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Electromagnetic Waves
Maxwell’s Equations for homogeneous media:
1)
3)
E

2)
òr ò0
B
 E  
t
4)
 B0
E 

  B  r 0  j  òr ò0


t


Where:
E ,B are the electric and magnetic field vectors,
 is the charge density,
j is the current (directed flow of charge),
ò0 , 0 are the dielectric permittivity and magnetic permeability, respectively, of free
space,
òr , r are the ratio between the permittivity and permeability in a medium and
space. (The relative permittivity and permeability.)
and the del ( ) vector operator is defined (in Cartesian coordinates) as:



i
j
k
x
y
z
These equations are too general for our use, because:
1. We are not going to consider wave propagation in any
media with a net charge density. (Hence, our conclusions
may not be valid for propagation in a plasma, for
instance.)
2. We are also not going to consider media with electric
currents.
3. The magnetic permeability arises due to the interaction
with atoms that have a net magnetic moment (unbalanced
electron orbitals). At optical frequencies (~1014 Hz), no
atoms can respond fast enough to afford significant
interaction.
(Electron orbitals can respond fast enough, however, so in
general, òr  1 )
Hence, setting 𝞀, j = 0, and r  1 , we get:
Where we’ve made the
substitution ò  ò ò
r 0
(1)
 E 0
(2)
 B0
(3)
B
 E  
t
(4)
E
  B  0ò
t
These are Maxwell’s Equations for the conditions
we are interested in for almost all optical design:
1.
2.
3.
4.
No free charges.
No currents.
No magnetic interaction.
The primary interaction is between the E-field
and the orbital electrons in the dielectric media.
The Electromagnetic Wave Equation
Since these equations are simultaneously true, we can combine them:
1) Take the curl of both sides of the 3rd equation (Faraday’s Law):
 E  
   B 
t
2) Substitute for   B from the 4th equation (Ampere’s Law):
2 E
    E   0ò 2
t
3) Applying the well known vector identity     E     E   2 E
and  E  0 (Gauss’s Law) , we arrive at the Electromagnetic Wave
Equation for homogeneous media:
2

E
2
 E  0ò 2  0
t
An equivalent derivation shows that the magnetic field also obeys a wave equation.
Comparing the EM wave equation to the generic wave equation:
2

E
2
 E  0òr ò0 2  0
t

2
1

ψ
2
 ψ 2 2 0
v t
We see that the velocity of electromagnetic waves, under our assumptions is:
1
c
c
v


0òr ò0
òr n
Where we have used the fact that the speed of light, c
And we have defined the index of refraction,

n  r
1
 0 0
A useful solution to the wave equation is the “plane wave”:
E  r, t   A cos  k r  ωt   
Or, for the case of an X-polarized wave moving down the Z-axis.
c 
 2 

E x  r, t   A cos   z  t    
n 
  

For use in optical modeling, plane waves have several important
characteristics:
1. Plane waves have a constant amplitude (A) over all space and time – the
only thing that changes with time and position is the phase. Hence plane
waves are trivially easy to “propagate” – just make the appropriate phase
change.
2. We will show, when we cover wave propagation, that the Fourier
Transform can decompose any Electromagnetic Wave field (defined on a
plane) into an equivalent set of plane waves. This reduces the problem of
simulating the propagation of the given field to simply re-phasing the
constituent plane waves and adding them up at the desired location.
3. A detailed analysis shows that plane waves travel in the direction
perpendicular to their wavefronts. Hence, rays (defined as lines drawn
perpendicular to the wavefronts) are perfect models for the propagation of
plane waves.
The law of refraction (Snell’s Law) from plane waves:
Consider a plane wave impinging on a plane interface with different indices of refraction
(hence wave speeds) on each side. The wave changes speed, but not frequency,
hence the phases (wavefronts) must meet from each side:
Angles a1 and a2 are called the “angle of incidence”
and “angle of refraction” respectively.
n1
a1
λ2
x
λ1 a
1
a2
n2
a2
Note that:
1 
0
n1
and
2 
0
n2
Where
0 is the free space wavelength
Looking at a close-up of the previous drawing:
1 
2 
0
n1
n1
n2
0
n2
Note: sina1
Hence:

1
sin a1
1
x

and
sina2 
sin a2
2
⇒
2
Hence,Snell’s Law is a
consequence of the
wave nature of light
and the physical
requirement for field
continuity at sourceless
points.
x
n1 sin a1  n2 sin a2
(Snell’s Law)
Total Internal Reflection (TIR)
Consider a ray traveling from a medium of high index to one of low index:
n1
i
n2
According to Snell’s Law, when
n1  n2
r
 n2 
 i  sin  
 n1 
1
Then
r  90o
And no light penetrates the second medium at all – all of the light must be
reflected.
This is called the ‘critical angle’ at or above which incident light will be totally
reflected. This is the way that optical fibers contain light.
Fermat’s Principle
Original: “Light travels along paths of minimum time” (1662)
Modern: “Light travels paths that are “stationary” w.r.t. all nearby paths.
Studing the following diagram shows why rays “obey” Fermat’s Principle:
1) Waves travel ⊥ to
wavefronts, hence rays
must also.
2) Any ray from ‘S’ to ‘P’
crosses the same number
of wavefronts (hence takes
equal time) and therefore
obeys Fermat’s Principle
3) Only one ray can go from
‘S’ to any point other than
‘P’. This ray is ⊥ to the
wavefronts, hence takes the
least time of any other path.