angle of incidence equals the
angle of reflection is known as the law of reflection.
3. The third one which is the most surprising one named after him is called snell’s Law is that
if we go from air to water.
If we go from air to glass that will be much higher, like 1.5. He introduced the idea of index of
refraction which we call n. Recall that for vacuum the index of refraction is 1, which is very
closely to the index of refraction of the air. In water the index of refraction is approximately
1.3 and in glass depending upon what kind of glass you have is approximately 1.5.
We can now amend this Snell’s law by writing
n1 being the index of refraction of the medium where you are and n2 the index of refraction of
the medium where you are travelling to.
Is the law of refracrtion
3.1.2 Fermat’s principle
The law of reflection and refraction and indeed the manner in which light propagates in
general, can be viewed yet another entirely different and intriguing perspective afforded us
by something called Fermat’s Principle. Fermat stated that the “The actual path between
two points taken by a beam of light is the one that is traversed in the least time”. As we
shall see in an little while this statement is somewhat incomplete. For the moment let us
embrace it but not passionly.
Fermat’s principle in the case of reflection
Fig. 3.3 Minimum path from
the source S to the observer’s eye at P
Fig. 3.3 depicts a point source S emitting a number of
rays that are then “reflected” toward P. If we simply
draw the rays as if they emanated from S’ (the image of
S ), none of the distances to P will have been altered.
But obviously, the straight-line path
S’BP is the
shortest possible one, which corresponds to
i   r
(3.4)
This is the reflection law. And point B must be on this paper plane, which makes the incident
beam SB, the reflected beam BP and the surface normal on the same plane.
Fermat’s principle in the case of refraction
As an example of the application of the Fermat’s principle to the case of refraction, refer to
Fig. 3.4, where we minimize t.
The transit time from S to P, with respect to the variable x. The shortest transit time will
then presumably coincide with the actual path. Hence
t
SO OP

vi
vt
or
b 2  ( a  x) 2
h2  x2
t

vi
vt
To minimize t (x) with respect to variations in x , we
set dt dx  0, that is
dt
x
 (a  x)


dx vi h 2  x 2 vt b 2  (a  x) 2
 0.
Using the diagram, we can rewrite the expression as
Fig. 3.4 Fermat’s principle
applied to refraction.
sin  i
sin  t

,
vi
vt
With
so
vi  c / ni and vt  c / nt
ni sin  i  nt sin  t
(3.5)
which is the refraction law or Snell’s law. Again SO,OP and the sample normal are on the same plane.
Suppose that we have a material composed of m layers, each having a different index of refraction, as in
Fig. 3.5. The transit time from S to P will then be
sm
s1 s2
t     
v1 v2
vm
or
m
t
i 1
where
si
and
si
,
vi
vi
are the path length and speed,
respectively, associated with the ith contribution.
Thus
1 m
t   ni s i ,
c i 1
(3.6)
Fig. 3.5 A ray propagating through a
multi-layered material.
In which the summation is known as the optical path length (OPL) traversed by the ray.
Clearly, for an inhomogeneous medium where n is a function of position, the summation
must be changed to an integral:
P
(OPL)   n(s)ds.
S
(3.7)
Inasmuch as
t  (OPL) c, Fermat’s principle can be restated : light, in going from points S
to P, traverses the route having the smallest optical path length.
3.2 Electromagnetic approach of laws of reflection and refraction
3.2.1 Boundary conditions at a surface of a dielectric
A dielectric is a substance that is a poor conductor of electricity but an efficient supporter of
electrostatic fields
Recall:
L
Gauss law-electric
Ke1 Km1
dz
Positive direction
Ke2 Km2
Where
=permitivity of the medium
Ampere’s circuital law
 
 
 B  dl    J .dS
c
A
Where  = permeability of the medium
Fig. 3.6 Boundary conditions at a
surface of a dielectric.
Recall Maxwell equations
Among them we have
Which means that


B
 E  
t

 
B 
cE  dl  A t dS
This is a closed loop and you can choose any loop you want
We are going to do the closed loop integral using figure 3.6
Now to make sure that we are at the boundary we make dz=0, that’s the trick!

B 
So if dz=0, then the surface equals zero so we have  
dS  0
A t
 
E  dl  0
and
Hence
and then
So for a dielectric the first boundary conditions is:
(3.8)
For the three Maxwell equations left, we can derive the other boundary conditions for the
dielectric which are
(3.9)
(3.10)
(3.11)
3.2.2 Wave at an interface
Suppose that the incident monochromatic light wave is planar, so that it has the form
 


Ei  E0i cos(ki  r  i t )
(3.12)

Assume that E0i is constant in time, that is, the wave is linearly or plane polarized. The
reflected and transmitted waves can be written as:
 


Er  E0r cos(k r  r  r t   r )
and
 


Et  E0t cos(kt  r  t t   t ).
(3.13)
(3.14)

Here  r and  t are phase constants relative to E and are introduced because the position of
the origin is not unique. Figure 3.7 depicts the waves in the vicinity of the planar interface
between two homogeneous dielectric media.
Fig. 3.7 Plane electromagnetic waves incident on the boundary between two
homogeneous, isotropic, dielectric media.
The law of electromagnetic theory lead to certain boundary conditions. For example, the
tangent of the electric field intensity to the interface must be continuous across it. In other
words, the total tangential component of the electric field on one side of the surface must be
equal to that on the other side. With û n being the unit vector normal to the interface, the
boundary condition can be written as



uˆ n  Ei  uˆ n  Er  uˆ n  Et
or
(3.15)
 
 


uˆ n  E 0i cos( k i  r   i t )  uˆ n  E 0 r cos( k r  r   r t   r )
 

 uˆ n  E 0t cos( k t  r   t t   t ).
(3.16)
Here û n is the unit vector normal to the interface. The above relationship must be obtained at



any instant time and at any point on the interface (y=b). Consequently, Ei , E r , and Et must
have precisely the same functional dependence on the variables t and r , which means that
 
(k i  r  i t )
y b
 
 (k r  r   r t   r )
y b
 
 (k t  r  t t   t ) .
y b
(3.17)
Inasmuch as this has to be true for all values of time, the coefficients of tmust be equal, so
i   r  t .
(3.18)
Clearly, whatever light is scattered has that same frequency. Furthermore,
 
(k i  r )
y b
 
 (k r  r   r )
y b
 
 (k t  r   t )
y b
,
(3.19)

where r terminates on the interface. From the first two terms we obtain
  
[(ki  k r )  r ]
y b
 r.
(3.20)
Notice that since the incident and reflected waves are in the same medium,
 
 
(ki  kr ) is parallel to û n , that is, uˆn  (ki  kr )  0,
we conclude that
ki sin  i  kr sin  r ;
Hence we have the law of reflection, that is
i   r .
ki  kr.
Again, from Eq. 3.19 we obtain
  
[(k i  k t )  r ]
y b
 t ,
(3.20)
  
 

and therefore (ki  kt ) is also normal to the interface. Thus ki , k r , kt , and u n are all


coplanar. As before, the tangential component of ki and kt must be equal, and consequently
k i sin  i  k t sin  t .
(3.21)
But because i  t , we can multiply both sides by c i to get
ni sin  i  nt sin  t ,
which is Snell’s law.
3.2.2 Derivation of Fresnel equations

E
We have just found that the relationship which exist amongst the phases of i


E
E r and t
 

E
E
at the boundary. There still an interdependence shared by the amplitudes
E 0t
0i , 0 r and
which can now be evaluated. To that end suppose that a planar monochromatic wave is
incident on a planar surface separating two isotropic media.

E
and

B fields can be resolved into two components parallel and perpendicular to the
plane of incidence and these components can be treated separately.

Ei 

B i ||

E i ||

ki


When you have an e.m wave coming in the direction of k i the E and

B fields can be
decomposed into two components one which is perpendicular to the plane of incidence and
the other one parallel to the plane of incidence.
perpendicular component

That means each vector E will have a


Ei  and a parallel component E i || .



Each vector E has an associated B vector, it is married to a B vector, and they go hand in

 



hand. E , B and k form a right handed system. The B component associated to Ei  is Bi ||

Case 1: E perpendicular to the plane of incidence.

Ei 

B i ||

E i ||

Er 

ki

kr

Br ||
θi
θr

un
θt

Et 

B t ||

kt


Fig. 3.9: An incoming wave whose E field is perpendicular to the plane of incidence B field is parallel to
the plane of incidence.
Now we can apply the boundary condition. The first
one if you remember is that:
represents the electric field in the medium one,
which is the sum of the incident and reflected electric
field.
Represents the electric field in the medium
two, which is the refracted electric field.
So
So now we can apply another boundary condition.
We are going to leave the Ks because they equal a as far as we are
concern. Km doesn’t in general deviate from one by any more than a few
parts in 104 for example Km for diamond is 1-2.2X10-5

B
The continuity of the tangential component of
requires that .


Bi // cos i  Br // cos r  Bt // cos t
(3.21)
Now we are going to use the relationship
So we can replace
by
Equation (3.21) becomes
So
Now we can substitute
so to have
Then rearrange to get
(3.22)
(3.23)
For the transmission, we again start with the equation
And eliminate
using
we then have
(3.24)

Case 2: E parallel to the plane of incidence.
(3.25)
(3.26)
Fig. 3.10 Reflection and refraction of an incoming wave whose E-
filed is in the plane of incidence.
Further notational simplification can be made by using Snell’law whereupon the Fresnel equation
for dielectric media become
(3.25)
(3.26)
(3.27)
(3.28)
3.2.3 Interpretation of Fresnel equation
This section is devoted to an examination of the physical implications of the Fresnel
equations. In particular we are interested in determining the fractional amplitudes and flux
densities which are reflected and refracted.
i) Amplitude coefficients
Let’s now briefly examine the form of the amplitude coefficients over the entire of θi –
values. At nearly normal incidence (θi ≈ 0 ), the tangents in equation(3.27) are essentially
equal to the sines in which case
(3.29)
(3.30)
In the limit as θi goes to 0, cosθi and cosθt both approach one and consequently
(3.31)
it follows from Snell’s law that
When
.
In contrast,
when
and so
is negative for all values of θi
starts out positive at θi =0 and decreases gradually until it equals zero
since
is infinite.
At normal incidence equation (3.24) and (3.26) lead rather straightforwardly, to
(3.32)
ii) Reflectance and transmittance
Recall that the power per unit area crossing a surface in vacuum whose normal is parallel
to
the poynting vector is given by
Furthermore, the radiant flux density (W/m2) or irradiance is then
This is the average energy per unit time crossing a unit area normal to
media
is parallel to
(in isotropic
). In the case at hand let Ii, Ir and It be the incident, reflected
and transmitted flux density respectively
Accordingly, the portion of the energy incident
normally on a unit area of the boundary per second is
. Similarly,
and
are the
energies per second leaving a unit area of the
boundary normal to either side. The reflectance R is
the ratio of the reflected over the incident flux (or
Fig 3.11 Reflection and
transmission of an incident beam
power) i.e
While the transmittance T is the ratio of the transmitted over the incident flux and is
given by
Since the incident and reflected waves are in the same medium
and
In the same way assuming
Where use was made of the fact that
and
Let’s now write an expression representing the conservation of energy for the representation
depicted in figure (3.11). In other word, the total energy flowing into area A per unit time
must be equal to the total energy flowing outward from it per unit time
Multiplying both side by c this expression becomes
or
But this is simply
Where there is no absorption it is convenient to use the component forms, that is
Furthermore, it can be shown that
3.3 Total internal reflection
4- 34
Suppose that we have a light
source imbedded in a glass, and we
allow  i to increase gradually, as
indicated in Fig. 3.12. As the angle
of incidence increases, the angle of
refraction increases. Moreover,
 t   i , since
sin  i 
nt
sin  t
ni
and ni  nt . Finally, when
 t  90  , the refracted ray points
directly along the interface. The
angle of incidence giving this
situation is called the critical
angle  c . It can be determined
by
Fig. 3.12 Internal reflection and the critical
angle.
sin  c 
nt
 nti
ni
(3.33)
For angles of incidence larger than
 c , there is no refracted ray and all the
light is reflected. This effect is called
the total internal reflection.
The critical angle for our air-glass
interface is roughly 42 .
Consequently, a ray incident normally
on the left face of either of the prisms
in Fig 4.15 will have  i  42o and
therefore be internally reflected.
Fig. 4.13 Total internal reflection of light.
Fig. 4.14 Total internal reflection